404 
PEOFESSOE J. CASEY ON A NEW 
Cor . The polar equation of the reciprocal of the nth evolute is 
P 
r w 
65. If in equation (182) we put £='$($), we find 
^)={i+(|) }4) ; 
F(<p)=F jm~^iwr 
( 183 ) 
(184) 
an equation which gives the intrinsic equation of a curve in terms of the polar equation 
of its reciprocal. 
G6. If in equation (178) we put g>=4{<p), we get 
JW— 4,( f )- 8 in ? * 
Hence if § = %|/(<p) he the polar equation of a curve, 
(185) 
'J'(f) sill <p v ' 
is the tangential equation of its reciprocal. 
Obs . — The problems we have solved in this section may be briefly stated thus : — 
Given 
Tangential equation of a curve, 
Polar „ „ 
Intrinsic ,, „ 
Polar 
To find 
Polar equation of its reciprocal. 
Tangential equation of its reciprocal. 
Polar „ „ 
Intrinsic „ ,, 
Examples. 
(1) Let it be required to find the reciprocal of the catenary. 
The intrinsic equation is 
s=c tan <p ; 
.\ F'(<p)=<? sec 2 <p, and, substituting in equation (182), 
|= C { 1+ (|)}-W 
Hence, from equation (170), w r e have 
k 2 
— =c sin <p Jsec <pd<p=c cos <p j sec <p tan <pdq>- J- C x cos <p + C 2 sin <p ; 
then performing the integrations, and determining the constants by the condition that 
