No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 151 



Turning now to the determination of e and r, let equations (235), (236) be WTitten in the 

 form (244), where 



^=-^+(4Tr3)-^^+g-+ 



Multiplying tlie first of these hy sin 9^, the second by cos (j; and adding, 



S sin ^ + Ccos </i= —5(2/ cos ^+z sin ^<) +oa;(y cos ij>+z sin ^) 



-3«oy(y cos 0+2 sin (j{r) +T— s(y sin ^-z cos 0) + . . . 



Here, again, the arguments and factors are functions of the elements a, e, tz, c, and the expansion 

 in a Taylor's series is necessary. 



Let 



S sin + Ccos =/()?, r„ e„ i, I) 



Then the form of Taylor's series is the same as the expression for w — w^,, (p. 148), with the 

 following modification. Within first order quantities, 



/(j;, r,, (9i, J, 2)= -2(j/cos0 + 2sin9!i) 



Hence, 



Ssin v'^ + f cos 0=/(,„, r, e,, J„, 2J+(^-^^-^^)^^, 



■*'VJ„'^ar'^5s„/2^'^ 5>)4^ 



4-j(l-,„ cos .0)^ + 2(^1-3 >J„ cos .„j^]-^ + 



The order of computation is : calculation of 



— 2^'/ cos y''+2 sin 0) 



by inspection of the table for W, in which the arguments are to be given the subscript zero, 

 differentiation of the first order terms, calculation of the necessary products of functions of y, z, 

 and the partial derivatives, and the addition of these products to the first calculation. The 

 required function is given in Table LVII. 



