112 
Dr. Waring on 
fluent. The firft term of the feries, in which e is fuppofed 
very fmall, will be the fluent of the given fluxion, when 
x — z. 
i r . If a fluxion Px, where P is a function of x, be transformed 
into another Qir, where Qjs a function of %, and they be 
reduced into feriefes A and B, proceeding according to the di- 
menfions of x and % refpedively ; find ct and 7r, correfpondent 
values of the quantities x and % ; then in afcending feriefes, if 
u has a lefs ratio to the leaft root of the equation P — o, than tt 
has to the leaft root of the equation Q=o, the feries A 
(exceptis excipiendis) will converge lwifter than the feries B. 
12. Dr. Barrow, in fome equations, exprefling the relation 
between the abfcifs x and ordinate y 9 found y in the two firft 
terms of x, viz. yz=.a + bx, which is an equation to the afymp- 
totes of the curves. Sir Isaac Newton, from an algebraical 
equation given, exprefling the relation between y and x, found 
a feries proceeding according to the dimenfions of x , exprefling 
y in terms of x. M Leibnitz performed the fame problem 
by affuming a feries Ax n ^Bx n + r -\-Cx fl + zr with general 
co-efficients, and fubftituting this feries for y in the given equa- 
tion, &c. from equating the correfpondent terms he deduced 
the indexes and co-efficients. M. de Moivre, Mr. Mac 
Laurin, &c. obferved, that when the higheft terms of the 
given equations have two or more (w) divifors equal; for 
example, (^ — ax n ) m ; to which we muft add, and when a 
value of y in this cafe is required nearly equal to A/, a feries 
Ax n -\-Bx m -J- &c. is to be aflumed, whofe indexes differ 
only by - , &c. if otherwife they would differ by r. 
This reduction feldom anfwers any other purpofe than find- 
ing the fluents of fluxions as J ' yx 7 &c. ; or the afymptotes, &c. 
of 
