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



Sa^ Sa^ 3^ 3 a^ 3 a^ _a_ , 



= 32^ ^5,0 — 3^3 05,2 — 16 « 4 %,i i- 3C2 a 3 «?5,o -h 32 « 3 05,2 "T 4 ^2 ^'s,! 



_ a , 3 a' , 3a^ , 



— T^ *'3.1 + 32 a,^ ^5,0 — 3£ fl 3 ^5,2 



2^ix— — 2da "^^^ 4 da 2^1 



— 2" ^3 — T ^1 — ~ 3 a 2 »3,1 — 32 af ^5,0 "T 32 « 4 05,1 "T 32 « 2 ^'a.l 



a . 3 a^ , 3 a^ , 



= ~ 32^ ^3,1 — i^3 05,0 + 32^ %,1 



— ~ "8^ ^5.0 + 8 a^ ^5,1 — 16 af ^3,1 

 2 ^ix = — 32^2 03,1 + ig ^^3 05,3 -h 32 a/ ^5,1 "T 32 a^ V^^ %,l — 2 ^5,0 — 2 \y 

 + i { $ (^ *7,0 - ^l) - -^ (^ *7,1 - T ^7,0 - i ^7,2) } 



3 a', . 3 a 



III i 



"■ 8^ ^5,0 + 8^ ^5,1 - 7g^2 ^3,1 



— 7a , 39 g^ 3a3 3a , 



— ~ 32^^ ^3,1 — ei4 ^3 05,0 + 4 a 4 ^5,1 — 5-4«^3 »5,2 



15/ o^ / g" + af , ^ , ^ ;. ^ _L _fL 7. ^'^ a 1 



+ 32 \ ~ a4 V a 2 ^7,1 " ^^ ^70 — ^^ 07^2; i" 2 a/ ^7,0 - 2 a/ ^7,2 J 



— ^^"^ A _L IL^ A 39 a^ , 3a^ 3a , ]5_^ 3 a^ 



— ~6-4 a/ ^5.0 + 64 a^ ^5,2 — 64 a^ %0 + 4 a/ ^5,1 — 64 a/ ^5,2— 32 a^ ^5,i + 32a*^5,i 



~ ~ 8 a,^ ^5.1 + 32^3 ^5,2- 

 Hence R contains the term | - jg-g b^^ + ^^3 b^^ \ e^ yf cos (2 r — 2 | + 2 ??^). 



Calculation of R^, or the Coefficient of cos (2 f^ ~ 2 vi), in the Development of R. 



Distinguishing the argument 2 ;?, — i, by the index 65, and the argument 2 ti^ by 

 the index 62, 



-^— ~ 2da, ~^^62 /{62 = - 8^ <53 1 



^65 - - 8 «^2 O3.I i- 15 ap \a, ^5,1 ~ ^g ^5,0 - Y 65,2^ + J^ ^3,1 

 — IG a* ^3,1 — 16 a 3 ©5,0 "i" i6^« %i 



