PRESENT PROBLEMS OF GEOMETRY 573 



responding in hyperspace to the simpler systems of invariants. (5) 

 Complete systems of orthogonal or metric invariants for the simpler 

 curves. 1 



Transcendental Curves 



To reduce to systematic order the chaos of non-algebraic curves 

 has been the aspiration of many a mathematician; but, despite all 

 efforts, we have no theory comparable with that of algebraic curves. 

 The very vagueness and apparent hopelessness of the question is 

 apt to repel the modern mathematician, to cause him to return to 

 the more familiar field. The resulting concentration has led to the 

 powerful methods, already referred to, for studying algebraic varie- 

 ties. In the transcendental domain, on the other hand, we have a 

 multitude of interesting but particular geometric forms, --some 

 suggested by mechanics and physics, others derived from their relation 

 to algebraic curves, or by the interpretation of analytic results - 

 a few thousands of which have been considered of sufficient importance 

 to deserve specific names. 2 The problem at issue is then a practical 

 one (comparable with corresponding discussions in natural history) : 

 to formulate a principle of classification which will apply, not to all 

 possible curves, but to as many as possible of the usual important 

 transcendental curves. 



The most fruitful suggestion hitherto applied has come from 

 the consideration of differential equations : almost all the important 

 transcendental curves satisfy algebraic differential equations, and 

 these in the great majority of cases are of the first order. Hence the 

 need of a systematic discussion of the curves defined by any algebraic 

 equation F(x, y, y'~) = 0, the so-called panalgebraic curves of Loria. If 

 F is of degree n in y' and of degree v in x, y, the curve is said to belong 

 to a system with the characteristics (n, v], and we thus have an im- 

 portant basis for classification. Closely related is the theory of the 

 Clebsch connex; this figure, it is true, is considered as belonging to 

 algebraic geometry, but it defines (by means of its principal coinci- 

 dence) a system of usually transcendental panalgebraic curves. 



Both points of view appear to characterize certain systems of 

 curves rather than individual curves. The following interpretation 

 may serve as a simple geometric definition of the curves considered. 



With any plane curve C we may associate a space curve in this 

 way: at each point of C erect a perpendicular to the plane whose 

 length represents the slope of the curve at that point; the locus of 

 the end points of these perpendiculars is the associated space curve 



1 Here would belong in particular the theory of algebraic curves based on link- 

 ages. Little advance has been made beyond the existence theorems of Kempe 

 and Koenigs. An important unsolved problem is the determination of the link- 

 age with minimum number of pieces by which a given curve can be described. 



2 Cf. Loria, Spezielle Kurven, Leipzig, 1902. 



