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



2 sin's , . , , 



_, 2 1 - cos « 1 , 1 , , , _ , . „ 



But :- = r— = -. - 1 = -jj- jr - 1 = i5- + AS< - ^s\ 



COS i C0S2 COS J V(l— S'') - " '" 



Therefore, 



^ = «1-i+'r + «H-§+i'r + '/)+^^(-§-iV'r-|^'/+K)Ki + 4A^). (12) 



Since 2 sm- - = 1 - cos « = 1 - /(I - s-) = y- + |i-^ + -^s'^. Putting this 

 value in (5), and developing the terms, it becomes 



^7 =^ W (i - -r) + ^'a + bf - rf) + 6" (tV + hf + W - n')l (13) 



We will now transform (2), so as to introduce a and A, or rather )/ and A. 



ds . di _ cos- i dR _ 1 — s' dR 



dt ~ dt h sin i d^ hs dS- ' 



rfS- cos^ i dR _ 1 — s' dR 



dt h sin i ds hs ds ' 



Or, putting sr) for a in R, and introducing the partial differentials of the new 



quantities, 



ds l-s' (dR dn dR dA\ • 1 



dt hs \d7] dB^ dA dS-^ 

 dB l-s-/dR (IRdA^ ^ ^^^^ 



dt hs \ds dA ds 



It may not be amiss to find 2 sin' - separately. Thus, {„ being the mean 



value of i, or its value when the disturbing force is nothing, and, therefore, a 

 constant quantity, we have 



2 sin^ r = '2 sin^ ^ + 2 J rf sin- „ = 2 sin- ^ + I sin idi = 



„ . .Jo [dt .dR „ . ,io [dt .fdRdn dR d, 

 2sm^-+J^cos.^- = 2sm^2+Jl^°^\l.7;^ + ;iA^: 



