RoGKKS — Orthogonal Trajectories of a Congruence of Curves. 93 



surfaces are not confined. Thirdly, the conception of a one-parameter family of 

 surfaces is replaced by the more general conception of a family of curves 

 (integral curves) satisfying the Pfaffian equation 



Idx + mcly + ndz = 0,* 



where l,m,n are funetionsf of xyz, and the condition of integrability is not 

 necessarily satisfied. 



This family of curves consists of the orthogonal trajectories of the 

 congruence of curves defined by 



dx dy dz 

 I m n 



The latter are illustrated by lines of force, lines of ilow, lines of displacement, 

 etc.; in some cases they might be defined by means of surfaces 



/, {x, y, z, a, h) = 0, ,/i {x, y, z,a,h) =<) 

 where a, b are variable constants, and the forms of fi and /a are determined. 



JVominal definitions. 



The family of integral curves of the equatrou of the form 



Idx + mdy + ndz = 



will be referred to as 11 or a n-family. When /,., m,-, n,. are used, the family 

 will be described as n,-. 



In this paper it is assumed that I- + ni^ + %• = 1, i.e. /, m, n are actual 

 direction-cosines. 



* A singly infinite system of curves of the family lies on any arbitrary surface (see Lie and 

 Scheilers, Gtometrie der Beruhrungstiansfonnationen, 1896, Band i, p. 203, or Forsyth's Oiffercnlial 

 Eqimtioiis, Arts. 150 ff.). This property may be regarded as a geometrical representation of the family ; 

 but a more inward geometrical representation — independent of arbitrary surfaces — is given by 

 means of what are hereafter described as normal curves. 



The pure differential method of treating the Pfaffian equation appears to have been used first by 

 Voss {Malhematische Annakn, xvi, 1880, p. 556, and xxiii, 1884, pp. 45, 359) and afterwards with 

 considerable variations by Lilienthal {ib. xxxii, 1883, p. 545), who also investigates the shortest 

 integral curve of a Pfaffian equation {ib. lii, 1899). Both discuss in different ways the normal 

 curvature of curves of the family (cf. D, below) as well as many other matters not referred to in this 

 paper, the leading idea of which is the use of normal torsion, and its relation to normal curvature. 



Lie has recognized that the general Monge equation, /{a.-, y, s, dx, dy, dz) = 0, has a geometry 

 of its own {Leipa. Berichte, 1, 1898, also Math. Ann., lis, 1904, p. 299) ; but he does not appear to 

 have published much on the subject, except in reference to the special case of complexes of right 

 lines. 



t It will be assumed that, if .•!; + .t', y + </', a + a' be substituted for x, y, z, the corresponding 

 values of I, m, n can be expressed by means of Taylor's theorem to any required degree of accuracy in 

 the neighbourliDod of the points x., ij, i. 



[13*] 



