390 
PROFESSOR J. CASEY ON A NEW 
Hence if v 2 , v 3 , See. represents for the successive evolutes what we have denoted by 
v for the curve itself, we have 
d . . . 
h 8m< P = d^( VSm $)> 
sin <P=^ (b sin »’ 
d? , . , 
Min<p=^0'Sin<p). 
d* , . . 
9 n sm<p=^(f sin <p). 
(96) 
similarly 
Hence in general 
44. Since 
v=f($), v s ^ n <P =/(<P) s i n <P J 
and denoting this by or(<p) and the corresponding functions for the evolutes by 7 r,(<p), 
7 r 2 (<p), &c., we have, from equation (96), 
’ r »( < P)=^iW < P)) (97) 
45. In art. 26 we have found the coordinates of a point on the curve v—f(<py . — 
x =/(<P) +f (<P) sin <f> cos <f> ; y= —f (<p) sin 2 <p. 
These assume, if we substitute from art. 44 for/(<p) the value the symmetrical form 
x=n(<p) sin <p+^(<p) cos <p, 
y=<7r(§) cos <p — 7r'(<p) sin <p 
(98) 
(99) 
and hence, from art. 44, if we denote by x,„ y n the coordinates of a point on the nth 
evolute, 
H“*(4)’ +o “*(4r , j* w - (100) 
^={co S $(^) -sin?(^y +1 }^) ; (101) 
46. By using Leibnitz’s theorem, we find, from equation (98), 
d n x . d v ir 
I m nr\cs 
dp 
d n *7 r n.n — 1 . _ d n ~' 2 Tt n.n—\.n—2 d n ~ ? ir 
Is C0S f + &°- 
df-= sm, fdf+ ncos ‘t‘dr^ ja~ s ' n <p^- 
d n+] 7T 
dcp n 
d” 7r n.n — I d n 'n n.n — \.n — 2 . d n ~ l n 
dr~— ]2“ ccr 
Hence, by equations (100), (101), we get 
<fri 
dip 
+ cos ? ir^~ n sin f dr — ~[2 cos f dr^<+- — sin f 
d n x n.n — 1 n.n — \.n — 2 
^ = ^„_ 2 £ y n _ 3 +&c. . . . (102) 
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