no. 
Dr. Waring on 
Ex.y, Let the fluxion bei (V«f a + (3 cfb + b ; )x + (^ab x + c / ')x % 
4- (^ 3 + d /s ) x 3 ^ m =(Px 9 where a\ b\ c\ See. are very fmall quan~ 
tities ; then will Px = ((# - 4 bx ) ■>** 4* nta 4- bx} ” 3 x (V + b'x 4- c'x % 
n fyi j — — ^ w— — 6 
4- d x ) 4 ~ til • - ^ a 4" bx x (^cd 4“ b x 4- c r x~ 4~ d ' x 5 ^ 2 4” Stc.^ x, 
of which the fluent is 
3W 4- 1 • b 
— — — 3?«4 x 
<2 4- ^ 4- m (a 4- 3«— * x 
x 3 4* 
2 a A b' — 2 aB 
X 4 - rrx 
■a C 
j 4- &c. where the 
\ ( 3 z »4 i)£ 3^b (2 > m—i)b (3^—2)^. 
letters A, B, C, &c. denote the preceding co* efficients* 
10. M. Euler and others, reduced the feries A^q-Rv^-p 
Cv r + a, 4-&c. into a feries A 7 fin. ra, 4-B fin. 4-^4- &c. &c. 
where a denotes the arc of a circle, whofe fine is ax , &c. 
It may be eafily reduced into infinite other feriefes proceeding 
according to the dimensions of quantities, which are functions 
of x ; but it is moil commonly preferable to reduce it into 
feriefes proceeding according to the fines, cofines, tangents, or 
fecants of the arcs of circle, which fines, See. can immediately 
be procured from the common tables. 
It has been obferved in the firft part, that to find the root of 
an equation, an approximate value much more near to one root 
of the equation than to any other mu ft be given. In this part 
it is further obferved, that feriefes deduced from expanding 
given quantities, fo as to proceed according to the dimenfions 
of the unknown or variable quantities, will not converge if the 
unknown quantities be greater than the leaft roots of the above* 
mentioned equations ; and that they will not converge much, 
unlefs the unknown quantities have a fmall proportion to the 
leaft roots : and if the given quantities be expanded into 
feriefes defeending according to the dimenfions of the unknown 
quantities, then the feriefes refulting will not converge if the 
greateft 
