12 INTRODUCTORY OBSERVATIONS. 



All the difficulty therefore reduces itself to finding a function 

 V which satisfies the partial differential equation, becomes equal 

 to the known value of F at the surface, and is moreover such 

 that none of its differential coefficients shall be infinite when p is 

 within A. 



In like manner, in order to find F', we shall obtain V, its 

 value at A, by means of the equation (a), since this evidently 

 becomes 



a^V'-'B, i.e. ~F'=~F 



Moreover it is clear, that none of the differential coefficients of 

 V = I- can be infinite when p is exterior to the surface A, 



and when^> is at an infinite distance from .4, V is equal to zero. 

 These two conditions combined with the partial differential equa- 

 tion in F', are sufficient in conjunction with its known value V 

 at the surface A for the complete determination of F', since it 

 will be proved hereafter, that when they are satisfied we shall 

 have 



the integral, as before, extending over the whole surface A, and 

 (p) being a quantity dependent upon the respective position of p 

 and da. 



It only remains therefore to find a function F' which satisfies 

 the partial differential equation, becomes equal to V when p is 

 upon the surface -4, vanishes when p is at an infinite distance 

 from A, and is besides such, that none of its differential co- 

 efficients shall be infinite, when the pointy is exterior to A. 



All those to whom the practice of analysis is familiar, will 

 readily perceive that the problem just mentioned, is far less 

 difficult than the direct resolution of the equation (a), and there- 

 fore the solution of the question originally proposed has been 

 rendered much easier by what has preceded. The peculiar con- 

 sideration relative to the differential coefficients of F and F', by 

 restricting the generality of the integral of the partial differential 

 equation, so that it can in fact contain only one arbitrary func- 



