GENERAL PRELIMINARY RESULTS. 37 



r being the distance between p and da-' an element of the surface 

 A, and ft a constant quantity dependent upon the quantity of 

 electricity originally placed on A. Now the value of F at p is 



by what has been shown (art. 5) ; (//) being, as in that article, 

 the density of the electricity which would be induced on the ele- 

 ment da by a unit of electricity in p ', if the surface A were put 

 in communication with the earth. This equation gives 



since S F = S =0; the symbol 8 referring to the co-ordinates 



x, y, z, of p. But we know that = S' F; where 8' refers in a 

 similar way to the co-ordinates x', ?/, 0', of j?' only. Hence we 

 have simultaneously 



where it must be remarked, that the function F has no singular 

 values, provided the points p and p are both situate within 

 the surface A. This being the case the first equation evidently 

 gives (art. 5) 



V=-j(p)daV; 



V being what F would become, if the inducing point p were 

 carried to da, p remaining fixed. Where F is a function of 

 x, y, z, and f, rj, f, the co-ordinates of da, whereas (p) is a 

 function of x, y, z, f , ?/, independent of x-, y f , z ; hence by the 

 second equation 



-j(p)da$V, 



which could not hold generally whatever might be the situation 

 of p, unless we had 



= SF; 



where we must be cautious, not to confound the present value of 



