From the equation (1) we see that (A) = 0, or (A) = (μ): in the former case the equation (2) becomes nugatory in order therefore to obtain the required results, we must differentiate this equation with regard to r. Thus, remembering that r = 0, (λ) = 0, we have

Again, supposing that (X) = (μ), we have, from (2),

so that there is a third asymptote represented by the equation

y' - x' = 3b.

Algebraical Method of finding Curvilinear and Rectilinear Asymptotes.

113. Let the equation to a curve be reduced if possible to the form y = f(x); and suppose f(x) to be developed in a series of descending powers of x, so that

Then, when x is indefinitely great, the terms involving negative indices will vanish, and the equation to the curve will ultimately be equivalent to

which is therefore the equation to a curve approaching the proposed curve asymptotically, that is, to a curvilinear asymptote. If m be equal to unity, which is the most important case, the asymptotic equation will designate a rectilinear asymptote. Should there be a rectilinear asymptote parallel to the axis of y, this method will fail to detect it, for the equation y = ax + α a will not represent straight lines parallel to the axis of y. In order to discover asymptotes parallel to the axis of y, we might obtain a development for x in descending powers of y. It is, however, frequently easy to ascertain by inspection these asymptotes; for, the value of y in the equation y = f(x) being infinite for the abscissa corresponding to such an asymptote, we have only to equate to zero the denominator of f(x): the corresponding values of x being ascertained, the indefinite ordinates belonging to them will be the required asymptotes. This method of finding asymptotes was first given by Stirling, in his Linea Tertii Ordinis Newtoniana, p. 48.

Ex. 1. To find the asymptotes of the curve

determine two rectilinear asymptotes.
Again, y when x is equal to a: hence

x = α

is the equation to a third asymptote.

Ex. 2. To find the asymptotes of the curve

If x be equal to b, y is equal to infinity: thus

is the equation to an asymptote parallel to the axis of y.

-1 b

b b2

Again, ay = (1-2) = x ( 1 + 2 + - - + ...),

x

whence it appears that the curve has also a parabolic asymptote, of which the equation is

ay = x2 + bx + b2.

For additional information on the subject of asymptotes, the reader is referred to a paper, by Mr. Gregory, in the Cambridge Mathematical Journal for November, 1843, and to a paper in the same Journal for February, 1841.

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CHAPTER III.

MULTIPLE POINTS, CONJUGATE POINTS, CUSPS, ETC.

Definition.

114. A multiple point is a point through which two or more branches of a curve pass. Thus, at the origin of coordinates of the curve of which the equation is

there is a double point; that is, a multiple point of two branches. The form of this curve is indicated in the following figure:

A conjugate or isolated point is a point, the coordinates of which satisfy the equation to a curve, while if to either x or y be assigned any value differing ever so little from its value at the point, the corresponding value of y or x respectively will be impossible. Thus, supposing a to be less than c, the curve

(y – b)2 = (x − a)2 (x2 – c2)

will have a conjugate point, of which the coordinates are (a, b). For, if we put x = a + h, where h is indefinitely small, (y - b)2 ±h, would be equal to a negative quantity, which is impossible so long as y is possible. In like manner, if we put y = b+k, k being indefinitely small, it appears from the equation to the curve that x cannot have any possible value nearly equal to a.

A cusp is a point where two branches of a curve stop abruptly and have a common tangent. Thus, the curve belonging to the equation

y2 = x3 (a2 + x3)

has a cusp at the origin, the common tangent of the two branches coinciding with the axis of x. The following is the form of the curve:

As another example we may take the equation

x* – ax3y – axy2 + { a3y2 = 0,

which has a cusp at the origin of coordinates.

There are two species of cusps: the ceratoid, so called from its likeness to the horns of cattle, the curvature of the two branches lying in opposite directions, and the ramphoid, so called from its likeness to the beak of a bird, the curvature of the two branches lying in the same direction. The former figure affords an instance of a ceratoid, the latter of a ramphoid.

Analytical Property of Multiple Points in Algebraical Curves. 115. If u = f(x, y) = 0 represent the equation to an algebraical curve cleared of radical and negative indices, the values of x and y, at a multiple point, will satisfy the equations