The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 57 



y = y^ cos i — z^ sin i, z = y^ sin i + z, cos i. 



Proceeding by exactly the same steps as before, we find 



<Py, dQ „ .dx . , . , . . ., 



-jTT + J 2aw cost-T- — a'K^ COS I {y^ cos « - ^^ sm 2) = 0, 



d'z^ dQ ^ . .dx , , . . , . . ., 



Or again, di-opping the distinctive marks of _y, and z, 



^ y , f^Q o •«'*■ 2 2 • / • ■ -x /^ 



-r5^ + -, ian cos I -= — a^n cos i ( w cos 2 — ^ sm i ) = 0. 



dt dy dt ^^ ' 



c?^^ rfQ - . .dx , , . . , . . .. „ 



-77 + -77 + -iaw sm I -T- + a-'n- sin i {y cos i — ^ sin ?) = 0. 



(3) 



Again, x, and y, lying in the sliding plane, let .7-, be the angle tt + Snt in 

 advance of the line of intersection of the two planes. Then € may be so de- 

 termined, that .r^ shall lie in the mean place of the apse when projected on the 

 sliding plane, or in any other position we please. And as before, 



X = x^ cos (tt + Qnf) — y^ sin (w + Qnt), 

 y = X, sin (tt + Snt) + y^ cos (tt + Qnt). 



Making for a moment, in order to abridge, 



A =2an -f — a^n^x, B = — 2an cos i -■= aV cos i (y cos i — zsini); 



dt dt ^^ ' 



we shall have from the first of (2) and the first of (3), by exactly the same pro- 

 cess as in the two former transformations, 



d'x d-v dQ 



cos (tt + Snt) -j-^ + sin (tt + tnt) -^^ + — + A cos {1^+ Qnt) +B sin (tt + Qnt) = 0. 



CIZ Clt CLX . 



- sin (tt -(- Qnt) ~ + cos (tt + Snt)-j^, + -J- -A sin (tt -|- ^nt) +B cos ( w -f Qnt) = 0. 



d'^x d-v 

 Or, putting for -p- and-^ their values, 



VOL. XXII. I 



