208 Mr, J. W. L. Glaisher on a Multiple 



If Dr. Kerr can show me what mistake I have made, no one 

 will be better pleased than myself, as the establishment of such 

 a relation as, according to Dr. Kerr, exists between electricity 

 and light will be a most important step in physical science. 



Fixliolme, Dorking, 

 August 5, 1876. 



XXVII. Note in regard to a Multiple Differentiation of a cer- 

 tain expression. By J. W. L. Glaisher, M.A., F.R.S.* 



CONSIDER the multiple definite integral 

 /'QO /»QO /»20 / 



) \ \ • • • exp ( -a^-a^l . . . -a/; 

 ^ o Jo Jo 



~ *~7V ■j)daide t ...dx t i . (1) 



where, as is convenient in printing complicated exponentials, 

 exp (a) is written for e a . 



By the aid of the well-known integral 



,--100/ I \ / 



j exp^-^ 2 ~^j^=^-^exp{-2 v / (^)}, 

 we readily see that the integral (1) 



= (f r )V(«T^„) exp( - 22 °" yva --^-'-^»-- 



V<H-Vq). ■ ■ (2) 



Now from an inspection of the integral (1) it is evident that 



if we denote by u the quantity (2), then, i being any integer, 



(£)'iiy '(£)"■ ••(0 ;= (j/ ? )'" ; • (3) 



viz. the differential expression on the left-hand side of (3) is 

 equal to the nth integral of u with regard to q. It is clear that 

 the quantity subject to the exponential sign in (2) need not be 

 negative ; so that, if we take 



and put i— 1 (whereby there is no loss of generality), the theo- 

 rem is that if 



v L a l a 2 • • • a n ) 



* Communicated by the Author. 



