624 SOLUTION OF THE EQUATIONS. 



above, that in the case of a spherical surface all except the principal 

 terms vanish, since 



2 [(x; x:^ w-] + 2 [( Y: Y:) W ] + 2 KZ: z;> ^ = o, 

 and 2 [(* ; x_) w-] + 2 [( r; F_ ) H + 2 [(z: z.-.) w ] = o. 



We have also seen that on a spherical surface 

 '_- ) ui] + 2 [( 7 -),] + 2 [(Z.-.)' ] -^ {2CW 



= n(2n + 1)2 



18. The solution of the final equations, stopping at equation (e) 

 inclusive and taking the equations corresponding to the polar segments 

 with radius 22 30', will give values of the magnetic constants in terms 

 of a, a', /B, ft', y, y which are simple functions of the forces at the poles. 

 These values of the magnetic constants for a given period, when equated 

 to the values of the same constants given in the table on p. 605 for 

 the same period, will give the values of a, ft, y, &c., from which, in the 

 absence of direct observations, we may derive a, b, c, &c., the forces at 

 the poles. The accuracy of the work may be tested by the close agree- 

 ment of the values of each of the quantities a, ft, y, &c., derived from 

 the several final equations which give the magnetic constants. The values 

 of these constants and the corresponding values of a, ft, y, &c., for the 

 periods 1845 and 1880 are here given. 



For 1845. 



g?= 6-166 +-0613y= 6'9808 , which gives y = 1 3'29 

 gr,= -2-5075 +'14925y = - "52399 , y=13-29 

 g?= -5-5700 +'4230y= '0513465, y=13'29 



g.?= -0545 +'0828y'=- '0275845, ,, y'=-'991 

 gf=- -4454 +'2302y'=- "67352 , y'=-'991 

 g= "3596 +'6657 /= - '30013 , y'=-'991 



gt= '58317 - "02269a = "602567, a = -'855 

 gt= "6034 -"0882 a = "678817, a = -"855 

 g}= -T0459 -'3059 a = - '784390, a = - '855 



g?=- "9900 - -0483 a'=- 1-065495 , a'= 1"563 

 gt=- "4403 -"1742 o'= - "712584 , a'= 1'563 

 gj= "6625 -"5980 a'= - '272348 , a'= 1-563 



