FLUXIONS. 
: g z Vl . 
tal Witha Vir ta sbeidgs, put for 7, and similarly 7’ 
: —l- 1 
= for wp and, consequently, ¥ for 
rm tome tome, 
Vat | elt na get Petes 
V'41 
Vi. 
7 Vp then the for- 
mula may be expressed Pate. ‘ 
1 1 
Famt7tt 1.) 
Let us now suppose that V’, V", V”, &. to V™ 
(m) denoting the number of accents over the last 
term) are a series of quantities each formed from that 
before it, as V’ is from V, or so that 
2V9=V+1,2 V7=V' 41, 2V"*=V" +1, &e. 
2 Vel V1 VM) 
be briefly denoted by ¢”, #”, &c. to ¢™, then from for: 
mula (1) we derive the following series of equations: 
1 1 t 1 
g=ptate 
1 1 fd 1 
ar Pprte tap 
1 1 #7 1 
fr aetatoe 
1 Me Slee ahh 4 a rt 
el ge Oa Ge” ome 
By adding the ng members of these equa- 
tions into two sums, and rejecting wliat’is common to 
both, we find = 
ae ae a ™ 
= EtG ecee te 
1 1 i 1 1 
T=) +et+ arta ar 
ky, 
ara 
Now, the numeral series being a geometrical pro- 
P 5 pe | 1 
Pent eee a gt ene — a, 
and common ratio, its sum will be --——7_; there- 
4 8, 3.4™ 
fore, after substitution and transposition, we find 
eer ene 
a+ 
xe YS 3.4" 
a Se + ( t a” “er ™m 
ttate te 
Let us now assume that, 
aV=ed—, 
Then 2(V+1)=0424—-=(e42)'; but 
2(V-+1)=4V”, therefore, ; 4 
2V'= 3 * 
v +7 . 
_In the very same manner, it may be proved, that 
s 
3895 
2V" =v +4: and in general, putting » = 2”, 
that ’ 
I 
2V) — v4 ca r 
v™_ 
and as 4™)= —_—_-;; therefore, 
yor) oe 
want g1_ (ora) 
ve pevrga (v% 41) 
By substituting this value of *) in the denomina~ 
tor of the first member of equation (2), and n? instead 
of its equivalent 4"; and also putting for > its value 
(ae 2 (v—1)*+40 
o—1/ ~ 15 
t= 
40 
ots oo =i” we have 
40 | 18 
_ GF 41)F aay tat on oa 
2 Ul t"’ uv" pi J 
lS GtatS--- +e. 
And this identical equation holds true, m being any 
whole number whatever- 
Because 2 V=04 =, and 2V’=v$ +, and 2V” 
Uz 
usp + &c, v must always be a positive quantity ;- 
. 
but it may be of an magnitede whatever: And as, 
whatever be the value of v, the terms of the series 
4 at : 
v, v®, v*, &c. approach continually to 1, and very 
fast, if v be a large number, or a small tion; there- 
fore, the terms of the series V, V’, V’”, V’’”’, &c. con- 
tinued indefinitely, approaches continually to 1, which 
is their Limit. 
To discover the ratio according to which the terms 
of the series ¢, t,t’, &c. Te ap let us take’ any 
— i me 
two succeeding terms, txargy at's vrai the 
latter divided’ by the former is 
tv vet Wel V1 VV +1 
Pes Vie * Vv ae ta (W"41)?V—7 : 
But because 2 V”?= V’ + 1, we have V’”* ~1 
4(V’'—1) ; therefore 
tv (W—1)(V'+1) V4 
 ~ 2V741)2 (VW 1)~ 200" 4-1)? 
We have ‘ate g thet the quantities V’, ee, &e. a 
Uy 
proach continually to 1; therefore the ratios oe 
&c. approach continually to 4 
: hence it follows, that 
when any term ¢’ is found to be nearly } of the term - 
immediately before it, the same will be more nearly 
true of each of the following terms. i 
From what has been shewn, it appears that the series 
of formula (3.), viz. , 
ee 4). 
. state: tay 
as it advances, approaches in its form to a geometrical 
rression, of which the common ratio of the adjoin- 
terms is ;;. It will therefore converge very fast, 
so dat lactweven tbat the number of terms may be, a 
vr 
Fundamen-. 
tal 
Principles. 
ehyeet 
