<p8 
JDr. Waring on 
-&c;=o,and confequently 7r.p,&c.nearly- - — £ f L . „t 
^ na~~ x ~-n— ipa n ~~ 2 4- &c. 
b n **-pb 
«— i , 
4 -?* 
&c. 
, &c. ; whence the convergency of the ap- 
—n—ipb -f&c. 
proximate values found by this rule depends on the principle 
before delivered. 
10. Let there be given (//?) equations, which contain m 
unknown quantities x 9 y r %, &c. ; and let a, /?, y 9 &c. be 
nearly correfpondent values of the unknown quantities x 9 y 7 
&c, refpeftively : afllvme n 4- i different values of the quantity 
x, viz. a, cu + 7r, a -f- tt / , u J r ' 7 r // r &c. &c. ; and in like manner 
a ffume n 4- i different correfpondent values of the quantity y 7 
which let be p , i G 4- f > P P + p /r , &c. ; and fo of the remain- 
ing; where 7T, rr f , 7 r'\ &c. p, p 7 , p /x , &c. &c. are very fmall quan- 
tities; fubftitute thefe quantities for their re fpe&ive values in the 
given equations, and let the refulting quantities be A, B, C r 
D, &c. in the firft equation ; P, Q, R, S, &c. in the fecond v 
&c< &c. : aflume from the firft equation the n Ample equations 
&7r 4 “ b^ 4 - &e>. —-A, 4 ~ b ^ 4 ~ &c. zz C * — A, ctir /r 4 - b ^ 4- &cc», 
= D~A, &c. ; and from the fecond equation the n Ample 
ones hw 4* 4“ &c. znQ — P, kir* 4~ k^f 4- &c. — R — P, hq? 1 ' 4- kpf- 
4-&C. — S-P, &c. From thefe equations can be inveftigated^ 
the co-efficients a 9 b , &c. h 9 k 9 &c. &c.; ultimately aflume the 
m equations A -^ae-ybi 4-&c. = o, P4-fe + &'b&c.ro, &c« 
from which can be deduced the values of the quantities e 9 
&c. ; and a 4-*, jS + /, &c. will be more near values of the 
quantities, x r y 9 &c. 
11. Sir Isaac Newton found the fum (A) of the 2 
power of each of the roots of a given equation, and then ex- 
tracted the 272 th root of Ay viz. X/ A for an approximate value, 
of the greateft root of the equation, and further added fome 
fimilar rules on the fame principle* 3 m. 
