Induction in Spheroids. 465 



We have also 



dc da = -il« r^S+^Sl <Zr<fr 

 L-vV-1 \/l-/t 2 -l 



= ~""9 1 A t V / l— yu- 2 cos _1 r + r \A' 2 — 1 sin -1 //- > 



s a (say) taken between limits. 



10. The potential, therefore, becomes 



-2wW['Q (r') [l + '[^y^coa i'V-r \£»=l] ^"^ } 



+ ? 2 (S^ p » ^ q ^'> { l% » 3lI » + lIX * 3% " } 



taken between limits. 



11. Calling this V and changing / into r and yJ into {a 

 and I'icg versa, let 



V = Go Q (r) + C, P, W Q,(r)+ ■•+... 



ing the potential inside the n 

 outside, let 



V^Ao + AjP^P.O-lf... 



And Vj being the potential inside the magnetic substance, 

 and V 2 that outside, let 



and 



V, = B Q (r) + B : P,( M ) Q,(r) + ... + .. . 



Then, since V 1 = V 2 at r = TL = -, where e is the eccentricity 



a 



of the spheroid, 



A = B Q (R), 

 A 1 P 1 (R) = B 1 Q 1 (R), 

 A 2 P 2 (R) = B 2 Q 2 (R), &c. 



12. Again from the equation of the flux, 



— (1 + 47TA;) ^— + ^ 47r/c^ = 0, 



but since 



a?i = — r ar, 

 we have 



0?* . O' 1 H)r 



