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PEOFESSOE J. CASEY ON A NEW 
^ =i ~sh Tf (see equation (64)); 
but by article 48 we have 
py^- J/Op) sin<p<fo . 
' ' sin <p ’ 
••• ^=f (<P) sin <p+f(<p) cos <p+Sf(<p) sin <p d<p : 
that is, 
^ (<P) sin ® +i’/( < P) sin <P ^ ; ( 125 ) 
s=/(<P) sin <p+jj (/(<p) sin <p r7<p) (7<p, (126) 
or, as it may be written, 
s ={ 1+ (i|) }(/(?) si “‘P) ( 127 > 
Hence we have the following theorem : — 
If v=f(<p) be the tagential equation of a curve, the intrinsic equation of its invo- 
lute is 
s ={ 1 + (4) }(/(<?) sin <P). 
ds 
Cor. 1. Since s is the length of the involute, ^ is the length of the given curve. 
Hence from equation (125) we have the following theorem : — If v=f(<p) be the tan- 
gential equation of a curve, the length of the curve is given by the equation 
s ={^+(^) '}/(?) 8“ (128) 
Cor. 2. The equation (127) is equivalent to the following: — 
s=§ f \<p) sm <pd<p+$(f'((p) cos <pd(p) dtp; .... (129) 
for we have proved, art. 30, that if v—f{<p) be the tangential equation, the intrinsic 
equation is 
s =f(?) sin <P+Sf'(<P ) cos <F> 
and we get the intrinsic equation of the involute from this by integration. 
51. From the intrinsic equation to find the tangential of the involute. 
Let s=f[tp) be the given equation, then the intrinsic equation of the involute is 
s=$f(?)d<p. 
Hence from equation (85), art. 38, the tangential equation of the involute is 
v =§ cosec 2 <p\f(tp) sin <p dtp] dtp (130) 
