Kesearches un the Üistrilmtion of the Meaii Motions of tho Asteroids. 9 



10. In this chapter I shall develop a simple theory of 

 planetary librations according to the method of Prof. Brown.'-* 



Assuming that the asteroid 7noves in the fdane of the orbit of 

 Jvpiter and that Jupiter inoves m a circular orbit, let n. a, e, s and 

 TIS be the elements of the asteroid as before, n', a, and s', the 

 circular elements of Jupiter and R the perturbative function, as 

 usual. Take the mass of the sun as unity and the unit of length, 

 so as to make 



(1) 7m'=l, 

 neglecting the mass of the asteroid, and also 



n'V' = l + w' 

 m' being the mass of Jupiter \Yhich is about 1/1047. 



11. Let Ho be the mean motion commensurable to n' and let 



(2) n=noil+x) or a=at(l-{-x)"^ 



The quantity x is supposed to be small, of the same order as e^ at 

 most. Let also 



(3) jL=jL 



n„ s 

 where s and s' are positive integers prime to each other. 



12. As a consequence of the assumptions mentioned above, 

 any argument of long period-terms, or critical argument as usually 

 called, takes the form 



is'l—isl'+jiv 



where i and j are any integers positive or negative, and / and I' 

 are mean longitudes. By the properties of R we have 



'is' — is+j=0 or j=iÇs — s') 



Hence the critical argument is 



ll Month. Notices of the R. A. S., Ixxii, p. 609. 



