1796+] 
doubles the labour of the computation, 
and ought therefore, if pollible, to be 
ayoided. While confidering this fubjeé 
fome years ago, a variety of new and ufe- 
ful theorems occurred, and among others, 
the following method of computation, to 
which the foregoing objeétion does not 
apply, and which, in practice, is at leaft 
as expeditious as any rule with which I am 
acquainted.— The method was firft invef- 
tigated in the following manner : 
Let N be the given power or number, 
whofe root is to be extracted; 2 the index 
4 
of that power; r the required root; N the 
‘ q Ul e 
affumed power, and its root. Then as 
Nosy, and Ney: by dividing the one 
Be 
by the other, we have oy ha 3 and 
N r 
N 
by the nature of logarithms, — log. ~ = 
N 
i; 
log. Now from this equation, by 
r 
means of any of the rons for the 
logarithms of numbers, tle value of r may 
be found in an infinite feries, and the con- 
vergency of this feries, it is evident, will 
a =am--bin3--cmis +-dmi, &c. 
ey 
3 
td ys — 
5 
I 
aes 
Mathematical Corre{pondence. 
depend upon that of the expreffions for 
N i 
~~ and i 
N y 
ries of f{wifteft convergency, expreffing 
the my bebalte logarithms of numbers, is 
243 2X7 
1 ee + $22, &c. =hyp. log. 
the logarithms of But the fe- 
iv 
— which feries we fhall, therefore, 
make ufe of. 
1 ’ 
N—N Y—yr 
Putting then m= t= and 
N +N r--r 
N 
taking the values of the hyp. logs. of ~ 
N 
and ~, the following equation willemerge : 
Secs ts 
I I i 
> ~ ( ne ee ee ? 
a+- 
3 
Hence, in order to determine # in terms 
of m and x, aflume *=am--bm?--cni-+- 
dm?, &c. and by De Moivre’s Theoyem 
(Phil. Tranf.) we have 
rota as += 47 &c, 
5 7 
3 
ne m3--arbm +-(ab?-+-a2c) m7, &c. 
5 : 
+e ns-basbnt, Se. 
7 
We e mi, &e. 
I I r¢ 
Se Cs ~ (m, &c. yet UT BUR Be. 
v/s 4 
32 
By equating the homologous terms, the values of the coefficients a, 4, c, &ec. 
termined as below: : 


5” 1 
are dee 

E 
aS = 
n 
I at nt—ty 
6=—-—-= 
Cas 23 323 
as H4——6 2-4-3 
— Le b— —_— — a2 2 at 
52 5 3-5 * 
ql n&— 98714 Onu—— 1 
ae yy) OAS ant ee, 
qu 
d &c. &c. 
7 
5-7-9 7 
7 hefe values then being fubttitated ; in the afymed equation, the yalue of #, and 
confe- 
303 
eo 
