94 Proceedings of the Royal Irish Academij. 



The line whose direction-cosines are I, m, n, drawn through the point P 

 (x, y, z) may be termed the normal, and the perpendicular plane through P 

 the tangent-jdaMC, to the Il-family at P. 



The angle of intersection at P of two families Up, Dg is defined as the angle 

 between the normals, and is given by 



± cos Qpj = Iplq + mpVij + iipn, 



A cvjrve of intersection ol two Il-families signifies an integral curve common 

 to both. These curves form a curve-congruence, one passing through each 

 point of space, the dii-ection-cosines (a, /3, 7) of the tangent line being 

 determined by 



Ipa + nip^ + npy = 0, l^a + m^^ + nqy = 0. 



A cm've whose principal normal at any point is the normal to 11 at the 

 same point will be tinned a normal curve of the family n. It is evidently a 

 member of the family. 



The normal toi'sion on 11 of any integral curve of n will signify the 

 torsion of the normal ciu've of 11 ha^nng the same tangent line at the point. 

 The normal curw.tv.Te is similarly defined. 



The normal toi'sion on lip at P of its curve of intersection (through P) 



with Ilj will be denoted by — • 



fdn dvi\ ■ fdl dn\ ( dmi dl 



\dy d,z j \d.z dx) \ dx dy 



For a reason which wUl appear, \ I will be termed the tnmn iarsion of 11 at 

 the point P, or the mean torsion of the normal curves. In tlie language used 

 in applied mathematics it is the magnitude of the component, along the 

 direction of the unit-vector (I, m, n) of the curl of the vector. 



_ , ^ dl dm dyn 



Let «/ 3 r=- + -;— + -;- • 



d.x d.y dz 



It will be seen that \J is fitly described as the mean curratv/re of II at the 

 point P. It is the ' divergence ' of the unit-vector {I, m, n) at the point. 



Noiinal Curves. 



A family n is geometrically determined by the orthogonal curve-congruence 



^ = -^ = — • But a more inward representation is by means of the curve- 

 l m n 



complex consisting of the normal curves. Any integral cui-ve of 11 is the 



envelope of some assemblage of normal cm-ves, and in this way the normal 



curves represent the whole family. 



