PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 103 



in which 

 and 



H2 = F2 (sTi, «r25 • • • ^3»J »?1J '?2» • • ^^Sn) — ^2 ('^U '?2» • ' ^^sJ* • • • • • (42.) 



the functions Fj Fg Ui Ug being such that 



Fi + F2 = F, Ui-}-U2 = U; (43.) 



the differential equations of motion (A.) will take this form, 



l!! - !iL a. Hi f^' _ iiii ?ii 



It '1 '» 



and if the part H2 and its coefficients be small, they will not differ much from these 

 other differential equations, 



dt "" 8«r.' rf^ — ~ 8,,. ' (H.) 



so that the rigorous integrals of the latter system will be approximate integrals of 

 the former. Whenever then, by a proper choice of the predominant term H^, a 

 system of 6 n equations such as (H.) has been formed and rigorously integrated, 

 giving expressions for the 6 n variables rii ^i as functions of the time /, and of their 

 own initial values e^ jo„ which may be thus denoted : 



^.- = Pi (t, ^1, e^,..e^^,p^,p2,'.p3n), (44.) 



and 



^i = '^i(t,e^,e2,'^€^n^Pl,P2^"Pzn)^ (45.) 



the simpler motion thus defined by the rigorous integrals of (H.) may be called the 

 undisturbed motion of the proposed system of n points, and the more complex motion 

 expressed by the rigorous integials of (G.) may be called by contrast the disturbed 

 motion of that system ; and to pass from the one to the other, may be called a Pro- 

 blem of Perturbation. 



9. To accomplish this passage, let us observe that the differential equations of un- 

 disturbed motion (H.), being of the same form as the original equations (A.), may 

 have their integrals similarly expressed, that is, as follows : 



Sj being here thQ principal function of undisturbed motimi, or the definite integral 



Si=/'(2--^-H,)'^<, (^6.) 



considered as a function of the time and of the quantities tji a- In like manner if we 

 represent by Sj -}- S2 the whole principal function of disturbed motion, the rigorous 

 integrals of (G.) may be expressed by (B.), as follows : 



8 S, . 8 Sa 8 S, 8 So /-gr \ 



