540 On the Inversion of the Independent Variable, 



Let now n— 1 be broken up without restriction in every pos- 

 sible way into parts, and let r, 5, / . . . / denote one such system 

 of parts so that r-\-s + t'\- .. . +/=«•— 1, r, s, &c. being all 

 actual positive integers. Then is 



£=2C{IT^, \Ts, rT7, ... TVi} 



X - 



L ^1 a7j ^, * * ' a?i J ' 



than which nothing more clear and simple can be desired or 

 imagined. And so more generally, if we make, as before, 

 r + s-{-t+ ... '\-l=n—ff, and give ^ in succession every different 

 value from 1 to n, we shall have 



iF,"\ a?, a?! • • • • • ^^ J Jf 



where ((1 +r, 1 +5, . . . 1 -f /), ^— l) means the number of ways 



in which l+r, + lH-5H-...+l + /+y — 1 elements can be par- 

 titioned off into groups of one kind containing respectively 

 (1 4-r), (1 +s), . . . (1 + /) of the elements, and into a group of 

 another kind containing the remainder (y — 1) of the elements. 

 This distinction of the groups into two kinds has no effect upon 

 the result except when g — 1 is equal to any of the numbers 

 (1 -}-r), (1 4-5), . . . (1 +/).' If we write, according to the nota- 

 tion above employed, 1 -|- r, 1-|-*, . . . 1-f / under the form. 

 a:a, b:^, ...c:y, {(] +r, 1 + 5, . . . 1 -I- /), ^—1} will represent 



fla + &yg+ ... +cy+g-\ 



G(«)(G«)" X G{b){(}P)' . . . G(c)(G7)^ . G(y- 1)' 



This more general theorem may of course be demonstrated by 

 a similar method to that employed in the text for the case of 

 ^= w, for which all the terms in the expansion vanish except those 

 in which ^ = 1. 



I have, since this paper was sent to the press, obtained a new 

 solution of the far more difficult and interesting question of the 

 change from one system of independent variables to another 

 system. I say a new solution, because one has already been 

 virtually effected, but under a form leaving much to be desired, 

 by the great Jacobi in his Memoir De Resolutione jEquationum 

 per series infinitas, Crelle, vol. vi. 1830. In my solution, a 

 remarkable species of quantities, to which I give the name of 

 Arborescent Functions, make their appearance in analysis for the 

 first time. 



