INTRODUCTORY OBSERVATIONS. 11 



Let us make B= I (Xdx -f Ydy + Zdz) whatever may be the 

 position of the point p, F= I - when p is situate any where 



within the surface A, and F'= I - when p is exterior to it: 



the two quantities F and F', although expressed by the same 

 definite integral, are essentially distinct functions of #, y, and z, 

 the rectangular co-ordinates of p ; these functions, as is well 

 known, having the property of satisfying the partial differential 

 equations 



\ rs=::v ' ^2 * j,,* T d^ > A, v ^* 



dx* Hh dy* " dz* ' 



If now we could obtain the values of F and V from these equa- 

 tions, we should have immediately, by differentiation, the re- 

 quired value of p, as will be shown in the sequel. 



In the first place, let us consider the function F, whose value 

 at the surface A is given by the equation (a), since this may be 

 written 



the horizontal line over a quantity indicating that it belongs to 

 the surface A. But, as the general integral of the partial differ- 

 ential equation ought to contain two arbitrary functions, some 

 other condition is requisite for the complete determination of F. 



Now since F= I- , it is evident that none of its differential 

 J r 



coefficients can become infinite when p is situate any where 

 within the surface A 9 and it is worthy of remark, that this is 

 precisely the condition required : for, as will be afterwards shown, 

 when it is satisfied we shall have generally 



the integral extending over the whole surface, and (p) being a 

 quantity dependent upon the respective positions of p and do-. 



