34 The Rev. Bkice Bronwdj on the Theory of Planetary Disturbance. 



Make eo = <?/(!- a), a being a small quantity of the order of the disturbing 

 force ; and we may write 



so flo flo 



To which wc may add from the third of (2) ; 



d^ _ ^ ^ _ 1 



A, B, and C being functions of e^ and a easily found. But p^o, a function of 

 \„, and, therefore, a function of Wot, must be converted into a function of np- 

 by Taylor's theorem. The results obtained are much more simple than those 

 obtained by M. Hansen in the same case, and I think also more convenient. 

 If we wish to make tt^ = tt^ + i/, we have 



cos (Ao — TTo) — cos i; cos (Ao — •^o) + sin >/ sin (Xo — tto). 

 — ^ = sin (Xo - TTo) = cos )/ sin (X„ - tt,) - sin ?/ cos (X„ - ttJ, 



cos (X„„ - TT^) = cos (X„ - ttJ = ( -^ - 1 ) - . 



Between these we may eliminate sin (X„ — ttJ and cos (X^— tt^,) and obtain 

 a result involving -p ; but I shall not pursue the subject fui'ther. 



"We may satisfy (2), and all the requisite conditions, by simply making 6',, = t?,, 



7r„ = TT^ , ^^ = — , and w^f „ =: n„T ; but this would not leave us any arbitrary con- 



stant except a„, and we might have an unsuitable term which we could not 

 get rid of 



We must now proceed to another class of formulas, some of which will be 

 wanted. But I shall not confine myself to these. 



Let [-J-), ( -T- ) denote that t is to be changed into t in these quantities. 



after the ojjeration of differentiation is performed. And thus this change will 

 be denoted in other cases. Thus from (2) of the first section we have 



