116 



<f F J du t d \djjj d\ t 



' dy d\, dy dr ' dy ' 



dz* du ' dz d\ ' dz dr ' dz' 



The expressions for all the quantities on the right hand 



dV 



are to be found in the last Article : those for -4- &c. re- 



dx 



quiring complete differentiation, with special notice 

 that -5- &c. are themselves subject to differentiation, 



as well as the more explicit quantities cos u, &c. Per- 

 forming all operations, this result is obtained: 



d 2 (rV) 1 cost* 



dr 2 r ' sin u ' du r' da 2 r sin 2 u 



7->+2 75 



Applying this to the general term ^-^ where P n is 



independent of r, 



n(n+I).P n 



_ 



'' du 



CORW 



or 



smw 



from which it is possible to find a general expression 

 for P n . The terms thus found are Laplace's Coefficients. 

 ]n the physical investigation now before us, we shall 

 not have occasion to use the general term. 



