222 Dr. H. Bateman 



on 



seem satisfactory for a fundamental principle, and is probably 

 a consequence of some deep underlying principles which are 

 the true equations of motion. These new principles should 

 indicate the reason for a similarity of design of the different 

 electrons. One of the fundamental facts of life is that a good 

 design is copied, and that there is a certain characteristic of the 

 design of an object and its surrounding medium, depending 

 perhaps on the closeness of fit of an imperfect correspondence, 

 which determines the extent to which the design of the object 

 is copied and preserved in the surrounding medium. This 

 may be called the value of the design in relation to the medium, 

 and it is a quantity which I feel must be taken into account 

 in the true equations of motion, and a number assigned to it 

 at each instant. As an example of standardization, the 

 Ford motor-car is not in it with the electron; and, according 

 to the above view, we must regard the design of the electron 

 as one of very great value in relation to the surrounding 

 medium. 



Returning to our generalized scheme of electromagnetic 

 equations, and looking at matters from the point of view of 

 physical optics, it may be remarked that the scheme of consti- 

 tutive relations mentioned above is not sufficiently general to 

 cover the case of a doubly-refracting crystalline medium*. 

 To remedy this defect we may use a biquadratic integral form 

 instead of a quadratic differential form to specify the con- 

 stitutive relations. The vanishing of the biquadratic integral 

 form may perhaps be regarded as the condition for action of 

 a moving curve on a particle, a type of condition that seems 

 natural if we regard moving Faraday tubes as fundamental. 

 With this generalized theory it is possible for the elementary 

 wave surface in a medium to be a general Kummer surface, 

 a surface of which FresnePs wave surface is a particular 

 case. It is doubtful whether this generalized theory is suffi- 

 ciently general for all purposes, and the above example is 

 given just to emphasize that the absolute calculus of Ricci and 

 Levi Civita can be used to develop a theory of generalized 

 relativity on many lines in addition to that adopted by 

 Einstein. 



Going back to the case in which a quadratic form is 

 sufficient to determine the optical properties of a medium, 

 we may remark that if Einstein's idea of the gravitational 

 equations is accepted, it is still by no means certain that his 



* Proc. London Math. Soc. ser. 2, vol. viii. p. 375. See also p. 251 

 of my first paper. 



