THE THEORY OF TERRESTRIAL MAGNETISM. 457 



equations for a n and fi n , &c. we must multiply each of the terms of the 

 respective partial equations of condition similar to the above equations for 

 a n and @ n by (1 e> 2 ) S//, or ( 1 t y2 + 2e 2 /A 2 ) S/i, and integrate between the 

 limits +1 and 1, i.e. over the whole surface of the Earth. 



The coefficient of a ni in the final equation for a n will be the same 

 quantity as the coefficient of a n in the final equation for a ni ; and similar!} 

 the coefficient of )8 ni in the final equation for /3 n will be the same quantity 

 as the coefficient of ft n in the final equation for /?_. 



10. Since H n ' is the same function of p.' that H n is of //,, it follows 

 from the results given above (p. 421) that 



< - + o^-o" <> 



and I* H n 'H nt 'dp' = 0. 



Hence we need only consider the terms involving e'pJ' in the above 

 expressions for the coefficients of the magnetic constants. 



The coefficient of a, h in the equation for a n will be 



Putting this under the form 





and putting /ii for /u/ in terms of the second order, we see that all the terms 

 are readily integrable by means of the above definite integrals. 



A. II. 58 



