Si/stems, and Planck 1 s Theory of Radiation. 951 



I£ /3 stands for d</> A /?)E 5 , a constant, the equations (11) 

 become 



in which the analogy with the Hamiltonian form is obvious. 



9. On giving different values to cf) and E as functions of 

 Q and R we obtain different systems of equations of motion. 

 Some of these may of course be incapable of representing 

 wave-motion at all. 



The simplest form which can be given to <f> is that of a 

 linear function of Q and R, and this may without loss of 

 generality be taken to be AR. The equations of motion 

 become 



Q=-A, R = 0, 



of which the integrals are 



H = cons. ; Q = B-Af. 



These will represent wave-motion if Q, R are taken to be 

 phase and amplitude or phase and energy respectively, but 

 will not satisfy the condition of the co-ordinates being deter- 

 mined uniquely from the state of the system, unless we 

 suppose the space limited to a range 2nr in the values of Q. 



10. The next simplest form which can be given to 4> i g 

 that of a quadratic function of Q and R, and this may 

 without loss of generality be taken to be -j(CR 2 + DQ 2 ). The 

 equations of motion become 



Q=-CR; R = DQ, . . .. - (13) 



of which the integrals represent simple harmonic motion. 



11. Equations (11) show that the motion of the fluid in 

 the generalized space is the steady motion of a homogeneous 

 fluid along the system of stream-lines cj> = constant. For 

 this to be capable of representing wave-motion the curves 

 cf> = constant must be a series of closed non-intersecting 

 curves. The mass of fluid and system of co-ordinates may 

 now be distorted so that these (or rather their projections on 

 the plane Q„ R s ) become a system of concentric circles, and 

 this may be done in such a way that the fluid remains homo- 

 geneous. On taking new axes the equations of motion 

 become identical with (13), so that to represent wave-motion 

 the co-ordinates must become identical w T ith the Lao;rangian 

 co-ordinates and momenta. 



It follows that, however far removed the general equations 



