MR. LUBBOCK ON CERTAIN TERMS IN THE DEVELOPMENT OF R. 75 



— 16 a,2 03^1 — 32 «^3 ^5,0 + 8 < ^5.1 — 32 a^ \ a, %l "" 2 ^5,0 " 2 ^5,2^ 



15 g^ 3a« _3«_ _3a2 ^3^ ^^ 



~* 32 « 4 t'M + 32 a,* ^5,1 - 8«f ^3,1 + g « 3 05^0 — 8 « * V " l6«,^ ^3,1 



+ 32^3 V«, ^5,1 — 2 ^5,0 — 2 ^5,2; -1- 16 «^2 ^3,1 



^5,0— 16 a,^ ^3,1 — 8«4 V 



~32«3'^5,0 16 a,^ ''3,1 8 a; 

 "" "" 8 a 4 ^5,1 + 32 « 3 05,2- 

 Hence i? contains the term j — j^^ b^ ^ + ^^^ J. 2 j ^i^ 7/^ cos (2 |, — 2 ^j,). 



Calculation of the Term in R^^ multiplied by e^. 



Distinguishing the argument r — ?j + ?j, by the index 1, r — ;? + ;?, — | by the index 3, 

 and r - ;? + P7, + I by the index 4, 



_ a^R ^ p __ «digi __ _ _a_ 



^3— --2dfl ^4--~2da~^3 ^1 — ~4«^2«3,l 



n «_ , 3a^ /^ , 1 JL ^ ^ 



^3 — 8 af ^^3.1 ~ 8 a,A a^ ^5,1 — 2 O5 q — g O^^^J 



— 16^^ ^5 — 16 «^s 05^2 — 8 a^ ^5,1 + i(j «3 05,0 + 16 af" ^^,2 



_ 3«^ _3a^ , 



— 8 a 3 ^5,0 — 8 « 4 %\ 



adi^i adl^g adi?4 a6.R^ aAR^ 



^ ^xi ~ 2da 2da 2da ~ 2da da 



3a^ , ,9a3 ,15 fa/a h \ ^(±h Lk ^h\\ 



= ~ 4^ ^5,0 + 8^ ^5,1 + T I ^4 V^ ^/,0 - bj^i) -^s \^^ ^7.1 - 2 ^7,0 - 2 ^7,2j J 



■"■ 8 a/ ^3,1 ■+■ 8^ \a^ ^5,1 — 2 '^^o — 2 ^^^V 



= - 4^ ^5,0 + 8^^5,1 +^ I - ^4 V"^;^ ^7,1 - ^^7.0 - «^ ^7,2; -I- 2 a,** ^7,0 - 2 ar7,2 j 



3a^ , 3^ 3_a^ 3 g^ ^ o^^ , 



■" 16^3 ^5 + 16 «3 05,2 + 8 g4 ^5,1 — 16 «3 05,0 — 16 a3 ^5,2 



3 «^ 7. I 9 g3 , 15g3 , 1^ 7. 3«^ 7 



— "" 4^ ^5,0 + 8 g 4 ^5,1 - 8 a/ ^5,1 + 3 a,^ ^5,1 " 16 a^ ^^5,0 



* For certain reductions which occur here, see the calculation of ^2^^^^. 



l2l 



