Partition of Energy between Matter and Radiation. 19 



The condition as to H ensures that these curves have no 

 branches at infinity and are therefore closed. (7) becomes 



jy^^=fPf^ 



(8) 



The integration is to be taken all over the areas enclosed 

 between H=H 2 and H=Hi. In the figure given the curve 

 H=H 2 has a double point. Now 



§ y <JH dx iy = j£ y W dx dxj £ y dH dx dy _ (g) 



In (9) the right-hand side represents the difference of: the 

 integrals taken over the area enclosed by H=H 2 and over the 

 area enclosed by H=H l . It is conceivable that the curves 

 may consist of two or more separated portions and that 

 H=H 2 may enclose M=If l in one portion and be enclosed by 

 it in the other. 



JJ y Ty dx dy = \\ dy^ yH)dy dx " jj Hdydx 

 = I H 2(VQ-yp) d a - I j Hdy dx 



=^_£f 



Hdydx (10) 



l/q and ?/p are the ordinates where PQ cuts H=II 2 . 



In (9) A 2 represents the area enclosed by the curve 

 H=H 2 . 



In exactly the same way 



^y^ dxd y^ H 2^-^Hdyd X . . (11) 



From (10) and (11) the truth of (8) follows. There is 

 equal partition of the kinetic energy. 



Further important results can be proved. In (8) we may 

 substitute for x and y any two of the 2n coordinates. Thus 



f dll , XT f dHj„ 



(12) 



and (12) is true for all values of r and s. For the proof it is 

 only necessary that // considered as a function of the o'a 



02 



