08 



PROCEEDINGS OP THE AMERICAN ACADEMY. 



a = Oj A\ + a„ X« + . . . + " A . 



ft = 6, A\ + />, A, + . . . + 6, A r . 



y = Ci X\ + c - A._> +•••+'' A . 



where the a's, Ps, and c's denote arbitrary parameters, aud by e a the 

 operator 



(1 + a + ^ + §-' + . . .)/ = /+ «/+ £/ + g^/ + • • • » 



wherea" + 1 /=o(a"'/)- 



Let 



and let 



i c 1 r,- 



= e" a:, 



= 1,2 . . . n). 



Since the a:"s are function? of /, any function of the x n s,f(x\ . . . x', t ), 

 is also a function of t. And we have 



/(A • • • *'„) 



, o + 3 



rf^ J,. 



+ . .. 



Hut 



(1 = 1,2 . . . «). 



df(x') » 9f(x f )dx' { _ » , (/) ?/(*') _ X > f(/) 

 ^/ / i d x | a t i c> x < 



where A"' denotes the result of substituting the accented for the unac- 

 cented variables in A'. Therefore 



= [/0O], o+<[ay,/,], «+Ux'V GO]. o+g-jC^ ' 03. Q+.- • 



= /(*) + ' -V. /■<■.-. : £ A-v '(«) + j 3", X»/<*) + ■ ■ • 

 [uently, if 



.-. • bave 



1. 2 n), 



1, 2 n). 



