1 08 Procpedings of the Royal Irish Academy. 



(h) The corresponding angles for the two directions of zero torsion are 

 given by 



1 1 









sin 29' 



T\ r^ I 



2</>', 







11 i) '°" 













T2 Ti 













T; D+I 







(") 



The curvatui 



•es for the 



1 r j_ 



directions of zero torsion 

 1 



are 







-P). 







.,JLy 



'- - i- {-J ± V -D-- 





(rf) 



The 



torsions 



for the directions of zero curv 



ature 



are 











1 







the last being a generalization of Enneper's formula,* to which it reduces 

 when 7=0. 



(«) To fiiid the radiics of mirvature of any curve of the U-family whose 

 normal is given, we use a theoi'em which is equivalent to Meunier's for 

 surfaces,! and is proved in the same way. The curvature of any Il-curve is 

 equal to the curvature of the normal curve having the same tangent line, 

 multiplied by sec <}>, where f is the angle between the normal to the Il-family 

 and the principal normal of the curve. 



Using the general notation of p. 97 and p. 102, and denoting the corre- 

 sponding elements of tlie non-normal curve by adding the suffix 1, we have 



cos </) = «, + ;»>»., + wit = pAl-x- + ni -j- + n -^ 1 



/ dl dm dn\ 



'^ \ dsi ^ rf.Si ' dsj 

 since la^ + wi/8, + ny^ = at all points of the curve, and at the point 

 considered a = ai, /3 = /3i, 7 = ji. Now /, m, n are functions of x, y, z; 

 therefore, if 'p is equal to /, m, or n, 



dp dp dx dp dy dp dz dp> dp dp dp 



dsi ~ dx dsi dy ds^ dz dsi dx ' dy ' ^ dz~ ds 



Hence the coefficient of p, is 



dl ,, dm. dn\ 1 



* Salmon, op. cit., p. 426. 



t Lie extends Meunier's llieoiem to iiny Moiige equation /(.r, i/, z, it.i-, dy, dz). Lnipz. Berichte, 

 •iO, 1898. 



