664 Proceedings of the Royal Irish Academy. 



virtual foci"' are situated upon every curve. The loci of virtual foci 

 are different for the different congruencies. 



III. — The congruencies in general are of the same class (M) and of 

 the same rank (E.) if we agree to represent the class of a curvilinear 

 cougruency by the number of its curves which osculate a given plane, 

 and to denote by its rank the number of tangents which can be drawn 

 to its curves so as to pass through a given point and to lie in a given 

 plane. 



IV. — Taking any small pencil of curves of a congruency it is possible 

 to draw through any point in the pencil a determinate element of sur- 

 face ultimately normal to all the constituent curves. The measure of 

 curvature, or the product of the principal curvatures of the element, 

 represents the characteristic known as the Density of the congruency, 

 or preferably of the pencil, at the point. Double the mean curvature 

 or the sum of the principal curvatures seems to deserve the name 

 Concentration of the pencil. It may also be described as the con- 

 vergence of the directions of the curves, that is >Sv ^^ "^here Ut is a 

 unit vector tangent to a curve of the pencil at the point Or, again, 

 the name is justified because it is proved that the concentration is 

 the coefficient of contraction of the normal cross-section as we pass 

 along the pencil. Related to a congruency we have in general sur- 

 faces of zero density and surfaces of zero concentration. 



V. — It is generally possible to determine one or more surfaces ortho- 

 gonal to all the curves of a congruency. Some curious relations 

 connect the various surfaces mentioned, for instance tlie locus of 

 Virtual Poci and the locus of Zero Density touch one another along a 

 curve situated upon this orthogonal surface. 



VI. — The transformation must obey certain conditions whenever one 

 of the transformed congruencies is orthogonal to a family of surfaces. 

 In fact it is shown that the system of parallel lines in the region 

 (w, V, w) must be parallel to an edge of a certain quadric cone, or that 



* In general, selecting any point on any assumed curve of a congruency it is 

 possible to find two adjacent curves and two adjacent points on the cuiwes, so tliat 

 the lines joining these points to the assiuned point are at right angles to the curves 

 through their extremities. "When these two lines coincide the assumed point is 

 said to he a virtual focus in analogy with the definition of a virtual focus of a 

 rectilinear congruency. 



