TO THE THEORY OF ELECTRICITY. 79 



will be completely determined, by satisfying the equation (b) 

 and the condition (c). Let us then assume 



DF(y cos ST x sin OT ; z) = hD(j> (y cos or x sin -or) ; 



7*. being a quantity independent of x, y, z, and see if it be pos- 

 sible to determine h so as to satisfy the condition (c). Now on 

 this supposition 



D V D' V hD<f) (y cos OT x sin CT) (# sin < ?/ cos </>) fo& 

 = Z></> {y (h cos ?r + & cos <) x (h sin or + b cos $)}. 



The value of Dp corresponding to this potential function is 

 (Art. 7) 



Dp = 0, 



and on account of the parallelism of the lines L, L', &c. to each 

 other, and to -4's equator da = da-^ . The condition (c) thus 

 becomes 



(c): 



Dp and Dp } being the elementary densities on A*s surface at 

 opposite ends of any of the lines L, L', &c. corresponding to 

 the potential function DVD'V. But it is easy to see from 

 the form of this function, that these elementary densities at 

 opposite ends of any line perpendicular to a plane whose equa- 

 i tion is 



= y (h cos r + b cos (/>) x (h sin r -f- b sin <j>) , 



i are equal and of contrary signs, and therefore the condition 

 (c) will be satisfied by making this plane coincide with that 

 perpendicular to L, L', &c., whose equation, as before re- 

 marked, is 



= x cos w + y sin CT ; 



that is the condition (c) will be satisfied, if h be determined by 

 the equation 



h cos VF + b cos (j) __ h sin -or + & sin $ 

 sin -cr cos r 



which by reduction becomes 



= h + J cos (<f> w), 



