670 ADDITIONAL NOTE TO Dr. WHEWELL'S MEMOIR ON THE 



The proof of this extension of Bernoulli's proposition easily follows from the mode of repre- 

 senting curves by their Intrinsic Equation, namely, the equation between the tangent and the 

 deflexion. 



Let AOB be any curve, and let the tangents at A and at B make with each other any angle 

 ma, a being a right angle. 



Let APB", B'O' A, A'P'B", B"0"A", &c. be the successive involutes, beginning alternately at 

 opposite ends. 



Let AB'= 6,, A' B"= b.,, &c. the whole arcs of the alternate involutes. 



Let the intrinsic equation to AOB be 



s = a^Kp + ^^ (p' + ^' + &c., which may express any curve. 



Hence AP = t, == — <b' + -^^— (h' + "^ 0' + &c. 



1.2^ 1.2.3^ 1 .2.3.i^ 



And B'P=b, - — <h' "^— d)' ^ 0' - &c.; 



1.2^ 1.2.3^ 1.2.3.4'^ 



.-. A'O' = s, = J PC/. d(p = fBP.d<p, beginning from (p = at A' ; 



^ 1.2.3 1.2^ 

 In like manner, if A'P" — t^, A'O" = Sj, &c. 



<2 = <i? + d) + &c. 



1.2^ 1.2.3.4^ 



So = fcod) d)' d)* + &c. 



^ ^ 1.2.3^ 1 ... 5^ 



^3 = A W>» h d,4 _ ^. ,6 ^ g^^ 



1.2^ 1.2.3.4^ 1 ... 6^ 



s, = 63d) '— (b' + — — d)' + — (b- + &.C. 



' ^ 1.2.3^ 1 ... 5 ^ 1 ... 7 ^ 



«„ = 6n<^ - -^^^^ 0' + -^ 0" - &c. ± ^ 0="+' ± &c. 



^ 1.2.3^ l...w^ l...(2n + l)^ 



Now as re becomes larger, the terms in a,, a^, &c. which have for denominators the factorials 

 1 .2.3 ... (w - 1) &c. become smaller and smaller, and thus the arc s„ depends less and less upon 

 the form of the original arc AOB. Hence we may ultimately omit those terms. 



Of the arcs <i, 4, ... ^„, each vanishes when (p = 0; and when (p = ma, they become respectively 

 6,, 62, ... b„. Hence, by the expression for t„, ultimately 



rn'o." . m*a , m'a' _ 



b„ = b„_, 6„_, + o„_3 -. - &c. 



1.2 1.2.3.4 1 ... 6 



This expresses a relation among the successive arcs, 6,, bs, 63 ... b„, which relation, it appears, 

 is ultimately independent of the form of the curve AB. But since a is a quadrant, and cos « = 0, 

 we have 



