CURVE TRACING
IN
CARTESIAN COORDINATES
BY


WILLIAM WOOLSEY JOHNSON
PROFESSOR OF MATHEMATICS AT THE UNITED STATES NAVAL ACADEM1
FIRST EDITION.
FIRST THOUSAND.
NEW YORK
JOHN      WILEY      &  SONS
53 EAST TENTH STREET
I895




COPYRIGHT, 1884,
BY WILLIAM WOOLSEY JOHNSON.




PREFACE
THIS book relates, not to the general theory of curves,
but to the definite problem of ascertaining the form of a
curve given by its equation in Cartesian coordinates, in
such cases as are likely to arise in the actual applications
of Analytical Geometry. The methods employed are exclusively algebraic, no knowledge of the Differential Calculus
on the part of the reader being assumed.
I have endeavored to make the treatment of the subject thus restricted complete in all essential points, without
exceeding such limits as its importance would seem to
justify. This it has seemed to me possible to do by introducing at an early stage the device of the Analytical Triangle, and using it in connection with all the methods of
approximation.
In constructing the triangle, which is essentially Newton's parallelogram, I have adopted Cramer's method of
representing the possible terms by points, with a distinguishing mark to indicate the actual presence of the term
in the equation.  These points were regarded by Cramer
as marking the centres of the squares in which, in NewilU




iv


PREFACE


ton's parallelogram, the values of the terms were to be
inscribed; but I have followed the usual practice, first
suggested, I believe, by Frost, of regarding them merely
as points referred to the sides of the triangle as coordinate
axes.  It has, however, been thought best to return to
Newton's arrangement, in which these analytical axes are
in the usual position of coordinate axes, instead of placing
the third side of the triangle, like De Gua and Cramer, in
a horizontal position.
The third side of the Analytical Triangle bears the
same relation to the geometrical conception of the line at
infinity that the other sides bear to the coordinate axes.
I have aimed to bring out this connection in such a way
that the student who desires to take up the general theory
of curves may gain a clear view of this conception, and be
prepared to pass readily from the Cartesian system of
coordinates, in which one of the fundamental lines is the
line at infinity, to the generalized system, in which all
three fundamental lines are taken at pleasure.
Lists of examples for practice will be found at the end
of each section.  These examples have been selected from
various sources, and classified in accordance with the subjects of the several sections.
w. W. J.
U. S. NAVAL ACADEMY,
November, I884.




CONTENTS


I
Equations solved for one variable.........
Diameters................
Limiting tangents.............
Asymptotes to an hyperbola...........
Parabolas........
Curvilinear diameters......
Employment of the ratio of the coordinates..
Points at infinity.........
Asymptotes -general method..........
Symmetry of curves.............
EXAMPLES  I...............


PAGE..  ~.  a  2.....  I.....  6... '   69.... 12.....13..... 14....'3......'4.. 


II
The analytical triangle.............  15
Intersections with the axes...............  16
The line at infinity...................  8
Asymptotes parallel to one of the axes........... 19
Parabolic branches...................  20
Parallel asymptotes...............  21
Tangents at the origin..................  23
Tangents at the points of intersection with an axis.. 24
Nodes..............26
Intersection of a curve with a tangent........... 28
Intersection of a cubic with its asymptotes...........  29
EXAMPLES  II................... 30
V




vi


CONTENTS


III
PAGE
Approximate forms of curves...............                 32
Approximate forms at infinity...............               33
Radius of curvature at the origin............ 36
Method of determining the equations of approximate curves.... 36
The analytical polygon..........                            42
Construction of the approximating curves........... 42
Sides of the polygon representing more than one approximate form.. 45
Imaginary approximate forms..............         47
EXAMPLES III..................    48
IV


Second approximation when the side of the polygon, gives only the
first approximation..................
Selection of the terms which determine the next approximation...


49
55


Successive approximation................. 59
Asymptotic parabolas..................  62
Continuation of the process of approximation.......  65
EXAMPLES IV...................66


V


Cases of equal roots..........
Cusps...........
Tacnodes......
Cusps at infinity.......
Ramphoid cusps...........
Circuits..............
Auxiliary loci......
Tangents at the intersections of auxiliary loci
Loci representing squared factors...
Points in which several auxiliary loci intersect.........  67......... ~  ~  ~,  69......... 72......... 73......... 74......... 75.........78......... 79.........8I...............83


EXAMPLES  V...................  85




CURVE TRACING
I
Equations Solved for One Variable
1. THE equation of a curve is in these pages supposed to
be given in Cartesian coordinates; and the curve is said to be
traced, when the general form of its several parts or branches
is determined, and the position of those which are unlimited in
extent is indicated. In the diagrams the coordinate axes will,
for convenience, be assumed rectangular; but the methods are
equally applicable to oblique axes. When it is possible to solve
the equation for one of the coordinates, so as to express its
value in terms of the other, the resulting form of the equation
affords the most obvious method of tracing the curve. This
may be done for either variable when the equation is of the
second degree in both variables, in which case the curve is a
conic. For example, let the given equation be
2X2 - 2Xy + Y - 4X + 2Y + I = O.     (I)
Solving for y, we have


y = x - I ~  (2x - x2).


(2)


I




CUR VE TRACING


[Art. I


Thus, for any given value of x, we have two values of y; and
if we put
y = x - I,                (3)
the equation of the curve becomes
y = y i V(2x - X).            (4)
Diameters
2. Equation (3) represents a straight line, and equation (4)
shows that the two ordinates for the curve may be found by
adding to and subtracting from the ordinate of the straight
line the same quantity. In other words, the chord joining the
two points of the curve which have the same abscissa, that is,
any chord parallel to the axis of y, is bisected by the straight
line. This line is therefore called a diameter of the curve.
The diameter represented by equation (3) is constructed in
Fig. i.
3. The radical V(2x - X2), which is half the length of the
chord, varies with x, and vanishes for the two values
x = o     and     x= 2;
the corresponding points on the diameter are therefore also
points of the curve. Writing the radical in the form
V[x(2 - )],
it is obvious that all values of x between o and 2 give real
values to the radical, since they render both of the factors




DIAME TERS


3


under the radical sign positive, and that all values exterior to
these limits give imaginary values to the radical. Hence all
the real points of the curve lie between the straight lines x = o
and x = 2, as represented in Fig. i. These 
limiting lines are evidently tangents to the 
curve at the points where they cut the diameter. Putting the radical in the form 
[i - (i -   )2],
we see that the maximum value of the rad-    Fig. 1
ical is unity, and corresponds to the value x = I, or the middle
point of the diameter. The lines passing through the corresponding points of the curve parallel to the diameter are
obviously tangents. The curve is an ellipse.
4. As a second example let us take the curve
y2 - xy+y +        I+  = o.      (i)
Solving fory,
y =  (X - i) ~  V(X2 + 2X- 3).
Putting x2 + 2x - 3 = o, we find that the radical vanishes
when x = - 3, and when x = I; the equation may therefore
be written in the form
= 2(X- I) ~ 2[(X   + 3)(X - I)].     (2)
The straight line
y =  (X - I)                (3)
is therefore a diameter, and the lines


x = — 3  and   x = x




4


CUR VE TRACING


[Art. 4


are tangents; but the radical is, in this case, imaginary for
values of x between the limiting values, since such values make
one of the factors positive and the other negative, and real for
values exterior to the limits. Hence the curve consists of two
branches touching the lines x = - 3 and x = I at their intersections with the diameter represented by equation (3), as in
Fig. 2.
Asymptoles
5. Putting the radical in the form
[ (x + I)2 - 4],
we see that its value increases indefinitely as x increases in
numerical value, but that it
Y    }  /      is always less than x + I.
Thus the   points of the
___  ~0 x~        > curve lie between the lines
A          LE      y          -) i      2(X+ xI), (4)
^ / >//  I ~ ~    that is to say, the lines
~/2      vI             y=x    and y= —I.
Fig. 2            The curve approaches indefinitely to these lines, since the difference between x + i and
the radical decreases without limit as x increases numerically.
The lines are therefore asymptotes. The asymptotes and the
two branches of the curve are constructed in Fig. f. The curve
is an hyperbola.




~ I]                 ASYMPTOTES                      5
6. It is evident from the above method of finding the
asymptotes that the position of these lines will not be affected
by a change in the value of the absolute term. For example,
if the absolute term in equation (I) be changed from I to 2, the
only change will be that the radical will become [(x -+ I)2 - 8],
and the asymptotes will still be represented by equation (4).
Moreover, the value of the absolute term may be such as to
make the equation identical with the equation which represents the two asymptotes. In the present instance, the latter
equation is (y- x) (y + I) = o, or y2 - xy + y - x = o, which
is equation (I) with the absolute term changed to zero. If, on
the other hand, the absolute term is changed to - I, the radical
becomes V(x2 + 2x + 5) which cannot be made to vanish as in
Art. 4, but is always numerically greater than x + I. Thus
the curve will no longer cut the diameter, as in Fig. 2, but will
lie on the other side of the asymptotes. It will, in fact, be the
conjugate of the hyperbola drawn in Fig. 2.
7. If we solve equation (I) of Art. 4 for x, we have
X = y + -
x= y + I
This form of the equation shows that for each value of y there
is a single value of x, but this value is infinite when y = - I.
The line y =- I is therefore an asymptote. The value of
the fraction     is positive for all values of y algebraically
greater than - i, and negative for all values less than- I;
hence the part of the curve above the line y = - I lies on the
right of the line y  x, and that below the line y = - I on the
left of the line y -= x. As the numerical value of y increases
indefinitely, that of the fraction decreases indefinitely, which
shows that y = x is also an asymptote.




CURVE TRA CING


[Art. 8


Parabolas
8. It follows from what is shown in Art. 6 that an equation
of the second degree will represent an hyperbola when the
terms of the second degree can be resolved into real factors.
In this case, they constitute the difference of two squares. On
the other hand, the equation represents an ellipse when these
terms constitute the sum of two squares, as in equation (i),
Art. I. The intermediate case is that in which the terms form
a perfect square. For example, let the equation be
x2 + 2xy + y2  2X - 6y + i = o.      (I)
Solving for y,
y + x- 3 =      ~2V(2- x);
hence
y+x -3 =                     (2)
is the equation of the diameter bisecting chords parallel to the
axis of y. The radical is real for all values of x algebraically
less than 2, and imaginary for all values
\Y         greater than 2; hence the curve lies on the
left of the line x = 2, which it touches at
\ I  the point (2, I) where it cuts the diameter.
'"',,s     If we solve for x, we find, in like manner,
+ y- - I =      i 2Vy,
0   \ i X
Xa hence
Fig. 3                  x +y- I =           (3)
is the equation of the diameter bisecting chords parallel to the
axis of x, and the curve touches the line y = o at the point
(I, o). The value of the radical in either form of the equation
increases without limit, but there is no asymptote. The curve
is a parabola.




CUR V~LINEAR DIAMETERS


7


Curvilinear Diameters
9. We pass now to an example in which the curve is a
cubic, although the equation is only of the second degree with
respect to either variable. Given the equation
2X2y- X2 + y2 + 2x =:         ()
solving the equation as a quadratic for y, we have
y = - X2 ~ V(X4 + X2 - 2X),
or, resolving the quantity under the radical sign into factors,
y = - x2 V[x(x - I) (x2 + x + 2)].    (2)
Putting y = o in equation (I), we find that the curve intersects the axis of x at the origin, and at the point (2, o); and
putting x = o, we find that it meets the axis of y at no point
except the origin. Equation (2) gives in general two values of
y for each value of x, and the corresponding points are equally
distant from that point of the parabola
y = -x2                   (3)
which has the same abscissa. In other words, chords parallel
to the axis of y are bisected by this parabola, which is therefore a diameter.  This diametral parabola is constructed in
Fig. 4.
10. The quantity which occurs under the radical sign in
equation (2), namely


X(X - I)(X2 + X + 2),




8


CUR VE TRACING


[Art. ro


changes sign only twice as x passes through all possible values,
since it vanishes only when x = o and when x - I. It is, in
fact, negative for all values of x
I\ t    between these limits, and positive
--  -------       for all other values of x. There
\ -,' ~~~is therefore no part of the curve
o- ~     '-. Jx between the lines x = o and
/ /        '          x-= I; and these limiting lines
~/ /~ I,      touch the curve at the points
/     I\ \where they intersect the diamei \        tral curve represented by equation
Fig. 4          (3); that is, at the origin and at
the point (I, -I). See Fig. 4.
11. If we solve equation (I) for x, we have
= -        (I +y      2y3)        (4)
2y- I
One value of x becomes infinite when 2y - I = o; the other
value takes an indeterminate form in equation (4), but, referring
to equation (I), it is found to be x -= -. Hence the line
is an asymptote, cutting the curve at the point (- -, ~), as in
the diagram. The expression under the radical sign evidently
vanishes for y =; and, putting the radical in the form
[(I -y) (I + y + 22)],
it is seen to be imaginary for all values of y greater than unity,
and real for all other values.  Hence the curve lies entirely
below the line
y= I;




~I]  EMPLOYMENT OF TIE. RATIO OF COORDINATES


9


and, substituting this value in equation (4), we find that this
line touches the curve at the point (- I, I).
Employment of the Ratio of the Coordinates
12. If we denote the ratio of the coordinates of any point
by m, thus
Y
m = _,
x
the value of m may be regarded as a new coordinate of the
point, which, in conjunction with one of the other coordinates,
will serve to determine the position of the point. The points
which have a common value of m are situated upon the straight
line
y = mx
which passes through the origin; and, when a point is determined by the values of m and x, it is virtually determined as
the intersection of a line of this character and a line parallel
to the axis of y.
If, now, we eliminate y from the equation of a curve by
means of the equation y = mx, we shall have the equation of
the curve in the form of a relation between m and x. When
this form of the equation can be solved for x, we can, by giving
values to m, determine any desired number of points on the
curve, and can trace its form with as much facility as when y is
expressed in terms of x.
13. As an illustration, let the equation of the curve be


y3- X3 + 2Xy + X2 = 0.


(I)




IO


CUR VE TRA CING


[Art. 13


Putting mx for y, this becomes
(m3 - I)X3 + (2m + I)X2 = o.       (2)
The factor x2 in this equation indicates that the curve passes
twice through the origin; for a straight line generally cuts the
curve in three points, and equation (2) shows that for all lines
of the form y = mx two of these points have the abscissa
x = o. Rejecting this factor, and solving for x, we have
I + -2                 (3)
1 - M3
in which x is expressed in terms of the variable m and denotes
the abscissa of the third point in which the line y = mx meets
the curve. The ordinate of this point, which we shall denote
by P, is
m(i + 2m) 
y = mx=;            (4)
I - m3
and, assuming the value of m, we may determine the position
of P by means of any two of the three coordinates x, y and m.
14. Beginning with m = o, equation (3) gives x = I, and P
is at A, the point (I, o), the line y = mx coinciding with the
axis of x; as m increases, the line rotates
Y,   about the origin, coinciding with the axis.  of y when m is infinite. When m = I,
_ o. —^y ^ x the values of x and y are both infinite;
j  A  ~thus, as m varies from o to i, x increases
from I to infinity, P describing the infinite
C     Fg         branch AB in Fig. 5. Passing the value
Fig. 5
m = i, x changes sign; and as m passes
from I to oo, P describes the infinite branch CO, arriving at the
origin when m = oo, since equation (4) then gives y = o.




POINTS AT INFINITY


II


Continuing the rotation of the line y = mx, m becomes negative, and P describes the loop in the second quadrant, returning
to the origin when m = - -, since this value makes x = o in
equation (3). As m passes from - j to o, x changes sign and
passes from o to I, P describing the branch OA and returning
to its initial position.
Points at Infinity
15. Every point of the plane except the origin has a definite
value of m; and the limiting value of m for a point which
recedes indefinitely from the origin along an infinite branch of
a curve is said to determine a point of the curve at infinity.
The points at infinity may therefore be found by putting
x = oo in the equation between x and m. Thus, if we divide
equation (2) of Art. 3 by x3, and then make x infinite, we have
m3 - I = 0,
of which the only real root is m = I. It is evident that in this
process all the terms except those of the highest degree will
vanish; hence, if we put the group of terms of the highest degree
equal to zero, the result, regarded as determining values of the
ratio of y to x, gives the points at infinity. In this case, the
equation is
y3 - X3 = 0,
or
(y - x) (2 + xy + x2) = o;
the first factor gives a real value, and the second gives imaginary values to the ratio - or m. The curve is therefore said
to have but one real point at infinity.




12


CUR VE TRA CING


[Art. I6


Asymptotes
16. A real point at infinity usually corresponds to an
a.ymptote whose direction is determined by the value of m.
To illustrate the method of finding its position, let us determine
the asymptote to the curve drawn in Fig. 5. The equation of
the curve, equation (i), Art. 13, may be written in the form
-    _ 2Xy + X2             (.I)
y2 + xy + X2
Now, when the point P recedes indefinitely on the branch AB,
x and y become infinite, but y - x may nevertheless have a
finite value. To find this value, put mx for
Y      /   y in equation (i), and then make m = i and
x -= oo; or, what comes to the same thing,
o A_        put y = x, and x = oo. The process may
/ A  x be expressed thus,
y//  - X =-  2Xy  2= -I,
Fig. 6                  y2+xy+~x2y=x=
in which the suffixed equation is to be understood to mean
that the ratio of y to x is one of equality when x is infinite.
The result shows that, when this is the case, the quantity y - x
approaches the finite limit - I. If, now, we draw the straight
line
y - x= - I,
in which this quantity has constantly the value - I, it is evident
that the point P moving along the curve approaches indefinitely to this line, which is therefore an asymptote, as represented in Fig. 6.




SYMMETRICAL CUR ES


13


17. It is obvious that, had the equation contained terms of
a degree lower than the second, they would have vanished in
the process of finding the asymptote. For example, the curve
y3 - X3 + 2Xy + X2 + X = O,
whose equation differs from that of the curve drawn in Fig. 6
only by containing the term x, will have the same asymptote;
for we have
-    2Xy + X2 +         _ 3X2 + x   = -.
y2 + xy + X2_jy==       3X2 Jo
Thus, if the equation of the curve is of the nth degree, the
asymptotes generally depend only upon the terms of the nth
and (n - I)th degrees.
Symmetry of Curves
18. In some cases, the form of the equation indicates the
symmetry of the curve in certain respects.  For example, if
the equation contains powers of y with even exponents only,
the curve is symmetrical to the axis of x; for, in this case, if the
point (a, b) satisfies the equation, the point (a, - b) symmetrically situated with respect to this axis also satisfies the equation. In like manner, if the equation contains powers of x
with even exponents only, the curve is symmetrical to the axis
of y. Again, if the sum of the exponents of x and y in each
term is an even number, or if it is in each term an odd number, the curve is symmetrical with respect to the origin as a
centre; for, in this case, if the point (a, b) satisfies the equation,
the point (- a, - b) will also satisfy it. If x and y are interchangeable, the curve is symmetrical to the line bisecting the
angle between the axes; since, if the point (a, b) satisfies the




14


CUR VE TRA CING


[Art. I8


equation, the point (b, a) will also satisfy it. These considerations will be useful in the case of some of the following examples.
Examples I
Trace the curves whose equations are given below:
I. X2 + xy - y2 + 3X = o.
2. 5X2 - 4xy + 4Y2 - 8y- 4 = o.
3. x2 - 2XY + Y + 2 = 0.
4. X2 - 2xy + y2 - 2x - 2y + I = 0.
5. 6X2 + 4xy + Y2 - 3x - 2 - 2 = 0.
6. x2 - 8xy + I6y2- 6x - I2y + 9 =o.
7. xy2- x2 - y2= o.
8. X3 -   -y = o.
9. x2y - a2y + a3 = o.
10. y3 = X2(X - a).
II. X3 + X2 - y2 = o.
12. x2y - 2axy + a2x - ay = o.
13. X4 = y2(4a2  X2).
14. 4y2(x + a) = a(5x + a)2.
15. x2y2 = a2 (2 + y2).
i6. (x- 2a)xy = a(x - a) (x- 3a).
17. X3 - xy2 + ax2 + ay2 = o.
I8.  4 - 3axy2 + 2ay3 = o.
19. y3 - X3 -y + 4X = 0.
20. X3 + y3 -  2 - y2= 0.
21. X3 + y3 - X2 + 4y2 = 0.
22. X3 - y3 + (2y - X)2 = 0.
23. (x + y) (X2 + y2) = 2axy.
24. y4 - y3x + x3 - 2X2 = 0.
25. y3 - X3 + 2y -   2= 0.
26. (I + x2)y = I + X ~ V(x - X3).
27. y3 = X2(X - a).
28. 6x(i - x)y =    + 3x.




THE ANALYTICAL TRIANGLEI


Is


II
The Analytical Triangle
19. LET two intersecting straight lines be drawn; and let
each term in the complete equation of the nth degree be represented by a point whose coordinates with respect to these lines
are the exponents of x and y respectively in that term: we shall
thus have a triangular arrangement of points, there being n + I
points on each side of the triangle. Thus, if n = 3, B
we have the arrangement given in Fig. 7, in which
there are four points on each side of the triangle.
The diagram thus formed is called the analytical o 
triangle; the lines of reference are the analyti-  Fig. 7
cal axes of x and y respectively. The points upon OA, the analytical axis of x, represent the terms of the equation which do
not contain y; the points on OB represent the terms which
do not contain x; and the points on the third side, AB, of the
analytical triangle represent the terms of the nth degree.
20. The equation of a given curve is said to be placed upon
the analytical triangle when the points which represent the
terms actually occurring in the equation are marked
B         in some convenient manner. Thus the equation
2x2y - X2 + y2 + 2X = o
X 0.
o      A  of the curve traced in Fig. 4, page 8, is placed on
Fig. 8  the analytical triangle in Fig. 8. In thus placing
the equation, no attention is paid to the coefficients, the object




CURVE TRACING


[Art. 20


being simply to indicate the presence of certain terms in the
equation, and the absence of others. When the values of the
terms are required, it is of course necessary to refer to the equation of the curve.
Intersections with the Axes
21. If we put y = o in the equation of a curve, and suppose
x to have a finite value, all the terms except those represented
by points on the analytical axis of x will vanish. Hence the
result of equating these terms to zero is an equation determining all the finite points in which the curve meets the axis of x.
This equation will generally have n roots real or imaginary;
thus, if n = 3, its form will be
A + Bx + Cx2 + Dx3 = o,           (a)
a cubic equation. If there be no absolute term in the given
equation, that is, if in equation (a) A = o, one of the roots will
be zero, and the curve will pass through the origin. On the
other hand, if the term Dx3 be absent, the equation reduces to
one of the second degree, and determines but two finite intersections with the axis of x; but, putting equation (a) in the
form
A B C
A    B + C + D = o,
X3   X2   X
which is a cubic equation determining three values of x, we see
that, when D = o, one of the values of - is zero, and therefore
one of the values of x is infinite. Hence, in this case, one of the
intersections of the curve with the axis of x is said to be at
izfiity. The geometrical meaning of this is that a straight




INTERSECTIONS WITH THE AXES


I7


line usually cuts the curve in three points; but, when the line is
brought into coincidence with the axis of x, one of these points
of intersection recedes indefinitely and disappears. An inspection of Fig. 8 shows that the curve whose equation is there
placed upon the analytical triangle cuts the axis of x at infinity,
at the origin, and at one other point.
22. In like manner, an inspection of Fig. 8 shows that the
curve cuts the axis of y at infinity, and twice at the origin; it is
therefore said to meet the axis of y in two coincident points at
the origin. The geometrical meaning of this is that two of the
three points in which a straight line cuts the curve come into
coincidence at the origin when the cutting line is brought
into coincidence with the axis of y. It is evident that this will
happen whenever the curve has two branches passing through
the origin, and also whenever the axis of y is a tangent to the
curve; but, since in the present instance the curve cuts the
axis of x but once at the origin, we infer that the curve has a
single branch passing through the origin and touching the
axis of y.
23. It is evident that there must always be at least one of
the marked points upon each analytical axis; for otherwise the
equation could be divided throughout by y or by x, and its locus
would not be a proper curve of the nth degree, but the combination consisting of the straight line y - o, or x = o, and a
curve of the (n - I)th degree. If there is but one marked point
on an analytical axis, it divides the side of the analytical triangle
into parts which indicate respectively the number of times the
curve cuts the corresponding geometrical axis at the origin
and at infinity, and the curve cuts the axis in no other point.
But, if there are two or more marked points upon the analytical
axis, the number of spaces between the most distant of these
points indicates the number of finite points of intersection with




CUR VE TRACING


[Art. 23


the axis distinct from the origin. When there are two or more
of these roots, a pair of them may be imaginary, or they may be
real and equal. In the latter case, the curve cuts the geometrical axis in two coincident points, which generally indicates
tangency to the axis, as in the case of the two zero roots fory
considered in Art. 22.
The Line at Infnity
24. It was shown in Art. 5 that, if we put equal to zero
the group of terms of the highest degree in the equation of a
curve, the resulting equation gives the values of m, the ratio of
y to x, for the points at infinity. These terms are those whose
representative points are situated upon the third side, AB, of
the analytical triangle; and it is customary to speak of this
equation as determining the intersections of the curve with the
line at infinity.  Each of the real intersections generally determines, as shown in Art. i6, the direction of an asymptote.
25. The equation in question is of the general form
Lxn + Mxn-Iy +.. + Ry= =o,*              (b)
which, being of the nth degree, determines n values of y; thus,
the line at infinity, like an ordinary straight line, is said to cut
the curve in n real or imaginary points.  If the term Lx, represented by the vertex A of the analytical triangle, is wanting
* If the equation of the curve is as usual rendered homogeneous by the introduction of the letter a, which may be regarded as denoting the unit of length, this
equation can be derived from the equation of the curve by putting a = o; just as
the equation for the intersections with the axis of x is derived by putting y = o.
Hence, asy = o is the equation of the axis of x, so the impossible equation a = o
is regarded as the equation of the line at infinity.




THE LINE AT INFINI TY


19


in the equation of the curve, L = o in equation (b), and y = o
is a root of the equation; that is, one of the points of intersection is that in which the line at infinity is cut by the axis of x.
We have indeed already seen in Art. 21 that, in this case, one of
the intersections with the axis of x is at infinity. In like manner, if the term represented by the vertex B of the analytical
triangle is wanting, R = o in equation (b) and x = o is a root
of the equation, the corresponding point at infinity being that
for which m = oo, the point in which the line at infinity cuts
the axis of y. An inspection of Fig. 8, page 15, shows that the
curve whose equation,
2X2y - x2 + y + 2X = o,         (I)
is there placed upon the analytical triangle passes through both
of the particular points at infinity considered above, accordingly, the curve, which is traced in Fig. 4, page 8, has an infinite branch in the direction of each axis, one of which has an
asymptote, while the other is a parabolic branch.
Asymptotes Parallel to One of the Axes
26. When the case considered in the preceding article
arises, the asymptote if it exists is very readily found. Thus,
dividing equation (I) of the preceding article by x2, we have
2y - I + -  +- = o.            (2)
x2  X
Let x be made infinite while y remains finite in this equation;
the result is
2y - I = o,                (3)


which determines the value of y when x is infinite, and is there



20


CUR VE TRA CING


[Art. 26


fore the equation of the asymptote. Compare Art. 1. It is
evident that, in this process, all the terms vanish except those
containing the highest power of x which occurs in the equation,
that is to say, the terms represented by the points adjacent to
the vertex A of the analytical triangle; hence the result is
arrived at by simply putting the sum of these terms, namely,
2x'y -  2, equal to zero, and rejecting the common factor x2.
Parabolic Branches
27. The attempt to find an asymptote parallel to the axis of
y in the case of the curve considered in the preceding articles
results in an impossible equation. Thus, dividing equation (i)
by y2, we have
x2  X2         X
2   -   + I +2- - = 0,
y   y2        y2
in which it is impossible to make y infinite while x remains
finite; we infer, therefore, that the curve has parabolic branches
in the direction of the axis of y; that is, infinite branches
which, like the parabola, have no asymptote. This is owing to
the absence from the equation of the curve of the term represented by the point on the third side AB of the analytical triangle adjacent to the vertex B. See Fig. 8, page 15. Now, in
consequence of the absence of this term, the equation for the
intersection with the line at infinity has a double root x = o,
so that the line at infinity is said to meet the curve in two coincident points where it crosses the axis of y. It is evident that this
would also occur if the curve had two asymptotes parallel to
the axis of y; but, since Fig. 8 shows that the curve meets the
axis of y but once at infinity, the diagram indicates the parabolic
branch.


28. In general, the line at infinity is said to meet the curve




PARABOLIC BRANCHES


21


in two coincident points whenever the equation for its intersections has a pair of equal roots, and this usually indicates a parabolic branch. For, let the equation of the curve be written in
the form
Pn + Pn-, + P_,2 +... = 0,           (I)
where Pn denotes the sum of the terms of the nth degree, P,_,
the sum of the terms of the (n - i)th degree, and so on: then,
if m, is a double root of the equation P, = o, equation (i) may
be written in the form
(y - mIx)2Qn_2 + PnI + Pn-_ +.. = o,     (2)
in which Q,,_ is an expression of the (n - 2)th degree. Putting y = mIx in the expressions Qn,,, P,_,, etc., the equation
takes the form
(y - mx))2xnq,-2q   + xn- p,_- + x7-2p,-2 +... = o, (3)
in which q,_,, p,_,, etc., are numerical quantities. Dividing by
xn-, and making x infinite, it is plain that this gives an infinite
value to y - mx, except when pn_, = o. Hence in general we
have a parabolic branch, and the line at infinity is regarded as a
tangent to the curve. This may be further explained as meaning
that, whereas the asymptote to an infinite branch is the tangent
whose point of contact is at infinity, a parabolic branch is one
for which the tangent whose point of contact is at infinity is altogether at an infinite distance.
Parallel Asymptotes
29. The exceptional case mentioned in the preceding article
occurs when the substitution y = mx reduces the expression
P,_, to zero; that is, when P,,, contains y - mx as a factor.




CUR VE TRACING


[Art. 29


When this is the case, the value of y - mx, when x and y are
infinite, may be determined by a quadratic equation.  For
example, let the equation of the curve be
X3 - 2x2y + xy2 + ax2 - axy - 2a2X + a2y = o,  (I)
which may be written in the form
x(x- y)2 + ax(x - y) - 2a2x + a2y = o.
Dividing by x, and making y = x when x is infinite, we have
the equation
(x - y)2 + a(x - y) - a2 = o,
which gives
x-y+~   (I +  5) =o     and    x-y+     (I -  5) =o,
2                              2
the equations of two parallel asymptotes.
The case here considered of course includes that in which
there are no terms of the (i - i)th degree; that is, when
P,,-I = o. It is to be noticed that the asymptotes, being determined by a quadratic equation, may be imaginary, in which
case there will be no corresponding infinite branches.
30. An inspection of Fig. 9, in which equation (I) is placed
upon the analytical triangle, shows that the curve
~B ~    cuts the axis of y at the origin and in two coincident points at infinity; and, since it cuts the line
0O 3     at infinity but once in the direction of the axis
c~       - A  of y, we infer that the axis of y is a tangent at
Fig. 9  infinity, that is to say, an asymptote.  The
method given in Art. 26 also shows that the asymptote is in
this case the axis itself.




PARALLEL ASYMPTOTES


23


31. The intersections with the axis of x are determined by
the equation
X3 + ax2 - 2a2x =- 
or
x(x - a)(x + 2a) = o.
The points indicated are constructed in Fig. io, together with
the asymptotes determined in the preceding articles. It is evident that a line
parallel to the parallel asymptotes cuts
the curve in two points at infinity, and
hence each of these asymptotes meets 
the curve in three points at infinity,     x7/
and in no other point. Thus the curve
consists of three disconnected branches,
as represented in the diagram.
Fig. 10
Tangents
32. The form of the curve is more precisely ascertained by
determining its inclination at known points. Dividing the
equation of the curve, equation (i), Art. 29, by x, we have
x2 - 2xy + y2+ ax - ay - 2a2 + a2  = o.   (2)
x
Now suppose the point (x,y) moving upon the curve to pass
through the origin; the direction of its motion at the instant
when it reaches the origin depends upon the value of the ratio
Y at this instant, which may be called the direction ratio of the
curve at this point. This value is determined by putting x = o
andy = o in equation (2). The result is
-2a2 + a2 -  =,
*O, O




24


CUR VE TRA CING


[Art. 32


in which all the terms of equation (2) have vanished except
those derived from the terms of the first degree in equation (i).
Hence the direction of the curve at the origin is the same as
that of the straight line whose equation is formed by equating to zero the terms represented by the points in Fig. 9
adjacent to the analytical origin. Thus,
- 2a2X + a2y = o,
or
y = 2x,
is the equation of the tangent at the origin.
33. The tangents at the other points where the curve crosses
the axis of x may be found by a method similar to that employed in the preceding article. Grouping together the terms
containing the same power of y, equation (I), Art. 29, may be
written in the form
x(x - a)(x + 2a) + y(a2 - ax  2X2) + Xy2 = o.  (3)
Now, when the point (x, y), moving upon a branch of the curve,
passes through the point (a, o), its direction ratio will be the
y                                       y
value of y-as the quantity which would be. denoted by - if
the curve were referred to the point (a, o) as origin. From
equation (3), we derive
y          x(x + 2a)
x - a    2X2 + ax- a2 - xy           (4)
whence, putting x = a and y = o in the second member, we
have
Y-  a  =3,
X - aJa,   2




TANGENTS


25


which determines the direction of the curve at the point (a, o).
Hence the straight line
2y = 3(X - a),
which passes through this point in the same direction, is the
required tangent. In like manner, the equation of the tangent
at (- 2a, o) is found to be
5Y = 6(x + 2a).
34. It was shown in Art. 17 that the position of an asymp.
tote generally depends only upon the terms of the nth and of
the (n - I)th degrees; that is to say, when the equation of the
curve is placed on the analytical triangle, the asymptote is determined by the terms represented by the marked points upon
the side AB of the analytical triangle and upon the adjacent
parallel line. The process exemplified in the preceding article
shows that, in like manner, the position of the tangent at a
point of intersection with the axis of x generally depends only
upon the terms not containing y and those containing the first
power of y, which terms are represented by the marked points
on the analytical axis of x and on the adjacent parallel line. If
the latter terms be wanting, the value of Y a  will be infix- a a,o
nite, and the tangent will be the line x = a, parallel to the axis
of y. For example, the curve
X3 - y3 + aX2 + ay2 = o
cuts the axis of x at the point (-a, o), and the equation does
not contain the first power of y; accordingly, we find
_      y     (y -   a) x2o,
x + a_-a, o y(y - a)J-a,o


thus the line x = - a is the tangent to the curve.




CUR VE TRACING


[Art. 35


35. If the equation determining the intersections with the
axis of x has a double root, the curve is said to meet the axis of
x in two coincident points. The numerator of the value of the
direction ratio, found as in Art. 33, will in this case contain
the factor x - a, supposing (a, o) to be the intersection in
question; hence its value will generally be zero, and the axis of
x will be the tangent. Thus, in the case of the curve
x3 + Xy2 - 2aX2 + a2x - a2y = o,
the equation for the intersections with the axis of x is
x(x - a)2 = o,
of which x = a is a double root, and we have
Y  1     x(x - ay: oo
X -   a,o  a2 - xy a,o
Hence y = o is the equation of the tangent, and the two coincident intersections indicate tangency to the axis of x.
Nodes
36. If, however, when x = a is a double root, the terms
containing the first power of y be wanting, or if they contain
x - a as a factor, the value of Y -   can be determined by
x -a 
a quadratic equation, as in the following example. Let the
equation of the curve be


x3 - y3- a(4x2 - 3xy + 2y2) +  (4x- - 6y) = o. (I)




NODES


27


The intersections with the axis of x are determined by the
equation
x3 - 4aX2 + 4a2x = 0,
of which x = 2a is a double root, and the terms containing the
first power of y are
3ay(x - 2a);
hence equation (i) may be written in the form
x(x- 2a)2 + 3ay(x - 2a) - 2ay -y3 = o.
Dividing by y, this becomes
FX - 2a2      x - 2a
x^   yj + 3a      -   - 2a - y = o.     (2)
z -- 2a
Denoting by P the value assumed by the fraction    when
Y
x = 2a and y = o, that is, the direction ratio of a point moving
on the curve through the point (2a, o), we have, by putting
x = 2a andy = o in equation (2),
2P2 + 3P- 2 = 0.
The roots of this quadratic are P = - 2 and P =; hence
we infer that the curve has two branches passing through the
point (2a, o), to which the tangents are the straight lines
x- 2a = -2y,        and     x- 2a =y.
37. In the case illustrated above, the curve is said to have a
node at the point (a, o). When, as in the illustration, the roots
of the quadratic are real and different, we have a crunode, or
point at which two branches, having distinct tangents, cross.




28


CUR VE TRA CING


[Art. 37


When the roots are equal, we have a cusp, or point at which
two branches meet with a common tangent. When the roots
are imaginary, there are no real tangents; hence we have an
isolated point or acnode. It will be noticed that the case of
parallel asymptotes, explained in Art. 29, is analogous to the
node; accordingly, a curve with parallel asymptotes is said to
have a node at infinity. If the asymptotes, or tangents at the
node at infinity, are found to be coincident, we have a cusp at
infinity; and if they are found to be imaginary, the curve is said
to have an acnode at infinity.
Intersection of a Curve with a Tangent
38. Since a tangent to a curve meets the curve in two coincident points at the point of contact, it can meet the curve in
only n -2 other points, the curve being of the nth degree.
Hence, if we eliminate one of the variables between the equations of the curve and a tangent, the result will be an equation
of the (n - 2)th degree. This circumstance facilitates the determination of particular points which may be useful in tracing
the curve. In particular, if the curve be a cubic, the equation
will be of the first degree, and the single point of intersection
determined will have rational coordinates. For example, the
equation of the tangent at the origin to the curve constructed
in Fig. IO, page 23, was found in Art. 32 to be
y = 2X.
Substituting in the equation of the curve, equation (I), Art. 29,
X3 - 2x2y + xy2 + ax2 - axy - 2a2x +- ay = O,
we have
X3 - ax2 = O.




~ II]  INTERSECTION OF CURVE WITH ASYMPTOTE


29


Rejecting the factor x2, which corresponds to the two coincident points at the origin, we have x- a; hence the point
(a, 2a) is a point of the curve.
intersection of a Curve with an Asympltote
39. Since an asymptote is a tangent whose point of contact
is at infinity, the remarks made in the preceding article apply
also to the intersection of a curve with its asymptotes. In particular, a cubic cuts each of its asymptotes in a single point.
When the cubic has three real asymptotes, it is convenient to
determine the three points of intersection at once by combining with the equation of the curve the equation which
represents the three asymptotes.  This equation will be a
cubic agreeing with the equation of the curve in the terms
of the third and of the second degree, for it follows from
Art. 17 that curves having the same asymptotes must agree
in the terms of the nth and (n - i)th degrees. If, therefore,
we subtract this equation from that of the given curve, we
shall have an equation of the first degree which must be
satisfied by each of the three intersections; in other words,
the equation of a straight line cutting the three asymptotes
in the required points.
40. For example, let the equation of the curve be
x2y + xy2 + 2X2 - 3xy - 2y2 = o.
The equations of the asymptotes are
X - 2 =,        y + 2 = 0,
and
x +   - 3 = o.




3o0


CUR VE TRA CING


[Art. 40


Multiplying these equations, we have, for the equation of the
three asymptotes in combination,.
xy(x + Y) + 2X2 - 3xy - 2y2 - IOX + 2)y + 12 = o.
Hence the equation of the straight line passing through the
VY           /               points of intersections
\s      i!                     is
\         1,                          5x -y- 6=o.
I\^ \ j,Combining this equa___ \ ____  Ition with the separate
a\/, |, x                  equations of the asymp----------- totes, we find the points
\Ij \ \~~~ \^of intersection to be
',,,"'      (          (2, 4), (4 -  2) and
{        \""i',  \     (a, a). The asymptotes
I 1             '\          and the line of inter'  |               \\       section are constructed
Fig. 11             in Fig. I. The curve
has a node at the origin, at which the tangents are
2x2 - 3xy - 2y2 = o,
or
x - 2y = o,     and      2x + y = o;
hence the form of the curve is that indicated in the figure.
Examples II
I. X(X2 + y2) + a(y2 - 2X2) = o.
2. y3 - x2y + xy + 2X2 = o.
3. X 2y- yx + x2 - 4y2 = o.




1 lj                    EXAMPLES                        31
4. X2y2 + 3Xy3 + X2 - xy - 2y2 = o.
5. X2y - y2X + X2 - 4y2 = o.
6. x5 - 2a3X2+ 5a3y - 2a3y2 + J-5 = 0.
7. X(X2 + y2) + X2 - y2 + x - y = o.
8. (x2 - a2)(y2 - b2) = a2b2.
9. x2(y - x) + xy + X2 + X + 2 = 0.
10. (X2 - y2)2 = a2(2 + y2).
I. x(y + x)2 + a2y = o.
12. X3 - xy2 + 3axy + ay2 = o.
I3- x2 + 4( + y) = 8.
I4. X3 - 4xy2 - 3X2 + I2Xy -  2y2 + 8x + 2y + 4 = 0.
Is. (y - 2X)(y y- X2) - a(y - x)2 + 4a2(X +y) - a3 = 0.
I6. (y2 - x2)(y - 2X) + a(4y2 - gx2) = o.
17. (y - x) 2(y + 2) + 4(Y - X)2 - i6(y + 2X) = o.
i8. (x2 - y)(x - a) + a2y = o.
19. y( -- x)2(y - 2X) + 3a( - x)X2 - 2a2X2 = 0.
20. X2y2  3Xy2 -- 4X2 + 2y2 = o.




32


CURVE TRA CING


[Art. 4I


III
Approximate Forms of Curves
41. WHEN the equation of a curve is placed upon the analytical triangle, the absence of the term represented by the
analytical origin indicates that the curve passes through the
origin of coordinates; and we have seen in Art. 32 that the result of equating to zero the terms of the first degree is the
equation of the tangent to the curve at the origin, that is to
say, the straight line which approximates most closely to the
curve at that point. Accordingly, if only one of the terms of
the first degree occurs in the equation of the curve, the tangent
is one of the coordinate axes; and it is now to be shown that
we can, in this case, with equal facility determine a closer approximation to the form of the curve at the origin.
42. For example, let the given equation be
x2y + ay2 - a2x = o,          (I)
which is placed upon the analytical triangle in Fig. 12. The
tangent at the origin is the line x = o, the axis
B
of y. The ratio y being infinite at the origin, the
ratio     may have a finite value when x and y
o      A        X
Fig. 12  vanish simultaneously. Dividing equation (I) by
x, we have


y2
xy + a --- a2 = o
X


(2)




APPROXIMA TE FORMS A T THE ORIGIN


33


and, making x and y each equal zero in this equation, we have
_- a = o;                  (3)
xJ 0, 0
hence the equation
y2 - ax = o,               (4)
in which this ratio has constantly the same value as that
assigned by equation (3), represents a curve approximating to the form of the given curve at the origin.
This approximating curve is the parabola y2= ax, _
which touches the axis of y and lies on its right; 
hence the form of the curve at the origin is as represented in Fig. 13.                            Fig.13
Approximate Forms at Infinity
43. The absence of the term represented by the vertex A
of the analytical triangle indicates, as explained in Art. 25, that
the curve passes through the intersection of the axis of x with
the line at infinity; and it is shown in Art. 26 that, when this
is the case, the result of putting equal to zero the terms containing the highest power of x which occurs in the equation is
the equation of the asymptote parallel to the axis of x. It will
be convenient to refer to this point at infinity as the point A;
thus, as in the case of the origin, the equation of the tangent
at the point A consists of the terms represented by the points
of the analytical triangle adjacent to the vertex A. Accordingly, when, as in Fig. 12, the term represented by the point
on the analytical axis of x adjacent to the vertex A is absent,
the tangent at A is the line y = o, the axis of x, as explained
in Art. 30; but, in this case, as in the analogous case at the
origin, we can with equal facility determine a closer approxima



34


CURVE TRACIZNG


[Art. 43


tion to the form of the curve at the infinitely distant point A.
Since, for a point P moving on the infinite branch passing
through A, y becomes zero as x becomes infinite, the value of
xy may remain finite. Dividing the equation of the curve,
equation (I), Art. 42, throughout by x, we have
y2
xy + a - - a2 = o.
Making x = oo and y = o in this equation, we find
xy]  -as = o.
00, 0
Hence the hyperbola


xy - a2 = o


(5)


approximates closely to
branch passing through


Y


the remote portions of the infinite
A. This hyperbola lies in the first
and third quadrants:
we infer, therefore,
______ x that the infinite branch
having the axis of x
as an asymptote lies
above the axis on the


Fig. 14


right and below it on the left, as in Fig. 14z


44. Fig. 12 shows that the curve passes also through the
intersection of the line at infinity with the axis of y, which we
shall call the point B; and the absence of the term represented
by the point adjacent to B on the side AB of the analytical
triangle indicates, as explained in Art. 27, that the line at infinity is the tangent at B; in other words, that the infinite
branch passing through B is parabolic. In this case also we
can determine an approximation to the form of the infinite




APPROXIMATE FORMS AT INFINITY


35


branch. When, for a point moving on this branch, y becomes
infinite, the ratio x becomes zero; but in this case x becomes
y
X2
infinite also, hence the ratio - may have a finite value. Dividy
ing the equation of the curve throughout byy2, it becomes
x2         X
- + a - a2 - = 0.
y         y2
Y         Y2
Making, in this equation, y and x both infinite while - is finite,
y
it is easily seen that - vanishes, and the equation reduces to
x2
- + a = o.
y
Hence the parabola
x2 + ay = o               (6)
approximates to the remote portions of the branch passing
through the infinitely distant point B. This parabola lies in
the third and fourth quadrants: we infer, therefore, that the
parabolic branches lie in these quadrants and tend to parallelism to the axis of y.
45. The approximating curves represented by equations
(4), (5) and (6) are constructed in Fig. 15; and, since the curve
represented by the given equation
x2y + a^y - ax = o
cannot cut either axis except at the origin, it is evident that it
is impossible to join the branches determined in Arts 42, 43
and 44 so as to form a continuous curve except in the manner
indicated in the figure.




36


CUR VE TRACING


[Art. 46


46. The general form of the curve being indicated-by the
positions of the approximating curves, it is of course unneces



Y
C


\.,


I\
//              \\
/
I                        I


\ I t^
II',\


sary to construct the latter,
unless an accurate drawing is
desired. It is, however, useful to notice that, when the
x approximating curve at the
origin is a common parabola,
the radius of curvature at the
origin is one half the coefficient of the coordinate whose
first power occurs, when the


Fig. 15


coefficient of the square is unity. For example, equation (4),
Art. 42, is
y2 = ax.
The equation of the circle whose radius is la, and which
touches the axis of y at the origin, is
a\2  a2
y 4-    - - = -,
2/   4
or
y2 + X2 - ax = o.


The method of Art. 42 gives for this circle the same approximating parabola as for the given curve. The circle is therefore
as good an approximation as the parabola is to the given curve.
General Method of Determining the Equations of
Approximate Curves
47. The process of finding the equations of approximating
Curves, illustrated in the preceding articles, consists in the




~:II I]  EQUATIONS OF APPROXIMATE CURVES


37


rejection of certain terms of the given equation, which vanish
at the origin or at one of the infinitely distant points A or B.
These three points, which are called the fundamental points,
may be regarded as corresponding to the three vertices of the
analytical triangle.  We shall now show that, two of the
marked points, when the equation of a curve is placed upon
the analytical triangle, being joined by a straight line, if
there are no marked points on the same side of the line with
one of the vertices of the analytical triangle, the terms represented by the points situated upon this line will constitute the
equation of aln approximate curve at the corresponding fundamental point.
48. We shall prove this proposition first when the vertex in
question is the analytical origin. Since  B
there is at least one marked point upon
each analytical axis, the line must in this  D
case cut both analytical axes, like the  s la
line CD in Fig. I6. Let p, q be the ana-  o     c    A
lytical coordinates of one of the two       Fig. 16
points, and p - r, q + s those of the other; so that the terms
represented by these points are of the form
LxPyq + Mx- ry + s.
Represent the analytical coordinates of any other point upon
the line by p - a, q + l3, in which a and / may be fractional.
Let us suppose that the equation contains a term represented
by this point; then the terms represented by the three points
will be of the form
LxPyq  Mxry      + Mx- yq+ + NP-yq+,  (I)
in.which 1N= o if the supposed term does not actually occur




38


CURVE TRACING


[Art. 48


in the equation. Now let the equation of the curve be divided
through by xtyq; these terms then become
L + MY + N.             (2)
xr     xa
From Fig. I6 we have, by similar triangles,
/   a
s   r
so that, putting y for the value of either member of this equation, we may write / = ys, a = yr, and the terms (2) may be
written
L + M    + N(x).               (3)
If, then, we suppose the ratio y to remain finite when the point
xr
P, moving upon the curve, passes through the origin, the
ratio y- will also remain finite.
Xa
49. Now consider any marked point not situated upon the
line: such a point is by hypothesis on the upper side of the
line; it therefore represents a term containing a higher power
of y than it would if, while containing the same power of x, it
were represented by a point upon the line. Hence, after the
equation is divided by xPyt, the term will be of the form
N
N HoYyt,
where t is positive; and, when x and y are each put equal to
zero, this term will vanish. The resulting equation will there



~ III]  EQUATIONS OF APPROXIMATE CURVES


39


fore contain only the terms which are represented by points
on the line; hence these terms constitute the equation of an
approximate form at the origin.
60. In the next place, the line having the same position as
in Fig. I6, let us suppose all the marked points not upon the
line to be upon the lower side of the line, B
as in Fig. 17. Then, using the same no- 
tation as before, when the equation is
divided through by xtyq, the terms represented by points on the line will take the 
same form as in expression (3), Art. 48. 0      A
Every other term contains a lower power     Fig 17
of y than it would if, while containing the same power of x, it
were represented by a point on the line; it will therefore take
the form
VN( X.,Yyt
YS
Now, if we suppose the ratio - to remain finite when x and y
Zr
both become infinite, this term will vanish. The resulting
equation will, as before, consist only of the terms represented
by points on the line; but it will in this case represent a form
at infinity.
51. Since the side AB of the analytical triangle must contain at least one marked point, it must meet the line CD
between its intersections, C and D, with the analytical axes.
If r = s, AB will coincide with CD, and the process is in fact
the same as that given in Art. 15 to determine the value of the
ratio of y to x when x and y are infinite. But suppose that s
is less than r, as represented in Fig. 17; in this case, when x
and y are both infinite and Y- is finite, Y is readily seen to be
Xr        X




40


CUR VE TRA CING


[Art. 5 I


infinite, so that the infinite point on the branch in question is
the fundamental point B. Accordingly we find that in this
case (see Fig. 17) B is the vertex of the analytical triangle
which is on that side of the line CD on which there are no
marked points, in conformity with the enunciation of the proposition in Art. 47. Since x and y are both infinite, the result
indicates a parabolic branch at the fundamental point B; and,
in like manner, if s be greater than r, we shall have a parabolic
branch at the fundamental point A.
52. In the third place, let the line cut one of the analytical
axes produced; for example, let it cut the analytical axis of y
below the origin, as in Fig. I8. Since the analytical axis of
B              y must contain at least one marked point,
c,    all the marked points not on the line are by
>^ I8  hypothesis above it; thus A is the vertex to
CQ, r -A- be considered, and the curve passes through
Fig. 18   the fundamental point A. Let the analytical coordinates of the points be p, q and p +- r, q + s; the
terms represented being of the form
LxfPyXP Mx+ ry +s.
Denoting the analytical coordinates of any other point of the
line byp + a, q + A, the term which would be represented by
that point is of the form
VNx + ay + 
Dividing the equation through by xPyq, the terms in question
become
L + Mxrys + Nxay,            (I)
which, since -= a, may be written in the form
s   r


L + Mxrys + NV(xrys)Y.


(2)




~ III]  EQUATIONS OF APPROXIMATE CURVES


Now, when the point (x, y) recedes from the origin indefinitely
on the branch passing through the fundamental point A, if
we suppose that, as x becomes infinite, y becomes zero but xrys
remains finite, xzyo will also remain finite.
53. Now consider the term represented by any marked
point not on the line.  Such a point is by hypothesis above
the line; hence the term contains a higher power of y than
it would if, while containing the same power of x, it were represented by a point on the line. Thus, after the equation is
divided through by xyq, the term will take the form
NV (xrys) Yy,
and, when y becomes zero, it will vanish. The resulting equation will therefore, as before, contain only the terms represented
by points on the line; and it will in this case be an approximate
form at the fundamental point A.
54. Finally, suppose the line to be parallel to one of the
analytical axes, for example to the axis of y, so that the curve
passes through the fundamental point A. Then, using the same
notation as in the preceding articles, we shall have r = o, and
expression (I), Art. 52, will become
L + Mys + Ny.
In this case we must regard y as remaining finite when x becomes infinite; and, since all the marked points not on the line
are on the left of it, the terms represented by these points contain powers of x lower than xP; hence, when x is made infinite
after dividing by xt, these terms will vanish as before. The
resulting equation will determine one or more finite values of
y when x is infinite; that is, one or more asymptotes parallel to
the axis of x, as in Art. 26.




42


CUR VE TRA CING


[Art. 5'


The Analytical Polygon
55. If, when the equation of a curve has been placed upon
the analytical triangle, the marked points are joined by straight
lines in such a manner as to form a convex polygon exterior to
which there is no marked point, the result is called the analytical polygon. This polygon will have at least one vertex lying
in each side of the analytical triangle, and it may have a side
lying in either side of the triangle. In the case of such a side
of the polygon, the result of equating the corresponding terms
to zero determines simply the intersections of the curve with
one of the axes or with the line at infinity. In all other cases,
the equation corresponding to a side of the polygon determines
an approximate form at one of the fundamental points, in accordance with the theorem proved in the preceding articles.
In the example given in Art. 42, the polygon reduces to a triangle having no side coinciding with a side of the analytical
triangle; the curve accordingly passes through each fundamental point, and each side of the polygon gives an approximate
form at one fundamental point.
Construction of the Approximating Curves
56. If a side of the analytical polygon when produced cuts
both axes, as in Figs. i6 and 17, and contains no marked point
except its extremities, the equation of the corresponding approximating curve will be of the form
LxPyq + Mx~- ryq+s = o,
(see Art. 48), or, rejecting a common factor,


Lxr + Mys = o.


(I)




~ III]  CONSTRUCTrON OP APPROXIMATE CURVES


43


Let us first suppose that r and s are prime to one another, so
that the side of the polygon does not pass through any point
of the analytical triangle, that is to say, any point whose analytical coordinates are integral. If r = I-and s = I, we have
simply the equation of a tangent at the origin, as in Art. 32.
In all other cases, equation (I) represents one of the family of
curves known as the parabolas; and it is obvious that when s
is less than r the parabola touches the axis of x, and when s exceeds r it touches the axis of y. When r and s have the values
I and 2, the approximating curve is the common parabola, as in
Fig. 15, page 36. Since we have supposed r and s to be prime
to one another, they cannot both be even; the parabola will
therefore always be possible, and it will consist of parts lying
in two of the four quadrants. These quadrants are readily
determined as in the following examples.
67. Let the equation corresponding to a side of the analytical polygon be
2X2 + 3y3 = o,
or
y3 = - X2.
Since x2 is positive whatever be the sign of x, y3 must be negative, hence y must be negative. The curve therefore lies in
the third and fourth quadrants; and, since it touches the axis of
y, its form is that indicated in Fig. I9, the 
curve being in this case a semicubical parabola. If this be an approximate form at the          x
origin, it indicates that the given curve has
a cusp at that point; but, if it be an approximate form at infinity, the given curve has
parabolic branches in the third and fourth
quadrants tending to parallelism to the axis  Fig. 19
of x; in other words, touching the line at infinity at the fundamental point A.




44


CUR VE TRA CING


[Art. 58


58. Again, let the equation corresponding to a side of the
polygon be
3X3 - a2y = o,
or
a2y = 3x3.
Since x has the same sign as x3, x and y must have the same
Y          sign, and the curve lies in the first and
third quadrants. The axis of x is the tangent at the origin; hence the form is that
o_0 A/_    indicated in Fig. 20, the curve being in
/  this case a cubicalparabola. If this be an
approximate form at the origin, the given
curve has a point of inflection at that point;
Fig. 20    but, if it be a form at infinity, the given
curve has parabolic branches in the first and third quadrants
tending to parallelism to the axis of y, and it is said to touch
the line at infinity at the fundamental point B.
59. If the side of the polygon when produced cuts one of
the axes produced, as in Fig. i8, and contains no marked points
except its extremities, the equation of the approximating curve
is of the form
Lxtyq + MxA + ry + s = o,
or
L + Mxry = o.               (I)
Supposing, as before, that r and s are prime to one another,
equation (I) represents one of the family of curves known as
the hyperbolas, having both axes as asymptotes. If r = i and
s = I, we have the common hyperbola, as in Fig. 15, page 36.
Since r and s cannot both be even, the curve will always be
possible; it will consist of two branches, and it is easy to




~ III]  CONSTRUCTION OF APPROXIMATE CURVES


45


determine the quadrants in which these branches lie. For
example, let the equation be
xy2 + 2a3 = 0,
or
xy2 = - 2a3.
Since y2 is positive whatever be the sign of y, x must be negative; hence the branches lie in the second and third quadrants,


as in Fig. 21. If this be an approximate form for the fundamental point
A, it indicates that the given curve has
branches approaching as an asymptote
the left end of the axis of x on both
sides; but, if it be an approximate
form for the fundamental point B, it
indicates that the curve has branches
approaching both ends of the axis of
y on the left side.
Sides of the Polygon Representing
Approximate Form


Y




Fig. 21


more than One


60. When a side of the polygon contains any other point
of the analytical triangle except its extremities, the corresponding equation represents
two or more approximate forms at the same
fundamental point. For example, let the
IGod. ~   equation of the given curve be


2y5 - 5xy2 + x5 = o,


Fig. 22
which is placed upon the analytical triangle in Fig. 22. The
side AC of the analytical polygon, which in this case reduces




46


SIDES OF THE POL YGON


[Art. 60


to a triangle, passes through the point whose analytical coordinates are (3, I). The corresponding equation is
- 5xy2 + x5 = 0,
or
Y2 = -X4,
which may be regarded as a quadratic equation to determine
Y       the ratio of y to X2. Solving, we have
-- <x                   y= ~VX2,
the equations of two approximate forms at the
Fig. 23  origin.  The form at the origin indicated by
this side is, therefore, that shown in Fig. 23.
61. The side BC in Fig. 22 gives the approximate form
y3 = 5x,
which indicates another branch passing through the origin,
touching the axis of y, and lying in the first and third quadrants. The side AB gives
V2.y + x = 0
for the only real point at infinity, and the absence of terms
represented by points on the adjacent
parallel line (see Art. 17) shows that 
the asymptote passes through the origin. The curve does not cut the axes.
or the asymptote except at the origin,
and it is symmetrical in opposite quadrants; hence its form is that represented in Fig. 24.                        Fig. 24




REPRESENTING MORE THAN ONE FORM


47


62. In general, whenever a side of the analytical polygon
is divided into segments by points having integral coordinates,
whether they be marked points or not, the number of these
segments indicates the degree of the corresponding equation,
and hence the number of approximate forms represented by
the side.
63. The equation corresponding to a side consisting of two
or more segments may have a pair of imaginary
roots.  For example, let the equation of the        C
curve be 
X2y4 + 2a2xy3 - a5y + a6 = o   (I) 
for which the analytical polygon is drawn in OD ---
Fig. 25. The side OC of the analytical polygon  F 25
gives, for the approximate forms at the fundamental point A,
x2y4 + a6 = o               (2)
which is impossible, because it gives imaginary values to the
product xy2. There are, therefore, no infinite
branches in the direction of the axis of x.
The side CD gives, for the fundamental
point B, the approximate form
xy = - 2a2,        (3)
o   x indicating branches in the second and fourth
quadrants having the axis of y for an
asymptote.  The side DE gives, for the
same fundamental point,
2xy2 = a3,        (4)
Fig. 26   indicating branches in the first and fourth
quadrants, which it is readily seen are much closer to the axis




48


CUR VE TRA CING


[Art. 63


of y than those corresponding to equation (3). The side EO
gives y = a for the intersection with the axis of y. The curve
has therefore the form represented in Fig. 26. The position of
the branches is more precisely determined by solving the equation for x and determining the limiting values of y; these are
found to be
- I ~:5
Y --          a.
2
Examples III
I. y4 + 2axy3 - ax3 = o.
2. x3y- 3Xy3 - X3 + 2y2 + y =.
3. y4 - y3x + X3 - 2X2 = 0.
4. y4 - i6X4 + x2 - 4xy = o.
5. x4 - y4 + a2xy = 0.
6. X6 + a2y4 + a2X3y + a3xy2 = o.
7. X4 - x2Y + y3 = o.
8. X4 + x2y2 - 6ax2y + a22 = o.
9. X5 + bx4- a3y2 -o.
10. y2x - x3 + aX2 - ay2 = o.
II. y4 + 2aXy2 X-  4 = o.
12. (X2 + y2)3 = 4a22y2.
13. 2X(X - y)2 - 3a(X2 - y2) + 4a2y   o.
4. x5 + y5 - 5ax3y = o.
15. x5 - 5x2Y - 3xy2 + y5 = o.
i6. x4 + y4 - axy2 = o.
17. X5 + 2a2X2y _a3y2 = o.
I8. y5 + ax4 - a2xy2 = o.
I9. x2y2(x2 + y2 + a2) = c6.
20. y4 + 2X2y2 + 8axy2 - 8ax3 = o.
21. (X2 + y2)(y - 2X)2 = 4a2y2.
22. y2(y2 - X2) = 2a2X2.
23. X3 -  y2 + ay2 = o.
24. 2X2(X2 + y2) - 4ay3 - 4a2x2 - 6a2y2 + a4 = o.
25. y4= a (y2 - X2)(y - 2x).




SECOND APPROXIMA TIONS


49


IV
Second Approximations
64. WE have seen that, when a side of the analytical polygon coincides with a side of the analytical triangle, the corresponding equation gives only the points of intersection with
one of the fundamental lines, that is with one of the coordinate
axes, or with the line at infinity; what is really determined in
the latter case being the direction of an asymptote. The
method of obtaining the position of the tangent line or first
approximation to the curve at one of these points of intersection
is explained in Arts. 16 and 33. But, when the intersection in
question is one of the fundamental points, we have a side of the
polygon which gives at once the first approximation or tangent
at the fundamental point.
We have also seen that, if the side of the polygon is parallel to one side of the analytical triangle, so that it corresponds to the same number of intersections with two of the
fundamental lines, it gives the first approximation only; but
otherwise the fundamental line with which the side indicates
the greatest number of intersections is itself the tangent, and
the side gives a second approximation, determining, in fact,
upon which side of the tangent the curve lies in the neighborhood of the fundamental point. We have now to show how
this second approximation can be determined in those cases
in which the side of the polygon gives only the first approximation,




CUR VE TRA CING


[Art. 65


65. Let us take for illustration the curve whose equation
is
X2y3 + xy3 - x2 + X -y =,         (I)
for which the analytical polygon is drawn in Fig. 27. The
side CD gives, for the tangent at the origin, the
G -F    equation
x - y = o.            (2)
C
D E    Now the value of the quantity x - y, when (x,y)
ig. 27  is a point of the curve, determines the position of
the curve with respect to the tangent; since, obviously, if this
quantity is positive, the curve is on the right of the tangent,
while if it is negative, the curve is on the left of the tangent.
In the process of finding the equation of the tangent, as in
Art. 32, we show that at the origin the ratio - has the value
unity; therefore --    o at that point, and --, which is
the ratio this last quantity bears to x, may have a finite value
when x = y = o. Dividing equation (I) by x2, we have
2-y      +   Y + y3   o
2           X
Y
in which, making y = x = o, we have
x7y L.3x
x - y                  r \
X2                    (3)    Fig. 2 
and, since x2 is essentially positive, we infer that x - y is positive on both sides of the origin. Thus the form of the curve
at the origin is that indicated in Fig. 28.




SECOND APPROXIMA TIONS


5 


66. The result expressed in equation (3) is equivalent to
the assertion that the equation
x - y =x2
represents an approximate form at the origin. This approximate form is a parabola; but it should be noticed that the
process does not necessarily give this particular curve. Thus,
if we had divided equation (I) by y2, and puty = x = o, we
should have found
x - y _
y2
or
x - y = y2
another parabola forming an equally good approximation at the
origin.
Moreover, if we put for the second member any expression
whose ratio to x2 is unity when x = y, we shall have a curve
having the same approximate form at the origin, that is, the
same tangent and the same curvature at that point. In particular, putting 2(x2 + y2) in place of x2, we have
X - y= I(X2 + y2),
or
x2 + y2  2x + 2y = o,
the equation of a circle. This circle furnishes the most convenient method of determining the form at the origin when an
accurate drawing is desired. Writing its equation in the form
(X - I)2 + (y + I)2 = 2,


we see that its centre is at (i, -  i) and its radius is V2.




CURVE TRACING


[Art. 67


67. The side EF in Fig. 27 gives, for the tangents at the
fundamental point A, that is to say, the asymptotes parallel to
the axis of x,
y3 - I = 0,
of which y =   is the only real root. Arranging the terms
according to powers of x, equation (I) may be written
x2(y3 - I) + x(y3 + I) -y = o.        (4)
Now, in finding the asymptote y -  = o, as in Art. 26, we
show that y - i, which measures the distance of the curve
from the asymptote, vanishes when x becomes infinite; hence
x (y - I) may have a finite value when x increases without
limit. From equation (4) we have
X(y - I)(y2 +  +     +y + +  -   = o,
in which, putting y = I and x = oo, we have
3X(y - I) + 2 = o,             (5)
which therefore represents a second approximation to the form
a~Y ~        of the curve when x is very
great. In equation (5), y - I
---   =-= is positive when x is negative,
Fig. 29          position of the curve with respect to its asymptote is that represented in Fig. 29.


68. In like manner, the side FG of the polygon gives the




~ IV]  SECOND APPROXIMA TIONS A T INFINITY


53


asymptote x = - i. Arranging the terms of the equation
according to powers of y, we have
y3x(x + I) -y - X2 + X = 0.        (6)
Dividing by the highest power of y which occurs except in the
first term, the equation takes the form
x2   x
y'x(x I)- i) --- +-= o,
y2X (X + I) _ I - + - = 0,
Y   Y
from which we see that, if we make y infinite and x = - I,
y2(x + I) will have a finite value: thus
- y2(x +  ) - I = O            (7)
is the second approximation. Since y2 is essentially positive,
this equation shows that x + I is negative both for positive
and for negative values of y; in other words, the branches to
which x --   I is the asymptote lie on its left at both ends of
the asymptote.
69. To complete the construction of the curve, we have
the side GC of the polygon giving, for another approximate
form at B,
xy2 = I,
which shows that the curve has branches asymptotic to the axis
of y in the first and fourth quadrants. The side DE gives the
intersection (I, o) with the axis of x, and the tangent at this
point is found to be the line whose equation is
x + y = I.


The results thus far determined are constructed in Fig. 30,




54


CURVE TRA CIVG


[Art. 70


70. The curve is of the fifth degree; but, since there are
three asymptotes parallel to the axis of x, although two of
them are imaginary, a line parallel to the axis of x meets the
curve three times at the fundamental point A, which is accord-,Y v          ingly said to be a triple point of
the curve. Hence the asymp\J', ^     tote y = I, which is a tangent
y   \   /     at A, can meet the curve in
/   only one other point. Accord-.          — " ----  ingly, putting y =  I in the
---  /o >  x equation of the curve, equation,' ",, )\   (4), Art. 67, we obtain
/'i
/     I   /                 2X - I = 0,
I  ~\ |I Ian equation of the first degree,
Fig. 30          giving the point (-, I) as the
intersection of the curve with the asymptote.
71. In like manner, a line parallel to the axis of y meets
the curve twice at the fundamental point B, and can cut the
curve in only three other points. But the axis of y itself cuts
the curve in only one other point, and meets the branch to
which it is the tangent at B in three coincident points at the
point of contact, like the tangent at an ordinary point of inflexion. The asymptote x = - I has the same character; and
accordingly, putting x = - I in the equation of the curve, we
find
- y-    = 0,
giving the single point of intersection (- I, - 2).
72. There can be no doubt as to the method of joining
the branches now determined, except in the case of those in the




THE BRANCHES DETERMIN ED


55


first quadrant. To remove this remaining uncertainty, let us
consider the intersection of the curve with the tangent line
whose equation is
x+y= I.
The result of eliminating x from the equation of the curve by
means of this equation is
3 - 3Y2 + 2y - I = 0,
an equation which is easily shown to have no negative roots.
But, writing it in the form
(y - )3 =,
it obviously has no positive root less than two. Its only real
root is in fact about 2.33. Hence the branches must be joined
as in Fig. 30.
Selection of the Terms which Determine the next
Approximation
73. It will be noticed that the additional terms employed,
when we proceed to the next approximation above that given
by the side of the polygon, as in Arts. 65 and 67, are those
represented by points situated upon the next adjacent line
parallel to the side of the polygon, exactly as in the case of
finding the first approximation when the side of the polygon
gives only points of the curve.  Compare Art. 34.  If this
parallel line contains no marked points, or if the terms represented are divisible by the factor corresponding to the side of
the polygon, we must employ the terms on the nearest parallel
line for which they are not so divisible. This is illustrated,
in the case of both classes of sides, in the following example.




56


CUR VE TRA CING


[Art. 74


74. Let the equation of the curve be
X2y2 + X3 - y3 + x2 - y2 = o,      (I)
for which the analytical polygon is drawn in Fig. 31. The side
CD gives the tangents at the origin
G                               a X - y = o  and  x +y = o.
c     F
The terms represented by the next parallel line
are
Fig. 31       x - 3 = (x - y)(X2 + xy + y2).   (2)
Now in the case of the tangent x - y = o, we have, after putting x = y in the coefficients of x - y in equation (I),
2(X -  ) + 3X(X- y) -+ X3 = O.
This equation shows that, when x = y = o, x - y has a finite
ratio to X3; for x(x - y) bears a vanishing ratio to x - y.
Thus the term arising from expression (2) vanishes, and we
have at the origin
x -y
2      +   =0;
X3
hence
2(y - x) =  3                (3)
is the approximate form at the origin of the branch whose tangent is x = y. Equation (3) shows that the curve lies above
the tangent in the first and below it in the third quadrant.
The second approximation to the branch whose
tangent is x + y = o is found, by putting y = - x 
in equation (I), to be                       _   '
2(X +  ) + 2X2 = 0,                 ' 
which shows that the curve lies below this tan-  Fig. 32
gent on both sides of the origin. Hence the complete form
at. the origin is that indicated in Fig. 32.




SECOND APPROXIMA TIONS


57


75. Consider next the side DE of the polygon, which by
itself gives only the point of the curve (- i, o). Writing equation (I) according to powers of y,
X2(X + I) +y2 (X2- I) -y3 = o.      (4)
There are no marked points on the parallel line adjacent to the
side DE of the polygon, which shows that x = - I is the tangent, as in Art. 34. Moreover, the expression y2(x2 - I), corresponding to the next parallel line, contains the factor x + I,
which vanishes at the point in question. Hence it is necessary
to include the term represented by the point G; thus, putting
x       I - I in the coefficients of y and x + i in equation (4),
X + I - 2y2(X + I) - y3 = O.
In this equation, x + I bears a finite ratio to y3 at the point
(- i, o); for y2(x + I) bears a vanishing ratio to
x +-. Hence the approximate form at this point  j
is                                           _
y3 = X +;                         x
and, since y and x + i have the same sign, the  Fig. 33
form of the branch is that indicated in Fig. 33.
76. To complete the construction of the curve, we have the
side FG, giving
X2 = y,
which indicates parabolic branches in the first and second
quadrants in the direction of the axis of y. The side EF gives
y2 = - x,
hence the curve has parabolic branches in the direction of the
axis of x in the second and third quadrants.




CUR VE TRA CING


[Art. 76


The side GC gives the intersection (o, -  ). Proceeding
as in Art. 75, the tangent at this point is
y = - I,
and the second approximation is
y + I = 2X2,
which shows that the curve lies above the tangent, and that
the radius of curvature at this point is j. See Art. 46.
77. The inflexional tangent y = x in Fig. 32 meets the
branch to which it is tangent in three coincident points, and
crosses the other branch at the node: it therefore meets the
curve in four coincident points at the origin; and, since the curve
is a quartic, it cannot meet the curve again. The inflexional
tangent x = -   (Fig. 33) meets the curve in three coin\  Y     cident points at the point of
contact, and meets it also at the
fundamental point B; hence the
curve cannot again cross this
tangent. It is therefore neces- {,  /// sary to join the branches deterI\,/    mined as indicated in Fig. 34.
/ _ _x    The tangent y  -x meets
>'/j    the curve in three coincident
points at the origin; and, putting
^I 'y = - x in the equation of the
Fig. 34         curve, the fourth point of intersection is found to be (- 2, 2), thus determining a point on the
infinite branch in the second quadrant. The line x =- 2
will be found to touch the curve at this point, and to meet it
again at (- 2, - I).




SUCCESSIVE APPROXIMATIONS


59


Successive Approximations
78. When the side of the polygon gives a point only of the
curve and the first approximation or tangent has been found,
a second approximation may, if desired, be determined in the
manner illustrated below.
Let the equation of the curve be


x2y2 - X3y -  - x2y + 2X2 - 2X = 0,


(I)


for which the analytical polygon is given in Fig. G
35. The side CD gives the point of intersection
(I, o) with the axis of x. Arranging the terms  \   E
according to powers of y, the equation is      C D
Fig. 35
2X(X - I) - y(3 + X2) + y2X2 - Y3 = o.  (2)


The first approximation is found by dividing this equation
by y, and then putting x = I and y = o, while it is assumed
that --   has a finite value at the point (I, o). The result is
By


X - I
2 --- 2 =0;
Y


(3)


whence we infer that
x - I -y = o
is the equation of the tangent. Now, since equation (3) is
equivalent to
x - I - y
--— y o,
By


the ratio which this fraction bears toy, that is, the quantity
x- I -y
t.




60


CUR VE TRA CING


[Art. 78


may have a finite value at the point (i, o). To ascertain this
value, divide equation (2) by y2, thus
2X(X - I)   X3 + X2
--       + x2 — y = 0.
y2         y
Before putting y = o in this equation, we must substitute
for x its approximate value I + y, derived from the first approximation, in the numerators of the fractions, and retain all the
terms which have a finite value when y = o. Thus we have
2(I + y)(X - I)   I + 3Y +... + I + 2y +....
Y2                      y
or
2(X - I) - 2y   2(X - I)
y2            y      5    I = o;
By
and, since      I     at the point (I, 0), this becomes
x —I —y(
X= I.               (4)
y2
Hence we infer that x - I - y is positive in the neighborhood
of the point (I, o), and the form of the curve is
q,/  therefore that represented in Fig. 36.
0o /  x     Equation (4) may be used to determine the
/ '      radius of curvature, as in Art. 66.
Fig. 36    79. We may, in a similar manner, determine a
second approximation when the side of the polygon gives only
an intersection with the line at infinity and the first approximation gives the asymptote. For example, the side EF in Fig.
35 gives
1o = I




SUCCESSIVE APPROXIMA TIONS


determining a point at infinity. The equation of the curve is
2y(y - x) - (y3 + yx2) + 2x2 - 2X = o,  (5)
and the asymptote is found by dividing by x3 and then making
y = x = oo, while y - x is assumed to have a finite value.
The result is
y -   - 2 =,               (6)
the equation of the asymptote. Now, since equation (6) shows
that the quantity y- x - 2 vanishes when x is infinite, the
quantity
x(y - x - 2)
may have a finite value when the point (x, y) recedes indefinitely upon the branch to which equation (6) represents the
asymptote. To ascertain this value, divide equation (5) by x2,
and put y = x + 2, as given by the first approximation, before
putting x infinite. Thus we have
3 + 6X2 +... + X3 + 2X2      2
(X + 2)(y - x)  -                       + 2   - = o;
X2                 x
and, rejecting terms which vanish when x is infinite, this
becomes
x(y - x - 2) + 2(y - x) -8 + 2 = 0,
or, since y - x = 2 when x is infinite,
x(y - x - 2) = 2.              (7)
Hence we infer that y - x - 2 is positive when x is positive
and great, and negative when x is negative and great; the
curve is therefore above the asymptote for distant points in
the first quadrant, and below it for distant points in the third
quadrant. See Fig. 37, page 64, in which the curve is traced.
80. It should be noticed that, in the processes illustrated in
Arts. 78 and 79, while the terms employed in the first approx



62


CUR VE TRAA CING


[Art. 80o


imation are those represented by points on the side of the
polygon and on the adjacent parallel line, the additional terms
introduced in the succeeding approximation are represented by
points on the next parallel line.
Asymptotic Parabolas
81. A process of successive approximation, similar to that
employed in the preceding articles, enables us to determine
more accurately the position of a parabolic branch at one of
the fundamental points A or B. Such a branch is indicated by
the side FG of the analytical polygon in Fig. 35. Let the
equation of the curve be written in the form
y2(X2 - y) - x3y- x2Y + 2X2 - 2X = o,  (8)
the terms included in the first expression being those corresponding to the side of the polygon. These terms alone
produce terms with finite values when the equation is divided
byy3; the result being
x2'                X2 - Y~
]or    x   Y]            (9)
y oo                 y,o
from which we infer that the parabola
x2 =y                   (Io)
is an approximate form at the fundamental point B. Now,
sifice it follows from equation (9) that  = oo, the product
Y x2   y      or     x2 X y
or
X    Y                 X
x    y                 x
may have a finite value at B. This will, in fact, be found to be
the case when the equation contains terms represented by
points of the analytical triangle situated upon the parallel line
adjacent to the side of the polygon in question.  In Fig. 35




ASYMPTO TIC PARABOLAS


63


there is one marked point so situated, namely the point E;
accordingly, dividing equation (8) byy2x, we have
x2 -y    X2   X    X    2
----  -  -  - + 2-  - = 0
x      y   y     y2  y2
in which the only additional term that does not vanish -at the
fundamental point B is that represented by the point E. The
result is that at B
X:    ] - I= o;               (II)
X   oo
hence we infer that the parabola
x2   y - x == o              (12)
is a closer approximation to the curve than that represented
by equation (io).
82. Equation (ii) shows that 2   - y  x = o at the fundamental point B; hence the product of this quantity by x,
namely
x2 - y - x,
may have a finite value at B. To find this value, divide equation (8) by y2, thus
X3   X2    x2   X
2 -y - - - - + 2- - 2- =.       (I3)
Y    Y y   2 y   2
In order to retain all the terms of this equation which have
X3
finite values at B, it is necessary in the infinite term -to substitute for x2 its value y + x as given by the second approximation, equation (12). The result is that at B
x(y + x)
X2 - y - X(       - I = 0,
Y




64


CUR VE TRACING


[Art. 82


or
hence the parabola


X2
c2 - y - x - - - I = o;
-2 y- - - 2 = 0


(I4)


is a still closer approximation than that represented by equation (I2). This curve is called the asymptotic parabola.
83. Putting the equation of this parabola in the form
(X -        + 2,
we see that its vertex is at (1 - 24) and its parameter is unity.
VV~ 3~ LIUL~ ~~ ~IL~~ rU e \2


Fig. 87
The parabola is constructed in Fig. 37, together with the
branches determined in Arts. 78 and 79, and those correspond



ASYMPTOTIC PARABOLAS


ing to the remaining sides of the analytical polygon in Fig. 35.
In joining the branches, it is obvious that the oblique asymptote must be crossed in the fourth quadrant, and also in the
second quadrant; and, since this asymptote cannot be crossed
again, the remaining branches must be joined as in the figure.
84. The process given in Arts. 81 and 82 may be continued
in order to determine the value of the quantity
X(X2 - y- x- 2),
which may be finite at the fundamental point B, since by equation (14) the quantity in the parenthesis vanishes at B. Multiplying equation (I 3) by x, we have the equation of the curve in
the form
X2 (X2 + X)   X3    x2
X(X2 - y)            + 2 -    2 - = 0.
y         y2    y2
The last two terms vanish at B; and, employing the value of X2
given by equation (14), we have at B
(  -  )  (y + x + 2)(X2 + X) 
y
x(X2 -y) _ (x2 + X) _ (X + 2)(y + 2X + 2)
y
X(X2 -y) - (X2 + X) - (X + 2) - -2(X + 3      _ + 2)  o, 
y
and, rejecting vanishing terms,
X(X2 - y- X - 2) = 4.           (I5)
By equation (14), the quantity y - x + x + 2 is the difference between the ordinate of the curve and the corresponding




66


CUR VE TRA CING


[Art. 84


ordinate of the asymptotic parabola. Hence we infer from
equation (15) that the curve is below the parabola in the first
quadrant, and above it in the second quadrant, as represented
in Fig. 37; the distance measured in the direction of the axis
of y being, when x is great, about 4. *
x
Examples IV
I. X2y3 + xy4 - y3 - X2 + X = O.
2. X4 - y4 + 2ax3y + a2y2 + a4 = o.
3. X4 - 2ay3 - 3a2y2 - 2a2X2 + a4 = o.
4. X3 + X2 - y2 + 4Y  = o.
5. x4 - 2ay3 - 2a2x2 + 3a2y2 = o.
6. (y2 - 4a2) (X2 + y2) + 4a4 = o.
7. X(X2 + fY)(Y + 2X) + 4ax3 + 2ay3 = o.
8. xy2 + yx2 + ay2 -    a2x = o.
9. (y - x)2(y + 2x) = 2y(y + IIX).
Io. y4 - 2ay3 4- 2a'2x - 4a2xy + a2x2 = o.
II. x2y2 = a(x3 + y3).
12. y(x + y)2    4ax(x + y) + 4a2y = o.
13. X6   x4y2 - 2a2xy3 - a4x2, + 3a5y = o.
14. (x -y ) (x  - 42) = 2ax2 -     oaxy + 20ay2 + 24a2x.
15. y4 - 2aXy2 -3a2y2 - 2a3X = 0.
16. X2f - 2xy3 - 2ax2y + 2axy2 +a(x- y)2 = o.
17. X4 + x3y + X2y2 - 2axy2 - ay3 - a2x2 + a2y2 = o.
I8. x3y - x2y2 - X3 - X2y + xy2 + y3 + X2 = o.
* This result, as well as that found in Art. 79, equation (7), gives a good approximation only when the value of x is considerable. For instance, when x = 5,
the ordinate of the parabola is i8, and that of the rectilinear asymptote is 7; the
values of y for the curve are about I5.5 and 9.4, differing from the former by 2.5
and 2.4 respectively, whereas the differences given by equations (i5) and (7) would
for this value of x be.8 and.4 respectively.




CASES 'OF PQAL ROOTS


67


V
Cases of Equal Roots
85. THE case in which an equation has a pair of equal roots
is ordinarily an intermediate case between that in which two
roots are real and distinct and that in which they are imaginary. Singular cases of equal roots, however, frequently arise,
such that the roots which become equal are real and remain
real, or are imaginary and do not become real and distinct.
For example, if the equation of a curve is regarded as determining the value of y, the case of equal roots ordinarily corresponds to a limiting value of x, which determines a tangent
to the curve, as in Art. 3; and the singular case of equal
roots occurs when, for a particular value of x, two values
of y become equal, but are either real for neighboring
values of x on both sides of the particular value, as in the
case of the real double point or crunode, or imaginary for
such values of x, as in the case of the isolated point or
acnode.
86. When the equation which determines the intersections
of the curve with the axis of x has a pair of equal roots, the
ordinary case is that in which the axis is a tangent, as shown
in Art. 35 by considering the first approximation; and, in like
manner, when the equation for the intersections with the line
at infinity has equal roots, this usually indicates tangency to




68


CUR VE Z'RACING


[Art. 86


this line, that is to say, a parabolic branch. It is to be noticed
that in these cases the process of approximation determines at
once the side of the tangent on which the curve lies, and in
the case of the line at infinity it determines in which of two
opposite quadrants the parabolic branches lie. For example,
take the curve whose equation is
X4 - 2x2y2 + y4 - 2ax2y + ay3 = o,     (I)
for which Fig. 38 gives the analytical polygon. The side AB
gives for the intersection with the line at
B.~ ~      infinity
Co                        X4- 2x2y2 +y4 = O
or
3D<                  (x + y)2(x _ y)2 = o,
0         A
Fig. 38
g.     an equation having two pairs of equal roots.
Now an infinite branch for which:  = I must lie in the first
coo
or in the third quadrant, and from equation (I) we have
( _  )2   2aX2y - ay3
(x + y)2
in which, putting y = x, we have, when x is infinite,
ax
(x - y) =.
4
The corresponding branches are parabolic because x - y is
infinite when x is infinite; and, since this quantity is real only
when x is positive, the branches are in the first quadrant.




PARABOLIC BRANCHES


69


87. The construction of the curve is readily completed; for
the side AD of the polygon
gives the form at the origin 
X2 = 2ay,
the side CD gives the tangents
y = ~V-2.x,
and the side BC gives the point
(o, - a). Moreover, the equation containing powers of x with        Fig. 39
even exponents only, the curve is symmetrical to the axis of y;
hence its form is that indicated in Fig. 39.
88. Supposing, as in Art. 86, that the equation giving the
intersection with one of the fundamental lines has equal roots,
the singular case, corresponding to a node, is indicated by the
occurrence of a quadratic equation for the first approximation,
as in Art. 36, the roots of this quadratic being real or imaginary according as the node is a crunode or an acnode. In the
case of the line at infinity, the singular case is that of parallel
asymptotes; and these may be real forming a crunode at infinity, as in Art. 29, or imaginary forming an acnode at infinity,
there being in the latter case, of course, no corresponding
infinite branches.
Cusps
89. Let us now suppose that equal roots occur in the quadratic equation determining the first approximation or tangents
at a node. This will usually constitute a case in which the
curve is real for values of x on one side, and imaginary for




70


CURVE TRACING


[Art. 89


values of x on the other side, of the particular value of x in
question. For example, let the equation of the curve be
X2(X2 + y) - 2aX3 + 2ax2y - axy2 + a2(x - y)2 = o, (i)
for which Fig. 40 gives the analytical polygon. The side CD
gives, for the tangents at the node at the origin,
D(          the equation
(X - y)D = o,
C   A
Fig. 40
Fig. 40  which has equal roots, showing that both tangents coincide with the line
y = x.                  (2)
Proceeding to the next approximation, by dividing equation (i)
by x3 and making x = y = o, we have at the origin
a2(X  y)2
X3
hence we infer that near the origin the quantity y - x is real
when x is positive, and imaginary when x is negative. Moreover, in the former case, this quantity, which is the difference
between corresponding ordinates of the curve and
the tangent, has two values, one positive and the
other negative. Hence the curve lies on the right  o
of the origin and on both sides of the tangent,  /  X
the two branches forming a cusp at the origin, as  /
in Fig. 41.                                   Fig 41
90. The side CA of the polygon in Fig. 40 gives, for the
intersection with the axis of x, the equation


X - 2ax + a2 = o,




CUSPS


7'I


which has a pair of equal roots.  Arranging equation (I)
according to powers of y, it is
X2(X - a)2 + 2ax(x - a)y + (x2 - ax + a2)y2 = o,
and the occurrence of the factor x - a in the coefficient of y
renders this a quadratic equation for the ratio of y to x - a at
the point (a, o), as in Art. 36; hence there is a node at this
point.
Putting x = a, the quadratic is
(x - a)2 + 2(x - a)y +y2 =0,
which has equal roots; hence the point is a cusp at which the
tangent is the line
x- a +y = o.                (3)
91. The tangent at a cusp, like a tangent at an ordinary
node, meets the curve in three coincident points at the point
of contact; thus, the curve being a quartic, the tangenty = x
can meet the curve in only one other point, which is found to
be (ia, Ia), and the tangent (3) passes through the same point.
The curve has no infinite branches;
for the side AE of the polygon gives  Y
x2 + y2 = o, 
indicating imaginary intersections with  ^
the line at infinity; and the side DE ~
gives, for the parallel asymptotes or
tangents at the fundamental point B,
X2 - ax + a2 = o0                Fig. 42
which has imaginary roots, so that B is an acnode.
The shape of the curve is therefore that shown in Fig. 42.




CURVE TRACING


[Art. 92


Tacnodes
92. When equal roots occur in the equation determining
the tangents at a node, we have seen in Art.. 89 that we ordinarily have a cusp; but here also a singular case, analogous to
that explained in Art. 85, may arise; that is to say, the quantity
which measures the distance of the curve from the tangent
may be real on both sides of the node, or imaginary on both
sides of the node. For example, let the equation of the curve
be
X2y2 + 2ay(x2 -y2) + a2(x - y)2 = o,  (I)
for which Fig. 43 gives the analytical polygon. The side CD,
indicating a node at the origin, gives an equation
F         with equal roots, and the line.\                y= X               (2)
is the only tangent at the origin, as in Art. 89.
Fig. 43  But, since in this case x - y occurs as a factor in
the terms of the third degree also, we no longer obtain a finite
ratio between (x - y)2 and x3; in fact, putting in equation (I)
y = x, as determined by the first approximation, we have
X4 + 4ax2(X -y) + a2(x - y)2 = o,
a quadratic equation for the ratio of x - y to x2. Solving, we
obtain real roots, namely                      y
a(x -y) = (-2     3)x2;   (3) 
0    X
hence the quantity  - y is real on both sides of 
the origin, having two values, both of which are
negative.  Thus the form of the curve at the   Fig. 44
origin is that represented in Fig. 44. The two branches having
a common tangent but different curvatures are said to form a




~V]


TA CNODES


73


tacnode. We have already had in Art. 60 an example of a tacnode in which the common tangent is one of the coordinate
axes.
Cusps at Infinity
93. The line DE in Fig. 43 corresponds to a node at the
fundamental point A; the equation is
y2 + 2ay + a2 = o,
which has equal roots; hence the curve has two tangents at A,
or asymptotes, coincident with the line
y + a = o.
Arranging the terms according to powers of x. the equation of
the curve is
x2(y + a)2 - 2a2xy - 2ay3 + a2y2 = o.
The next approximation is found, as in Art. 67, by dividing by
x and then making x= oo
and y = - a as determined
by the first approximation.
The result is
x(y + a)2 + 2a3 = 0;
whence we infer that for the                0       x
infinite branches x must be
negative, and that y + a will..-. -
have two values, one positive
and  the   other  negative.            Fig. 45
Hence the curve has two infinite branches approaching the
left end of the asymptote, one on each side of it, as in Fig. 45,
forming a cusp at the fundamental point A,




CUR VE TRACING


[Art. 94


94. To complete the construction of the curve, we have the
side. EF giving parabolic branches in the first and second
quadrants, and the side FC giving the point (o, -a). The tangent y = x meets the curve in four coincident points at the
origin, and therefore cannot meet it again. The asymptote,
being a tangent at a cusp, meets the curve in three coincident
points at infinity, and the fourth point of intersection is found
to be at (- 3a, - a).
Ramphoid Cusps
95. When a quadratic equation presents itself for the second approximation at a node where the tangents are coincident, if the roots are real and distinct, as in Art. 92, the two
branches having the common tangent have different curvatures; and these branches are real upon both sides of the node,
forming a tacnode. If the roots were imaginary, there would,
of course, be no branches, and the node would be an isolated
point of the curve.
96. Suppose, now, that this equation has equal roots; the
two approximate forms determining the curvature come into
coincidence, and it will usually be found that the curve is real
on one side and imaginary on the other side of the node, as in
the case of the ordinary cusp. For example, let the equation
of the curve be
X4 - 2x2y - 2xy2 2+  = o,       (I)
which is placed upon the analytical triangle in Fig. 46. The
side AC, which represents a second approxiC AD          mation at the origin, the axis of x being the
tangent, gives the quadratic
F A             x4 - 2X2y + y2 = 0,
Fig. 46
which has equal roots, and


X2 _-y = O


(2)




RAMPHOID CUSPS


75


is the approximate form determining the curvature at the origin. Now, writing equation (I) in the form
(X2 - y)2 = 2Xy2,
it is obvious that the curve has two real branches when x is
positive, and is imaginary when x is negative.
Hence the form at the origin is that represented  Y
in Fig. 47.                                        A 
Two branches thus lying on the same side of   o    x
the common tangent and not passing through the 
point of contact are said to form a ramphoid cusp.
97. To complete the construction of the curve, we have the
Y   l I!   side CD, Fig. 46, giving the asymptote.~/            x = x,
IJ  /    and the side DA giving the form
x~ =   X ~x3 = 2y2
at the fundamental point B.
Equation (2) shows that the radius of
curvature of both branches at the origin
to \    is 1. The form of the curve is shown in
ig   \     Fig. 48.
Fig. 48
Circuits
98. The branches or portions of a curve corresponding to
the sides of the analytical polygon form, when properly joined,
one or more geometrically continuous branches, which are
either closed curves or else lines extending to infinity at each




CUR VE TRA CING


[Art. 98


end. A branch of the latter character may be regarded as an
arc of the curve terminated in each direction by 6ne of the
points in which the curve meets the line at infinity. But we
may regard the infinite branch as continuous at one of these
points with the other infinite branch which terminates at that
point, thus making the curve to consist entirely of reentrant
branches or circuits, which may or may not be cut by the line
at infinity. For example, the conic consists always of a single
circuit, which in the case of the hyperbola is cut into two arcs
by the line at infinity.
99. A circuit is odd or even according as it is cut in an odd
or an even number of points by a straight line. Obviously an
odd circuit must cut every straight line in at least one point,
and in particular it must cut the line at infinity in at least one
point. A curve of odd degree must contain at least one odd
circuit. Thus a cubic contains always an odd circuit which
cuts the line at infinity at least once, and it may in addition
contain an even circuit, which last may be a closed curve or
oval not cutting the line at infinity.
100. A circuit containing a crunode consists of two parts
or loops, the two extremities of a loop being at a common point
but not having a common tangent. For example, in Fig. 48
both extremities of the infinite branch in the first quadrant are
at the fundamental point B; but it does not form a complete circuit, because the tangents at the extremities are the asymptote
and the line at infinity respectively. The branch in the fourth
quadrant is the other loop completing the circuit. An acnode
is a point at which an oval vanishes. Thus the nodal variety
of a curve is intermediate between varieties which differ in the
number of circuits. As an illustration, take the curve whose
equation is
(X2 + y2 _ a2)2 = c4 - 4ay2.




CIRCUI TS


77


The curve has no infinite branches, and is symmetrical to both
axes. For its intersections with the axis of x, we have
X2 - a2 _= ~ C2
or
x = ~ V(a2 ~ c2);
and for its intersections with the axis of y,
(y2 + a2)2 = C4,
or
y =    /(~2- a 2).
The curve always cuts the axis of x at the distances ~ \(a2 + c2)
from the origin; and if c < a, it cuts this axis also at the
distances ~ V(a2 - c2), and does not cut the axis of y. In
this case the curve consists of two ovals. If c > a, the curve


cuts the axis of x only at the
distances V(a2 - c2) and it cuts
the axis of y at the distances
V(c2 - a2) from the origin. In 
this case it consists of a single
oval. In the intermediate case, 
when c = a, there is a crunode at
the origin. The three varieties
of the curve, which is known as
the Cassinian, are drawn in Fig. 49.
lemniscata.


Y


0


x


Fig. 49
The nodal case is the


101. It is evident that a circuit which does not cut either
of the fundamental lines, namely the coordinate axes and the
line at infinity, will not be detected if we employ only the
method of the analytical polygon. The portion of the plane in
which such a circuit may possibly lie may generally be greatly
limited by the consideration of the intersections of the curve
with a straight line moving in such a manner as to sweep over




78


CUR VE TRA CING


[Art. IOI


the entire plane. For example, in Fig. 48 the curve is a quartic and the origin is a node; hence a line passing through the
origin meets the curve twice at the origin and in two other
points. If the line revolves in the positive direction from
coincidence with the axis of x through 180~, it sweeps over the
entire plane, and we notice that the two points are real until
the line arrives at a position in which it is tangent to the
branch in the fourth quadrant. Hence, although there can
obviously be no branch in the fourth quadrant other than that
drawn, we might still suspect the existence of an oval in the
second quadrant.
But, if we put y = mx in the equation of the curve, and solve
for x, as in Art. 13, we find
x = m(I + m) ~ m \m(m + 2)t;
whence we see that the two values of x are real until m = - 2,
and imaginary for all values of in between - 2 and o. Hence
there is no other circuit. The limiting value, mn = - 2, gives
x = 2; therefore (2, - 4) is the point of contact of a tangent
passing through the origin.
Auxiliary Loci
102. When the equation of a curve is in such a form that
each of its members is readily resolved into simple factors, the
loci of the results of equating these factors separately to zero
can frequently be used in constructing the curve in the manner
illustrated below.
For example, let the equation of the curve be
y(y2 - ax) = (y2 + a2)(x - a).     (I)
The locus of y = o is the axis of x, and that of y2 - ax = o is
the parabola constructed in Fig. 50. It is to be noticed that
the value of the expression in the first member for the point




AUXILIARY LOCI


79


(, y) vanishes and changes sign whenever the point crosses


either of these lines, being positive in
two of the four regions into which the
lines divide the plane, and negative in the
other two.
The locus of x - a = o, corresponding
to a factor in the second member, is for
distinction constructed by a dotted line in
the figure. The factor y2 + a' is always
positive, and there is no corresponding
locus; thus the expression in the second


-I.


0/+


x


+


I\
N
Fig. 50
of the dotted


number is positive for all points on the right
line, and negative for all points on its left.


103. Since each member of equation (I) reduces to zero at a
point where the dotted line intersects a full line, it is evident
that the curve passes through every such point; it is also
evident that the curve cannot meet either of the lines at any
other point. Moreover, if we mark each of the eight regions
into which the lines divide the plane by the sign + or -,
according as the expressions in the first and second members
of the equation have like or unlike signs in that region, it is
evident that no part of the curve can lie in a region marked -.
Thus at each of the points of intersection a branch of the curve
passes from one of the vertically opposite regions marked + to
the other.
104. The points of intersection are (a, o), (a, a) and (a, - a).
The inclination of the curve at either of these points is readily
determined in a manner equivalent to that which we employ
when a curve passes through the origin. Thus, for the point
(a, o), making x =-a and y = o in the coefficients of the factors which vanish at this point in equation (i), we have


- a:y = a2(x - a);


(2)




80


CUR VE TRA CING


[Art. I04


hence
y = a -x
is the equation of the first approximation or tangent at this
point.
105. Again, at the point (a, a) we have, in like manner,
y2 - ax = 2a(x - a),            (3)
which is in fact the equation of a parabola approximating to
+         the given curve.  If we desire only to
determine the inclination at (a, a), we,//  +  write equation (3) in such a form that
-  /^ ^  y - a or x- a is a factor of each term,
thus
0              x    y     + yX  - a2 + a(a - x) = 2a(x - a);
+     and, again putting x = a andy = a, we
have
-+  ig               2(y - a) = 3(x - a)
Fig. 51
for the equation of the tangent. The process is equivalent to
transferring the origin to the point (a, a) and retaining in the
result only the terms of the first degree.
In like manner we find the tangent at (a, - a) to be
2(y + a) = x - a.
106. When, as in the present case, only two branches enter
a region, it is evident that these branches must be continuous,
as represented in Fig. 5I. The construction of this branch,
which forms a complete circuit, is readily completed after determining the asymptote, which is found to be the line x y.
The fundamental point A is an acnode; and, considering straight




AUXILIARY LOCI


8I


lines passing through this point, that is to say, lines parallel
to the axis of x, it is evident that the curve contains no other
circuit.
Loci representing Squared Factors
107. In the preceding example each of the auxiliary loci
represents a single factor, and separates regions oppositely
marked. If, now, one of the loci represents a squared factor,
the expression containing this factor vanishes but does not
change sign, when the point (x,y) crosses this locus; hence
the adjacent regions separated by it will be similarly marked.
Such a locus may be regarded as formed by two auxiliary loci of
the same system coming into coincidence. Hence, at the point
where it crosses a single locus of the other system the curve
meets the single locus in two points which have come into coincidence; that is, it touches the single locus; in fact, it lies on
both sides of the double locus, and on that side of the single
locus where the adjacent regions are marked +.
108. To illustrate, let us take the curve whose equation is
4y2(x - a)2 (2a - x) = X(X2 _y2 - 4ax + 5a2)2. (I)
The auxiliary loci for the first member are the axis of x, the
line x = a and the line x = 2a, which are drawn as full lines
in Fig. 52. The loci for the second member are the axis of y
and the rectangular hyperbola
x2 - y2 - 4ax + 5a2 = o,
or
(X - 2a)2 _ y2 + a = o,         (2)
which are drawn as dotted lines. The origin is a point at which
a single factor meets a double factor, and accordingly the curve




82


CURVE TRACING


[Art. 0o8


touches the axis of y, lying on its right side where the regions
are marked +.   In fact, if we put x = o and y = o in the coefficients of y2 and x in equation (I), we have
8a3y2 = 25a4x,
the process being equivalent to that of finding the approximate
form at the origin.
Again, the single locus x = 2a cuts the hyperbola, which is
a double locus, in the points (2a, + a); hence the curve touches
the line x = 2a at these points.
Y                       109. At a point where two double
-      \             loci, one belonging to each system,
-\          + [  meet, the four regions are all simi-/\  -  larly marked; and, supposing them
_-~ ^,        marked +, we have two branches
f/+   +    -       crossing each of the loci and forming
-~- -    --- a a crunode. Thus, in the present ex+  +     -      ample, the double locus x = a cuts
- \;  ""     the hyperbola, equation (2), in the
'/      ^A /points (a, ~ aV2); hence these points
/;, " are nodes. To find the tangents at
/-    /.            the upper node, we put x = a and
si /~ y = aV2 in the coefficients of the
Fig. 52        factors which vanish at this point in
equation (i); the result is
8a2(x- a)2 = (X2 -Y - 4ax + 5a2)2,
or
2V2. a(x - a) = ~ (2 -y - 4ax + 5a2).
Proceeding as in Art. I05, we have


I: 2.2.a(x- a) = (x - a)2 - 2a(x - a) - (y2 - 2a2),




~ V]    LOCI REPRESENTING SQUARED FACTORS


83


and, for the tangent lines,
(2 + 2)(X - a) = - 2( - 2. a)
and
(2 - V2)(X - a) = 2(y - /2.a).
110. The single locus x  o is a tangent at its point of
intersection with the loci x - a and x = 2a, that is, at the fundamental point B. Joining the branches which have been
determined, we have the curve drawn in Fig. 52.
Points in which Several Auxiliary Loci intersect
111. When two distinct auxiliary loci of one system intersect
a locus of the other system in the same point, the case is the
same as if the two loci were coincident as in the preceding example; that is to say, the curve touches the single locus. The
six regions which have a common vertex at the point will be
alternately marked +  and -, and the curve 
lies, as in Fig. 53, in the regions marked +, ad- + 
jacent to the single locus. It is to be noticed 
that the two loci of the first system may be im- 
aginary straight lines; for example, if the point  Fig. 63
were the origin, the factor X2 + y2 would be regarded as representing two such loci.
112. In general, if m linear loci of one system and n of the
other pass through the same point, and m > n, each of the n
loci is a tangent to the curve. This is evident if we suppose
the origin transferred to the point in question; for the factors
corresponding to the loci will then contain no absolute terms,
and the product of the n factors will constitute the group of




84


CUR.VE TRACING.


[Art. I 12


terms of the lowest degree present in the equation. Hence
the result of equating this product to zero gives the n real or
imaginary tangents at the origin.
Furthermore the result extends to curvilinear auxiliary loci
if we count every such locus as equivalent to as many linear
loci as it has branches passing through the given point. For
example, suppose the equation of the curvilinear locus, after the
origin is transferred, to be
x3 + a(x2 - y2) = o.
In finding the terms of the lowest degree this factor is equivalent to x2 - y2, that is, to the two linear factors x + y and
x-y; and, if these occur among the it factors where n < in,
the lines
x+y= o         and     x-y= o
are among the tangents to the curve; but these lines are the
tangents to the given curvilinear locus.
113. If in - n, the curve will have m tangents, real or imaginary, at the given point, which may be found as in Art. o19.
For example, in the case of the curve whose equation is
a(2y- a)2 = 4y(2x- a)(x + y - a),
the loci 2x - a = o and x + y - a = o intersect the double
locus 2y - a = o in the point (a, a). Putting y -  a, we
have
(2y - a)2 = (2X - a) (2x - a) + (2y - a)},
and solving, we find
2y -a 
2X - a-    ~ Xi 5,




V   P] POINTS IN WHICH AUXILIARY LOCI INTERSECT  85


the equations of the tangents at the node. The curve is constructed in Fig. 54.  The asymp- 
totes are the lines
x = a     and    x +y = ra,        -       I
and the axis of x; the liney =- 2a         /
touches the curve at (ja, - 2<).
114. When the m - n auxiliary 
loci are all real and distinct, we    ',
have 2(m + in) regions alternately 
marked + and - meeting af the 
point of intersection, of which there 
are m + n lying on each side of any 
given locus. Hence, if m  -+ is           Fig. 54
an even number, the vertically opposite regions are similarly
marked, and if m + n is an odd number, they are differently
marked, as in Fig. 53. This is still true if two of the loci
become coincident or imaginary, because in that case two
vertically opposite regions disappear simultaneously. It follows
that, supposing m > n so that the n loci are tangents, when
m   -+ is odd, the branch to which the locus is tangent lies
on one side of it, as in Fig. 53.  On the other hand, when
in + n is even, the branch crosses as well as touches the
auxiliary locus, and if the latter is a straight line, it is a tangent
at a point of inflexion.
Examples V
I. y4 + 2x9y - (2X - y)2 = o.
2. y'(x- + y2) = ax3 - 3ax2 - 2ay3.
3. (X2 + y2) (y-  ) + a(y + x) = o.
4. X(X2 + y2) + a(x + y)2 = o.
5. x3 + xy + 4aX2 + 4axy + ay2 = o.
6. e (x2 + y) = a2(x -y)2




86                  CURVE TRACING               [Art. 114
7. X4 + a(x + y)(x -y) - 2a2 (x - y)2 = o.
8. 4 - ax2y - axy2 + a2y2 = o.
9. x5 - 4ay4 + 2ax3y + a2xy2 = o.
10. x4 - 5ax2 - 2axy2 + 4a2y  = o.
II. (y2 - X)3 = 4a 2Xy.
I2. X2y2  2Xy3 - 2y2axxy +     a2(xax  - y)2 = o.
13. x4y + X4 - 2X2y + y2 = o.
I4. (y2 - 2aX)2 + 4a(a - b)y2 + 8a2(bx - cy) = o.
I5. y2 (X + 3a)2 = 4a3(x - a)2.
6. y(y2 - a2) = X2(X - a).
17. (y2  a2)3 + X4(2x + 3a)2 = o.
I8. (a - y)2(a - 4y) = 4x2y.
19. (y - x)2(y - 2X)(y + x) = ax(y2 - 9X).
s0. (X - y)2(y- X) = ay2