644 Prof. J. H. Jeans on the Analysis oj 



orbit. Eliminating the time, the equation of the orbit is 



d 2 u _ /jl 



71-2 



(4) 



where u as usual is 1/r. If it is the value of m at the apse 

 nearest the origin, integration with respect to u gives : 



H 2 (rc-1) 

 If we now put u = u cos ^ and 



2yuw;;" 3 



(6) 



H s (n-1)' 



this has the integral : 



e-[ sin %^ m 



Jjsin^ + atl-cos-^)}!- •••<<; 

 From (3), the time is given by 



~JHw 2 "Hw 2 Jcos 2 % {sin 2 x + a(l-cos»- 1 X )p<• W 



The components of acceleration are 



/xr ~ n cos 0, }jLr- n smQ, 



from which the radiation can be written down. 



Resolved into its constituents by Fourier's Theorem, the 

 radiation from the complete orbit is 



3^f rP + J2 ^' W 



1— \ /j,r~ n cos Q cos pt dt (10) 



J =\fjir- n sin Q sin ptdt, .... (11) 



in which the limits are t= — go to t= -f oo for an open orbit, 

 and are taken through a complete revolution if the orbit is 

 closed. On substituting the values of and t from equations 

 (7) and (8), and effecting the integrations with respect 

 to Xi we obtain : 



I = ^- 2 H- 1 d>(a^/Hu 2 ) J 



J = ^-"H-^&'fo p/Ku 2 ), 



where 



