INVERSE METHOD OF DEFINITE INTEGRALS. 359 



Again, if we represent the distance PQ by r, and the law of force 

 by y(/-) and put AQ = k the initial value of r, the total action is 



r rr ds „, ^ ON 



2ATJ;r,y^./(r).^, 



which by the property of F"o is equal to S\traRf{k), or to E .f{k). 



Let us now suppose an equal quantity of fluid, but of a contrary 

 nature in its action, and therefore represented by — E to be collected 

 in a single point C in the axis produced to a small distance AC- a. 



The total action of the compound system on Q will then be 



E{f(,k)-f(k + a)}, 



which tends to vanish as C approaches A. 



Lastly, suppose a unit of fluid when distributed over the surface 

 according to a law expressed by {t), which depends on the figure of 

 the solid, will exert no action on any point Q in the axis; then if 

 the law of distribution of the fluid be expressed by X V^ + c (p {t), the 

 total action on Q including that of C, will be still E {f{k) - f{k + a) ^ . 



From which it follows that when an electrical spark -Eh in- 

 finitely near to the vertex of a conducting solid of revolution charged 

 with a quantity of electricity E', the distribution of the latter under 

 the influence of the former is expressed by the law 



pi 



\Va + c<i>{t) where \ — ^ ^, 



otraH 



and where c is determined by the equation* 



Having thus given the geometrical and physical interpretations of 

 Vo, it will not be necessary to discuss the transient functions V^, V^, 

 &c., of which the properties are very analogous. 



* Vide First Memoir, Art. 35, the expression there obtained for a sphere being in- 

 cluded in that obtained above, when the influencing point is infinitely near the sphere. 



