PRESENT PROBLEMS OF GEOMETRY 581 



but if three pencils are given, it is necessary to distinguish the case 

 where the three base points are in a straight line from the case where 

 they are not so situated. We thus have two protectively distinct 

 cases, which may be represented canonically by: (1) z=const., 

 y=const., x+y=const., and (2) z=const., y = const., y/x=const. 

 The first type may, however, be converted into the second by the 

 transcendental transformation x l =e c , ?/ 1 =e y , so that, in the general 

 group of point transformations, all sets of three pencils are equivalent. 

 The discussion for four or more pencils yields the rather surprising 

 result that the protective classification remains valid for the larger 

 group. 



Dropping these special considerations on algebraic systems, let us 

 pass to the theory of arbitrary systems of curves, or, what is equiva- 

 lent, the geometry of differential equations. While belonging to the 

 cycle of theories due primarily to Sophus Lie, it has received little 

 development in the purely geometric direction. Most attention has 

 been devoted to special classes of differential equations with respect 

 to special groups of transformations. Thus there is an extensive 

 theory of the homogeneous linear equations with respect to the 

 group x l =$(x), y 1 =yi](x) which leaves the entire class invariant. 1 

 A special theory which deserves development is that of equations of 

 the first order with respect to the infinite group of conformal trans- 

 formations. 



As regards the general group of all point transformations, all 

 equations of the first order are equivalent, so that the first case of 

 interest is the theory of the two-parameter systems. The invariants 

 of the differential equation of second order have been discussed 

 most completely in the prize essay of A. Tresse (submitted to the 

 Jablonowski Gesellschaft in 1896), with application to the equiva- 

 lence problem. A specially important class, treated earlier by Lie 

 and R. Liouville, consists of the equations of cubic type 



y" = Ay' 3 +By' 2 +Cy' +D, 



where the coefficients are functions of x, y. It includes, in particular, 

 the general linear system and all systems capable of representing 

 the geodesies of any surface. While the analytical conditions which 

 characterize these subclasses are known, little advance has been 

 made in their geometric interpretation. 



Perhaps the simplest configuration belonging to the field considered, 

 that is, having properties invariant under all point transformations, 

 is that composed of three simply infinite systems of curves, which 

 may be represented analytically by an equation of third degree in 

 y' with one-valued functions of x, y for coefficients. In the case of 

 equations of the fourth and higher degree in y', certain invariants 



1 The elementary (metric) theory of curve systems has been too much neglected ; 

 it may be compared in interest and extent with the usual theory of surfaces. 



