LAPLACE'S COEFFICIENTS. 115 



and T n is the coefficient (in terms of u, U Q , X,X ) of f J 



in the development of -. 



48. Incidental introduction of Laplace's Coefficients 

 (not further used in this Treatise). 



If we differentiate twice the expression 



with respect to x, also with respect to y, and with 

 respect to z, we find 



i 



And since F= jfyt-, and since the application and 



limits of this integration do not depend on x, y y z, it 

 follows that 



_ 



' 



da? ' dy 2 ' dz* 



Now, by the same principle which we have used in the 

 last Article, for F, and which we shall here use succes- 



, , dV dV , dV 



sively for -=- , -j- , and -j- , 



dx dy dz' 



(dV\ j (^\ fJ i^ 7 \ 



<7 2 F V dx ) du \dxj d\ \dx) dr ^ 



dx* ~ du ' dx d\ ' dx dr ' dx' 



82 



