THE THEORY OF TERRESTRIAL MAGNETISM. 411 



5. To adapt the preceding investigations on Legendre's and Laplace's 

 coefficients to the theory of Terrestrial Magnetism, there are certain rela- 

 tions of the functions H and the functions X, Y and Z which still 

 remain to be developed and which will be found useful and will greatly 

 facilitate the determination of the magnetic constants of Terrestrial 

 Magnetism from the observed values of the magnetic elements at places 

 regularly distributed over the Globe. 



We proceed now to the development of this Theory of Terrestrial 

 Magnetism. 



m 



Assuming, as in Section I., that (?" = (! j"-' 2 ) 2 D m P n , it has been 

 proved in Section III. (see p. 363), that 



except when n = n 1 ; and that when n t = n, we have 



'1 o 



(n m) ! ' 



Also from Section I. we have 



(n-m)l 



ri 



hence we have I H^H^d^ = 0, except when n = n l ; and when n, = n, we 

 J-i 



, 7 2 (n-m)l(n + m)l 



have 



6. From equation (4) above (p. 246) we have 



therefore by successive substitutions, 



DP n+l = (2n + l)P n + (2n - 3) />_, + (2n - 7) P n _ 4 + &c., 

 the last term when n is even being P , and when n is odd being 3P,. 

 Putting n for n + 1 and differentiating ml times, we get 



JD-P.= (2n - 1) ZJ-'P,., + (2n - 5) D~>P n , + &c. + F^f ^ n " dd ' I 



1_3J? P! when n is even.J 



522 



