FORM OF TANGENTIAL EQUATION. 395 
Observation. — Under each of the heads Evolute and Involute it will be observed we 
have solved three problems, which may be stated briefly as follows : — 
Given To find 
Tangential equation of a curve, Tangential equation of its evolute, involute. 
Tangential „ ,, Intrinsic „ „ „ 
Intrinsic „ „ Tangential ,, „ „ 
We have omitted the problems given the intrinsic equation of a curve to find the 
intrinsic equation of its evolute and involute, because these had been previously solved 
by Whewell (see ‘ Cambridge Philosophical Transactions,’ already cited). 
Examples. 
Examples 1-3 are illustrations of art. 48, 4 and 5 of art. 50, and 6-8 of art. 51. 
(1) Let j'=^tan”<p be the equation, it is required to find the involute. 
From equation (118), art. 48, we have 
v_ x sin <p=/rj tan 71 <p sin <p d<p. 
We can get a formula of reduction Tor this integral as follows : — 
Put P=tan” _1 (<p) sin <p ; 
dV 
.'. tan 78-2 <p sin <p-\-(n— 1) tan” <p sin <p, 
.-. J tan” <p sin <p d<p =— - n ^W sm< P _ -TL- jtan”" 2 <p sin <p <?<p, . . (131) 
which is the required formula. 
Cor. If v=wtan” -2 <p+(w— 1) tan” <p be the equation of a curve, the equation of its 
involute is 
J'_i=tan” _1 <p (132) 
Compare equation (116). 
(2) Find the involute of the curve 
v=a log tan <p 
(that is, of the logarithmic curve), we have 
v_ x sin (f>=a Jsin <p (log tan <p) d<p 
=C —a jcos <p . log tan <p +log tan || (133) 
Cor. v_ x sin cos <J3=C— «log tan 2 (134j 
(3) Let the curve be 
’=(l+cot*<p) 3 , 
