102 APPLICATION OF THE PRELIMINARY RESULTS 



because - = 7 and - = dy. Writing now in the place of i its 



Cl CL 



value ay, and neglecting infinitesimal quantities, we have 



4 



D 



Hence the value of the total potential function becomes 



where the integral relative to 7 is taken from 7 = to 7 = GO , 

 to correspond with the limits and co of i, seeing that i = ay. 



The preceding solution is immediately applicable to the 

 imaginary case only, in which the inducing bodies reduce them- 

 selves to a single point P, but by the following simple artifice 

 we may give it a much greater 'degree of generality : 



Conceive another point P', on the line PQ, at an arbitrary 

 distance c from P, and suppose the unit of positive fluid con- 

 centrated in P' instead of P; then if we make r = Pp, and 

 & *. pPQ, we shall have u = r' cos 0', v = r sin 0', and the 

 value of the potential function arising from P' will be 



Moreover, the value of the total potential function at p due 

 to this, arising from P and the plate itself, will evidently 

 be obtained by changing u into u c in that before given, and 

 is therefore 



Expanding this function in an ascending series of the powers of 

 c, the term multiplied by c* is 



