A CURVE, AND ITS APPLICATION. 669 



36. The following proposition may be proved by the intrinsic equation. 



Fig. 25. Let any curve be evolved, and its involute evolved, and the involute of that evolved, and 

 so on, beginning always the evolution with a rectilinear tail, AA', extending beyond the curve, 

 and all these tails being equal. The curve tends continually to the form of the equiangular spiral. 



Let s, s', s", «"', &c. be the successive curves, d) the angle, which is the same for all, be- 

 ginning from for each. And let each of the tails AA', A A", A" A"', he, = a. 



Let s = ai(p + a-^cj)^ + a,(p' + &.c., which may express any curve. 



Then s' = f(a + s) d<p = a<p + - cj)' + ~ (p'' + - (p* + &c. 



2 3 4 



s = j(a + s ) d(h = ad) + — 0-+ — (f)^ + (h* + gjc. 



^ ^ 1.2^ 2.3^ 3.4^ 



rn • // a a at d)^ 



s = l(a + s ) d<h = ad) + 0° + dy' + ^-— + &c. 



^ ^ 1.2^ 1.2.3'^ 2.3.4 



And as the operation goes on, the terms in a,, a,,, a^, &c. being divided by the factorials 2.3.4, 

 &c. indefinitely, may be neglected as to their influence on the curve. Therefore ultimately 



s = a(h + -- 0= + — — 03 + &c. = a fe* - l], 



which is (Art. 7) the equation to the equiangular spiral. 



Of course, from the nature of the construction, the curvature of the original curve is throughout 

 towards the same side*. 



Additional Note to a Memoir on the Intrinsic Equation of Curves. 



Trinity College, April 12, 1849. 



In the Memoir on the Intrinsic Equation of Curves, I gave a proof of the following Proposi- 

 tion, which was enunciated by John Bernoulli, and demonstrated by Euler. (Novi Comm. Petrop. 

 Tom. X.) 



Fig. 26 and 27. If AB be any curve, AB' its involute beginning from A, B'A' the involute 

 of AB' beginning from B', A'B" the involute of A'B', beginning from A' ; and so on, alternately 

 and indefinitely : the successive involutes approach indefinitely to the form of the common cycloid, 

 provided ffie tangents at A and B in the original curve are perpendicular to each other. 



The question naturally offers itself. What is the curve to which the successive involutes tend, if 

 the original curve do not conform to the condition above stated, that the total deflexion is a right 

 angle ? 



I am now able to state that in that case the curve will be an epicycloid or a hypocycloid as the 

 total deflexion is greater or less than a right angle. 



• AUo it in neccsAary, as has been remarked to nic, that the i ginal curve, or any of its cvoUltCB in infinitum. For if it were, 

 point where ';i = 0, is not a point of contrary tiexurc from the ori- some of the iiuantities «,, «a, &c. woiiKl be inlinitc. 



Vol. VIII. Fart V. 4U 



