Equipartition of Energy and Radiation Theory. 65 



and " momentum " need, in the above, have no other sig- 

 nificance than that they are quantities which make the 

 Hamiltonian equations true for the system. 



Now let us inquire as to how much of the theory is a 

 result of pure mathematics, and how much involves other 

 considerations. In the first place, suppose that, following 

 Jeans, we consider a generalized space of 2n dimensions so 

 that each point in the space has 2n coordinates. Further, 

 let there be a number of quantities, E l5 E 2 , E 3 , &c, each a 

 function of only one of the coordinates, and suppose that * 



E 2 + E 2 + E 3 + &c. = constant = E, say. . . (1) 



Let us imagine a large number of points distributed with 

 uniform density in the generalized space, and let us confine 

 our attention to a region of the space corresponding to values 

 of E between E and E + ^E . Consider any point in the space, 

 and from its coordinates pick out a large number p. These p 

 coordinates need not be all of the same type, but let us suppose 

 that the quantities E 1? E 2 , E 3 , . . . E p , corresponding to them 

 are all of the same mathematical form f. In a similar way 

 pick out another group of q coordinates from the coordinates 

 of the point, and again, let all the E's be functions of the 

 coordinates of the same mathematical form, though the form 

 for the q group need not be the same as that for the p group. 

 Let us in a similar way pick out a group of r coordinates, 

 and so on ; then it follows as a purely algebraical fact that 

 relations of the type 



8n p = A p e- m ^d% (2){ 



8n q = A q e- hE< *d%, (3) 



&c, 



* It is to be noted that no dynamical significance is as yet attached 

 to the coordinates or to the quantities Ei, E 2 , &c. 



t So that by altering the scale of the coordinates we can reduce the 

 E's to exactly the same function of the coordinates. Thus for example 

 the condition would be satisfied if, in the p group, 



Eiswi!**! 2 , E 2 =m 2 2 2 , &c, 



but would not be satisfied if 



E 1 =w 1 ^ 1 2 , E 2 = a9 2 3 . 



% The equations are usually given in a form expressing the number 

 of systems having values of the different coordinates between £ x and 

 £i+d£i, £ 2 and £ 2 +d£ 2 , &c, a sort of differentiation in kind being 

 maintained in this way between the coordinates. So long* as the E's 

 for a particular group, for example the p group, are the same function 

 of their respective coordinates for all coordinates of the group, mathe- 

 matically the coordinates -may be treated as of the same type, even 

 though physically they may be different. 



Phil. Maq. 8. 6! Vol 33. No. 193. Jan. 1917. F 



