PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 107 



tiirbed motion, provided that besides thus altering the initial velocities and directions 

 we alter also the initial configuration, by the formula 



^'i=Jo -f^/' (Z.) 



It would not be difficult to calculate, in like manner, approximate expressions for the 

 disturbed directions and velocities at any time t ; but it is better to resume, in an- 

 other way, the rigorous problem of perturbation. 



Other Rigorous Theory of Perturhation, founded on the properties of the disturbing 

 part of the constant of living force, and giving formulae for the Variation of Ele- 

 ments mot^e analogous to those already known. 



13. Suppose that the theory of undisturbed motion has given the 6w constants 

 e. p. or any combinations of these, x^, x^, . . . x^^, as functions of the 6n variables 

 fj. w. and of the time /, which may be thus denoted : 



^• = %ift'?i. '?2. • •^3n'^i'^2' • -^sJ' • ••.•.. (54.) 

 and which give reciprocally expressions for the variables ri. rs. in terms of these ele- 

 ments and of the time, analogous to (44.) and (45.), and capable of being denoted 

 similarly, 



\ — ^i {*> ^U »2J • • • «6 J' ^i = "^i ft ^1' -^25 • • • ^6n) ' • • • • (^^'^ 



then, the total differential coefficient of every such element or function «., taken with 



respect to the time, (both as it enters explicitly and implicitly into the expressions 

 (54.),) must vanish in the undisturbed motion ; so that, by the differential equations 

 of such motion (H.), the following general relation must be rigorously and identically 

 true : 



o = -^ + 2(^ia_ -EJi) 56. 



In passing to disturbed motion, if we retain the equation (54.) as a definition of the 

 quantity «., that quantity will no longer be constant, but it will continue to satisfy 

 the inverse relations (55.), and may be called, by analogy, a varying element of the 

 motion ; and its total differential coefficient, taken with respect to the time, may, by 

 the identical equation (56.), and by the differential equations of disturbed motion 

 (G.), be rigorously expressed as follows : 



» -- 2 ( I ^ — i l±h) (A'.) 



14. This result (A^.) contains the whole theoiy of the gradual variation of the ele- 

 ments of disturbed motion of a system ; but it may receive an advantageous trans- 

 formation, by the substitution of the expressions (55.) for the variables ri. -a. as func- 

 tions of the time and of the elements ; since it will thus conduct to a system of 6 w 



p2 



