CONTEMPORARY ADVANCES IN PHYSICS 663 



equation (13), and to a peculiar feature of that equation — to the fact 

 that in it the magnitude of the gradient of W stands equated to a 

 function of the coordinates. This indeed is the feature which rendered 

 it possible to imagine flowing waves. Now this feature occurs because 

 the system to which equation (13) relates — the particle voyaging in a 

 force-field — has a kinetic-energy-function which is the sum of the squares 

 of the momenta (multiplied by a constant). Had we presupposed a 

 system possessing a kinetic-energy-function not capable of being so 

 expressed — two particles of different masses voyaging in a force-field, 

 or a rigid rotating body of irregular shape, for example — the equation 

 which we should have obtained in lieu of (13) would not have had the 

 peculiar feature aforesaid; the wave-picture would not have offered 

 itself, much less the equation (20) which was superposed upon the 

 wave-picture. It is precisely at this obstacle that the mode of thought 

 known as non-Euclidean geometry proves itself useful. It proposes 

 equations of a general type, which can be written down for every 

 system of which the kinetic-energy-function is preassigned, and which 

 for the single particle floating in a force-field become the equation 

 (13) and (20). In the language of non-Euclidean geometry, even the 

 words and the symbols for ivave and ivave-speed and gradient and 

 Laplacian are preserved; but whether they are advantageous to any- 

 one not already versed in this subject may well be doubted. Suffice it 

 to say, that non-Euclidean geometry provides a general equation ^ 

 of which (20) is a special case, and that the general equation has 

 already justified its existence by its successes in dealing with certain 

 atom-models and molecule-models such as the rigid rotator used in 

 the study of band-spectra. But the question as to what the waves 

 "really are" becomes in these cases all the darker and more perplexing. 



One further step, and we attain to the idea on which the calculation 

 of the energy-values of the Stationary States reposes. 



It is very well known that a medium capable of transmitting waves, 

 and hounded in certain ways, may develop what are variously known 

 as standing waves — stationary wave-patterns — the phenomena of 

 resonance. Air enclosed in a box, a string pinched at the ends, a 

 membrane clamped around its circumference, the mobile electricity in 



^ Let the kinetic-energy-function of the system, expressed in terms of the co- 

 ordinates and velocities, be written 



r = SS QiMi 

 i j 



and let A stand for the Laplacian operator in the non-Euclidean configuration-space 

 of which the metric is ds^ = Xl/Qijdqidqj] then the general wave-equation of de Broglie 

 and Schroedinger is: 



h^A4' -H 8^2 (£ - V)^p = 0. 



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