288 LIPKA. 



general problem of dynamics; using the geometric language of natural 

 families, this theorem may be stated as follows: 



The 00 "-1 curves of a natural family which pass out normally to any 

 hypersurface, F„_i, form a normal hypercongruence, i. c. admit of oo' 



normal hypcrsurfaces, lohich are the loci of eqtial values of j F ds 



measured from any one of the system of hypcrsurfaces. 



This theorem is true for space of any dimensionality or any curva- 

 ture.^ 



Thomson and Tait * had given this theorem for dynamical trajec- 

 tories in a conservative field of force in a euclidean space of three 

 dimensions. We quote: "If from all points on an arbitrary surface, 

 particles not mutually influencing one another be projected with the 

 proper velocities [so as to make the sum of the kinetic and potential 

 energies have a given value] in the direction of the normals; points 

 which they reach with equal actions lie on a surface cutting the paths 

 at right angles." The ooi orthogonal surfaces appear as surfaces of 

 equal action. A similar theorem may be stated using the language 

 of brachistochrones, or of optical light paths; in these cases the ooi 

 orthogonal surfaces appear as surfaces of equal time. 



The question arises whether the orthogonal properties stated above 

 are characteristic of the trajectories under the principle of least action, 

 or the brachistochrones, or the optical light waves, or general cate- 

 naries, or, in short, of ariy natural family? In other words, does the 

 orthogonal property belong exclusively to natural families of curves? 

 This question may be answered in the affirmative. Using geometric 

 language we may state the theorem : 



// a system of co2(»-i) curves in space of n dimensions {euclidean or 

 curved) is such thai oo"-i curves of the system meet an arbitrary hyper- 

 surface {space of n —1 dimensions) orthogonally, and always form a 

 normal hypercongruence, i. e. admit of c»i orthogoncd hypcrsurfaces, the 

 system is of the natural type. 



Thus the system may be considered as any one of the types, dynami- 

 cal or optical, discussed above. This converse of the Lipschitz theorem 

 or of a generalized Thomson and Tait theorem was first proved for 



3 For a detailed discussion of these theorems and of the general problem of 

 dynamics, see G. Darboux, Lei^ons siir la th6orie g($n6rale des surfaces, vol. 2, 

 chapt. 8. See also, P. Appell, Traite de mecanique rationelle, vol. 2, chapt. 25. 



4 Thomson and Tait, Treatise ott, Natural Philosophic, vol. 1, part 1, p. 353, 

 1879. 



