I08 JUPITER AND SATURN. ^ 



wherein M =0-43429 (9"63778)* and D=a-j- 

 and F may be taken = 



+ \eei cos [w—wi) (2— D — D^) hix 

 or with sufhcient exactness, more simply 



Hog. a=-Jw'MaD6„_,. ^ log. a' = ^ mM (i+D) 6„ ,. 

 or if the planets are very distant from each other 



^log. a= -Im'Ma^. c log. a'=|wM (14- fa-'). 



The last equation shows us that this effect of an inner planet on 

 a distant outer planet is nearly equivalent to an increase of the 

 Sun's mass by that of the inner planet, viz. : — 



fl3, w^, = / (i + w + w, ). 

 Some elaboration on this point seems necessary as even in so 

 modern a work as Charlier's Mechanik des Himmels, 1902, the 

 values of log a given for the major planets are all erroneous. f '* 'f 

 The following table exhibits the value of 3 log a for each of the 

 major planets. It is mainly due to Newcomb and Hill. 

 Table of <? log a (in units of the 8th decimal). 



The approximate calculation for two cases may be given : — 



Mercury by Jupiter. 

 Loers. 



M 



,.3 



9-222 

 6-980 

 9-638 

 6-615 



1 

 3 



m 

 M 



Neptune by Jupiter. 

 Logs. 

 = 9-5229 



= 6-9799 

 = 9-6378 

 — 0-0097 



n 2-455 = — 3 6-1503 = 14136 



From the tables of the planets by Newcomb and Hill we get 

 the total c log a (unit of 8th decimal place). 

 Mercury — 8 



* The number in brackets is the logarithm of the preceding number, 

 f Band i, Tafel i, p. 439. 



