96 Proceedings of the Roj/al Irish Academy. 



that a 'doubly' orthogonal system of surfaces should belong to the same 

 triply orthogonal system, is that the curves of intersection of any two 

 surfaces of the two different families should always be lines of curvature on 

 either surface. 



Joachimsthal's theorem : — If two surfaces cut at a constant angle, their 

 curve of intersection is a line of curvature on both or on neither ; and if the 

 curve of intersection of two surfaces is a line of curvature on both, they cut 

 at a constant angle. 



By the principles expressed in the following simple formulae, these 

 theorems arc snvivied up, and extended to cdl triads or pears of one-parameter 

 families of surfaces, emd further, to all triads or pairs of H-families : — 



l_l = ±i!^', ■ (1) 



- + -=/.> (2) 



Tp Op 



where Spq signifies the arc of the curve of intersection (through F) of V\p and 

 n,y, — and — represent the norvicd torsions on Tip for any two orthogonal 



Tp tTp 



curves of the family n^,, intersecting at P, and the other letters have the 

 meanings explained on p. 94. 



Before proving these formulae we may deduce the special theorems 

 mentioned. 



Joachimsthal's theorem follows at once from (1) ; for if dpq is constant, 



then — = — , and therefore both or neither of these normal torsions vanish 



Tpq Tqp 



at every point on the curve. And if both of those normal torsions vanish 

 (or are equal) along a curve of intersection, the Il-families intersect at a 

 constant angle along this curve. When Ip and Ig are both zero, 11^, and U,j 

 represent one-parameter families of surfaces, and normal torsion becomes 

 geodesic torsion, which vanishes for a line of curvature. But it will be 

 noticed that the disappearance of Ip and /, is irrelevant to the proof, so that 

 the correct statement of Joachimsthal's theorem even in its limited form 

 is — If two n-families intersect at a constant angle along any curve, the 

 normal torsions of this curve vanish on both or on neither, and if the normal 

 torsions are equal, the curves intersect at a constant angle. 



Dupin's theorem is equivalent to the statement that if Ii = I2 = Iz = 0, 



and 023 = fl,„ = 9y, = -, then the six torsions — , — , — , - , - , — vanish, 



2 r^s T32 T31 Ti3 Tlj Tji 



