The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 27 



The two theorems now found are fundamental. The latter is not no- 

 ticed by M. Hansen ; the former, or one equivalent to it, is employed by 

 him. But there is another fundamental theorem whicJi he has not found ; for 

 we have necessarily 



as in an undisturbed orbit ; because, since d operates on t only, the elements 

 and h may vary in any manner whatever, and this equation will still hold. 



mw^ = p$ + ^±'-, ^ + ,<ip,d^_d^ h dp,. ,^^ 



dr ^'dr + ^ dr-P' dr + ^d^Tr- P' dr + ^"1 T^' ^^ (^) ' ^°^ 

 ^^p'<^.±,^d^_±^dp,d^k dy, d^ d'^ ' h d'p, dr , 

 d-^ P'd^^d^drdr Kel^dr^h;^TCTr = P'd?^UWidr'^y'<'^y^ 



d'-^ ^ ¥ d% , , ' ^ 



~ ''' d? h'i' dF-' ^ ^^'^*^- 



^"* 'dC~fi^?,"^- ^'^<^^ef°^'=' substituting for |^ and j^ their values, 

 /~^' ^^^^^3-^. we have 



^'_M_ d}^ h- (h] p. 



„3 „2 — P, J_2 + 



or, smce e'p,' = p^^ 



_ /x __ d-^ h- fi 



P'~'''d?~¥;'i?/ 

 which easily reduces to 



d'^ f^ fh' \ 



d^~?\f^,~V' (*^ 



There are still two other theorems to be investigated, which are given by 

 M. Hansen ; but found by him in a manner very different from that which I 

 shall employ. 



- = ^, +g cos {V - ^) gives ^ -|; = J cos (v - ^) = 



Tj (COS TT cos y + sm 7r sm v). 

 e2 



