358 



+ 



PROCEEDINGS OF THE AMERICAN ACADEMY 



J_ dP^ (a,) dP,(a.;) ^^^ 



2ai dcosai d cos a^ 



1 r'-i dPi (ui) dP,(a,) 



i (i -\- 1) a^i d cos flj d cos a^ _ 



[XXV.] 



If now the axes are inclined, making an angle 6 with each other, we 

 have as before (page 352) only to introduce the P (^) of the corre- 

 sponding order, and a^ in place of r, which gives 



3fz= 47r'^sin^a^ sin"a2a2 



1 d Pj (aO d P, (a,) 



P,(e)-\-&c. 



2 a^ d cos Qi d cos og 



I 1 (f2~^ d Pj ( ai) dPiJa.^) p 1 rxxVI.T 



~^ i{i-\-V) Qj^ dcosoi dcosa^ J 



as the expression for the mutual action of two circular currents of unit 

 strength, the axes of the coils making an angle 6 with each other at a 

 point c distant Oj and a^ from the circumference. 



It will be observed that equation [XXVI.] may be written 



(^ . o 1 f?Pi(ai)) ( • o odP.(a.)) r^.^s , o t 



sin-oi 1 dPi{a^ 



i a^ dcos 



-^ M 27r sin^ a, t^T , F ^. (^). 



The quantity within the first pair of brackets is precisely what we 

 made Ql equal to in equation [XVI.], and the second quantity is what 

 we made q^ equal to in [XXI. B] ; therefore equation [XXVI.] may 

 be written 



M= Q,'q,'P,(e) + &c. + Q/q/P,{6). [XXVII.] 



This is for one turn. If we have coils of a number of turns of wire 

 whose sectional areas are | »;, the Q' and q^ will be replaced by the cor- 

 responding values of G' and g' as given in [XXI. A] and [XXIII.]. 

 This gives the general expression for the mutual action of two coils in 

 the form 



M= G,'g,'P,(d) +G,'g,'P,(6)-\-&c.-{-G/ffJP,{e). [XXVIIL] 



