lovene 
Methad. 
Of the recti- 
fication of 
curves. 
Fig. 10. 
rig. 
450 FLUX 
equal but with opposite In fact, 
>= 1, weed that the area BC =venid the 
fluent | s*— 4 2* 4 ¢ between the limits x= AH=1 
and 2 => AP =4/}{, we get the area HDF = —4.. In 
like manner, the area taken from~K to'H comes out 
= 0 because the space KOA below the axis is equal'to 
the space ACH above it. 
155. Although it is not 
the abscissa x, then if we assume the ordinate y = =, 
we have ydx=du, and fyde=w pe. For ‘ex 
ample, let w = aa", ee ett aren Hence 
every curve of which the co-ordinates x and 
related, a fy Ba pi is quadrable in 
terms, and has its areas-aa2"+1+4¢. The case of 
pate Bs A ag te tga Pe 
Of the Rectification of Curves. 
157. We have found, (art. 75.) that 2 and y being 
the co-ordinates, and 2 any are of a curve dz = 
dx + dy*); we shall this formula 
v(d oa : we now apply to 
Exampe 1. a ee 10.) 
Q=2, parameter = 2a. The equation 
of the curve 2adx = y', gives ade=ydy, anddz= 
dy “2 / (y+0*) ; the fluent (art, 128, formula (B), and 
art. 123.) is 
see¢ Ee v@+y) tbat fy+ vier}: 
If the fluent begin at A, then, when y= 0, z=0, in 
this case the general formula becomes 0=c + 4 a1.(a), 
and hence c= —}al. (a); therefore s=¥¥ (+2) 
+ hal fetes te}. 
fais Re rae eis Se Alia the sea 
cul parabola, which the equation is 93 = aa%. 
In this ‘case: the general: formula gives d=" 
ayy (1492); hence, by art. 126, 
t= tral (14 92)" +c. 
This curve is ‘ectly rectifiable, and is remarkable on 
account of its eg owes thet cotrve thet tanning 
This dcovery wat de y Neil, and after- 
bra, chap. 77. 
Des Cartes’ , end of 
Ex, 3. Let the curve be a circle 
the radius OA =a, then, ; ig St wat 
OQ = z and QP= y from the centre, ep+yca’ 
KIONS. 
ewds sinh ee 
. —, Method. 
Wehr dey ala ea, 
e have 
an; inlining eles 4g rah inet a 
rad = 1. 8 a we have only to sub- 
ative = for x and = for a, we thus find: 
If the are. AD be a "and the fluent begin 
at D, then when x = 0, = 0; therefore in this case 
e=0,, 
“We have given a d ‘series in art. 148. Ex. 2. 
for an arc, in terms of the t ; and othersmay be 
found which shall. express it’ the cosine, cotangent, 
&c. from the formule of art. 35. But in no case what- 
ever can an arc be expressed by trigonometrical lines 
in finite terms. 
Ex. 4. Let the curve be be ge clipe t (Fig. $1. es Me SL 
ig oh the fiche (Ione FS 
ite =, ,w is 1l—c*),=e: 
(oe eee at ee eee 
CQ =eund PQ=y, and put the are BP (reckoried 
from the extremity a the empyema By 
the nature of the curve 
cy =4/(1— 2°), hence d y= exe 
vay 
vfi—asewe}ae a lemae 
VU Bs 0 AO 
The fluent of this’ cannot be found aaa 
tray eve withthe help of ciel reso ga 
therefore it can only be expressed by an infinite series. 
By-the binomia) theorem, we get 4/U'——*0= 
sH) 1.1.3: 
Lape oe ot oF eres Be 
pea ciao en grirr sp lp series by 
Faas amd take the uents, which will all be of 
eed 
the form A Va—ay and may be found by art. 130. 
and 
de" 
Thus, putting for the arc whose sine is x, we 
+= elie te BP = m et 
Vag!) os geal 10 
pss 1A aoe gai ~aaases—**) 
-ihereey 
che Ut? + a3 s*)va—=} 
1S 1.3.5 . 
4 ER f(t Fee +355" a. 
+ &e. “im 
This expression vanishes wera 2D, a tought, 
therefore it wants no correction. a=1 
all the terms containing ona) mak , as in 
this case 9 = } x, we Mi cniptis gy 
AB=ja(1— hem 1 LS pea 
This expression eeeeas 
Kargdre-mpar pestering 
have a complete solution, Eisley of ny wa Te 
