MR. LUBBOCK ON CERTAIN TERMS IN THE DEVELOPMENT OF R. 65 



— 64 a 3 ^5,1 - 32 a 3 05,2 — 54 ^3 05,3 - 32 | "" «/ V «/' ^'^ ~ "^ ^7,1 " "^ O7 3 j 

 "■ 2 a 4 07,0 -t- 2 a^4 07,2 i- 2 «/ ^7,1 - 2 a/ ^7,3 J 



— 64 a 3 ^5,1 32 ap ^5.2 04 « ^ %3 + 32 «; ^5,2 + 32 ap ^5,1 — 1 g a* *5,2 



___ 15 ff^ , 3 a 



— 32ai^^^A — 1 6 a/* ^5,2- 



Hence i? contains the term 



r 15«3 , 3a , "1 _ , ^ . ^v 



'^' \ 32^^ Ki - 16^^ h,2 j e^ e3 cos (r - 1 + ^ . 



Calculation of the Coefficient of e ef cos (r — | + !/)• 

 If /?i9 denote that part of the coefficient of cos (r + 2 1^) which is multiplied by ef, 



Ri COST e,o, 



R'l . e,2, 



^6 COS (r - I/) e^, 



^7 cos(r + y e^, 



iJ'_ . ... e^ 



"^^7 — 2da, ■" 2da, "*" ^19 "T 16 da, + 8 ^1 ~ 4 da, "" 4 ^6 "" 2da, ~^i 



n __ 3a , ^ , g 7 P _ g fl^ 7 g 7 



^7 — 4 a^3 03,0 — 2 a/ ^3,1 — 4 «^3 »3,2 ^19 — 8 af "" 16 a/ ^3,0 " 16 a,^ ^^3,2 



ij __ 2a a -L 7. j_ ^^ A P' _ ^ I « r g 7 



^6— a,^ ~" 4 0/^^^3,0 - 2 a, 3,1 "r 4 ^^2 »3,2 ^1" ""2a/2 "T" 4 a ^ ^3,0 "" 2 a^^ ^3,2 



^1 = :$ - :^ *l.l 

 2 ~ 2 ■*" 16 "1 ■" 4 ^6 — 2 — — 4 a/ "T" l6a/ ^^3,0 T- 8 a^ ^3,1 — 8 a/ ^3,2 



^19 -r g ^1 — 4 Itg — ^1 — — 2 a; ~ l6a/* ^3,0 i" 8 a^ '^3,1 — iQa,^ ^3,2 



Q »' — '^^ 3a ^ 7. I « t , 9«^ M A 7, "^ 



•^ ^7 — 2a2 ~ 8^ ^3.0 - 8^^ »3,l + 4^ ^3,2 + Hf^ Va^ ^5,0 - ^S.l^ 



+ ¥ { < \:^ ^5,1 - -2 ^5,0 - -2 ^5,2) - ^ (^ ^5,2 - -2 ^5,1 - -2" hs) j 

 ~ 2^ ~ 76^ ^3,0 + 8^ ^3,1 — W^f ^3,2 



— — 16"^* ^3,0 -1- 2«/ ^3,1 - i6fl^2 ^3,2 + 8 I a/ V af ^^,2 - ^^ %\ «^ ^3,3; 

 a^ a* a a ") 9 a^ 9 a^ 



+ "2^ K\ "" 2^ ^5,3 — 2< ^5,0 + 2^" ^5,2 I + re'^^ ^5,0 " 16^ ^5,1 

 MDCCCXXXV. K 



