Laplace's theorem, &c. 419 



^~dJ ~ d(? "'" dc" 



d^V d'Y cf U _ rf^V ^ cFV' fV 

 rf^ + ^F + "^ ~ t^a'' "*" db' '^ dc' ^ da? 



Now Poisson supposes that 



^.^+^-0 ..(22) 



da' + da^ ^ dc' —^ ^ ^ 



d'Y . rf^V , ^^V _ rf^U , ^^U ^ .^„v 



•■• -^ + "rfF + "^ - rf^ "*" "rfft^ "^ ^c^ ••• ^^"^^ 



The equation (22) is not, I believe, correct ; 

 it is established by Poisson from the commutability 



of the independent operations of c?, & i{x^a) dx 



where the integral sign j refers to dx\ and the 



diflFerential sign d refers to a ; this principle only 

 obtains when the limits of integration with respect 

 to X are independent of 



a. 



dY rfV dV 

 Thus, -d^r - ~d^ '^ ~d^ 



