FORM OF TANGENTIAL EQUATION. 
389 
••• ^= 2 /(<P) sin < P-/"( < P) cos <P ; 
and, as in the last art., we find 
scos<p=-~(f(<p)cos 2 <p. ........ (94) 
41 . The Tangential Equation of a Curve being given, to find the Tangential Equation 
of its Evolute. This problem is solved by articles 38 and 39. 
For if v- =/'(<p) be the tangential equation, the intrinsic equation of the evolute is 
s=2y / (<p)cos<p+ t / >,, (0)sin<p. (Art. 39.) 
Let this be denoted by F(<p), and, by art. 38, the tangential equation is 
{/=Jcosec 2 <p|JF'(<p) sin <pd<p}d<p. 
Now we have, from the value of F(<p), 
F'(<p) sin <p = 3/"(<p) sin <p cos <p +/'"($) sin 2 <p — 2/'(<p) sin 2 <p ; 
and integrating by parts we easily get 
JF'(<p) sin <pd<p=f"(<p) sin 2 <p+/'(<p) sin <p cos <p— /(<p). 
Multiplying by cosec 2 <p, and integrating again, we get 
/'(<P)+/(<P)cot<p. 
Hence the tangential equation required, 
►=/W+/(f))cot<p (95) 
42. The foregoing result may be obtained very simply from geometrical considerations 
as follows. In fig. 4 (art. 26) the line PS is a tangent to the evolute, and the angle 
OSP=<p ; then we have 
OS =ON+NS 
=LQ +OLcot<p 
=/( < P)+/(<P) cot <p. 
Hence if OS be taken as the directing-line, the tangential equation of the envelope 
of SP is v=f (<P)+/(<P) cot <p ; but the envelope of SP is the evolute, and therefore we 
have the same result as before. 
43. The right-hand side of equation (95) may be written 
^(f(<p)sin<p) 
sin <p 
