THE THEORY OF TERRESTRIAL MAGNETISM. 451 



proportional to its breadth (8/i), and multiplying by S/u, and integrating 

 from 1 to +1, we get 



-2nre- (a n + /B,,) I p.H n ,-- dp + terms involving a,,,, /3 ni , &c. 



ri ri 



X n x m dp,+ Y n y m dp. 

 J-i J-i 



Hence referring for the values of the above definite integrals to Section V. 

 Art. 9 (p. 417), we get 



P (Hrf dp \[(a n + (3 n )+ l -r [na n -(*+!) A,]] " (n+l) 

 J -i IL * J 



= J X n x m dit + I ^ Y n y m dp. 

 In the same way from the equation for z m we get 



(2n-l)(2n + 3) 



|1 /^Q _ 777,) ^ (7?, -4- 77i) t 



Since J_ 1 (g)'^-2| 1 . 8 ; 5 ... (2 / ;L 1)} .( 2n+1) (as proved above, p. 411), 



we see that the coefficients of a n and /B n in the above final equations are 

 determined. 



572 



