MR. LUBBOCK ON CERTAIN TERMS IN THE DEVELOPMENT OF R. 67 



^' r] {cos (X' — X^) + * *J = r r^ j cos^ y cos^ ^ cos (X — \ — g + g^ + ^ — ^j 



+ Sill2 y cos Y cos (X -f X^ — g — g^ — »; -I- ,.^) 



+ cos2 ~ sin2 -^ COS (?i + X^ — € + I' — i/^) 



+ sin2 Y sin2 -|- cos (X — \ — g + €^ — v + j^^) 



, tan * tan i, , <=, « . 



+ — 2 — <=°^ (K - X, - € + g,) 



cos 



2 



Iftan/ = y, cos2y = 1 - ^ + -^ / sm2y = ^--^/. 



If ?2 ^ — w^ # + g — 2/ — g + g^ + " — ^ be called r, 

 and if ??^ + s — g = ^ w^ ^ + g^ —• g^ = p?^ since when the eccentricities 



are neglected X = w ^ + g, X^ = w^ # + s^ r =. a, r^-= a^ 



r r/ {cos (V--X\) + ^^^} = a «^ j cos2 y cos^ y cos r + sin^ y cos^ y cos (r — 2 ??) 



+ cos2y sin2 -^ COS (r+2 ri) + sin^y sin^ -^ cos (r~2??+2;j^) 

 + ^' cos (;? — ^^) — ^' cos (?j + ;?^) 



+ ^ - ^') '=*'« ("^ - 2 ') + ^ - t) ''"s (t H- 2 ,,) 

 + 4r cos(r- 2, + 2;,,) +^(l - ^ - ^') cos(, - „) 



In order to have the terms required depending upon the squares of the inclina- 

 tions, it is sufficient to take 



R— — ^ |y ^1,0 + ^1,1 COST + ^i2COs2r -I-&C. j 



+ l< { 2" ^3,0 + ^3,1 COS r + ^3^2 COS 2 r + &c. j 



{ (^^"I-^) COS r - ^ COS (r - 2 ;?) ~ ^%os (r 4- 2 ;j,) 



— 7 y^ COS (;? — rt) -\- 7 y, cos (?? + ?7^) > 



k2 



