54 Proceedings of the Royal Irish Academy. 



Curvature of a Complea: Curve. 



39. If we write a = S-, k"' = a"-, ec^uation (21) becomes 8 = ay-i. 

 Now dyjcds = ya = - dyzjads, 



c 



dyi d 



hence -=--!.'=-« t^v/I - T^' - Vs"- 



(T dy3 dS ' ' 



Again ay, = - dS/ads - - dS/ds, where de = ads = E''ds = the elementary- 

 angle of torsion. Hence 



d 



r-|j--(SJ-«-. 



where S' = a, di - ads. 



Hence the curvature is a given function of the toi'sion arid of its first tuv 

 dAfferential coefficients ivith respect to the arc. 



Applications. 1. Curves of Constant Torsion. 



40. The right-hand member of the equation (22) is a function of 

 S, d^jds, and d^ljd.f. Hence if S is constant, so is c. The only curves of 

 constant torsion which belong to a linear complex are circular lielices. 



2. Helices on a Cylinder of any form. 



41. For these eja = const. = h. Integrating (22), we get 



= -J«^-a^- 



6S+ J, =_ /^. .ga./'^V 



\,di 



f^J =a'-S'-{bS + hy = h' [A, - S) (8 - B,), 



where 7i- = 1 + 6- = (c^ + CT=)/(T^ and Ai and Bi are constants. 



Let us now take the s-axis parallel to the generators of the cylinder. If 

 dxp is the angle of contingence of the section of the cyHnder by « = const., 

 we get 



dxfj = ^c- + (7^ ds = had.s = hcU. 



The cUfferential equation becomes therefore 



d^ = x/(3i - Sj {B - B,) d-4,. 



Let 2S = ^1 + ^1 - {Ai - B,)eosd; we get dd = rfi/- ; and we can 

 ake = i//. 



