Therefore, 



56 The Eev. Brice Bkonwin on the Theory of Planetary Disturbance. 



X = Xi cos {6 — ant) — y^ sin (6 — ant), 

 y — X, sin (6 — ant) + y^ cos {6 — ant). 



We form the two equations, 



cos (6- ant) [-^+-^) + sm (0 - ant) [J+-^)= 0, 



. , , /d'x dQ\ ,^ , fd'y dQ\ - 



-sm(e-anO(^ + ^j + cos(e-anO(J + ^j = 0. 



But 



N 'IQ ■ /. sdQ dQ 



cos (6— ant) —j — 1- sm ((? — ant) -j- — -j—, 



• /. sdQ ,^ , da dQ 



— sm {6 — ant) T — \-cos (0 — ant) -t- = -j~-- 



, d^x . ,„ , d^y dQ . 



cos (e-ant) -^ + sm {0 - ant) -^ +— =0. 



■ .„ X d^-x ,„ , c?V dQ . 



- sm (e - ant) -^ + cos (6 - ant) -^ + ^^= 0. 



Substituting for -jj and -j^ , their values in x, and y^ , we have 



d'a;, dQ „ rfw, , , . 

 —J. + -p + 2an^'- aVj;^ = 0, 

 dt' dXi dt 



d^y, dQ a dx, , , . 



-^' +-, 2an—r- arn-y, - 0. 



de dy, dt ^' 



Or dropping the distinctive marks of «, and y,, as tending to produce con- 

 fusion, and as no longer necessary : 



d^x dQ ^ dy . , n T 



d> dQ dx „ , I 



We now take the new co-ordinates y^ and z, , y, lying in the sliding plane, 

 and z^ being perpendicular to it ; and, consequently, we have 



