VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 301 



V z= y, X, which is now become x -{■ xv •{■ — — |- &c. must become equal to 

 nothing. 



The root v in the equation ■a? + -y + 7-5 + t-t^ + &c. = O, is to be 

 found on the supposition of its being very small with respect to z, as it must 

 be, if z be taken tolerably exact; by which means the terms -^ 1- 



1.2.3 ' 1.2.3.4 

 -[- &c. may be neglected, on account of their smallness with respect to the 



other terms, so as to leave the equation jc + --+— = O, for finding the first 



approximation of v. 



By extracting the root of this equation, we have v = \/^ ^ — *. 



That is, 



First, V J. — -^ - J» if ar + XV + ~ = o. 



, , /F' , 2x X .^ , • xv"* ^ 



H \/j;+^_--j,'f-^ + .«* = — = 0. 



3d, I - s/ 'i - % , i(x - .VV + f , &c. = o. , ,: , , ^ 



4th, i_v/i: + 2f,if-x-.™ + f,&c. = o. 



This approximation gives v exact to twice as many places as there are true 

 figures in z, and therefore triples the number of true figures in the expression 

 of 3/ by z + t;, which may be taken for a new value of z, for computing a 

 second v, seeking other values of a?, i', x, &c. Though when z is tolerably 

 exact (which it may be esteemed when it contains two or three or more true 

 figures in the value of 3/, according to the number of figures the root is pro- 

 posed to be computed to,) the c alculation may be restored without so much 



trouble, only by taking V j. ± -- - jjr.v - yjj;^^v' &c. mstead of 



\/\ 



2x 



^2 ± ^; taking every time for v its value last computed. 



X 



XV 



From the same equation x + xv + — -\- j~ + &c. = 0, may be gathered 

 also a rational form, viz. i; = -H-! For, neglecting the terms -—^ &c. 



XX * ■ ' 



X — — : 



2x i-i.' 00 ' :-r: 



we have v = ~ ^ which is nearly — ~. Therefore in the divisor, in- 



X + T^l' 



stead of v writing —.^ we have more exactly v = Z. , that i& 



* , XX 



*^ "~ 2J 



