GENERAL PRELIMINARY RESULTS. 27 



dU dU % -J _ -I 



dr z ~ a 2 



becomes 4?r F' 



when the radius a is supposed to vanish. Thus, the equation (2) 

 becomes 



jdxdydzUSV+jda U^= jdxdydz FS U+jdo- V^~-4ar V ... (3); 



where, as in the former equation, the triple integrals extend over 

 the whole volume of the body, and those relative to cfor, over 

 its exterior surface : V being the value of Fat the pointy/. 



In like manner, if the function F be such, that it becomes 

 infinite for any point p" within the body, and is moreover, 



sensibly equal to -, , infinitely near this point, as U is infinitely 



near to the point p , it is evident from what has preceded that 

 we shall have 



jdxdydz m V+jda- U^-lv U"=jdxdydz F8 U+fd<r V^-lw F. . . (3'); 



the integrals being taken as before, and U" representing the 

 value of Z7, at the point p" where F becomes infinite. The same 

 process will evidently apply, however great may be the number 

 of similar points belonging to the functions U and F. 



For abridgment, we shall in what follows, call those sin- 

 gular values of a given function, where its differential coefficients 

 become infinite, and the condition originally imposed upon U 

 and F will be expressed by saying, that neither of them has any 

 singular values within the solid body under consideration. 



(4.) We will now proceed to determine some relations ex- 

 isting between the density of the electric fluid at the surface of a 

 body, and the potential functions thence arising, within and with- 

 out this surface. For this, let pdar be the quantity of electricity 

 on an element da- of the surface, and F, the value of the potential 

 function for any point p within it, of which the co-ordinates are 



