[ 402 ] 
l!LP„±«0C J ^6j*+ i sB + A &c where the co - 
efficients of B, C, D, E, F, &c. in the values of 
O, E, F, Sec. are the 2, 3, 4, &c. higheft coeffici- 
ents in a -(- b \ 7 , a -|- b\\ a 4- ^ 1 ", Sec. expanded; 
affixing in every particular value the leaft of thefe 
coefficents to B, the next in magnitude to the fur- 
theft letter from B, the next to C, the next to the 
furtheft but one, the next to D, the next to the fur- 
theft but two, and fo on *. 
With refpeCt to the value of the feries ^2 — 
Sec. he has obferved that it may be 
tn 3 z 3 . «— 2 
3 ‘ ~ 
X 
m z 
calculated direCtly when mz is lefs than 1, or even 
not greater than \/y: but when mz is much larger 
it becomes impracticable to do this ; in which cafe he 
fhews a way of eafily finding two values of it very 
nearly equal between which it’s true value muft lie. 
The theorem he gives for this purpofe is as fol- 
lows. 
Let K, as before, ftand for the ratio of the qua- 
drantal arc to its radius, and H for the ratio whofe 
hyperbolic logarithm is - — - — - — 4 - ^—r~i — ■ 
yr & 2 n 0O0 » 3 1 1 loon 5 
2 — 1 
1680 n 1 
See. Then the feries m z 
m i is 
Sec, will be 
2 
greater or lefs than the feries — r— x 
0 »-j-i V2 ” + 2 
n 
1 — 2 tn 
? + 1 
2 mz 
+ 
2 'i 
2 m z 
n 
? + 2 
n+ 2 
n + 4 X 4 m z z 3 
X 
+: 
* Thi9 method of finding thefe coefficients I have deduced 
from the demonftration of the third lemma at the end of Mr. 
Simpfon’s Treatife on the Nature and Laws of Chance. 
a 3 * 
