98 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



a ^ Oi Xi + a^ X2 + . . . + a,. X^, 



ft ^ b, Xi + b. X, + . . . + b, a;, 



y = Ci A'l + Co A^2 + • • • + C^ A,., 



where the a's, b's, and c's denote arbitrary parameters, and by e" the 

 operator 



a" a ^ ^ ^ « «." . a „ 



(1 + « + ^, + :r, + • • •)/ = /+ «/+ :71^ + .1/+ • • • ' 



5i! 



where a'" + i/= a (a™/)- 

 Let 



2!-' 3!- 



1 C/^i 



and let 



x', =: X, + tXx, + f_ X^'X, + ^X^x, + . . .= e'^x, 

 2 ! o ! 



(i = ],2 . . . «). 



Since the x"s are functions of t, any function of the x''s,f(x'i . . . x'„), 

 is also a function of t. And we have 



f{x\ . . . x'„) 



+ 21 



■dy(x'y 



But 



(/=1, 2 . . . .0- 



+ o, 



df 



+ 



<=n 



dt 1 Sx', dt 1 5 «'« 



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

 cented variables in X. Therefore 



e 



f{x I . . . X „) 



- [/(^')].^o+^[A-/(x')]..o + ^[X'Y(x')]<.o + ^;[^'y(:^')]<-o+. . . 



= f(x) + tXf{x) + 1^ X^f{x) + |j AV(x) + 



— e'V(^i • • • ^«)- 

 Consequently, if 



we have 



(^- = 1, 2 . . . 7i) 



/ (x'l . . . x'„) = e-f, (xi 

 (; = 1. 2 . . . «). 



x„) 



