PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



133 



may comprise the differential equations of relative motion in the following simplified 

 formula, 



in which 



And the integrals of these equations of relative motion are contained (by our gene- 

 ral method) in the formula 



in which a^ y a V d denote the initial values of | ;j ^ ^ y' %', and S is the principal 



function of relative motion of the system ; that is, the former function S, simplified by 



the omission of the part which vanishes when the centre of gravity is fixed, and which 



gives in general the laws of motion of that centre, or the integrals of the equations (181 .). 



Second Example : Case of a Ternary or Multiple System with one Predominant Mass ; 

 Equations of the undisturbed motions of the other masses about this, in their seve- 

 ral Binary Systems ; Differentials of all their Elements, expressed by the coeffi- 

 cients of one Disturbing Function. 



32. Let us now suppose that the w — 1 masses m are small in comparison with the 

 wth mass M ; and let us separate the expression (F^.) for H into the two following parts, 



H,= 



+ 



M 

 mi mi, 



{W.) 



of which the latter is small in comparison with the former, and may be neglected in 

 a first approximation. Suppressing it accordingly, we are conducted to the following 

 6w — 6 differential equations of the 1st order, belonging to a simpler motion, which 

 may be called the undisturbed : 



d^_±in,_( m\ d^__ l^Hi^ivi^- 



dt — m S^ — V^ "T" M/^ ' dt —~ m 8^ —^^^d^ 

 rf>, _ 1 SHi/, , m\ , dl_ 1 g Hi _ ^^ df 



^llMl- (i ill),/. !1Z.___1.1^_M 



^_ 1 ^H,_ 8/ 



=:M 



(P.) 



di — m ^z' —V '^ U)^ '' dt—~~m S? — "'"8^ J 

 These equations arrange themselves in w — 1 groups, corresponding to the w — 1 

 binary systems (m, M) ; and it is easy to integrate the equations of each group sepa- 

 rately. We may suppose, then, these integrals found, under the forms, 



^ = x^'^ (t, I ^, K, ^, y\ ^'), " = x^'^ {t, I, n, Z, ^', y. ^X 



X = x^^) {t, I, ,, ^, of, y\ ^\ r = x^^) (t, |, n, ^, ^, y\ z% V . . (K^.) 



^ = %(3) {t, I, n, ?, a!, y\ %^\ « = >:(«) {t, I v, ^, a!, y, ^') J 



