RoGKRS — Orthogonal Trajectories of a Congruence of Curves. 95 



The normal curves are evidently generalized geodesies, since they become 

 geodesies on surfaces of a one-parameter family of surfaces when the condition 

 of integration / = is satisfied. In fact a normal curve is an integral curve 

 whose osculating plane contains the normal to 11 at each point. If a string 

 under constant tension is in equilibrium in a field of force X, Y, Z, where 

 X : Y : Z = I : vi : n, it must lie along a normal curve, and if a particle can 

 move with constant velocity in such a field, its path is a normal curve. 



The normal curves are simply curves whose principal normals {I, m, n) are 

 given functions of x, y, z. They form a curve complex in the sense that those 

 passing through a point form a singly infinite system (and therefore generate 

 a surface associated with the point), just as the lines of a complex of right 

 lines through a point generate a cone. To prove that these curves form a 

 complex, it is only necessary to show that a normal curve through a point P 

 is, in general, uniquely determined if its tangent line at P is known. 



This can be proved by means of the Frenet-Serret formulae.* a, /3, 7, 



I, m, n, being Imown at P, -j is known since 



dl _ dl ^ dl dl 



ds dx dy ' dz' "' 



and I, m, 11 are given functions of x, y, z. Using the formulae it will be found 

 that — , — , and all subsequent diflereutials are known. Therefore, 



since a = ^-, /3 = -7, 7 = -r, x.ii.z and all their differential coefficients 

 dJ ^ ds ' ds '' 



are known. But we may assume that there camiot be two different con- 

 tinuous functions of s, possessing the same differential coefficients and the 

 same value, for the same value of s. Hence, x, y, z are definite functions of s, 

 and therefore the curve is unique. 



B. GENEKALIZATIO^^S OF THEOREMS OF DUPIN, Da]{BOUX, AJ^D JoACHDISTHAL. 



The special theorems referred to are — 



Dupin's theorem : — The curves of intersection of any two sirrfaces of 

 different families of a triply orthogonal system are lines of curvature on 

 these surfaces. 



Darboux's reciprocal theoremf : — The necessary and sufficient condition 



• Salmon's " Geometrj' of Three Dimensions," tilth edition, vol. i, p. 3S7. 

 t Davhons, " I,e(;ons siir les SystcniPS Orthog"naiix,'' j 6. 



