86 Sir Oliver Lodge on 



So in like manner the astronomical solution for orbits of 

 small excentricity, with v reasonably constant and both w 

 and v small compared with c, is 



[A / id 2 4- v 2 wv cos X a . \ , . 

 u= jp\l 2^ — +e cos (6-ot) — 2 0sin0J, (3) 



instead of: the usual 



u= 0(1 + 6 cos (0-*)). 



Now being defined with reference to the sun's way we 

 cannot make a zero; in fact a must be practically the angle 

 between the major axis of the orbit and the line of reference; 

 for, save for a minute correction term, it represents the 

 value of at the perihelion apse. The extra constant term 

 in (3) only matters in cases where w or v is beginning to be 

 comparable with c ; but the progressive term containing 0, 

 an angle which steadily increases with the time, is important. 

 For in a century the whole angle swept through by Mercury's 

 radius vector, at the rate of 4 revolutions per annum, is 800tt. 

 The progressive term shows either that the constants of the 

 orbit must change, or that the orbit must revolve in its own 

 plane. 



To find its rate of revolution, consider the apses as places 

 where du/d0 = O, and where = a. Then in general 



du ii \ . ir . , , vw cos X . Q a ns \ ... 



dd = ~k*\ e sin ( } + ^~2? — < sm0+ 6cos ^ } • ( 4 > 



So at an apse, writing k for ^ivv cos X/c 2 , 



e(sin6 cos <x — cos 6 sin a) + 7c"(sin 0+0 cos #) = 0, 



or (e cos a -f k) sin 0=(e sin a. — k0) cos 0. 



So /i esina — kd ^ e sin a k0 ,_,. 



tan (9= ——j- ^ , . . (5) 



e COS a + k e COS a -t- h £ COS a v y 



A being very small; so that if O is the initial value of at 

 an apse 5 and <9 72 , its value after n revolutions, 



2irnk 





e cos c 





but ^ tan a = sec 2 a t/a ; 





so 2'iTnk cos a 



rrnvi 



e 



ec 2 



cos X cos a. . . (6) 



