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



d-y, dQ ,, ,, 

 dhdy^'''='- 



And again dropping the distinctive marks, 



d-x dQ T _ n 

 dt'^ dx 



d-y dQ ,, ,, 



— - -I — - 4- M = 

 df ^ dy^ 



(4) 



L — -j-\ 2a?icos^^— 2e?i+2ansiu'*-cos2(7r+en<) l+2anT-sin-^ sin2(7r+to) — 

 x\ oW+ e-n- - 2 a^n^ cos^^ — ^"^'i^ sin^ i+ ( Ja^n^ sin^ i— 2aen^ sin^ h ) cos 2 (wH- en<) i+ 

 y ( ^a?r^ s\.v?i — 2ae?i^ sin^ ^ J sin 2 (tt + ^nt) + a-n" sin i cos /> sin (tt + Znt). 



JVz=^|2e?i-2aw cos- |+2ansin=|cos2(w+e?iO |- 2ara -J^ sin- ^ sin 2 (tt +??«)•) + 

 yl 2aera-cos^s-"''*'~^'"'+i"^'*'SUi"i+(5-"^'*"sin-i— 2aen^sin-^jcos2(7r+eM;) 1+ 



X {^aV sin' i — 2ae«- sin^ ^ ) sin 2 (tt + €7i<) + a-?«^ sin i cos /z cos (tt + ^nt). 



The terms in Z and M containing z are of the order of the third power of 

 the disturbing force multiplied by sin i, and will not be wanted in these cases 

 in which i is large, and, I think, cannot be wanted when it is small. Most of 

 the other terms are of the second order; and as / is constant, they are all of a 

 very simple character. 



We must add from the last of (3) 



dt- dz 



dx 

 N—'2an sin /-y- + aWsini(?/cosJ — 2rsini). 



(5) 



