394 Mr. Beverley on the Rectification of Curve Lines. 
- dx —ad 
cos TAQ, AT = seas, anithe MG 2S te 
dz 
. dAQ= t> x { (yde ~ 2dy)dz—(yde —xdy) a} 
and d TAQ = eee. 62 .” But since im any case either 
dz.dy 4 . 
x or y may be supposed to flow equably, their second fluxions 
will be equal tu zero. Hence, in order to abridge the two 
‘ last expressions for operation, let us suppose d*r = 0, we shall 
1 
then have dAQ =—d’y.dx x (w.dz+y.dy) x er and 
dTAQ = —d’y.du x a Draw another polar ordinate 
Ap, indefinitely near to AQ, and with centre A and radius AQ 
describe the indefinitely small are Qr; then is pr = d AQ, and 
Qr =AQx dTAQ; and consequently, Qp = 7 pr*+Qr* 
—d2y.da(rdr+ydy)\? —dy.dxr(ydr+ady) ate 
eat ee 
dy? dx ye ts Sie dy? d x? (29+ ¥?) } 
= {26 xara} = ee 
= ev 2+ y= ACxad TAQ. Q. E. D. 
Example 1.—Let ACM be a circle whose radius is 7, and 
put TAQ = 24; we have 1(= rad.) :2r::sin$: 27 sin g= 
AC, and fAC x dTAQ = 4rfsin ¢d$¢=— 47 cos ¢, which, 
between ¢ = 0, and 90°, becomes 47 for half the length of 
the curve, whence 2 x 47 = 87 = 4.x diameter, for the whole 
‘length of the cardioide. 
Example 2.—Now let another curve be similarly described 
upon the curve we have just investigated, we shall then have 
to find AQ, and a second angle T’AQ!. 1(= rad.):7:: cos2$:r 
cos 24, and r —r cos 26'= r (1— cos 2¢) = 2r sin? = AC, 
Also 1(= rad.) :2r sin® $:: sin 2¢: 27 sin*® >.sin 26 = y, and 
1(= rad.): 27 sin? ¢::cos2$:2r sin*¢ . cos2¢ =a = ab- 
2rd¢ sin 2¢(1—4 sin? 9) 
: Ve di 
scissa of the first cardioide .-. a = 
4rd@sin¢sin3 9 
a = cot 3¢, and we therefore obtain fAC' xd T’AQ! = 
Qrfsin®> x 3do=6r(4o — } sin2$) = 3r(o — $ sin 29), 
which, when ¢ = 90°, becomes 37 x 1°5708 = 4°71247, for 
half the length of the second cardioide. et 
Remark.—It almost appears from what has already been 
done, that the angles TAQ, T’AQ’, T"AQ", &c. will always be 
equal to 2¢, 39,4, &c. and that we shall have 27 fsin x 2d¢, 
arf sin*¢ x 3d¢, 2rf sin?¢ x 4d, &c. for the rectifica- 
tions of the several curves respectively; but whether it is uni- 
versal or not I have not yet had leisure to determine. 
Example 
