274 Mr. Woodhouse on the Integration 
more precisely, it would be extremely incommodious to compute 
A and B from the series ascending by the powers of -p if b were 
nearly = a . 
The method of Lagrange, given in the Berlin Acts for 1781, 
p. 252, has been followed by Laplace, Mecanique celeste , p. 268; 
and, in that part which relates to the derivation of A, B, from A', 
B', by Lacroix, Calc, diff. p. 120; and by Mr. Wallace, Edinb. 
Transactions, Vol. V. p. 256. But the great difficulty of the pro- 
blem does not consist in deriving the coefficients from one another, 
but in computing the value of the first and second ; and, for this 
end, a series that simply expresses the expansion of 
2WZ-pl 
( 1 — e 2 .(cos. e) 2 )— must be inadequate, at least, it cannot be com- 
modious and general. 
Clairaut has given a peculiar method for finding A, Mem . 
de VAcad, 1754, p. 54b. Arbogast, Calcul des Derivations, p. 359, 
has given a form for the expansion of (1 — c.cos. G)~r", which 
agrees with Lacroix’s, Calc. int. p. 121 ; but the expansion is 
inconvenient, for reasons already stated, for the purposes of arith- 
metical computation. 
If we join together certain parts of Legendre’s Memoir, we 
shall obtain a complete solution of the problem of the expansion of 
(a?-\-b 2 — sab (cos. Q)~~v~ ; for he shows, that E, the integral of 
dQ . v/(i — e"~ (cos. 0) a ) may always be resolved into similar integrals 
E', E", or V E, "E, or, by continuing the resolution, into E", E w , or 
"E, V "E, &c. and, consequently, he shows how E may, in all values 
of e, be computed ; and moreover, he shows that the integral of 
(cos. 6)”. (1-4-a. cos. 0)~T ~ dQ, may be always reduced to that of 
cos. 0) 2 “^“ d 9 , and therefore to the integral of 
