54 Mr. S. B. McLaren on the 



A result similar to Planck's can be obtained by putting 

 a =0, 



a H = Ki : ) 71=1, 2.. .00, 



E\=87rR6\-±{l + K i \0{e^ e -l)}-\ . . (40) 

 For large values of \6 this gives 



-£ t x = 8ttRi9X- 4 (1+^ 1 )- 1 (41) 



For small values of \0 it gives 



£ , \ = 87rRA- 5 ^-^(«i)" 1 (42) 



Thus if we abandon the principle of least action a wave 

 theory of radiation is at least a formal possibility. But 

 nothing has been done to show what laws of motion lead to 

 the results (36), (37), (38), or, indeed, that they are con- 

 sistent with any continuous laws of motion whatever. 



Rayleigh's formula for the complete radiation is inde- 

 pendent of all particular properties of matter. It is enough 

 if the laws of motion are deduced from the principle of least 

 action. Once abandon that and various results can be reached 

 inconsistent with each other and with the truth. But because 

 an object can be imagined without logical contradiction it does 

 not therefore exist. The distribution of energy depends only 

 upon such properties of matter as are common to all forms 

 which actually exist. It does not follow that these are found 

 in matter as it is by us imagined. This seems the only answer 

 to the difficulties raised by Poincare. The mechanical inter- 

 change of energy leads to equipartition. But it is not shown 

 that any such process exists, or at least that it alone exists in 

 any actual species of matter. 



§ 6. Theoky of a Simple Wave-Train. 



The main object of this paper is to show that the effect of 

 radiation on matter is calculable although the laws of motion 

 may differ widely from those known to ordinary dynamics. 

 The behaviour of a simple harmonic wave-train advancing 

 through any material system is easily determined. 



The vector potential F 1 satisfies the equation 



( v2 -?£) Fi+4,r<ri ' m=o ' • • • - (43) 



