PRESENT PROBLEMS OF GEOMETRY 583 



with the question whether an arbitrary transformation belongs to 

 a continuous group. The problem deserves treatment not merely for 

 the analytic transformations, but also for the algebraic and for 

 the continuous transformations. 1 



Aside from such fundamental questions, further development 

 is desirable both in the study of the general properties (associated 

 curve systems and contact relations) of an arbitrary transforma- 

 tion, and in the introduction of new special types of transformation, 

 for instance, those which may be regarded as natural extensions of 

 familiar types. 



The main problems in the theory of point transformation are 

 connected with certain fields of application which we now pass in 

 review. 



1. Cartography. A map may be regarded, abstractly, as the point 

 by point representation of one surface upon another, the case of 

 especial practical importance being, of course, the representation of 

 a spherical or spheroidal surface upon the plane. As it is impossible 

 to map any but the developable surfaces without distortion upon a 

 plane, the chief types of available representation are characterized 

 by the in variance of certain elements, as angles or #reas, or the 

 simple depiction of certain curves, as of geodesies by straight lines. 

 Most attention has been devoted to the conformal type, but the 

 question proposed by Gauss remains unsolved: what is the best 

 conformal representation of a given surface on the plane, that is, 

 the one accompanied by the minimum distortion? The answer, of 

 course, depends on the criterion adopted for measuring the degree 

 of distortion, and it is in this direction that progress is to be 

 expected. 



2. Mathematical theory of elasticity. As a geometric foundation 

 for the mechanics of continua, it is necessary to study the most 

 general deformation of space, defined say by putting x i} y l} Zi equal 

 to arbitrary functions of x, y, z. The most elegant analytical repre- 

 sentation, as given for instance in the memoir of E. and F. Cosserat 

 (Annales de Toulouse, volume 10), is obtained by introducing the 

 elements of length ds and ds l before and after deformation, and the 

 related quadratic differential form ds* ds 2 =2e l dx 2 +2e 2 dy 2 +2e 3 dz 2 

 + 2y l dydz + 2y 2 dxdz + 2y 3 dxdy. The theory is thus seen to be ana- 

 logous to though of course more complicated than the usual theory of 

 surfaces. The six functions of x, y, z which appear as coefficients 

 in this form are termed the components of the deformation. Their 



1 This problem is not to be confused with the similar (but simpler) question 

 connected with Lie's division of (analytic) groups into demokratisch and aristo- 

 kratisch. In those of the first kind all the infinitesimal transformations are 

 equivalent, in those of the second there exist non-equivalent infinitesimal trans- 

 formations. Lie shows that all finite groups are aristokratisch, while the groups 

 of all (analytic) point and contact transformations are demokratisch. Cf. Leip- 

 ziger Berichte, vol. XLVII (1895), p. 271. 



