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



Calculation of the Coefficient of ^e^ cos (r — | + I) in the Development of R. 



If R^ denote that part of the coefficient of cos {r — 21) which is multiplied by e^ 



jRj COST e^ 



i?! e2 



R^ cos(r + |) e 



i?3 cos (r + 1) e 



^3 ^ 



•^^3 — "adT ~ 2d« ■^^s'T- 16 da "T- 8 ^1 "" 4 dfl ~ 4 ^4 — gda "" ^i 



„ _3_fl ^ , c^ J « - J? _ _?_ g , ^ ?> 



^3 — — 2^3 + 4 a,2 ^3,0 - 2 a^2 ©s,! — 4^2 ^3^2 ^9 — 8 af " Uu^^ ^3,0 " Jg^ h,2 



I? — J?_ -^ ^ -^ A _J_ -i^ h R' — -^ _l_ -^ /; « 7 



^4 — 2 a 3 4 a^3 »3,0 — 2 a 2 03,1 i- 4 ^^3 %,2 ^ 1 — ~ 2 a ^ i" 4 «^2 «'3,0 — 2 «/ ^3,2 



^1 ""a; ~ a, 2 "" 2 ■*" l6^i ~ 4 ^4 — 2 " "" 8a/ "•" l6a/ ^3,0 i- g^^a O31 — ^^^2(^3^2 



^ -I- 8 ^1 ~ 4 ^4 — ^1 — 8 a/ "■ 16< ^3,0 i- 8 a3 %i — jg ^^2 03,2 

 ,5 /t3 — — 8 a/ ■*■ 16 a/ ^3,0 "t" 4 «/ ^3.1 — s a/ -^.2 — ] g ^^ \^a^ ^5 ^ — Or^-^J 



■" ¥ {^ ("^ ^5,1 ~ 2" ^5,0 - "2 ^5,2) - ^3 (— *5,2 - "2 ^5,1 - "2 ^5.3) j 



, 3a 3a , \_^ ^ h ^ ^ h 



+ 8a/ ~ 16a/ ^3,0 + s^^s »3,l ~ Hj^^ ^3,2 



— l^h 1^ h 1^ (^ k I. \ 



— 8 a/ ^3,1 - 16 a/ ^3,2 — 16 a 3 V"^ ^5,0 — %l) 



3 f a^ / a^ + af a_ «. \ _^. , «' , , «" , «% 1 



8 \ a/ V «/' ^'1 " «, ^5,0 - -^ 05,2; - 2 a/'^5,l + 2^^5,3 + 2^4^5,0 - 2^405,2 | 



8 a/ ^3,1 16 «/ ^3.2 - 16^ ^5,0 + 16^3 05^1 



«/ a_ . , 3a^ 3 a» 



^3 — 16a/ O3.2 -r 16 a/ ©5,1 — J6^ 65,2. 



If /2'i5 denote that part of the coefficient of cos (r-^ + y which is multiplied by e^e^, 

 ^15 2da^ -^3 



- 16 a/ ^^3,2 -1- 32 «^3 ^5,1 - 8 a/ ^5,2 + 32^3 (^ — ^'5,2 ~ -i" ^5.1 - "g ^5,3 J 



"" 52 {.< (^ *7,1 - 2" *7.0 - "2 ^7,2) - ^5 (^ ^7,2 - 4" ^7,1 - i f'7,^ I 



, _o , 3g« , 3 rt3 



T" 16 a/ ''3.2 ~ 16^3 ^'s.i + 16 a/ *5,2 



