270 Mr. Woodhouse on the Integration 
two preceding coefficients "N, 'N ; and the method of deduction 
nearly the oldest, that of Clairaut,* seems to me the best. 
It is, in substance, nearly as follows. 
l-f e'*— 2.e r . cos. 0 = (1 . cos. 6) 
=1+^(1 —c .cos. 0 )=(i -f * / 2 ) V% putting c= -^,^=1 — c . cos. 0 . 
Hence, "N i = (1 fV zm+ \. cos. (72— 2) 6 . d$, 
'N £■ = (i 4 -^ i ) 2 - f f i /V 2W+I . cos. (72—1) 0 . d% 
N ~ = (1 . cos. nb.S; 
consequently, it is necessary to determine 
fVzm+i . cos. w 9 . dQ (F") from fV*m+\ . cos. (77-— 1) 0 d § . (F'),and 
y"V 2?M + I . cos. (72-^- 2) 0 . . (F). 
Now, -§• . cos. 720 + . cos. ( 72 — 2 ) 0 = cos. (n— 1) 0 . cos. 0 , 
A ^F"+ i dF = V zm + 1 2/0 . COS. (72— l) 0 . cos. 0 
= Vzm+i J0 . COS. (77 — l) 0 (— 7 -} 
dF 1 cos. («— 1) (78 y 2 m + 3. 
but, V 2m+3 . sin. (77 — 1)0}== (77—1 ) cos. (77—1) 0 . V 2 m+ 3 d 0 -f 
,.( 2 «+ili. . CO s. ( W __ 2 )0 v 2m + 1 _ .Wil f . cos. 770 V 2m + 1 d 9 . 
F' sin, (w— 1) 6 .V* ffi + 3 (20143) 
Hence, - -f 7 ” T7;f (»— 1) fcT 4 (»— 1) 
when sin. (72—1) 0=o, 
p// 4(w— 1) F / 4-(2m4-3 — z ( n — i)) cF 
(2042014 1) c f 
or, 
]vj ( 4 « — 4) V N+ (2m-45 — 2W) 
(20142041)1: 
Let 77 = 2 'N = B, "N = 2A, N = C, 
4B + (2m+i)2cA 4B , (20141) 2A 
(20145) c (20145) c ' 20145 
* M?'oi. V AcademU, 1754, page 55a. 
F" 4 - 2m+3 F 
r + 4(0-1) 
.-. c 
