FLUXIONS. 
Direct Infiniment petits, L’Hopital; and in most works 
_ Method. Asa coaae expressly of the fluxional or differential 
Fig. 14. 
Fig. 15. 
Of the Fluxions of the Area and the Arc of a Curve. 
71. In any curve, we may consider 2, one of the co- 
ordinates, as an dent variable tity, and 
then the other, also the area, the arc, aid avaey ob 
area CEQP, and it as a function of x, let 
s=f (x); then the area CEQ’P’ will be f(z+A) ; and 
ee oe will be the in- 
crement of s, ing to h increment of 2. 
Let P’F, the increment of y, be denoted by 4. 
The curvilineal is less the rect- 
GQ’=( y+), but than the es 
PO my be theceteney ie wil equal to (y+k')h, k’ 
ing a ity between 0 and & Hence, f(x-+4)— 
S@=Cyt¥ and 
Let —Se) =yt+e. 
now / to decrease continually, then k and ’ 
continually, and the limit of the ratio 
S@+H—IE) will be y; therefore, the fluxional ratio 
d {re} ord! 
a2,°. ds 
= ¥y, art, 23. and hence 
ds=ydz (1) 
From which it that the fluxion of a curvilineal 
Pp. a ordinate multiplied by the fluxion of 
72. When the nature of the curve is by a 
polar equation pie. ap eprdaen high cay apreecd 
position, which meets the curve in C, and A a given 
point in that line, round which the variable radius AP 
revolves. Draw another radius AP’, so that all inter. 
mediate radii may form with them an increasing or a de- 
ing series of quantities, and on A as a centre de- 
scribe the circular ares PG, P*F. Put PA=r, the an- 
gle PAC=», and the curvilineal space ACP=s; and 
Gre Or peel dg 
Put A for the angle PAP”, the increment of v.. 
employing the usual notation s=CPA=f(v), and area 
CP’A=/(v-+h), therefore area PAP’= OF Se). 
Put  =P’G, the increment of 7, then’ the arc =rh, 
the arc P’F=(r—A)h, the sector APG=} r*h, the sec- 
tor AP’F=}(r—k)*h. Now the curvilineal space PAP’, 
is less than the sector APG, but greater than the sector 
AP’F ; therefore it will beequabto 4(r—¥’ }*h; supposing 
/ some quantity greater than 0, and less hk; hence 
seashore yh, and f a bet 
ROAD SO) — irk: 
423 
as h decreases, k and k’ decrease, so that the limit of the 
expression for the ratio is }7?; therefore ; 
a ONE ae ea 23,) and 
dv dv 
ds=}eede * (2) 
Hence it appears, that the fluxion of the curvilineal 
sector CAP + is half the product of the square of the 
revolvi: 
with the ficed axis AC, 
73. Archimedes, and all geometers since his time, have 
admitted as an axiom, that if two lines of any kind have 
the same extremities, and their concavities turned both. 
the same way, the shorter of the two is that which is 
in the apane bounded by the other line, and the straight 
line which joins their common extremities. Hence it 
follows, that an arc of a curve, which has its concavity 
turned all one way, is greater than its chord, but less 
than the sum of two tangents drawn at the extremities 
of the are. Proceeding from this principle, we may de- 
termine the Limits of the ratios of the arc, the c , and 
the sum of the two ts to one another, su 
the arc to be diminished indefinitely, as follows : 
Direct 
Method. 
Oo . 
radius r, and the fluxion of the angle it makes ° 
In Fig. 16. let ADB be any arc of'a curve, AB=c the Fig, 16 
chord ; AC=d, and BC=a, the tangents at its extremi- 
3 A a sin. A 6 
ties. By trigonometry, — = an, (AqBY and >= 
sin. B a+b _ sin, A + sin. B 
in (aE) ee oe aA tap ee 
sin.A + sin, B=2sin. }(A 4 B)cos.}(A—B), (ArrrHMe~ 
ric of Sines, art. 12,) and sin, (A4+-B)=2 sin. 4 (A+B) 
x cos, § (A+B), (art. 13, form. G), therefore 
a+b _cos. }(A—B) - 
c ~~ eos, (A+B) 
the ; roped ae yadhg the aha 
int ing to A; angles _A-and.B 
wil imenifestly both arene and they co become 
less than any ay aoe angles whatever ; therefore 
A—B, and A+B, proach continually to 0; and 
cos. }(A—B), and cos. 3(A +B), h to 1, which 
is their common limit. Hence the limit of the ratio of 
a+ to c is the ratio of 1 to 1, that is, a ratio of - 
lity ; and as the are ADB is always.of an initevinedixts 
itude between a+4 and c, much'more is the limit 
of the ratio.of the are to the chord, also the limit of its 
ratio to the sum of the ts a ratio of equali 
74. In the curve CPD, (Fig. 17.) let PQ, 
two ordinates dicular to the axis AB; and su 
pose that the intermediate ordinates go on colstiandally 
increasing, or else decreasing. Draw the tangents P 
P’H’, meeting the ordinates-in H and HM’. The are PP’ 
is less than one of the two PH, P’H’, but 
than the other; for draw the chord PP’, and let 
tangents PH, P’H’ meet in I, then because, from 
the nature of the figure, P’H’, one of the tangents, must 
make a acute angle with the ordinates than the 
chord makes with them, it will be less than: the chord, 
and therefore it will be less than the are, And again, 
because the acute angle, which the other tangent PH 
makes with the ordinates, is less than the acute. angle 
made by P’H’, the line HI must be greater than P’I, 
and HP greater than P’I+-IP, and therefore HP must 
be greater than the arc P’P, ‘ous 
ence also it is-easy to infer, that the limit of the ra- 
"tio of an arc to its chord is a ratio of equality. ‘For the 
lines PH, P’H'’, are manifestly to one another as the 
PQ! be Fig. 17- 
