Mr. Beverley on the Rectification of Curve Lines. 395 
Example 3.—When ACM is the common or Apollonian 
parabola, whose equation is az = y*, or y = at yt, a being the. 
ss ————> yd 
parameter. AC= f#+y= Var + 2’, A 
subtangent TD, and = = Var+4xe= CT, the tangent. 
Consequently, 
ee 
Ota PP: 2 yen TAQ, and CT:1::CD:; 
: a/ art 4x 
gr S = cos TAQ; whence dTAQ = and 
Ja+de “i Hat42) 
is eae Cy atde Di gett sate 
AC x dTAQ = V axz+ 2? x eae gadis 25 
whose integral is a*(4 Vatuex—t¥ 8a x hyp. log. 
a/3a+2 asB 
Va+4e 7 
Example4.— When ACM isthe cissoid of Diocles whose equa- 
~.. FirstACe / @ py =/e4 = 
tion is y? = ss 
J a— a—@r 
I 
a2. 
rt also the subtangent TD = a = aon), and 
the tangent CT = i =3 — x (4=* ® consequently 
OP ores TD 36k = ge TED = sin TAQ, and 
a 4/ da—3e j : 
¥ 3axr — 222 
CT:1::CD: PPS rane tee cosTCD = cos TAQ; whence 
a 
wy 3dx(ax—a*)* 4 ate 
dTAQ= Se aa? and fAC xdTAQ =f = x 
3dz(az=x2)* ia 3a2 x de (oe s da 
fone SS 4 (when corrected) 75% hyp. 
log. ee — 2a2 22. And when «x = a, it becomes 
aegr 
aE x hyp. log. (2 + /3)—2a = 3:041385 a—2a =1°041385a, 
or 2 x 1:041385a¢ = the whole length on both sides of the 
axis. ~ 
Example 5.—When ACM is the lemniscata whose equa- 
tion is (a? ab yy a (2* — y’) =.0, Let AC — g, and DAC 
= 6; we have y = sin 9 &, (2° + y*)’ = &, and a? (2® — y?) = 
(e+ y—2yY)=aCeP—-2eeY=ave —Va'sin® h &, 
whence &* = a*? — 2a’*sin’ $2 and & =a /f1—2 sin? 6=a 
oD 2 cos 
