io4 Dr. Waring on 
Caf. i. To this queftion an anfwer, I believe, was firft 
given in the Meditationes , viz. reduce the fun&ion to its loweft 
terms ; and alfo in fuch a manner that the quantities contained 
in the numerator and denominator may have no denominator: 
make the denominator Q = o, and every diftin£t irrational quan- 
tity contained in it = o ; and alfo every diftinft irrational quan- 
tity H contained in the numerator ~o ; then, let a be the leaft 
root affirmative or negative (but not=o) of the above-men- 
tioned refulting equations, the afcending feries will always 
converge, if the value of x is contained between a, and — oc ; 
but if x be greater than u or -a, the above-mentioned feries 
will not converge. 
If the above-mentioned feries (S) be multiplied into x, and 
its fluent found ; then will the feries denoting the fluent con- 
tained between two values a and b of the quantity ( 'x ) con- 
verge, when a and b are both contained between a and — a : 
the fluent always converges fafter than the feries S, the un- 
known quantity x having the fame values in both. 
Ex. Let r = Ax -f i Bx z 4 - f Cx 3 + &c. and 
J a + bx+cx 2 • • x 
the roots of the equation a + bx + cx l + . . x n = o be a, 0, y, 
See. ; then 
a + bx -\-cx 2 
a — @ 
j- 4. &C. = A# + Bxx + 
y—x 
Cx 2 x + &c. ; but — = — + ™4-^- + &c. in infinitum , = 
" a a, “ — - 
a x 
0 
+ ? ~+ e — 
T e 1 ^ p 
+ &c. See . ; the former feries converges when x is 
contained between « and - «, the latter when x is between 
/3 and ■/ 3 , and fo on. In the fame manner the fluents 
!£+**+ See. See. a fortiori con- 
& 20 Z J 
verge when x is between u and - a, 0 and — 0, See. refpec- 
tively, and fo on : hence the feries Ax + iB.ri + iCx 3 + &c. 
4 
