OF ARTS AND SCIENCES. 



357 



If now, instead of finding the potential at a point, we wish to find 

 the force exerted by two coils, the one npou the other, we may replace 

 the point by a coil and calculate the mutual 

 potential of the two coils from the formula; al- 

 ready deduced. Let us suppose, at first, that 

 the two coils are co-axial. Then we may 

 replace the coils by spherical magnetic shells, 

 which are concentric. The radius of the larger 

 shell will be Oj and of the smaller a.^ ; a^ and 02 

 will be the angular radius of the larger and 

 smaller coils respectively. Let O be the po- 

 tential due to the first shell at any point within 



it, then the work required to carry the second shell to an infinite 

 distance is given by the equation 



M: 



-lit 



dS 



(see Maxwell, § 423), extended over the smaller shell. Hence, since 



\dS^2TTa.^ j dcosd, 

 *J *J cos a.2 



pi do, 

 M= I ~2na,^ dcosd, [XXIV.] 



*J cos 02 



6 being the varying angular radius of the smaller coil. 

 In equation [XIV.] 



Q = — 277 + 27rC0Sa, — 27rsin"a, - ^p^^ PJ6) + &c. 

 ' '■ ^ a^ d cos oj i ^ ^ I 



s_in^ ^ rfP^(a,) 

 t Oj' a cos a^ 



differentiating Q with respect to r, and substituting in [XXIV.], 



1 dP,(a,) /^l 



M=z 4:7rHm^a.a,^ [- —^^^ f P {6)dcos6 + &c 

 i 2 a, rf cos a, / ' ^ ^ ' 



*- ^ ^ O cos a.. 



+ a,' dcosa, / ^^^^^ 

 i i »/ cos a.. 



dcosQ 



Substituting the value of / P(d) dcosd from [IX. 1>] page 352, 

 »^cos a.. 



we find 



