( 772 ) 



TiCl non' «r)^, aiid if'„ lio tlio iiicliiialioii and tlio lonjjitnde (connted 

 f!'om the first point of Ai-iey) of tlie descending node of the plane 

 which I wisii to adopt as the fundamental plane, I'eferred to the 

 plane of .Inpiter's orbit. Longitudes in the fuuduniental ]ilane are 

 counted from the node i|'„ as zero. 



Then if / and f^ are the inclination and node of the orbit of one 

 of the satellites referred to the fundamental plane, we have, neglecting 

 quantities of the third order in i, I and tu„ : 



* ■"■'" SI = ^«'« i^^—V\) 



i cox Sh — ^f"*" (^— V'o) + «'A. 



If furllicr we introduce the notations 



l-=^^^-6i x,. = i,,„_é'„ + 180° 



Xi=z yiain^ri ,i\^vimii)j \ . . . (2) 



;/,■ z=. Yi cos Fi )/„ = to cos \p — io„ 



then the expressions for y; and q become: 



/ 

 qi = 2 a; J Hj + (1 — ^,) tü„ — Hi !/„ 



(3) 



Marth has adopted 



co„ = the value of m ) ^ 



^ I lo/.o Ironi SoriLLART s theory') 

 V'o = >, ,. >. (^, + J80° ) •" ' 



and has computed the values of p and q by the formulae (3), taking 



•^'o = y, = *^- 



The unknowns y, , r-,, .v„ and ?/„ must be determined from the 

 equations (3). This is, of course, onl}' possible if the coefficients 

 Ojj and Hi are known. I have adopted these coefficients from 

 Souillart's theory, as being the best available. They are very com- 

 plicated functions of tl)e masses, the compression of Jupiter, and the 

 mean motions. As a rough approximation, we can say that the 

 coeflicients <Ty are proportional to the mass m.j. Since the masses are 

 very imperfectly known, the same thing is true of the coefficients of 

 the equations (3). Therefore the results of the present discussion cannot 

 be consideretl as final, but the discussion will have to be repeated 

 when better values of the coefficients are available. The results here 

 derived will however doubtlessly represent a very fair approximation. 



It may perhaps be mentioned that the uncertainty of these coeffi- 



1) Martii has made one or two mistakes here, which will be duly mentioned 

 in the detailed publication, but as they have no inthicnce on the result they can 

 be ignored at present. 



