Definite Integrals, with Physical Applications. 397 



above causes are known, it* f represent the electro-motive force 

 at a distance /3 from the centre, then — f$f must be the internal 

 tension produced by the action of the sphere, and the application 

 of the second rule will then give the law of accumulation. 

 We must however always put the internal tension under the 



•»£/©■■ . 



Thus in Fig. 4. if the point P separates by influence, the 



combined electricities of the sphere, and Q be a point within at 



a distance /3 from the centre, then if we put OP=ha the force 



c 

 on Q = ( o_i ^ to destroy which the force exerted by the elec- 



— c 

 tricity on the surface must = 7-5 — 7— r- > and therefore in this case 



[p—nay 



1 f //3\ c _ 1 c ~c fl 1/3 1^ /? \ 



a J \a) ~ (i-ha ~ a ' /3 __ _ ~ a \k + ¥ ' a + h 3 ' a" KC '/' 



a 



and therefore the law of accumulation is 



-c fl 3 d(tt') 5 d".(tt') 



47ra s 



fl_3 <W2, _5_ d*.(tf) 1 



U h 2 ' dt "^1.2. A 3 " rft 2 °""J 



chi £ fl Id 1 rf 2 1 



" 2™ 2 <//> 1*J hi' dt Kt)+ 1.2.M- dP {tt > 4XC -| 



cte d i\ du\ , . 1 ,, 



fe-rf-J' when »-#- j. «.(!-«) 



2ira 2 ' f/A 



C& rf^ fl_ A 1 



" 27ra 2 'd/i \&i' \l+2h.(l-2t) + /i J \i] 

 c 1-k* 



~ 47ra s ' {1+2A.(1 - 2*) + #}»' 



3 E 2 



