646 Prof, J. H. Jeans on the Analysis of 



At the apse, from equations (2) and (3), 



so that for the electron to come to its apse within an interval 

 dt, the value of r at the beginning of the interval must lie 

 within a range (c 2 u + /lmi|[) dt of zero. Giving this value to 

 dr in expression (16) we find 



NAr* ffl& r 4 sin 2 dr d6 d$ dd d<j>(c? Q u Q + (tuty dt 



for the number of orbits of a certain type described per 

 interval dt. On integrating with respect to 6 and <j>, and 

 with respect to all values of 6 and (/> which give a range dc 

 to c , we obtain 



STT^Ae-^^rldrQCodco^Uo^-fjuu^dt, . . (17) 



as the number of electrons which, per time dt, describe orbits 

 having r and c within ranges dr and dc . 



The orbit may be more conveniently specified by the con- 

 stants G and H, given by 



G = i+ (n -l )r n-i> H = r c . 



We find, in the usual way, that 



dGdK = 2drdc(c 2 + f*u^ 1 ), 



so that expression (17) becomes, omitting the factor dt, 



±7r 2 ?$Ae- hm mdGdR, .... (18) 



giving the number of orbits per unit time for which G and 

 H lie within ranges dGr dK. 



4. On multiplying expressions (18) and (14) we obtain 

 the total radiation per unit time. Transformed to the variables 

 G and a, the new expression becomes 



^i^J^^ [ , * r {«^((«-l)«/2/4"- 4i (G/(l+"))"^} 



X2n 2 NAe- hmG G^{2ii/n— l)^i^\* »-i(l + a) n-i\dGd*. 



We obtain the radiation from all possible open orbits on 

 integrating this expression from a = to a = + 1, and from 



