FLUXIONS. a 
if 4 
448 
Taveree 
Method. 
Fig. 12. 
Pig, 27. 
'@= 4AOxarec AH—} 0Qx QH = circ. 
safyde= & fas Vee’) 
We may proceed with this fluent, as with that in exam- 
ple 2d, or else by formula (B), art. 128, which gives at 
once 
d 
faz J/(t—o) = ha (a) —4 “fT5aey 
Now, by art. 123, 
Senay eh fre v(e—o)tte 
Therefore, 
oo ss Vena — 1, fe + Vat — = 
Now, from the nature of the figure, when r=a, then 
¢ = 0; therefore, in this case, the general equation be- 
comes 0 = — “71. (a) —e3 
hence ¢ = — “1. (a), and 
6 cm 
s =i yee) — FL pti ve ot 
ors =4zy—" 1. {-+4} 
If we join CP, it is evident that 42y expresses the 
area of the triangle CPQ, therefore 
ab a omy 
Sector ACP = $l. {~+5}- 
Ex. 6. Let APD bethe common cycloid, (Fig, 12.) 
of which AB is the axis, AHB the generating circle, 
having its centre on the axis, AK a perpendicular to gree 
the axis at the vertex, and PR a to AK 
Se B tare pated ha tha coca, coll tcl a eee 
to find the external area APR. 
et leg che eqntee 2° the eaee Tend 
Pe perpouioae to , meeting the circle in H, 
join OH, Put AR=z, RP= y, AO =a, the angle 
AOH=v. Then AQ =a(1—cos.v), QH = asin. », 
arc AH = av; and since, from the nature of the curve 
(see Ericycioiw), PQ=AR = arc AH + HQ, there- 
fore 
x=a(v +sin.v), y= a(1 —cos. v), 
‘de=adv(i +cos.v), fydx=a’ {dv (1—costv), 
Now this last fluent, or f‘d v sin.’ », is found, by art. 
144, to be — 4 cos. v. sin. v. + $v, therefore 
2 
s= 5 (o—sin.v. C03. v) + c. 
When v = 0, then s t to vanish, therefore c= 0, 
ion putting for v, sin. v and cos. v their values, we 
. AQH. 
This agrees with what was shewn in the article Ep1- 
CYCLOID. 
Ex, 7. As an example of a polar curve, let us take 
the spiral of Archimedes, (Fig. 27.) Let A be the pole, 
AC the position from which the revolving radius AP 
ins its motion. Pat AP =r, the angle PAC = 0; 
let a be a given line, and # = 3.14159, &c. The 
nature of the curve is the ion 2aer= 
av. Now ifs denote the area AP’P, we have found 
(art. 72.) that in curves expressed by a polar equation, 
ds=jridv. Inthe present casedu= “Far, there. 
fore 
sah frdvatfrdra Zeige 
If we suppose the fluent to begin when r = 0, then 
c=0; therefore, when r has made a complete revolu- 
tion, so that v = 2, and r = a, the area generated will 
be “ = } of a circle, of which @ is the radius, To 
find the space which r passes over in the next revolu- 
tion, the fluent must be taken between v= 2, and 
v = 4-, that is, between r= a, andr=2a. Corre- 
sponding to the first value of r, we have s = = © and 
to the second = 5 at; the difference of these, which 
is the area required, ig 
151. As the area ECPQ (Fig. 14.) of any plane curve fig, 14. 
is expressed by the fluent f'yd, in which y (PQ) is 
some function of the abscissa « (AQ), on the other 
hand, every fluent /"y dz may be represented geome- 
stan fi Sapa ne eto the ab- 
sci inate. geometrical represen 
tation of a uent shews distinctly wherein it differs from 
a common analytic function, such as a + 6.2", or a*, or 
sin. «, &c. These last have determinate Cone Fe 
ding to any assi value of x, and the valueof = 
cach function ia i of its preceding 
of magnitude to another. 
152. The analogy 
of nearness. Let the fluent / y dw be required between 
the limits of = aanda= 6, Let CPD (Fig. 28.) be 
ig i Fig. 28. 
a curve, such, bag ye etc _ is 
begin at a given point A; ma x of 
the axis A, take AQ =a, : ig een re Se 
AQ” = 4, its greatest value, and draw the ordinates 
PQ, PQ”; then the area PQQ” P” will be the geome- 
trical expression for the fluent f(y dx, between the li- 
mits of « = a, and « = 6: and by whatever means that 
area can be found, the same will apply to the determi- 
eatin oe be divided ber of 
Let ivided into any num parts 
QO OO, he. and let the ordinates P’Q’, P"Q” fe. 
be drawn ; these will divide the figure into the curvi- 
lineal trapeziums PP’Q’Q, P’P"Q’Q’, &c. Let a series 
of rectangles PQ’, P’Q”, &e. be constructed, each ha- 
ving the shortest of two adjoining ordinates for its 
height; these will fall pains! within the figure, suppo- 
sing the curve to be entirely concave or convex to- 
wards the axis. Let another series P’Q; P’Q’, &c. 
be constructed, each having the lo of sroginns 
ordinates for its height; and all these will be- 
yond the figure. Because AQ = a, and = b—a, 
and the number of parts into which QQ’ is divided 
is known, the abscisse AQ’, AQ", &c. will be 
known, and from the nature of the curve; the cor- 
responding ordinates P’Q’, &c. will be known. Hence 
we can find the inscribed PQ’, PQ", &e, the 
sum of which will be less than the curvilineal sp 
PP"Q"Q;; also the circumscribed rectangles P’Q, > 
&c. the sum of which will exceed that space. Thus two 
