Mr. Woodhouse on the Integration 
272 
Consequently, 
(2^-j- 1) ( e 2 — 2) A+2.(2m+i) (1 — e 2 )a 2 .( l-f e') e A', 
— gW- B = ~ (a, " + 1 ^W A + 2 - (^+») («-»)■*, 
ab t-j , «*+&* A 
or 
or,A'= 
2W + 3 
B 
2 m+i '(a 2 —# 2 ) 
jain, since ( 2 X 2 — 1 ) R 2 ” 1 * -1 d6= 
a 1 -\-b x 
d9 
(since x*= — — ) 
B' («+*)•= • (^+^)* »A' - (,hV 
and substituting for A', 
T (a*-^ 2 ) 2 
2 — gl t;> zm—i 
e 2 A 
(*+&)* 
2 R iw +* 
za 6 
B' 
2m + 3 a 2 -f& 2 
B 
zab 
- 2 A; 
R 2W + 1 ^, 
A. 
2 m + i ’(a 2 — & 2 ) 2 1 («*— 6*)* 
The method of deducing the coefficients, by a direct process of 
integration, from R 2WI + I . efo, cos. 9 . R 2m +* «? 0 , &c. differs, when 
examined, scarcely at all from the method of determining A and 
B (index 2 W+ 1 ) from A' and B' (index (sw+i ) — a) ; for, in 
the first method, 
/R ,m V dH=ufR 2m ~' dt+a'fR 1 " 1 -! <ft+&c. +'a/Rdt+"ccf- f- 
(x=i); 
or, by continued reduction, 
= /3/Ri0 + yj~3T 5 Greek characters denoting constant 
coefficients ; ) 
but, since JR 2m ~' dt, JR 2m -i dt 8co.,fRd», 
multiplied into certain constant quantities are respectively equal to 
the coefficients, A', A", A'", &c "A, 'A, the indices being 
2m— 1, 2m — 3, 2m — 5, 1, —1, 
it is clear, that by determining J Rzm+i d9, from d9, &c. 
we, in other words, determine A by A', A" ...... "A, 'A, or, when 
A', A ", &c. are reduced to depend on "A, 'A, by "A, 'A. 
