544 
Mr. Hellins's improved Solution of 
— 2 
b (a + b)l 
X < 
32 —10 CC 
<35 
16—9 CC 
_ + 1/(1 — cc ) (p -j- <r r r -J- r c 2 * 4 ) for the other 
of the fluent sought And since, by Art. 14, this fluent is 
= B 7T, we have, by dividing both sides by tt. 
B 
2 a 
A 
x 
32 — I o cc 
05 
16—9 CC 
-+v/(i' — cc) (p + <r cc-{- rc 4 ) s which 
^ w 6 (a + 6)i 
is its value when n — 4. 
17. We are next to find the value of this coefficient, when 
n — \\ which, for the sake of distinction, I denote by B'. 
With this value of n , the fluxionary expression in Art. 15, be- 
comes 
2 a 
2 W 
2n.1V 
&(« + &)£ A/( l—w)v'(yv—cc) b(a + b)i V(i —vv)^{vv~cc)' 
which being compared with the fluxions in Art. 5 and 10, it 
will appear that the fluent of the former part, when v = 1, is 
= ~ A V, and that the fluent of the latter part is = Att; 
which fluents, taken together, are, by Art. 14, — BV. There- 
fore we have B' = y A' ■ 
2 
T 
A = t (A'# — A), 
Thirdly, to find the Values of C, D, E, 
18. The values of the coefficients A and B being now found, 
corresponding to the values of n f and f, we might proceed in 
the same manner to find the value of C. For, if the equation 
in Art. 2, be multiplied by 2 cos. 2 z, and cos. (m — 2) 2 + 
cos. (m •-{- 2) % be written for 2 cos. 22 x cos. m%, it will become 
2 5 — . £ ( 2 A x cos. 2 s- + B (cos. * + cos. 32:) + 
(fl-6XC03.z) n v 
C (1 -f- cos. 4%) -f- D (cos. % cos. 5 2), &£•) And the sum 
