98 



Proceedings of the Royal Irish Academji. 



Now the direction-cosines of the tangent-line to a curve common to 

 the families IIi and lis are 



sin ' sin ^ ' sin ' 

 where [riiiih] = nhni - iihih, and 6 = ^,2, s = s^, and the signs are fixed by 

 convention. 

 Thus 



sin0 



'•12 



h 



m. 



h 



/, 



chh 

 Is 



nil 



cJli chill 

 els ds 



chix 

 h ' 



In like manner, since a, /3, 7 have the same value as before, and ?,, 7h,, «, 

 are interchanged with l^, m^, n^ in the rest of the expression for torsion, 



dl. 



dm, 



= h — T~ + llh —rr- + il^ , 



ds ds ds 



sin dU dm^ dn^ 



To. ds ds ds 



Hence 

 1 



Tn 



sinfl 



Therefore 



d ,, , , "■ , n\ • /> dv 



-=- (Ixh + riii'in-i + 'ihrh) = + -y- (cos Q) = + sm ^— 

 ds ds ' ds 



— = + — r-^ , the suffixes being restored. 

 r-.i dSi. 



This formula may be verified by spherical representation. Let T, Ni, Bi 

 and T, Ni, B2 be the points on the unit-sphere representing the common- 

 tangent line, the normals, and the bi-normals of the normal curves on 

 the two n-families. Let iV,', Bi and N^', B.^ be ' consecutive ' positions. 

 The points NiN^iB^B^ lie on the same great circle. The preceding equation 

 is now equivalent to pro\diig 



N.Ni' - n^n; = BiB: ~ b,b:. 



This will be found to follow from the fact that the angles Bi'B^B^ and Bz'BzB,, 

 or their supplements, are small quantities of the first order. 



Proof of Equation (2). 

 The torsion for any curve (C) is given as before by 



1 dl dm dn 



T ds ds ds 



where 



= X («ia + aj/S + ttsj) + fJ. (bia + hfi + bsy) + v (cia + ftjjS + Csy) 



dl dl , d7)i dn 



«! = — , ch = -^, Oi = y-, Ci = -J-, etc. 



aa; dy dx dx 



