516 
ON THE DEVELOPMENT OF THE DISTURBING 
[479 
The process for doing this is explained, First Part, pp. 205—207, [214], viz., writing 
r = a(l+x), r' = a( l+x), and restoring j' (instead of its value —j, ...—j + 6, as the 
case may be), we have a general term 
ITalla 
, a a a 
. D ( j, j'). x a x' a ' cos [j (II — 
®+f) +/(11'—®'+/')], 
where JD(j,j') now denotes the value obtained by writing a, a' in place of r, r and 
f f’ are the true anomalies elta (e, L — II) and elta (e', L' — IT). And the second 
factor, x a x a ’ into the cosine, is given as a series 
22 ([cos] 4 + [sin]*) ([cos]*' + [sin]*') cos [i(L — II) + ï (L' — II') +j (II — ©) —j'(IT — ©')], 
where [cos]*, [sin]* are functions of e, [cos]*', [sin]*' functions of e'. Or, what is better, 
the term x a x' a into the cosine may be written x a x' a ' cos [j (L — © + y) +j' {II — ©' + y’)\ 
and the expansion then is 
22 ([cos]* + [sin]*) ([cos]*' + [sin]*') cos \i (L — II) + i (L' — II') +j (L — ©) +j' (L' — ©')], 
where as before [cos]*, [sin]* are functions of e, [cos]*', [sin]*' are the same functions 
of e', viz. the e-functions are those given in the two “ datum-tables ” («°... x 7 ) cos jy 
and (x°... x 7 ) sinjy, taken from Leverrier, which I have given in my “Tables of the 
Developments of Functions in the Theory of Elliptic Motion,” Memoirs R.A.S. vol. xxix. 
(1861), pp. 191—306, [216]. In order to better show which are the symbols referred to, 
we may, instead of [cos]*, &c., write [¿c a cosjyf, &c., the formula will then be 
x a x a ’ cos [j (L — © + y) +j' (L' — ©' + y')] = 
22 (\x a cos jyf + [x a sin jyf) ([x' a ’ cos j'y'Y + [x' a ’ sin/y']*') 
x cos [i (L-U) + i' (L' - IF) + j (L - ©) + j' (L' - ©')]; 
and if we attribute to i, % any given values, that is, attend to any particular multiple 
cosine, 
cos [i (L — II) + % (L' — 11') +j (L — ©) +j' (JJ — ©')], 
the coefficient hereof will be 
2 —— a 0 
w Ilalla 
(la) D U’ ^ ^ cos mY + t xa ([«'“' cos j'y'Y + [x’* sin j'y'J), 
where a, a.' each extend from zero to infinity, but to obtain the expression up to a 
given order p in e, e!, we take only the values up to a+a'=p. 
Particular Case. 
Thus, for instance, in cos [j(L — ©) — j'(L'— ©')] the terms independent of e' are 
D (j. - j ) {[«° cosjyp + [x° sin jy] 0 } 
+ l a (jfa) D (j> -j') {W cos jy]° + [x' sin jy] 0 } 
+ 1^2 ft2 (¿) D (j, - j ) {[x 2 cos jyf + [x 2 sin jy] 0 }, 
+ &c.