FORMATION OF THE EQUATIONS OF CONDITION. 589 



Regarding the Earth as a spheroid of revolution, we have seen that 

 the values of \i! or cos 6', where 6' is the geocentric colatitude, have 

 been determined for every 5 of geographical colatitude. Also the values 



a 



of cosifj, shit//, -j G r and H' have been determined for every 5 of 



geographical colatitude and have been tabulated for all values of n and 

 m from to 10. The weights of the observations of the magnetic elements 

 for these belts of latitude have also been determined on the assumption 

 that the weight is proportional to the area of the corresponding portion 

 of the Earth's surface. From the values of H' the values of X, X m n , 

 Y, Y n , Z and Z\ have also been determined and recorded, and from 

 these have been determined the values of X' or ( Jf cos i// + 2T sin /;), 

 X' n , Z' m n or (-J:; i sinr// + Z;"cos^) and Z' m _ n , which are the resolved parts 

 of the expressions for the horizontal and vertical forces in the plane of 

 the meridian on the spheroid, and so are the coefficients of the magnetic 

 constants in the equations of condition. 



2. In the preceding investigation Section VI. (p. 449) a ;l has been taken 

 to represent any magnetic constant depending on the action of magnetic forces 

 in the interior of the Earth, and ft n has been taken to represent any 

 magnetic constant depending on magnetic forces outside the Earth's surface. 

 Thus a n will represent the Gaussian constant </" in the expression g cos mX 

 or the constant /i in the expression h sin m\. These Gaussian constants 

 eft and A will have the same coefficients X' or Y"" or Z' in the equa- 

 tions of condition for X, Y and Z respectively ; and the corresponding 

 external magnetic constants, which may be denoted by g n and h n , will 

 have the same coefficients X' n , Y' n and Z' m _ n in the same equations of 

 condition. 



As in the equations in Art. 8, p. 454, these equations will be of the 

 form 



x-'lgl + x im - n gn + X'igi + x\ g \ + & c . = x' m , 



Y':g m n + Y'" n g m _ n + Y& + Y'\g\ + &c. =y' n , 

 Z' m n g m n + Z' m _ n g m _ n + Z'g + Z\g^ + &c. = z' m , 



with similar equations for h, h n &c. 



where X' m n = X m n cos ^ + Z^ sin ^, 



y/>_ ym 

 -* n * J 



Z' w = - X sin tf + Z cos . 



