Definite Integrals, ivith Physical Applications. 389 



In fact, when the electric accumulation is constant the ex- 

 ternal attraction will evidently be =£ , and the integral with 

 respect to /3 which represents the tension of the fluid at Q is 

 evidently then = '| . 



28. The law of accumulation in the case considered above 

 is the simplest possible, but when the sphere is subjected to the 

 electnc influence of other bodies, the function which expresses 

 that law may have any form (subject to vanishing at infinite 

 distances), the indication of which form is to be had in the law 

 of the attraction on external points, or its integral, the tension 

 of the electricity in the infinitely thin rod ABx. 



To generalize the preceding investigation we shall take the 

 following example, which leads to some curious results. 



The tension in the external rod, varying as any inverse power 

 of the d.stance from the centre of the sphere; to find the law 

 of electric accumulation on the spherical surface. (Fig. 3.) 



Adopting the same notation as in the last article, and repre- 

 senting the law of external tension by 



o" 



m - 7f7T-4™\ we have when «</3 - 



and therefore for internal points where .> A ,ve may pnt i i„ 



tt 



this expression for J ; and thus we find the law of internal 

 tension to be .. £ . w : that is> when ^ ^^ ^.^ 



.; i- ? 



