AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 613 



Ada" + A!dx a ' + (Bdx" + B'dx b )p + (Cdx c + C'dx c ) p* + ... - o, 



where a is not < 1, since <p (x, y) = ; J is not < 1 ; c is not < 2 ; and so on. Let the lowest 

 power of dx in the whole expansion occur for the first time with p k , and for the last time 

 with p l : this lowest power must have one of the unaccented exponents. We then determine 

 the finite values of y, which are the values of p in Kp k + ... + Lp = 0. But p may have 

 infinite values. Suppose, for instance, that the lowest power of dx is dx 7 , occurring for the last 

 time in the coefficient of p*. It is only by casual vanishing of coefficients that p 6 , p s , and p 7 are 

 wanting in the final equation : the powers of dx are not necessarily above the 7th, and their 

 effects do not necessarily vanish in the final result, until we have passed p 1 . The final equation, 

 then, is Kp k + ... + Lp 1 + .p 5 + ,p 6 + .p 1 = 0. In confirmation, observe that if the lowest 

 terms be of the seventh order, and LdaPdy* be the last of them, then, dx being qdy, Lq s will 

 be the lowest term of the equation for determining y'~ l . Hence, the lowest power o? dx deter- 

 mines the number of values of y : the final equation determines the number which are not infinite. 



If the final equation, which write Kp k + ... + Lp 1 + 2 . 0p m = 0, have no equal roots 

 whatsoever — and infinites here count as equals, so that 2 must not have more than one term — 

 or if all equal roots be sets of imaginary pairs, the problem is solved. If /u of the roots be 

 real, and n' imaginary, the point (x, y) is in /u ways on a real branch of the curve, and in 

 lp! ways an isolated point. There is no need to introduce the terms A'dx" , &c. : for infinitely 

 small alterations in the coefficients of an equation do not affect the character of its single 

 roots. But if two or more roots be equal, the effect of the second terms may be of either of 

 a threefold kind. First, a pair of roots may become real and unequal, for both signs of dx : 

 this gives a pair of branches with contact passing through (x, y). Secondly, a pair of roots 

 may become real and unequal for one sign of dx, and imaginary for the other sign : this gives 

 a cusp. Thirdly, a pair of roots may become imaginary for both signs of dx : this gives an 

 isolated point with a real tangent. 



The curve <p (x, y) = c, c being infinitely small, adjacent to the point (x, y), is defined 

 by Kdx h ~ k dy k + ... + Ldx h ~ l dy l = c, dx h being the lowest power of dx already mentioned. 

 Let c : dx h — H, where H may be any quantity whatever. Then p, derived from 

 Kp k + ... + Lp 1 = H, indicates all the directions in which <p(x, y) = c has branches cutting 

 the vertical at a distance dx from (x, y). Let Kp k + ... + Lp 1 =fp. Then for every real 

 root of fp = for which fp .f"p is positive, there are values of H limited or unlimited as 

 the case may be, between which imaginary roots of fp = 0, have corresponding real roots 

 in fp — H = 0. And hence may be detected the character of the parts of <p {x, y) = c, which 

 vanish in the isolated points of (p(x, y) = 0. The case in which fp = has equal roots may 

 be similarly reasoned on. 



It thus appears that the theory of equations is the proper instrument for considering the 

 singular points of curves. A few months ago, I saw in the notes to the second edition of 

 M. Serret's excellent Algebre Superieure, an account of a process for eliminating between two 

 equations, attributed to Mr Minding, with a reference to the sixth volume of Liouville's journal, 

 into which it was translated from Crelle's journal. This process of elimination depends upon 

 an elegant and powerful theorem, which is demonstrated at full length by M. Serret, without 

 the slightest hint of any other author except Mr Minding. Perceiving the importance of this 



