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PEOFESSOE J. CASEY ON A NEW 
right-hand side rendered unnecessary, for the process then will be performed by differ- 
entiation. 
Cor. 3. The equation (169) may be written 
£=sin 0 J cos <pf((p)d<p — cos <p J sm <f> f(<p)d<p 
-f- Cj cos <p + C 2 sin <p 
59. Since (l+^" , )~ 1 =^(l-f x) if we put for x the symbol , we get 

(171) 
(172) 
.-. if the right-hand side of equation (169) be differentiated twice with respect to <p, we 
get the right-hand side of equation (168). Hence equation (168) may be written 
g = cos <p J sin q>f(q>)d<p — sin <pj cos <p f(<p)dq> 
+/( < P) + C i cos <P + C 2 sin<p (173) 
60. From equation (160) art. 56 we have at once the following theorems. 
If we have three polar curves given by the equations 
f=F(<p), £=F,(<p), ^=mF((p)+wF,(<p); 
then, 1°, if the corresponding lengths of their negative pedals be denoted by 
s, s„ S, 
w'e shall have 
S=ms-fw5, (174) 
2 °. 
then 
If the corresponding lengths of the %th evolutes of their first negative be 
<r, and 2, 
2=w?<r+w<r 1 . 
(175) 
61. To find the curve whose length bears a constant ratio to the radius vector of its 
first positive pedal. The given condition is expressed by the equation 
Hence 
where m=1c J r^ ; 
A/(<p) sin <p=f'(<p') sin cos <p d<p ; 
k(f(<p) sin <p+/(<p) cos <p)=2f(<p) cos <p+/* ,, (<p) sin <p. 
gmQ 
sin <p ’ 
(176) 
This curve is the equiangular spiral ; and we infer from the form of its equation that 
