the Inversion of the Independent Variable. 539 



each part being affected with the factor ( — 1)^"' derived from 

 the differentiations performed upon 



Xp.x^, x^_ ^. x^ : x^ 

 &c. 

 2nd. Suppose p, the lowest index, is 2, then the term 



r 8 t 

 ^p-\ 'OCp.Xfj .X^ 



must be rejected, because Xp_^ becomes a?,, which is excluded 

 from appearing in any numerator. But then, per contra, in this 

 case there will be a portion of the coefficient derivable from the 

 differentiation of the denominator of the term 



((r-l):2, S'.a, tir) 





where 



(N-l) = l4-(r-l)2 + 5.o-4-^T + &c. 

 This portion will be 



(_)N-..(N_i)^C((r-l):3,s:<7, t-.T-), 

 or, which is the same thing, 



C(l, (r-1) :2, 5:0-, tir, &c.), 

 and therefore the portion of the coefficient corresponding to 

 Xp^i .x^ . a?^ . x^, &c. is supplied from another source, and the 

 expression d remains good for all values of /o, cr, r, &c., and con- 

 sequently, by virtue of the second lemma, is equal to C{r:p, 

 sia-j t-.T, &c.) ; and thus we see that if the law assumed is true 



IT d'"''^^ . u 



for — '—, it remains true for -7-r— , as was to be shown. And 

 dx"- dx''-^'- 



as it is evidently true for r=l, it is true generally. 



Lincoln's Inn Fields, 

 October 13, 1854. 



Postscript, The formula expressing Burman's law may be 



d^x 

 exhibited as follows : Xr will still be understood to denote -^, 



and C[p, q, . . . m} will, as before, denote the number of distinct 

 modes of combining p-\-q-\- ... + m things in sets oi p,q,, . . in 

 at a time ; so that, ex. gr. C{2, 2, 4, 4, 4} will denote 



1x2x3... X 16 1 _1 



(1 . 2)^ . (1 . 2 . 3 . 4)3 ' 1 . 2 * 1 . 2 . 3' 



2N2 



