tal 
Principles 
—— 
$98 
AA iastead of hand ¥ for a certain quantity between 
0 and 1, we get es 
‘hy Aye 
(1 +) —i=A'(1+> h; 
but since (: +S)'= a’, the second member of the a- 
, s 
bove equation mE oat therefore after 
14— 
n 
Aha® - 
annie) 
Now, as we have it in our power to make n as great as 
we please, let us suppose it a very great number ; then 
of?" = 
=h may be reckoned =0, and a* =1, and ARS 0, 
” 
thus we have amply a'—1=Aha™. (I) 
and as a***_a* = a* (a*—1), it follows, that 
+ 
ae Aa*a**, (K) 
2’ being, as already stated, a quantity between 0 and 1. 
This ion for the ratio of the increments of a” 
and x agrees in its general properties with that for the 
ratio of <* and z, (art. 7.) Its boundaries found by 
making 2'=0 and 2'=1 are Aa‘a” and Aaa"; there- 
= an put 4’ for some quantity between 0 and h, we 
8 ve 
that : 
« =Aa*a". 
. (K’) 
And as spin, by equation (I) of this article, a’ = 
14A kh’ a™, if we put H=A h’a*, where H denotes 
a quantity that vanishes when h’ or h=0, we shall also 
h, 
have this fonmule ee =Aa*4H. (K") 
15. We come now to the circular functions sin. «, 
and cos. z, that is, the sine and cosine of x, an are of a 
circle, the radius of which we shall assume =1 ; and 
we are to investigate in each case an expression for the 
ratio of the increments of the function, and of the are, 
which we consider as its variable basis. 
ing x and a to be any arcs, from the fourth 
of formule (C) An:rumetic or Sines, Art. 12. we de- 
rive the following series of equations, ‘ 
Sin, (7-+4+-2a)—sin. z=2 sin. a cos. (ra), 
Sin. (2-4-4 a)—sin. ae a)=2 sin. a cos, (143 a), 
Sin. (r4+64 24-4a)=2 sin. a cos. (x45 ay 
Sin.(z42na)—sin. {42(n—1)a} C— 3 
2 sin. a cos. {2+ (2x—1)0}; the number of equa- 
tions being'n. 
By taking the sums of the corresponding members . 
* In the series of ares t+4-a, -4-3a, x-+4-5a, &e. the cosines of those less than a quadrant will be positive quantities ; and the. 
FLUXIONS. 
Pondemen- therefore, putting in this expression 1 instead of x, and of these equations, ont: celoction the quantities found F 
wisin. aa ae 
@sin.a { cos. (24-0) -008.(24-3 @) cos. (2-45 a) + 
Are yer +con(24[2n—1] a) }. 
sot pees eee 
raultipli num terms, 
be a quantity evidently ter than their sum ; and if 
by their number, ‘the pro- 
series ; it must therefore be less than 2n: We have 
now : bis 
Cos. (4-4) + cos. (r-+$a) ... C08. {e4(en—1)a} 
=n Cos. GN and hence 
Sin. (242 n a)—Sin. z = cos, («#+4+N a) 2 nsin.a, 
Let us now put A=2 na; then a= 5» and Na= 
hs as N is always less than 2x, 2 will be a posi. 
tive fraction less than 1, let us denote it by z, and also 
let us put n’ or rather n instead of 2n, and upon the 
whole we chall have = 
Sin, (24h) —sin.emcos. (2--2h)nsin. ~, 
and in this formula n may be any positive number 
whatever. , 
16. By the second of formule (C) Anrrumetic of 
Sings, art. 12. we have the following series of equa 
tions, 
aie Cos, « — cos. efi) 2sin, asin.(x-+-a), 
Cos. (x-+4-2a)— cos. (x44) = 2sin. asin.(z43a), 
Cos, A aan (+62) = pemipte i i so 
Cos. {e+ 2(n—1)a } — cos, (#-4+2na)= 
2sin. a sin, 2+(2n—1)a } ; 
the number of equations being x. Hence, by adding. 
as in last article, we : 
C08, 2-—COS, (-- 2n oe 
2sin. a { sin. (s-+0)+ sin.(x 48a) sin.(-45a) + 
wae $sin.(e+ [2n—1 Ja) i 
As the sum of this series of sines will ‘be less than the 
greatest term multiplied by the number of terms, and 
greater than the least term multiplied by the same. 
number, it must be exactly equal to some quantity of 
an intermediate magnitude multiplied by that num- 
ber. This quantity may evidently have the form 
sin. (a4-Na), where N denotes a positive quantity less - 
than 2n; we have therefore 5 
sin, (2-4-@)-+4-sin.(4-43a) 4 ..+++ 
+sin. fr (2n—1)abn sin.(2-++Na); 
sosines of those greater than a quadrant, but less than three quadrants, will be negative. Here we reckon that to be the greatest term 
which is nearest to +1; and that to be the least which is nearest to = 1. : 
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