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



= - ^ {t *i.o + \i COS r + *i,2 COS 2 r + &c. | 



+ 1^ (7^ + y/') *3.i + 8^ (^3,0 + M (7' + r>^) COS r 



m.a ^ „ , ^ X ^/« 



- 1^ «3,o y^ cos (. - 2 ,) - ^ «3.o r,^ cos (r + 2 ,,) 



- ^ *3,o 7 r, COS (, - ri,) + ^ A3.0 COS (, + ,,) - ^ b,^^ yf cos (2 r - 2 ,) 



- 1^ *3,i y^ cos 2 , - ^ ^3., 7,^ cos (2 r + 2 ,,) - 1^ A,,, y,^ cos 2 ,, 



"~ 4i?"^3,i yy/cos (r - ?j + J7,) + 4^ ^3.iyr/Cos (r + ;? + ?7j 



+ 4^?" ^3,1 77/ COS (r - JJ - ^,) - 4^ ^3,277, COS (2r - ?? + ^^) + &c. 



, , , di? d72dr , diidA 

 As before, "dl = d7 dl + 7u d7^ 



but in this form of development 



dR __dR dR dR __ _dR dR 



dA'~"dT'T'd»j dx^ dT'd>j/ 



Calculation of the Term in R multiplied hy e^ y^. 



^ — d^ ^0 "" 8a2 %,l 



^2 — — 8 af ^3,1 i- 8 a/ \«/ ^'^ "~ 2 ^^,0 — 2 ^5,2 j 



"" "" 16 af ^5,0 + 16 af ^5,2 + 8 a/ ^5,1 " 16 a^^ 65 — ^g ^3 05,2 



— "" 8fl3 ^5,0+ 8 a* V* 

 If the term in R^ multiplied by e^ y^ \^q called R\, 



„ _ adRp adR^ 

 "^^0— 2da "T 2da. 



__ 3a^ , 3a» , _3_a^ _9^ 



— 16 a3 ^5,0 — 16 a/ ^5,1 + 8 a,^ %o " iq ^ 4 ^s.l 



- tI { ^* (I; ^7.0 - ^7.1) - ^ (^ ^7.1 - i ^7,0 - Y ^7,2) } 



— I6a8\0— 4a/%l - 16\~ a4\^ a,^ 67,1- a/7,0 — -^07.2 j +2^507,0-^6^7,2/ 



~ I6fl» ^5.0 — 4 a; ^5,1 + i6a/ ^5,1 "" 16 a/ ^3,1 



_ 9a^ , 



— 16 a/ ^5.0- 



