the Radiation from Electron Orbits. 645 



in which <l>, <£' are functions of which the form is not at 

 present required. If <!> 2 -{-<I> /2 be denoted by ^, the integrand 

 of expression (9) for the complete radiation becomes : 



r*u*-*K-*9(a, p/Rv%) (12) 



Let c be the velocity at the apse, so that Hw = c . The 

 value of a becomes (cf. equation (6)) 



- 2/ ^~ (13) 



e*(n-iy ' 



so that a is the ratio of potential to kinetic energy at the 

 apse, and expression (12) becomes 



i(7i-l¥**#P(a, p/c u ) (14) 



At this stage the treatments appropriate to open and closed 

 orbits diverge. 



Open Orbits. 



3. Open orbits will be described by electrons which are 

 free except when in encounters with the centres of forces, 

 and the law of distribution of these electrons is known. We 

 require lirst to investigate how many orbits of any specified 

 kind are described per unit of time. The orbit may be 

 specified in time by the instant at which the apse is passed, 

 this instant being specified analytically by the condition 

 r=0. 



The probability at any instant of finding an electron within 

 a given element dx dy dz of volume, having its velocity com- 

 ponents within a range dx dy dz is 



NAe~ hmG dxdydzdxdydz, .... (15) 



where N is the number of electrons per unit volume, and A 

 is a constant determined by the condition that expression (15) 

 integrated through unit volume shall be equal to N ; k has 

 its usual meaning in kinetic theory, being given by 

 l/2/i = RT, and G is twice the total energy of the electron 

 per unit mass, given by 



Gr = s» + y» + i» + 2 I* 



In polar coordinates expression (15) becomes 



XAe- hm6 r* sin 2 ddrddd<j)drddd<j>. . . (16) 



