418 
ME. C. W. MEEEIFIELD ON A NEW METHOD OE APPEOXIMATION 
Section I . — Approximants to the Square Root. 
1. Ml’. Sylvester gives, for the approximants to the square root, the following state- 
ment ; — 
“ Let r be an approximate value of \/N ; then by that mode of apphcation of Newton’s 
method of approximation to the equation ^^=N, which is equivalent to the use of con- 
tinued fractions, we may easily establish the following theorem, viz., that 
r^ + N r^-l-SrN r‘* + 6?’^N + N^ r*-[- 10r^N-j-5?’N^ 
2r ’ 3r^-l-N’ 4?'^-f-4rN ’ 5r^+ lOr^N + N^ ’ 
will be successive approximations to v^N.” 
2, Their general form is 
(r+ 'v/N)» + (r-'v/Nh ^ 
( 1 -) 
which is always rational. In this form the approximation to -/N as ^ increases is 
obvious. The method of my previous memoir is simply the particular case of ^=2h 
3. If we wish to approximate to N“*, we may take the reciprocal of (I.), or, what is 
simpler, we may divide (1.) by N, thus obtaining 
(r-f •\/N)*-f (?’— v^N)* 1 ,c) \ 
v'Nh-lr- a/N)' VN ^ 
Before we can integrate these formulae, we must reduce them by means of the method 
of rational fractions ; the simplest and most general way is as follows : — 
4. Let § be an «th root of unity ; then, obviously. 
log (1— .i?')=log(l-g>^)-flog(I— -flog (I 
Multiplying the differential coefficient of this by ( —x), we obtain 
IX' _ gx s „ 
1 — x' 1 — gx‘l — g^x' 1 — g^x 
g'x 
1 —g'x 
j ■ 1 11 ^ 1 l+a?‘ , 2x' 
and since ■ = l-f:; and .z=14-:, 
l—x' ' 1—x' 1—x' ' 1 — a 
1 — x' i—gx"^! — g^x~^ \ — g^x~^ — g'x' 
. 1 +x' 1 + gx l^g^x 1 + g^x 1-i-g‘x 
1 — x‘ 1 — gx'\ — g^x'l — ^^x' 1 — g'x 
T — a/N 
Making x——_^ we may thus divide into ^ fractions, each of the form 
1 (r+ A/Nl+g^fr- a/N) 
i {r+ a/N) — g* (r— v'^N) ’ 
k being any integer not exceeding 
5. If we add the pairs k and i — we obtain for the sum of the pair, 
2(r-FA/N)^-2(r- a/N)^ , 
{r + a/N)^-)- [r — a/N)^ — (g* — N) 
SrN 8r 
2(?’‘^ + N) — -f — N) same denominator’ 
