1796.] 
2” 1 10 616 
3.17) 815173 397953 
t--e 69. 2 
pe ee ee 
I—x- 382337 382337 
4.32664, the value of the quantity fought 
nearly. 
_ From thefe few exampies, the accuracy 
and ufefulnefs of this method are fufh- 
ciently evident. 
As the coefficients depend upon the 
value of z alone, they will remain the 
fame in the extraétion of the fame root, 
whatever the values of N, N, and r, fhall 
be. It will, therefore, facilitate the cal- 
culation, to have thefe coefficients pre- 
vioufly computed, and arranged in tables 
like the following, which contains the co- 
efficients of the three firft terms of the 
feries for the 2d, 3d, 4th, sth, and 6th 
roots + that.is, if A, B; °C, &c. be 
£2 3 2d | 3d | 4th | sth | 6th 
os root. {root. {root, froot. {root 
RoW 
I | 1 1 1 
A [L.term]] => 4 1 1 1 
| SER 1 8 lal 8 34 
B iti. - 8 BI 64 | 125-4 648 
af a o | 23. | _sa4 | 1995 
Cc III, 16 725 5’2 | 15625 E370 
&e.] &e. 

equal to the fraétions in the fame lines, 
correfponding to the given root, and 
#=Am-+Bi34+Cms, &c. then will r= 
From this general method, an infinite 
number of approximating theorems may 
be derived. If the firft term of the feries 
enly be taken for the value of #, and if 
this value be fubftituted in the equation, 
I 
4 “ {——-7 
Sd, T tf 3 
a A we fhall have== == 
4 
r 2 
N—N 
a24+-— p: 
atm. _N+N_n(N4N)4(N—N)_ 
ey . Cn ee, 
aw NIN 2(N+N)—(N—N) 
N+N . : 
- MontHiy Mae. No, IV, 
Masbeniticnl Correfpondence. 
395 
CDN rarer a) or (2——1) N+ 
(n—1)N+(r-+ 1) N : ee) 
(n-t1)N: (n+1)N+(2— 1)N bur iiry 
This is the general theorem, given by Dr. 
Hutton, in his Traéts, Mathematical and 
Philofophical, which he there invettigates 
in a very different manner, and illuftrates, 
by avariety of examples. From this, an 
idea may be formed of the great conver- 
gency of the feries, expreffing the value 
of w, feeing, from the firft term only,.a 
rule is derived, which is confidered as the 
mof{t convenient one for practice, that has 
yet been difcovered. If the two firft terms 
be taken, atheorem much more accurate 
will refult. aS 
From the fame fource, many other ap- 
proximating theorems may be derived, in a 
different manner; one of which I fhall 
here take notice of. It is fhewn, in Simp- 
fon’s Mathematical Differtations, that the 
value of the feries am—-bm3, 8c. is nearly == 
a~m 
a—bm? 

I 2%——I 
- therefore —m —-+- —— m3, &c. 
2 378 
i | te that is, 4 
is nearly, =—————__; that 1s; \47=— 
V9 37? (tat a™ 
e 
I> 
— nearly : whence cae 
a 
30m 
pep Tae a Ra 
3n2—(2*>—1 mn 
mann )\—(A%——1 m2 4 
32( ——)—( 22mm 1 ore | 
quired root, which value is much more 
accurate than the foregoing. 
Lonzdon, Feb. 15. gp. CX¥GNI,, 
—— eS 
vEsTTon I. Propofed in No. l—— 
ee Anfwered by Mr. T. Hackman. 
Dr. Hutton, in his new Mathematical 
Diétionary, vol. i. p. 11, and vol. il. 
p- 726, has given the refults of fome ex- 
periments he made at the Royal Military 
. Academy, at Woolwich, one of which ts, 
that “a plane furface, of a foot fquare, 
fuffers a refiftance of 12 ounces, or 3 of a 
pound, from the wind, - when blowatig 
with a velocity of 20 feet per fecond ; an 
that the force is nearly as the. fquare of 
the velocity.’’-—-Now, putting a = the 
fuperficial feet contained in any plane fur- 
face ere€ted perpendicular to the dire¢tion 

of the wind; 4 = the force of the wind 
againft that plane, in pounds ayoirdu- 
r ie eee ‘poife 5 
