FORMATION OF THE EQUATIONS OF CONDITION. 591 



The coefficients a m of cos mX and b m of sin mX in this series are the 

 absolute terms of the equations for g and /*,"' respectively in that belt 

 of latitude. 



Let the forces whether X, Y or Z, when X = (T, X=10, X = 20, &c. 

 to X = 350, be denoted by 



F , F 1} F,, &c. to Fy, respectively. 



Let 2 (F) denote the sum of these quantities, and let S 9 (F ) denote 

 the sum of every ninth of these quantities beginning with F a , and similarly 

 let 2 9 (F^ denote the sum of every ninth of these quantities beginning with 

 F 1} and so on. 



Thus 



Then we shall have 



+ (F 3 - F u - F n + F a ) cos 30 + (F 4 - F u - F.,, + F. M ) cos 40 + (F, - F 1:i - F.,. + F n ) cos 50 

 + (F. - F N - F + F)cos60 + (F, - F u - F a + F w )cos70 + (F s - F 10 - F. a + F, s ) cos80; 

 =^ + F a - F n - ^)sin 10 + (F, + F w - F., - F :u )sin20 + (F, + F u - F,, - F a )smSO 

 + (F. + F u - F a - FJsin 40 + (F s + F n - F,, - FJ sin50 + (F, + F,, - F, t - F. M ] sin 60 

 + (F 7 + F u -F m - ^)sin70 + (F, + F M - F. x - F 28 )sin 80 + F, - F^. 



Also let ^(F )^F -F a + F ls -F a , i.e. the sum of every ninth 

 beginning with F and changing the sign of the alternate terms, and let 



* t (F J )=F 1 -F w + F w -F a , &c. 

 Then we shall have 



9 ( F 7 )~] sin 40 



