FLUXIONS. 399 
P eal 'g wh Megat > i! hi ~ Fundamen- 
tat Conmemetat oa) ealetauiN ayes sin.a, Segue ped bree, oy (= +3) sing Principles. 
i’ —— 
y ip : hy. 
=~ Asia last article, pat h=2na, then a=>> andNa ss Gin. (24h)=sin. x4 2 cos. («+-5)sin ak. 
Bee tet Put z toe: so that z will always be a po- 
. ae 2Qn , 
sitive quantity less than 1. Then ing the proper 
substitutions, and also putting instead of 2, we get 
from last equation ; ; 
: Cos.) —cos. 2=—sin. (r+-z h)nsin. ca 
and in this expression, may be any positive number 
whatever. 
17. It is an axiom in geometry, that any arc which 
does not exceed a quadrant is of an intermediate magni- 
tude between its sine and t. Now, z being = 
for some fraction between 0 and 1, and v for an are less 
than a quadrant, the expression Gray may denote 
any quantity between sin. v and tan. v ; use when 
==0 it becom es and when z=1 it becomes “= 
= tan. v; therefore while z increases from 0 to 1, the ex- 
pression will increase from sin. v to tan. v, and will have 
successively every degree of intermediate magnitude. 
Hence we may assume that w= en)? z being a 
fraction between 0 and 1. . Then sin. v=v cos. (zv). 
Instead of v put 4 and we have sin. A= cos.(2 i) 
and n sin —=h cos. (+). Now we are at liberty 
to suppose as great as we please, therefore the frac- 
tion — may be as small, and cos. (= 1) may differ 
from unity by as little as we please, and so n being in. 
definite, in respect of magnitude, we may express 
nsin. “by the are h, We then get from the two 
formule, ; 
Sin, (¢-+h)—sin. « = cos, (2-42h) n sin.” 
” 
Cos. (z-++h)—cos. « = —sin.(x-+zh) n sin. 
these others : 
Sin. (zh) —sin. x 
h 
= cos. (x-+-zh), 
(L) 
Cos. (x4-h)—cos.r ; 
et = —sin.(x+zh). 
Observing always that = is a positive fraction less 
than unity. And these expressions are the boundaries 
to the ratio of the increments, which we proposed to 
investigate, : 
If we put z=0, we get cos. x and —sin. x for one 
w is. to indefinitely ; and 
if we make z=1, we get cos. (w--h) and —sin.(2-4-h) 
for another boundary : so that the first of the two func- 
ba always yale 5 Palo a (eth); and 
Serened cos, (x » h’ being 
ome are. wher fy peg Ch Sey henna: 
—sin..a, and —sin, (4h), and may similarly be ex- 
pressed by —sin, (x4/’): And as, by the Anirumeric 
or Sines, art, 12. formule (D). _ 
. boundary to the functions which express the ratios ; 
: hihlante 
_will appear that the property belongs to ev 
If we put H to denote briefly the expression 
—2 sin.(« +5)sin. Pe and H’ to denote 
’ : , 
— 2 cos. 2+5) sin. ay where it is manifest that H 
and H’ are quantities which vanish, when A, and con- 
uently when 4’=0, we have also 
a ad —sin.t _ 24H \ * 
‘) 
= on TF = sin. t4+H’ 
18. We shall now bring into one point of view the 
different expressions for the ratio of the corresponding 
increments of the five simple functions considered in 
this section. ; 
hy—. 
is wey (t4-2')" =n +H; 
log. (2 + 4) —log. x 1 - 
A ~ Be@+#') 
PB LT Aa*t¥ Ag 4H; i 
sin. (x--h) — sin. x 
cos, rey —cos. x 
1 
1h Fate 
‘ 
= cos. (2 +h')= cos, « + H; 
=>—sin. (rh’) => —sin. cx +H; 
h 
and in each of these expressions, }’ is some uantity 
greater than 0 and less than h. : 
By comparing these formule, it appears that the five 
functions, 
a", log. 2, a®, sin. x, cos, 2, 
have the following common p: ies :—~ 
1. The expression for the ratio of the increment of 
each function to the increment of the variable quantity 
x, which serves as its basis, is always a finite quantity 
contained between two determinate ies, 
2. The ion for each ratio may be resolved in- 
to two parts, one of which is independent of the incre- 
ment h, and the other.is a function of x and h, of such 
a nature, that it is always finite while 2 is finite, and 
vanishes when h=0. 
3. From this last pro , the function which ex. 
presses each ratio has always a limit to which its value 
approaches, as / decreases, and to which it may come 
nearer than by any assignable quantity ; and it appears 
that in the case of the function 2”, this limit is m 2’—1 ; 
in the function log. z the limit is 5, B being theNa« 
pierean logarithm of the base. In the function at the 
limit is A a*, A being the Napierean ithm of a; 
and, lastly, in as functions sin. ye oy cos. z, the 
limits are cos. 2 and —sin. « respectively ; so that each 
limit is a new function of z, peculiar to the original 
function from which it is derived. We have seen, 
(art. 8.) that the of the limiting ratio extends 
to the algebraic functions A «4 Br’4, &c. and 
Ac™+Brv 4+, &e. .. é 
ax bas 4, Ke.” which are composed from the sim- 
ple algebraic function 2”; and in the same manner it 
expres= 
sion, composed in any manner whatever, by the opera- 
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