XLIX. Of the Intrinsic Equation of a Curve*, and its Application. 

 By W. Whewell, D.D., Master of Trinity College, Cambridge. 



[Read February 12, 1849.] 



1. Mathematicians are aware how complex and intractable are generally the expressions 

 for the lengths of curves referred to rectilinear coordinates, and also the determinations of their 

 involutes and evolutes. It appears a natural reflexion to make, that this complexity arises in a 

 considerable degree from the introduction into the investigation of the reference to the rectilinear 

 coordinates (which are extrinsic lines) ; the properties of the curve lines with relation to these 

 straight lines are something entirely extraneous, and additional with respect to the properties of 

 the curves themselves, their involutes and evolutes ; and the algebraical representation of the former 

 class of properties may be very intricate and cumbrous, while there may exist some very simple 

 and manageable expression of the properties of the curves when freed from these extraneous append- 

 ages. These considerations have led me to consider what would be the result if curves were 

 expressed by means of a relation between two simple and intrinsic elements, tlie length of the curve 

 and the angle through which it bends : and as this mode of expressing a curve much simplifies the 

 solution of several problems, I shall state some of its consequences. 



2. Let s = f {(p), any function of d), when s is the length of the curve, and (p the angle of 

 deviation of the tangent from the tangent at the origin. 



Then using the common notation we have ds = f ((p) . dxp. The curve may be constructed 

 npproximately by taking small finite differences o{ (p, and determining the corresponding rectilinear 

 elements of the polygon or approximate curve, by the equation 



As =/' {(p) . A0. See Fig. 1. 



ds 



3. It is evident that is p, the radius of curvature. Hence p =/' (rf)), and the curve mav 



d(p 



also be contracted approximately by taking small finite difPerences of (p, drawing a line perpen- 

 dicular to the curve at first, setting off the value of p, drawing a circular arc to radius p for A0, 

 then setting off p„ and drawing a circular arc with radius p^ for A^i, and so on. See Fig. 2. 



4. The cvolute of a curve is easily found from this equation. For if (Fig. 3) JP be the 

 curve, BQ the cvolute, AP = s, RQ = s', it is evident that (p is the same for both s and .9', if s in 

 IIQ, (p be measured from BA, perpendicular to A, v. 



And QP= QB - BA, or p = s' + C. Hence s =/' (^) " C = ^X " C- 



If the curves have the forms represented in Fig. 3a, the formula- arc nearly similar. 



• After writinK this paper, I found that Kulcr had, in the solu- 

 tion of a particular problem, expressed curves by means of an 

 et|uation between the arc and the radius of curvature. This equa- 

 tion is, as is shown in the paper, the difterential of my "intrinsic 

 f'luation," and has an erjually f^ond right to the name. .My eijua- 



tion being the intcRral of Euler's, has, of course, one more arbitrary 

 constant tllari his. There may very possibly be other modes of 

 cxprerthinjj curves which may l)c fitly described as '' intrinsic eqiui- 

 lions" to the curves. I was not able to iind any other name for the 

 etjuation which I have employed. 



