112 



SECULAR VARIATIONS OF THE ELEMENTS OF 



m 

 na 



na 



na 



n a 



n'a" 



&c. sin 



(408) 



(409) 



And since, for the root # 4 , we have N t =N^=:N t "=N t m , &c., the system of equations 

 similar to equations (133) will be divisible by JV 4 NJ &c., we shall have the follow- 

 ing equations of condition : 



na 



a 



n a 



&c .=o 



&c -= ; 



'na 



na' 



n' a " 



&c. 



(410) 



Now putting the first members of equations (408) and (409), equal to x and y 

 respectively, and the coefficient of sin or cos of (gt-\-(3), in the same equations 

 equal to 2, we shall have 



x=z sin (gt+(3) ; y= cos (gt-\-@) ; 1 

 whence tan (gt-\-@)=x-i-y. j 



(411) 



6. In order to find the values of x and y, which are to be used in equations (411), 

 we shall assume the following values of <>, $', $", &c., 6, 6', 6", &c., corresponding 

 to the beginning of the year 1850, at which epoch t0. 



Now, since ^=tan <> cos 0, and p=:tan $ sin 0, we find, 



p =+0.0891690, log.jp =8.9502138; 



p' =4-0.0573577, 



p" = 0. 



p m =+0.0241597, 



log. p' =8.7585915; 



log. p" = QO 



log. p m =8.3830911; 



