MR. LUBBOCK ON CERTAIN TERMS IN THE DEVELOPMENT OF R. 79 



In order to obtain the term in R depending upon y^, y"^, yf and y *, it is sufficient to 

 take 



^ = '^\k ^3,0 + «'3,iC0sr + ^32COs2r +,&c. j 



I - (|- y2 + ^ + |- y/) COST + y 7^ (y2 ^- y2) COS (>j - n) 

 -5^'cos(r-2?? + 2^^) +, &c.| 

 "" "8^3" I ~2 ^5,0 + 05,1 COS r + <^5 2 COS 2 r + , &c. j 



{ (^ + ^) ^^® '' ~ ^ ^^^ (r - 2 ;;) ~ ^'cos (r + 2^,) 

 — y y^ cos (>; - ??^) + y y< cos (;? + n) | 

 = ^ { Y ^3,0 + ^3,1 cos T + ^3^2 cos 2 r +, &c. j 



{-(|/ + ^' + |y;')cosr + yy,(7" + y;^)cos(;j-0 



-5^cos(r-2^4-2;?^)+,&c.j 



- ^^ {4" *5,o + &5.1 cos r + \^^ cos 2 r +, &c. j 



|^ + i4^ + ^-5^'(y2 + y^2)eos(r + ,-^,)-^'(y2 + y,2)eos(r-., + ,^) 

 + (t +^' +^') cos r + 4y'y/' cos (2pj - 2;j,) + ^'cos (2r - 2H- 2;?,) j 



~ I "■ 64 af ^5,0 + tj4 a 3 %.2 — 64 « 3 05,0 54 «^3 ^5,2 J y 



r 3 a2 , 3a« , i >''> «^ , ' 15«^ j, 1 2 2 



+ I "^ 64^3 05,0 + 64 af ^5,2 - (j4 «^^ ^5,0 " 64 af \'^ J ^ ^^ 



I I A^ A I ?-^ A Ai A -1^ 7, "I 4 



"^ I ~ 64 a^^ ^5,0 + 64 a^3 ^5,2 " 64 af '^^.o "" 64 af ^5,2 J y/ 



+ { 1$ *5.o - ^3 ^^5.2 + 1^3 ^^5,0 + 1^3 ^5.2 } y y. (y^ + y.^) co^ 



+ {-^3^5,0 + ^s\2 - ^3^5,0 - 64f3*5,2}y'y;'cos(2r - 2 ;, + 2 ,^) 

 = (- T6^3 ^'s.oH- 32^3 &5,2J/ + |- 32^3*5.0- I6^3^'5.2|y y/ 



+ { " ll^^ ^5.0 + ^3 ^^5.2 } y/ ^^f hfs y y/ (y' + y?) cos (r - »; + ^,) 



- ^3*5,0 y'y;'cos(2r~2pj + 2^,). 



