FORM OF TANGENTIAL EQUATION. 
393 
Section II. — Involutes. 
48. From the equations in art. 43 for the successive evolutes of a curve, we can 
conversely infer the equations of the successive involutes : thus, let the tangential 
equation of a curve be 
the tangential equations of the successive involutes are 
v_y sin v sin <p d$, 
or, as it may be written, 
9-1 sin v sin <p, 
y_ 2 sin v sin <p, 
y_ 3 sin <p=fff d<p v sin <p ; 
and in general, for the «th involute, 
y_„ sin v sin <p (118) 
Mathematicians have recognized it as legitimate to interpret the symbol of differen- 
tiation with a negative index, as denoting integration ; therefore we may write the 
equation (118) as follows : — 
y_ n sin <p= (./(<?) sin <p). ' (119) 
Hence the equation (96) includes the formulse both for evolutes and involutes, 
according as n is regarded as positive or negative. 
By an extension of the notation of art. 44, the last equation may be written 
*-*m=(4)~V(‘»> ( 12 °) 
49. If y_ x denote the coordinates of a point on the first involute, a*_ 2 , those 
of a point on the second involute, &c., we have 
-£_, = cos <p(7r(<p)) + sin 7r(<p), ....... (121)- 
y_ x -— sin<p(7r(<p))+cos<pJ^x(<p) ; ...... (122) 
and, in general, 
*_„=jsin<p(^) +cos<p(^) ’}»•(*),. .... (123) 
y_.= (co S $(^) -sin(p(^) ‘ \($). .... . (124) 
50. The Tangential Equation of a Curve being given, to find the Intrinsic Equation 
of its Involute. 
This problem is solved by articles 30 and 48. Thus, if v=F(<p) be the tangential 
equation of the involute, 
3 k 2 
