80 



MR. LUBBOCK ON CERTAIN TERMS IN THE DEVELOPMENT OF R. 



In order to give another example of the employment of this method, I propose to 

 calculate the coefficient of 



e / cos (13 r — i — 4 ri), 

 the argument of which occurs in Professor Airy's inequality of Venus, nt and n^t 

 being the mean motions of that planet and of the earth. 



It is easily seen from the preceding pages that R contains the term 



- ll -$- y" ^Mi c«« (13 r - 4 n). 



If the coefficient of / cos (13 r — 4 ;?) be denoted by R^ 

 e/cos(13r — I — 4?j) . . . R>, 



adR^ 



R,= 



2d« 



-9iRi 



3 C «2, 5a^ (a , \ , }_ . \ 9«_S \ 



= - 128 I "" ^ ^5,11 + -^ V^ ^7,11 - 2 ^7,10 - 2 ^7,12 ) - « 3 ^5,11 | 



3 riOa«, 5a^(a, \_ , 1. h W 



= 728 I "<" ^5,11 - -^ V^ ^7,11 - 2 ^7,10 - 2 ^7,12 ; I 



And R contains the term 



3 ri0a2 5aWa 1 1 h \\ a 



128 I "^ ^5,11 - « 4 V a, ^7.11 - 2 ''7,10 - 2 «'7,12 J j ^ r- 



Professor Airy has 



{?(0>0) + ^(0,l)}c;'eA* 



2 



In Professor Airy's notation 



2 2 



and substituting my notation in Professor Airy's expression, that which I have found 

 results. 



The method I have given of developing the disturbing function in terms of the 

 mean longitudes may also be employed with advantage in procuring tlie development 

 in terms of the true longitudes. In this problem 



dR rdRdr adRdr 



de drrde da;di 



d r d , log. r 

 rde de 



log. r = log. a 4- log. (1 — e2) ~ log. (1 + e cos (X — tsr) 



= log. a - e2 — 



ecos 



(>^-^) + 2T2{l+2cos(2X-2m) I 



* See p. 89 of Professor Aiky's paper. 



