finding the Value of an infinite Series , Sec. 2x7 
_ i-i- 3-3-5-5-7 : 7;9-9 _ 8tc. in which the law of 
the continuation of the terms is very manifeft, 
every new term being derived from the preced- 
* 1 ) 1 ) 
irig term by multiplying it into the fraction — , and 
likewife into a numeral fraction, whofe denominator is 
the fquare of the index of the powers of v and s in the 
new term, and whofe numerator is the fquare of the odd 
number that is lefs than the faid index by an unit. 
Art. 18. Let the numeral co-efficients of the terms 
j 
of this feries be denoted by the capital letters of the 
alphabet, a, b, c, d, e, f, g, h, See. in their natural 
order, fo that a ffiall be equal to 1, and b ffiall be = 
, and c = , and d — 7 > and fo on of the reft. 
7 2. 2. 4.4 7 2. 2. 4.4.O. O 7 
1. 1 
2.2 
And we fhall have 
0.0 II. II 
Ej F, H~ 
B =S A > C= H B > D= ^6 C ’ E= H D > F - 1 o.io-> 
~~ g, and fo on ; and confequently the feries 
1. 1 VV 1. 1.2.3 v 4 I.I.q. 2.5.5 V 6 o 
I- x — + —x-r- — f ‘:xt + &X. 
or a 
2.2 ss 2. 2. 4.4 r 2.2.4«4.6.6 
EVV CV 4 DV 6 Ev 8 FV 10 Gv‘ 
ss r s s 
.6 
+ s' 
KV 
+ 8cc. will be a 
1 , 1 , aw 3.3-B‘z; 4 5.5.C7; 6 7.7 D-y 8 9.9. E v l ° ii.n.Fi; 12 13.13. G7/ 14 
H : .-7 — 7T-+ c tt: — — — — — ttt~ -f 
2.2. ss 4.4. s 4 6.6. s ( 
8 . 8 . 
IO.IO. s 1Ci 12.12. s' : 
I2IFV 
o IAVV QBi; 4 2ZCV 6 4Q D*y 8 8lE7/ c 
+ 8cc. or A + Vr- 4 t+ 2 7 - 5 nr + ; 
4 ss i6j 4 36/ 64 i s 100/ 144* 
14.14. r- 
I69G7/ 14 
I96F 4 
+ &c. or, if we convert the co-efficients of the terms 
into decimal fractions, 
Vol. LXVII. Ff t- 
