SOLUTION OF THE EQUATIONS. 601 



The final equation for any magnetic constant derived from the series 

 for X, Y and Z combined will be formed by adding together the co- 

 efficients of the same magnetic constants in the three final equations for 

 X, for Y and for Z, each taken separately. As there is a separate final 

 equation for each magnetic constant, there will be as many combined 

 final equations as there are magnetic constants in them to be determined. 



Let us illustrate the mode of solving these final equations by taking 

 the case given above (see p. 596) in which m = 4 and n is odd, taking the 

 equations up to latitude 77^ inclusive, and combining the equations for 

 X, Y and Z, supposing the magnetic constants corresponding to negative 

 values of n to be non-existent. We will include the terms involving n = 7. 



The coefficients for g^ and h* being the same, the final equations for 

 g* and h s * for the period 1845 may be written thus : 



(for g) (for h) 



from (X) 3-4034960 a 5 4 - "3898572 a 7 4 = "2416593 or -'0159063, 



(Y) 9-4158541 a 5 4 + -4092903 a 7 4 = '0589245 or '3418323, 

 (Z) 15-3871472 a 5 4 + -0223528 a 7 4 = -4657356 or '1824818. 



Adding these together we have 



28-2064973 a 5 4 + '0417859 a/= '7663194 or '5084078 (l). 



Similarly the final equations for g 7 4 and h 7 * may be written thus : 

 from (X) - '3898572 a/+ '2637326 a~ 4 = '0204205 or '0140404, 

 (Y) '4092903 a 5 4 + '3081774 a/= '0454171 or "0373065, 

 (Z) -0223528 a 5 4 + -6536612 a. 4 = '0056358 or '0882180. 



Adding these together we have 



0417859 a 6 4 + T2255712 o 7 4 = -0714734 or '1395649 (2). 



Eliminating a* from the equations (l) and (2) we get 

 1-2255093 a, 4 ='0703382 or '1388117. 



Hence g 7 4 = '0573951 and hf = '1132686. 



Substituting in the first equation, we get 



<7 5 4 = -0270832 and V = '0178567. 

 A. ii. 76 



