The Rev. Brice Bronwin on the Theory of Planetary Disturhance. 49 



J^here some small quantities are retained, which, perhaps, are not wanted 

 In taking the partial differential coefficients of A, it will be better to find 



-J-, and not —rr 

 ds di 



dA _ 2r\H dr^ .^ , 



ds 



"Write this for a moment ^ - M j. i a2 ''^^ t ., , 



°^' (^s ~ "*" 2^ ^j- Transpose the last term of the 



second member. Then (1 - • A=) ^ = M ; whence ^ = M {I - l a^)-. ^ 

 ^/(1 + |A2); and "^^ ^ 



fi?A _ 2r5 di] 



ds-~hTt '2'' + ^' ('/ + hf)+s^ (l^'? + i^';' + T%>/)| (1 + iA=). (11) 



Preparatory to finding '^ , we may observe that -' = cos {v-b)'t = h. 



at (If J.2 



cos (v-^); and, therefore, ^ =. A sin (v - ^) = ^ ,, Also ''" - rr, 



dMt r- ^ ^ ,,2 V- Also, -/^ — — cos 



*" ^^--II?^'^"'^A2^ = cos^(«-&) = l-sin^(„-5)=.l_,,2. We 

 now find 



'^^ ^ M , . 2sin2^ 



^- = .r ,; j 1,; + .- (1, + I ,/) + s^ (_l_, + _._,^s + ^ ,^ , 2 ^^. ^ _ 



COSJ 2 j^ 



By further reduction, and treating the equation with regard to A as in the 

 last case, 



dA _ 



2sin^^ 



(l + lA=) 1 (1 + Un. 



cos I ^ ^ ■' 



VOL. XXII. jj 



