THE THEORY OF TERRESTRIAL MAGNETISM. 437 



Similarly, the final equation for a, h is found by multiplying the above 

 equations by X, Y, and Z respectively, and we get 



x:x:dp + , 1 (x$>dp + & c . = 



Similarly, if y m denote the coefficient of sin m\ or cos m\ in the 

 value of the force Y as derived from observations, we have 



2 (a n Y n ) = y m , 

 and the final equations for finding a n and a n> respectively will be 



and a n r; Y' dp + a,,, j ( y-)"^ + & c . - 



Combining the final equations for a n from Jf and F together, we have 



a n 



since the coefficients of a, lt and all the other terms on the left-hand side 

 of this equation vanish when the integration is taken all over the Earth's 

 surface. 



Hence a n . n (n+ l) /^N/^ ^ + 



., 



Similarly, by putting n, for n, we may get the value of a ni . 



In the same way the final equation for finding a n from the equations 

 for Z would give us 



.= ! 1 z; 

 j -i 



or a n (n+ i) \ La. 1*0/4= I "z m au, since 



ie a 2(n+lY- (H ~ m) 



l.C. t* l'*l-'-fr^ r* r- /^* 



