52 Mr. S. B. McLaren on the 



The variables x x v n become the momenta pi . ..jj>,/and 



the coordinates q x q% • • • q K > The number n is here even being 

 2k. In (30) 



_ dq r dH 



dt dp r ' 



and <I> — X cim 2 ?/ s 6>,9. 



J»=l S — l 



<X> is a linear function o£ fhe velocities, so that o) sm is here a 

 function of qi q 2 ... <?„ only (see Phil. Mag. April 1912). 



Arranging the 2k variables with all the momenta first, it 

 follows by comparing (16) and (30) that for values of r from 

 1 to #, 



S ^ K dco sm S = K d(o rm S = K dH /da) sm da> rm \ 



X(r,m)= *< u s—, — Z —j U s = Z j—l-j —~j 



s=i dq r , =1 dq s 8= i dp s \ dq r dq s J 



X{r,m)——<0(r,m)- 



And for values of r from k + 1 to n, 



Hence 



r=n ^ d r= K c i 



S ~ J" (PX'rm) =— t j~ («l rai ) = 0, 

 r=l p dx r ^ r= i dpr 



p = l and co rm is not a function of the momenta. 



r g* 1 d , . _ S = K d 2 /I / dco sm dco rm \ _ ^ 



r= i pdair ™ r ' s =i dp r dp s \ dqr dq s J 



(29) reduces to the first term only. 



§ 5. Complete Radiation. 



In this case the formula for complete radiation becomes 

 by equating (29) and (25) 



»(£)'-S C3D 



This represents equipartition of the radiant energy. (31) is 

 equivalent to Rayleigh's formula 



Z a = 8ttR6>a- 4 (32) 



If p is not unity, that is if the laws of radiation are not 

 deduced from the principle of least action, the second and 



