944 Prof. J. H. Joans on Non-Newtonian Mechanical 



raised by Larmor's suggestion and is in brief as follows : — 

 Can any system of physical laws expressible in terms of con- 

 tinuous motion (or of mathematical laws expressible in terms 

 of differential equations) be constructed such that a system 

 of matter and aether tends to a final state in which Planck's 

 law is obeyed ? It will be found that the answer obtained is 

 in the negative. 



General Dynamical Investigation. 



2. We shftll assume a law of causation — namely, that the 

 state of the system at any instant is determined by its state 

 at the previous instant, and that this state can be specified by 

 the values of certain definite quantities pi, p 2 > • • • Pw which 

 we shall call the co-ordinates of the system. We sliall first 

 examine the consequences of assuming that time is con- 

 tinuous and tluit these co-ordinates vary continuously with 

 the time. 



3. I f we construe! an ^-dimensional space, a single point 

 in this Fpace, namely the point whose co-ordinates are 

 p\. /'o, . . . /'„, will represent the state of the system at any 

 instant. A knowledge of the dynamical or kinematical laws 

 obeyed by the system would lead directly to a knowledge of 

 the paths or trajectories traced out in this space by the 

 representative points as they follow the different possible 

 motion- of the system. We must not, in the present investi- 

 gation, nssuine any special dynamical laws, but the general 

 law of causation enables us to suppose that through every 

 point in the generalized space there is one and only one 

 trajectory, find that as a point moves along a trajectory, and 

 so follows the motion of a system, its velocity at any point 

 depends only on the co-ordinates of the point and not on the 

 time. 



In the usual manner, we imagine every region of the 

 generalized space which represents a physically possible state 

 of the system to be filled with so many representative points 

 that the whole collection of points may be regarded as 

 forming a continuous fluid. The law of causation now states 

 that this fluid moves along fixed stream-lines and that the 

 velocity at any point remains constant. 



The initial distribution of density of the imaginary fluid in 

 the generalized space remains entirely at our disposal. Since 

 the motion is along fixed stream-lines with velocities fixed at 

 each point, this initial distribution of density can be so chosen 

 that, as the motion progresses, the density at every point of 

 the space shall remain always equal to the initial density at 

 the point. We elect to arrange the initial distribution of 



