584 GEOMETRY 



importance is due to the fact that they vanish only when the trans- 

 formation is a rigid displacement, so that two deformations have 

 the same components when, and only when, they differ by a dis- 

 placement. The case where the components are constants leads to 

 the homogeneous deformation (or affine transformation of the geo- 

 meters), the type considered almost exclusively in the usual dis- 

 cussions of elasticity. It would seem desirable to study in detail 

 the next case which presents itself, namely, that in which the com- 

 ponents are linear functions of x, y, z. 



In the general deformation, the six components are not inde- 

 pendent, but are connected by nine differential equations analogous 

 to those of Codazzi. The fact that a transformation is defined by 

 three independent functions indicates, however, that there should be 

 only three distinct relations between the components. This means 

 that the nine equations of condition which occur in the standard 

 theory are themselves interdependent; but their relations (analogous 

 to syzygies among syzygies in the algebra of forms) do not appear 

 to have been worked out. 



3. Vector 'fields. From its beginning in the Faraday-Maxwell 

 theory of electricity until the present day, the course which the 

 discussion of vector fields has followed has been guided almost 

 entirely by external considerations, namely, the physical applications. 

 While this is advantageous in many respects, it cannot be denied 

 that it has led to lack of symmetry and generality. The time seems 

 to be ripe for a more systematic mathematical development. The 

 vector field deserves to be introduced as a standard form into geo- 

 metry. 



Abstractly, such a field is equivalent to a point transformation of 

 space, since each is represented by three scalar relations in six variables. 

 Instead of taking these variables as the coordinates of corresponding 

 points, it is more convenient to consider three as the coordinates 

 x, y, z of a particle and the other three as components u, v, w of its 

 velocity; we thus picture the set of functional relations by means 

 of the steady motion of a hypothetical space-filling fluid. This image 

 should be of service even in abstract analysis; for its role is analogous 

 to that of the curve in dealing with a single relation between two 

 variables. The streaming of a material fluid is, of course, not suffi- 

 ciently general for such a purpose, since, in virtue of the equation of 

 continuity, it images only a particular class of vector fields. 



In addition to the ordinary vector fields, physics makes use of 

 so-called hypervector fields, which, geometrically, lead to configur- 

 ations consisting of a triply infinite system of quadric surfaces, one 

 for each point of space. In the special case of interest in hydro- 

 dynamics (irrotational motion), the configuration simplifies in that 

 the quadrics are ellipsoids about the corresponding points as centres. 



