j jo Dr. Waring o« 
integrals by converging feriefes either afcending or defcending, 
of which the given increments are either finite or evanefcent. 
7. 2. It may be obferved, that generally the particular cafe of 
which the increments are nafcent or evanefcent may be deduced 
from the general, in which the increments are finite ; and con- 
fequently in many cafes the general will, mutatis mutandis , 
correfpond to the particular ; e. g. 1. the integral cannot be 
expreffed in finite algebraical terms, when any fadtor in the 
denominator of the increment has not a fucceffive correfpon- 
dent one; which is analogous to the cafe of the fimple divifor 
in the denominator of a fluxion publifhed in the Quadrature of 
Curves. 2. Nor can it be expreffed by the above-mentioned 
terms, when the dimenfions of the variable quantity in the 
denominator exceed its dimenfions in the numerator by unity, 
which correfponds to a fimilar cafe in fluxions firft given in 
Medit. Analyt. To thefe may feveral others be added. 
8. Let the fluent of the fluxion (A + B*” + C* 2 " + &c.)" X 
sc*—x = ax e + bx 9 +" + cx f +^+ &c. = (A + Bx” + C* 2 " + See.) 1 ”* 1 x 
«« 4. |Sa; 9 +" + y* 9 + 2 " ■+ &c~ = a V + &'x*— -)- cV -1 ' + &c. =» ' 
(A + B*” +Cx 2 ” q- &c.)" +I x k x x ‘ + ( 2 ' x 7 '~ n -{-</' x x '~ 2 " + &c. 
1, If the infinite afcending feriesowc 9 + jG* 9+ '’ + y* 9+2 ” -1- &c. con- 
verges, then will the feries ax 9 + bx s +” + c* 9 + 2 ” -j- &c. converge, 
and nearly in the fame ratio ; and vice verfd, if the former 
diverges, the latter will alfo diverge. In the fame manner if 
the infinite defcending feries «V + |S V - ' + y + & c. con- 
verges, then will the feries a 1 x x + b’ x L ~~ n + c' x‘~ zn 4- &c. con- 
verge ; and if the former diverges, then will the latter alfo 
diverge ; and in all the cafes nearly in the fame ratio, except 
only when their convergency is the leaft. 
n 2. If 
