96 Dr. Waring on 
each other, as the arc and the fine, it is necefiary that both 
ihould be correfted in every approximation to fuch a degree as 
the fiabfequent approximations require. 
6. In the Meditationes it is obferved, that in any algebraical 
equation x” — ax 1 + bx n ~ z — cx v ~> + dx n ~^ — ex”~s . . ^gx "— m + 1 — 
hx H ~ m ± kx"~ m ~ l =p lx "-*™- 1 — &c. zr o, if a be much greater than 
and - has to e - a much greater ratio than a : - ; and in the 
Cl M 0 d 
fame manner c - has to ^ a much greater ratio than b - : and 
ifo on ; then will a be a near approximate to the greateft root 
of the algebraical equation ; b - a near approximate to the fecond ; 
a near approximate to the third, and k - a near approximate to 
f \ 
a roof, which is much lefs than m roots of the given equation, 
but much greater than the remaining ( n~m •— i) roots. 
If the equation above-mentioned r±^'“ I + &c.^o, or 
which is the fame, i rt= - 4-4-— & c * = o be infinite ; then will, 
in like manner, all its roots be poffible and their approximate 
values a , ~ , 7 , &c. as before. 
a b 
This eafily appears by fubftituting for a> b , r, &c. their 
walues in terms of the root of the equation. 
7. A nearer approximate to the above-mentioned root will be 
8. Equations, of which the fluxions of the quantities con* 
tained in the given equations can be found, may be reduced to 
infinite algebraical equations, in which is involved no irrational 
function of the unknown quantities contained in the given equa- 
tions by the incremental theorem ; ws.letArto be the given 
5 equation, 
