Differentiation of a certain Expression. 523 



(on which see Liouville's Journal, ser. 2, vol. i. p. 82, 1856). 

 Eeplacing x, y, . . . t by ax, by,... Jet, and taking q=r n -~(ab ...&). 

 we find 



j j ...j ex.v(-ax-by...-kt--^—Jxn~ 1 y" 1 



dxdy...dt- Vn ^/(^V.-.F- 1 ) 5 



so that the expression on the right-hand side of this equation 

 is such that, if we denote it by u, 



£? =f-Y^ 



dadb...dkdq ^ ' U > 



giving a theorem which may be enunciated as follows : — 



exp { n \Z(a^a 2 . . . a n ) } 

 V ~ ^(a^J.-.a--') ' 

 then 



d n v ( 



da 1 da 2 . . . da n ' ^ ' 



This can be readily verified by actual differentiation, if the 

 order of the performance of the differentiations be that of the 

 letters %, a 2 , ... a n . 



If we start with the nple integral 



Jo «^o Jo 



exp{— X— y ...— t — CL\/{xy...t)}yn; 



...* » dxdy...dt=^n.(2iry (n - 1) , ; \ n * 



^ v 7 (n + a) 



(Liouville, ser. 2, vol. i. p. 289, 1856), and therein replace 

 x,y,...t by a^, by,... Jet, taking p = x%/(ab ... &), then we 

 find 



•»<» ,-»CO x-i CO 



| I ...1 exp{— a#—ty. ..-**—;> ^(ay ...*)} 



Jo Jo Jo 



1 1_ n—\ 



xynzn...t n dxdy ... dt 



, rra _ n 1.2.3...Q-1) 1 



so that if w denote the right-hand side of this expression, 

 d n w d n w 



dp n dadb ... dk' 



■ ■ (2) 



