56 Proceedings of the Royal Irish Academy. 



Curvature arid Torsion at SiTU/vIarUies on a P-Curve. 



43. Taking a point M on the P-curve as origin, the equations of the eiu've 

 in the neighbourhood of M can be written 



X ^ aV -^ a't"*^ + . . . , y ^htfl + . . ., z = ct^ + . . . {p = q + r, q> r). 



If the axes are not rectangular, we may make them so by a transformation 



x = \x + fiy , y = Xy ■¥ fiz, z = z 



without altering the fh-st teims in the series for x, y, z. Suppose this done : 

 then the axes fonn the trihedi'on MTNB akeady described. 

 Curvature. — At a point M, [xyz) on the curve we have 



'^^ = ^ = F dt V 



Let us estimate the order of the terms in this equation when t — >- 0. 

 Clearly /S^ — >- 1 and s — >- x = art"-''- -f . . . Also ijl's is of order 

 {q - 1) - (r - 1) = q - r. The right-hand side is therefore of order 

 q - r - 1) - (/• - I) = q - 2r. This is the order of c. The value of p has 

 not come into the discussion, except that of course it was impheitly supposed 

 greater than q. Hence for any curve, at a point with the characteristic 

 numbers 'p, q, r the infinitesimal order of the cui'vature is q - 2r m terms of t, 

 or qjr - 2 in terms of the arc. 

 Torsion. — We have 



ayz = -^ + 07,, and 73 — ^ I . 



The term cyi on the right is of order {q - 2r) + (p - r) = 2 (g' - r], since 

 p - r = q. Again, yz = c^dyjcis = c^dh/ds-, and is therefore of order 

 2r - q + p - 2r = p - q = r. Hence dyzjds is of order r - r, and is therefore 

 finite, while cy, is zero (g - r > 0). Hence ol every point of a P-curve in the 

 finite portion of spaxe the torsion has a finite Tum-zero value. 



This apphes of course only to real points on a real portion of the curve. 

 It is an extension of a well-known property of cui'ves of a hnear complex 

 dedueible from Lie's expression for the torsion. 



The method by which we have deduced these results seems at fii^st sight 

 open to objection. We have assumed in fact that if a function F {t) is of 

 order v, F" {t} will be of order v - 1 in t. This is only true when v ± 0- 

 Our investigation, however, only requires that this shaU be so for the functions 

 X, y, s, for each of which v > 0; and therefore x, y, "z will each have a leading 

 term of the proper order with a non-zero coefficient even ia the most 

 unfavourable ease where p = B, q= 2 v -= 1. 



