Definite Integrals, with Physical Applications. 387 



OQ from the centre of the sphere ; and the law of electric accu- 

 mulation at the surface of the sphere is required. 



Make AB the axis of x, the extremity A of the diameter 

 being the origin; the element of the surface included between 

 two planes at the indefinitely small distance Sar, and perpendi- 

 cular to AB is then = 2Tra.$x, a being the radius of the sphere; 

 and the distance of any point in this annulus from Q 



= {a 2 +/3 2 + 2/3. («-*)}* 



putting OQ = /3, and if we represent the electric accumulation by 

 A, which is in this case a function of x, the expression for the 

 tension on any point Q of the rod AB is 



.( {«' + /* + 2/3.(a-*)}4 fr ° m X = to * = 2a ' 



neglecting the indefinitely small action due to the electricity on 

 the infinitely small surface of the rod. 



Put x = 2at to make the limits and 1; the expression is then 

 transformed to 



, s r A 



'J, {a 2 + /3 2 + 2a/3.(l-2*)}4 ; 



and therefore by the proposed condition we have 

 m r A 



= f ± 



Jt Ja 2 + /3 2 + 2a/ 



/3 Jt 5a 2 +/3 2 + 2aj3. (1-2*)^ 



A 



when a</3, or 



/{l + £. <!-«) + |f 



equal a constant quantity m, j> being any proper fraction ; from 





 Vol. IV. Part III. 3 D 



