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XXIX. 



SOME PROPERTIES OE THE GENERAL CONGRUENCT 

 OE CURVES. (Abstract.) By CHARLES J. JOLT, M.A., 

 F.T.C.D., Andrews' Professor of Astronomy in the University 

 of Dublin, and Royal Astronomer of Ireland. 



[Read June 26, 1899.] 



I. — The general equation of a congruency of curves may be 

 represented by three equations of the type — ■ 



X = f{u, V, tv), y = g {u, V, to), and z = h {u, v, tv), 



where f, ff, and h are functions of three parameters u, v, and to. Of 

 these parameters two, ti and v, serve to select an individual curve of 

 the doubly infinite system, and to specifies the individual points upon 

 that curve. JN'ow these three equations establish a transformation or 

 correspondence between three variables, w, v, and w, and three others, 

 X, y, and z, and this transformation may be considered as producing a 

 congruency in the region {xy%) from a system of parallel right lines in 

 the region (tivw). Corresponding to any assumed direction of this 

 parallel system we have in the region {xyz) one of a doubly infinite 

 system of congruencies. All these are of the same order, and all have 

 the same focal surface. The order is the number of points in the 

 region (iiviv) which correspond to a given point in the region (xyz), 

 and the focal surface is tlie locus of points in the latter region for 

 V5^hich two of the correspondents in the former region coincide. This 

 focal surface is represented by combining the original equations with 

 the result of equating to zero the Jacobian of x, y, and z with respect 

 to u, V, and tv. 



II. — It is shown, moreover, that every curve belonging to any of the 

 system of congruencies touches the common focal surface in a certain 

 definite number (A) of points, and that a determinate number (B) of 



