Differentiation of a certain expression. 209 



then 



lfift*X( '2- )*• • • (£f •- -^ - 



\dp/\da 1 /\da2/ \da 3 / \daj 2 2(2 " ,} 



As an example put n = 2 ; then if 



1 /f i i i, 



as 62 

 the theorem is that 



Writing, for brevity, w for exp (6 ? a*p*), 

 dv __ /l pi IIS 



(jd\fd\ _ (I j>*' _Jl j>J , 1 1 >j 



(d\(d*?_ (±-pl 3 jri 3 jpj _ 3 1 \ 

 UJUJ ^~ W U6 at 6 16 a6l 8 a f tf» 8 a * &f/' 



which differentiated with regard to /> becomes 



~ JL J_ _ J_ 



The theorem is by no means an obvious one ; and if the 

 differentiations be performed in the order in which they stand, 

 the verification even for a moderate value of n is tedious. If, 

 however, the differentiations be performed in a certain other 

 order, each of the intermediate results consists of only a single 

 term, and we thus obtain an elementary and very interesting 

 proof of the theorem, and have, as it were, its raison d'etre 

 exhibited. The process will be best understood by taking only 

 three letters, a, b, c, besides p. Let 



v = i,i iexp(c^£a£jt>8): 



and for the sake of brevity write w for exp (c*b*($p E ), and 



dv d v 

 Lp]> [P c ]? & c * f° r T~ ? 7 7 , &c. ; then, omitting all numerical 



factors, 



Phil Mag. S. 5. Vol. 2. No. 10. Sept 1876. P 



