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PROFESSOR J. CASEY ON A NEW 
and subtracting this from equation (a) we get 
2 fW cos 9 +f"W - 9=0^--^ 
dS 2C, 2C 2 cosip 
dtp sin 3 <p sin 3 <p ’ 
S = ClC s Tn^ C j + Cl l0g ^ ^ + C4 ' 
This is the intrinsic equation of B, and therefore the tangential equations of A and B 
respectively are 
,=g T fn^ ~ C * + c - lo g ta ° if' + C ‘ («) 
C, — C 2 COS<p 
sin 2 <p 
+ C 2 log tan \ <p-fC 3 
(92) 
CHAPTER III. 
Section I. — Evolutes. 
39. If the tangential equation of a curve be 
we have proved, in art. 30, 
dtp sin <p 
Hence if g denote the radius of curvature, we have 
§sin <P=^(/W sin 2 <p) ; 
if v=f(<p) be the tangential equation of a curve, the intrinsic equation of its evolute is 
d 
ssin<p = ^(/(<p)sin 2< p) . , 
40. If our movable line had been given by the equation 
y=x tan <p4-/(? 1 ), 
(93) 
Hence 
we get in the usual manner 
x= —f((p) cos 2 <p, y=/(<p)-/(<p) sin <p cos <p. 
^=cos <p(2 f'(<p) sin <p-/"(<p) cos <p), 
^=sin <p(2/(<p) sin <p— /"(<p) cos <p ) ; 
