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



9 o T I 1 i I « r 3« I 1 ^ i 



= ~ 16^2 h^ + 2^, ^3,1 + l^f «'3,2 - 8^ ^3,2 i" 4 a,^ ^^3.2 



" 87^ ^3,1 + 16 a4 «'5.o — 16 af ^5,1 



3 , 3 a« , , 3 « A '^« , I _?_ / _L ^^^ A ^«^ / 



= " 8^/3,1 - 8^ ^3,1 + I6< ^3,2 - i6a,^^3,2 + 80,^3,1-1- i6tf/^5,o i6«3^5,i 



_ 9a» , _L_9f^ A 

 — ■"■ 16" a 3 ^5,1 -t- 16«4%2' 



u 3 «9 _3«^ , 



If i?'i5 now denote that part of the coefl&cient of cos (r — ^ + |J which is multi- 

 plied by e ef, 



— WJp ^5,1 — 32 fl 4 05,2 -t- 32 a/ ^5,2 -t- 32 a ^ O5 1 > 



3o3 3tf^ 3a" , 



■""16^^5 2+ J6«f^5,i"" 16 < ^5,2 



— 32 « 3 ^5,1 — i6a;^5,2 

 Hence R contains the term 



^/ 1 32^ ^5,1 - T6 < ^0,2 j ^ ^/' cos (r - I + IJ. 

 I have found that the disturbing function contains the term 



As I have given elsewhere the details of the calculation of this term, it is unneces- 

 sary to repeat them here. 



In order to obtain the terms depending partly upon y^, y/^, &c., from the same 

 equation, the following transformations are necessary : 



cos X' = cos (V — v + ") = cos {\' — v) COS V — sin (V — j*) sin v 

 tan (>C — v) = cos / tan (X — €) 



,^y . 1 ' /^\ \ cos I tan (X — S) 



^ '^ ^1 + cos«»tan2(x — 5) ^ ^ ^ 1 + cos^ 1 tan^ (A — §) 



-, . cos^ — COS (A — ^ + v) + sin- — cos (\ — S — v) 

 ,_ cos V — cos I tan (X — 6) sinv __ 2 ^ ^ 2 ^ 



"~ ^ I + cos* » tan^ (X - g) "" ^ 1 — sin^ j sin^ (X — f) 



-. . cos^ -— sin (X — g + y) — sin^— sin (X — ^ — v) 



. , cos I tan (X — b) cos v — siny 2 2 



^^^ ~" ^l"+~cos«Ttan2{x^rg)~ — ^1 - sin^ sin2 (X - g) 



r" = r ^i - sin« » sin« (X - g) 



