54P 
Mr. Hell i ns's unproved Solution of 
Secondly, to find the Coefficient B. 
14. Multiply the equation in Art. 2. by 2 cos. z = 2 x, and we 
shall have — = % ( A x 2 cos.% +B x 2 cos. 2 xcos.x + C x 2 cos . % 
x cos. 2 % + Dx 2 cos. z x cos. 3 z, &c.); which, because 2 cos .z 
x cos. mz is = cos. [m — 1 ) z -f- cos. (m - f- 1)2:, will be = z 
( 2 A. cos. z + B ( l-j-cos. qz) -j- C (cos. 2; -j~ cos. 3 2;) -f- D (cos. 2 2; 
+ cos. 4 2), And, by taking the fluents, we have 
/ * 2,X-Z, • 
Ja—bxy = g A.sin.g -j- B£-{-iB.sin. 22:-(-C (sin.2;-J--jSin.32;) 
+ D (i sin. 22 -|- i sin. 4 3), &fr.; which equation, when z = 
3 * 1 4 1 59 > &c. = 7T, becomes J-~~„ = barely B 2; = B tt, the 
sines of 2;, 2 2;, 3 z, &c. being then = o. 
15. Now it appears, by the notation in Art. 4, that 
[a+bj 
X 
z-vv 
.1—2 n 
\Z(i—xx) ( a—bx) n 1 ^ J ^ V C — vv) \/ [vv — cc) 
we therefore have, by proper substitution, 
, and that x 
b 
ZXZ 
’ ZXX 
za 
zmv 
1 — 2H 
1 
( a — bx) n v' (1 — xx) ( a—bx )’ 
~ b(a-t-b)” 
— 2 
*/[\ — vv)^/{vv — cc) I 
2T V 
,3 — 2 « 
>1 
b(a + b) 
n—i 
V [i—vv) (vv—cc) 
of which two fluxions the fluents may be found, when n has 
any particular value. 
16. First, let n be then the last expression in the pre- 
ceding article becomes 
2 a 
2W 
X 
2*V 
V^( I — vv) \/VV — cc) 
b(a-\-b jT y^(i — vv)\/{vv — cc) b{a + bY'2. 
, Now, the fluent of the affirmative part of this 
expression is evidently = y x the fluent of the fluxion in Art. 5, 
See Simpson’s Miscellaneous Tracts, lemma I. p. 76. 
