582 GEOMETRY 



may be found immediately from the fact that when x and y undergo 

 an arbitrary transformation, the derivative y' undergoes a fractional 

 linear transformation (of special type). The invariants found from 

 this algebraic principle are, however, in a sense, trivial, and the real 

 problem remains almost untouched: to determine the essential 

 invariants due to the differential relations connecting the coefficients 

 in the linear transformation of the derivative. 



General Theory of Transformations 



Closely connected with the geometry of differential equations 

 that we have been considering is the geometry of point transform- 

 ations. In the former theory the transformations enter only as 

 instruments, in the latter these instruments are made the subject- 

 matter of the investigation. The distinction is parallel to that which 

 occurs in projective geometry between the theory of projective 

 properties of curves and surfaces and the properties of collineations. 

 (It may be remarked, however, that although a transformation is 

 generally regarded as dynamic and a configuration as static, the 

 distinction is not at all essential. Thus a point transformation or 

 correspondence between the points of a plane may be viewed as 

 simply a double infinity of point pairs; on the other hand, a curve 

 in the plane may be regarded as the equivalent of a correspondence 

 between the points of two straight lines. 1 ) 



We consider first two problems concerning the general (analytic) 

 point transformation which are of interest and importance from the 

 theoretic standpoint. The one relates to the discussion of the char- 

 acter of such a transformation in the neighborhood of a given point. 

 Transon's theorem states that the effect of any analytic transform- 

 ation upon an infinitesimal region is the same as that of a pro- 

 jective transformation. This is true, however, only in general; it 

 ceases to hold when the derivatives of the defining functions vanish 

 at the point considered. What is the character of the transformation 

 in the neighborhood of such singular points ? 



A more fundamental problem relates to the theory of equiva- 

 lence. Consider a transformation T which puts in correspondence 

 the points P and Q of a plane. Let the entire plane be subjected to 

 a transformation S which converts P into P r and Q into Q'. We thus 

 obtain a new transformation T' in which P' and Q' are corresponding 

 points. This is termed the transform of T by means of S, the relation 

 being expressed symbolically by T' =S~ 1 TS. The question then arises 

 whether all transformations are equivalent, that is, can any one be 

 converted into any other in the manner defined. The answer de- 

 pends on certain functional equations which also arise in connection 



1 Geometry on a straight line, in its entirety, is as rich as geometry in a plane 

 or in space of any number of dimensions. 



