FORMATION OF THE EQUATIONS OF CONDITION. 593 



n = F 9 -F U -(F 1 - F v - F w + F.) cos 70 - (F t - F u -F x + F M ) cos 40 



(F 3 - F u - F n + F a )cos30 + (F t - F u - F 13 + FJcosBO" - (F t - F 13 -F. a + F a 



U =(^ + F a - F u - F^sm 70 - (F t + F u - F M - F) sin 40 - (F t + F u - F 3l - 



(F t + F u -F - F a )am 8 - (F. + F 13 - F n - F 31 )sin 1 - (F t + F K - F - F M ) sin 60 



The same groups of terms enter into the expressions for a, & b t , 

 5 & fe c , a- 7 & 6 7 , a u & b n , a 13 & b a , and a n & b 17 , only multiplied by 

 different coefficients in each case. 



Also the same groups of terms with different coefficients enter into the 

 expressions for c< 2 & b. 2 , for 10 & 6 ]0 and for a u & 6 14 ; and the same groups 

 enter into the expressions for a, & &, a s & b s and a !6 & 6 16 . 



The above equations give the values of a m and b m for different values 

 of m for each belt of latitude, and it will not be necessary to go beyond 

 the terms above found. 



Formation of the Equations of Condition. 

 4. Take 2(X'*g + X' n g n ) to denote the series of terms 



m \ "V^ m *- m i $m . 

 + A _f_ + &C. ; 



and employ a similar notation for Y and Z as well as for the h constants ; 

 then the equations of condition for each belt of latitude are 



'^h~ n )= b m , ' 



S(Y': 9 ;+Y'-M=b., and 2 (F';/C+ Y'' n h^) = -a m , 



S (Z': ;/: + Z' m _ n g_ n ) =a m , and 2 (Z'; A; + Z'r n A_ M B ) = 6.. 

 A. n. 75 



