374 Rev. B. Bronwin on certain Definite Multiple Integrals. 



Then, separating the symbols of operation from those of 

 quantity, 



D* + D x 2 + D 2 2 + ... = o Dl = - (D 2 + D, 2 ... + DJU,). 



I><=-D*(D 2 + D 1 2 +...) = -(D 2 + D 1 2 + ...)D* 



=, (D 2 + D 1 2 + ..,)«, B 6 n = - (D 2 + D x 2 + ...) 3 , &c. 

 Therefore 



{« 2 D 2 + &D*... + a 2 D* } s = (a 2 D 2 + g 2 D! 2 ... + 2 DLi) s 

 + s {a? D 2 + £ D x 2 ... + 2 DLi)- 5 " 1 *? D* 



+ ^i"^ (« 2 D 2 + 6*I>i a - + ^DJi-O-'x^i + &c. 



= (a 2 D 2 + ^D^... + fl*D|[_ 1 )' 



- 5 ( a 2 D 2 ... + fi 2 DLi) s - 1 (^D 2 + x 2 D I 2 ...+x 2 DL 1 )+&c.. 



= {a 2 D 2 + g 2 Dj 2 ... - x 2 D 2 - x 2 D x 2 ... y 



= {(a 2 -x 2 )D 2 + (&-k*)D*... + p-Wjptiiy. 

 In this case therefore 



#s$ 



('.) 



+ (g 2 -x 2 )D 1 2 ...}^(o-, /,...) 

 Change in this last a, 6 ... A into «, b ... £, but so that 

 « 2 - x 2 = a* -1% & - A 2 = 6 2 - / 2 , &c. ; 

 and let this change vj/ into $, we have obviously 



, « § ... 



+ = ^r.* w 



This is an extension of Laplace's theorem relative to the 

 attraction of ellipsoids on a point exterior. And if a, £, &c., 

 a, b, &c. be independent of g, /i, &c, we have also 



To give an example or two, let 



R = {(g-^+ik-y)*...}*, 



and let R x stand for the same quantity when a, £, &c. are 

 changed into «, Z>, &c. ; then, since 



(&_ d*_ \_J_ 



W 2 + ^ 2 "7R' 1 - 2 " ' 



w being the number of variables, we have 



