AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 



611 



The multiple point, the cusp, and the isolated point, must be looked for only in 

 places where (p x =0, (p y = 0, or where both the subordinates meet the curve and each 

 other. The following proof applies with ease to a multiple point of real branches, a cusp 

 which is not in any other branch, and an isolated point which is not also on a real 

 branch of the curve. At a multiple point, either co-ordinate being constant, the other 

 may pass from = to <p = 0, infinitely near the multiple point, and during this 

 passage cp x and <p y change sign ; hence <p x = and (p y = at the multiple point. The 

 same of any cusp near which are small deflexions parallel to both axes. But where such 

 deflexion can only be parallel to one axis, as where y is a maximum or a minimum at a 

 cusp* of contrary flexures, we see that points on different sides of the cusp are similar, 

 while y" has one sign for both, whence <p y has one sign. Nevertheless, vanishes at 

 the cusp, since y" is infinite, and <p y 2 <j> xx — ... is not. As to the isolated-f-, or conjugate 

 point, we know that (p(p y changes sign in passing through the point, while d> does not 

 change sign, as may be proved by going round it : hence <p y changes sign at the point, 

 or (p y = passes through it. And similarly for <p x . 



A cusp is an evanescent loop ; and an isolated point is an evanescent oval, or 

 bounded portion of the curve. In one view, it is so, or is not so, at our pleasure. It 

 is always possible to make (p(x, y) a particular case of yj/(x, y, a), say when a — 0, 

 in such manner that \|/(#, y, 0) and y\r(x, y, da) shall both have cusps, or both isolated 

 points. But, looking at the fact that in our present subject we are only concerned with 

 the relations between <b = and its subordinates <p, = 0=0, it seems obvious that 

 we are only concerned with = as being one of the family (p(x, y) = c, belonging to 

 the pair of subordinates (p x = <f> y = 0, which are common to all. This being granted, 

 the assertion follows. Let 0=0 have an isolated point (x, y). Let c be infinitely small : 

 then, since <p{x, y) is always real, we cannot proceed in any one direction through points 

 (x + dx, y + dy) without <p(x + dx, y + dy) passing through c in value. If the isolated 

 point be not on any real branch, so that has one sign in all parts of the neighbourhood, 

 it is clear that the isolated point marks the passage from a real oval to an imaginary 

 branch, as c changes sign through 0. But if the isolated point be on a real branch, 

 so that has different signs in different regions of the neighbourhood, it is indicated 

 that = c has one or more ovals (according to the branches which pass through the 

 point) when c has either sign. And similar reasoning for the evanescent loop of the cusp. 



It has been asserted that if the isolated point were always an evanescent oval, 

 y would admit an infinite number of values. Now first, this does not happen explicitly 



• A cusp may be said to be of similar or contrary flexures, 

 according as the tangent does not or does divide the branches. 

 I have long ceased trying to keep in memory which is the cusp 

 of the first kind, and which of the second kind. Nor do I 

 admit the uncouth terms ceratoid and ramphoid; for the cusp 

 called ramphoid is as much like the point of a horn as that of a 

 beak, and the cusp called ceratoid is not the point of a horn at 

 all, but that of a spear. 



f The idea implied in the word conjugate is that the point 

 is algebraically conjoined to the curve, but not geometrically 

 part of it. In geometrical algebra, the word isolated is prefer- 

 able, since, by definition, the point is really on, or part of, the 

 curve. Nor is it any objection that an isolated point may 

 be on a real branch of the curve; we mean that a point 

 on a real branch may be the only real point of an imaginary 

 branch. 



