AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 615 



(a + A) (u + U) a x a + ar + ... + (k + It) (u + U) K ^ Kr + ... + (t + T) (u + U) T x'-^ r + ... = 0. 



We have then to find in how many and what ways r can be so taken that two or 

 more of the set a + ar, b + fir, &c. shall be equal, and lower than all the rest, or higher 

 than all the rest, according as we are seeking lower dimensions or higher dimensions. 

 These cases both come under one rule, the difference of arrangement making all the 

 difference required. For the demonstration I refer to the paper in the Quarterly Journal 



(No. I. p. 1) already cited. Take the succession and from it form the fractions 



b — a c — a k — a t — a % — a 



a — fi a — y a — k a — t a — £ 



take the greatest (the furthest from — oo ) or if several be equal and greater than all the 



rest, take the last of the equal maxima. Let it be p = {k - a) : (a - k), and, to shew 



the case of equal maxima, say that p = (e — a) : (a — e) also. Then there are roots of 



the form y = x p (u + U), and u is derived from au a + eu e + ku K = 0, every value of u 



finite both ways giving one form, and the final equation being always derived from the 



first term of the succession, with all the terms which contribute to the equal maxima. 



k ... t 

 The next step is, to treat the succession ' in the same way ; and so on until the 



K *■■ 7* ••• 



last maximum contains £. If all the set o, /3, &c. be integer, the number of solutions 

 is a — £ or £ — a, the whole number of roots which are not in y = : the forms obtained 

 having their multiple character duly augmented, if a, b, &c, or any of them, be fractional. 

 And the successive values obtained for r are always descending in algebraic magnitude. 



Having thus obtained y = x'(u+ U), substitution gives an equation for determining 

 U, which may thus be found as x r {u + U), and so on. Confining ourselves to integer 

 data, a, a, &c. I may remark that, when r is fractional, w r must be interpreted univocally, 

 the equivocality of u supplying the requisite multiplicity of forms. For if, /n being a 

 proper root of 1, we use /j. r af for <v r , this is equivalent to assuming y = (/ua?) r (u + U), 

 which substitutes n r u for u in the final equation, and all the combinations are but 

 repetitions of those which we obtain from y = x r (u + U). Next, since there can only be 

 X, — a or a — £ values of y, the process for determining U f , U u , &c. is bound to shew 

 this : and the mode in which it succeeds is an instructive example of the use of the 

 theorem. 



Having obtained a value of r with, say aw" + ... + kw" = for determining u, write 

 x'u in place of y, and collecting powers of x arranged according to the problem, let 

 Ay a + ... become fM .x m + vu . x" + wu .x p + ..., where /xu = determines u. 



The equation for U is found by substituting u + U for u in the last. And m is k + kt, 

 where k and k are integers from the data, while n, p, &c. are of the same form, with 

 integers usually not found among the data. We have then, since /xu = 0, 



{vu.x , ' + 'sru.xf+ „.) + (n'u.x m + v'u. x" + vr'u.x?+ ...) U+ U'u.x m + v'u.x" + -sr"u.xP + ...) — 



2 



+ ... = 0. 

 Vol. IX. Part IV. 79 



