106 Proceedings of the Royal Irish Academy. 



extreme curvature, and is equal to Imlf the magnitude of the comiionent, along tlie 

 direction of tlie unit vector I, m, n, of the ' curl ' of the vector. 



Let the axes of the torsional indicatrix, i.e. of the directions of extreme 

 torsion, be taken for axes of r and y. Let ti and n be the extreme or 



principal radii of torsion. For these axes we have is -«! = —;-—,= 0, 



and if we put 



^ dl dm ,1111 



ax ay p, p-z p, ,0, 



we have, replacmg 6' by tp, 



1 cos- d) sin-d) 



- = ^ + ^ (o 



1 1 ,- /'I 1 \ • 



— = iJ + sni d» cos (h. (6) 



P ' yr-i T,J 



Since for the axes used — = 0, the quantity J is equal to the ' divergence' 



of the unit-vector (/, ///, n), i.e. for general axes, 



dl dm dn 

 d.r dy dz 



The sum of the ciu'vatures at any point for two perpendicular directions 

 is constant ami is equal to ./. Thus 5 J may be named the mean curvature at 

 the point ; the mean curvature is equal to the curvature for either direction of 

 txtreme torsion, and is equal to half the divergence of the unit vector I, m, n. 



The extreme torsions and the extreme curvatures are not independent, 

 and the relation between them may be expressed by saying that the 

 ' deviation ' of torsion is equal to the ' deviation ' of curvature, ' deviation ' being 

 taken to mean the difference between the mean and either extreme. For it 



is easy to see, from (4) or (6), by linding the limiting values of — or — , 



T p 



that 



r, -2 \pi p. 



Also the directions of extreme curvature and of extrejiie toraion bisect 

 eacli other. For 



'1=1+ (- --]sm29\ 



T \pi pij 



o 



therefore the limiting values of r are given by 6 , 



