104 



Proceedings of the Royal Irish Academy. 



The IndAcoArix of Torsion. 



By the Frenet-Serret foiiuulae 



r 



TJsiDg the special axes, for which 



a = COS a, |3 = siu ff, 7 = 0, / = III = 0, n = 1, 

 this gives 



= 0... sm-,Q - 6, cos- 9 + (ai - h) sin d cos 9. 





a 



P 



7 



dl dm dn 

 ds ^ '' ds ^ '• d, - 



I 



dl 



in 

 dm 



n 

 dn 





ds 



ds 



ds 



I = 



hi - «2, 



Tlie conic aif - hijr + {a.., - i^) j.->j = const, (with its conjugate if a 

 hyperbola) may be named the indicatrix of torsion, since the normal toi-sion 

 in any direction is proportional to the square of the corresponding radins 

 vector of the conic. 



Since, for the axes used, 



dm dl 

 dx d]j 



tlie condition that the curves of thefo.mily II may lie on a one-pcLrameter family 

 of surfaces is equivalent to the condition that the imlieatrix of torsion may he am 

 equilateral hypcrhola. 



If the indicatrix of torsion be referred to its axes, it may be written 



— + ^^— = const. 



Ti -2 



where — and — are limiting torsions, or principal torsions at the point. 



There are two directions of zero normal torsion at each point, correspoudiug 

 to the asymptot€s of the indicatrix of torsion. These diiections may be real 

 or imaginary according as the toi-sional indicatrix is an hyperbola or ellipse. 

 The curves along which the noimal torsion is zero form a congi-uence, two 

 passing through each point. 



These ciu'S'es are a second, fjeneralization of lines of curxoiure* with which 

 they coincide when 7=0. But in general they do not cut at right angles and 

 they may be imaginary. 



• Voss {Math. Ann., zsiii, 1884, p. 70, ff.) calls the curves of extreme curvature ' Hauptkriim- 

 mungslinien ' and the curves of zero torsion ' Kriimmungslinien,' but he does not use directly the 

 conception of torsion. 



