66 APPLICATION OF THE PRECEDING RESULTS 



and as the electricity is in equilibrium upon A, the force with 

 which a particle p, infinitely near to it, would be repelled, must 



dV 



be directed along dw : but the value of this force is -j-j , and 



consequently its effect in the direction of the radius r, and tend- 



dV' 

 ing to increase it, will be -jr cos (f>. This last quantity is 



equally represented by -7 , and therefore 



dV' dV' 



-- -j = - -j-r cos 6 ; 

 dr dw 



the horizontal lines over quantities, indicating, as before, that 

 they belong to the surface 

 duced from this equation, is 



they belong to the surface itself. The value of f -jr J , de- 



/i 



_ ._ 



dw ~~cos$ dr cos< 



this substituted in the general value of p, before given, there 

 arises 





47T dw' 2?rr 2 cos 



Supposing Q is the quantity of electricity communicated to the 

 surface, the condition 



-^ = hV (where R is infinite), 



before given, becomes, since r may here be substituted for E, 

 seeing that it is measured from a point within the surface, 



Q^aTi Q 



7" : ~7" "2^' 



We have thus the rigorous value of p for the surface A whose 



k z / Q\ 



equation is r = ( 1 + ^2 cos - ) when the quantity Q of elec- 

 Q, \ 2/ 



