479] FUNCTION IN THE LUNAR AND PLANETARY THEORIES. 515 
(consequently E~ { — E\ D~* — G\ D~* = H\ L~ i+2 = If, S~ i+2 = S\ T~ i+i = T'), and that the 
terms in question, putting in the coefficients e—e' = 0, are with him 
{(1Y + (11) 4 rf 4- (17 )*' f + (20 y cos (iV - i\), 
{(212)* rf + (218)* rf + (221)*’ rf\ cos [il' - (i - 2) A - 2r], 
{(372)*' f + (375)*' f) cos [iV - (i - 4) A - 4t], 
{(449)* rf] cos [iV — (i — 6) X — 6t'], 
where, substituting for (1)*, (11)*, &c., their values, the coefficients are 
4 * - T 4 E i + rf. £ № - -1 IT*-, 
= 1 ^4* - 77 2 . i (D*- 1 + D* +1 ) + 7? 4 . ^5 (O*- 2 + 4O' + D* +2 ) - ?7 6 .3 5 ¥ (D*~ 3 + 9D*" 1 4- 9D* +1 + D*+ 3 ) ; 
rf. D* -1 — f. L l + rj 6 S l , = rf . \ B l+1 — rf (f 0*~ 2 4- G l ) + v 6 . ^ (D* -3 + 3-D* -1 4- D* +1 ) ; 
t; 4 . f Ü* -2 - t; 6 D*, = t; 4 . § C*“ 2 - 7?«. (D*" 3 4- D*“ 1 ) ; 
and 
Writing herein y in place of i, and for Af D-?' -1 , &c., the equal values A~f B~i +1 , &c., 
we have precisely the foregoing coefficients D(j, —j), ... D(j, — j + 6). 
The Development in Powers of e, e. 
The complete expression of the reciprocal of the distance is obtained from 
-j) cos (jU-jU') 
4- 2D (j, - j + 2) cos (jU + (-j 4- 2) U') 
+ 2D (j, ~j + 4) cos (jU 4- (-j 4- 4) U') 
4- 2D (j, - j 4- 6) cos ( jU + (-j + 6) U'), 
by writing therein for r, r', U, U', instead of the circular, the elliptic values, that is 
the values 
r — a elqr (e, L — II ) , = a (1 4- x ), 
r' — a! elqr (f, L' — IT) , = a’ (1 4- x'), 
U = II — © 4- elta (e, L —II), =11 — 0 +f, 
U' = IT - ©' + elta (e', L' - IT), = IT - 0' +/' ; 
L, II, © the mean longitude in orbit, longitude of perihelion in orbit, and longitude of 
node; and the like for L', IT, ©' ; “ elqr ” = elliptic quotient radius, “ elta ” = elliptic true 
anomaly ; or, what is the same thing, if we write elta (e, L — II) = L — II 4- eltt (e, L — II), 
and the like for elta (e, L’ — IT), then 
U =L -© 4- eltt (e, L — II ), =L — © +y, 
U' = L'-&' + eltt (e\ L' - IT), = L’ — ©' 4- y'• 
65—2