The Rev. Bkice Bronwin on the Theory of Planetary Disturbance. 53 



serve, liowever, that making cp, cq. Is, l^, the alterations produced in />, y, &c., 

 by the disturbing force, we have for the first power of that force, 



lp=^is- qhb, lq = l Is +pl^ ■ 

 whence, observing that s' = p- + q^, we have 



S ' ' 5- 



Tlius we find the alterations produced in s and 5 from those produced in p 

 and q. And in like manner we may find the alterations depending on the se- 

 cond power of the disturbing force. 



In this section we have taken no notice of the development of r in the for- 

 mula; where it has appeared. Making r = r, j3, we may either let it stand thus, 



or put from (3) section (1) its value in terms of--, which we easily find to 

 be 



Then we shall have only r, and v„ besides the terms depending on the latitude, 

 to develope. These are given functions ofz = t + to, and may, therefore, be de- 

 veloped together by Taylor's theorem. 



We might have developed the value oiS in (11), section (2), relative to | 

 and <p, differently. 



Since the difference only of X and v enters into the composition of 5, we may 

 change these quantities into \ and i\, or into the anomahes X,- tt^ and r,- tt . 

 Using for simplicity X, and t;,, and making 



jj^_ /l (A^ 2 co s v^ dE ^ 2 sin v, dR 



r ' h'J k dv "*" // dr 



Y _ /I , m\ 2 sin V, dR 2 cos v, dR 



' ~\T-hy h AT h 'dF' 

 we have 



