102 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS, 



F(if)=Fw=:-^^=i^, (3«-) 



we easily transform the first equation (C.) to the following, 

 which gives rigorously 



supposing only that the two parts S^, Sg, like the whole principal function S, are 

 chosen so as to vanish with the time. 



This general and rigorous transformation offers a general method of improving an 

 approximate expression for the principal function S, in any problem of dynamics. 

 For if the part Si be such an approximate expression, then the remaining part 83 will 

 be small ; and the homogeneous function F involving the squares and products of the 

 coefficients of this small part, in the second definite integral (E.), will be in general 

 also small, and of a higher order of smallness ; we may therefore in general neglect 

 this second definite integral, in passing to a second approximation, and may in general 

 improve a first approximate expression Sj by adding to it the following correction, 



^S.=/'{-^ + U(''.,--0-F(|J.--||t'''"--0}<''' (I^-) 



in calculating which definite integral we may employ the following approximate forms 

 for the integrals of the equations of motion. 



expressing first, by these, the variables jj^ as functions of the time and of the 6 n con- 

 stants €i Pi, and then eliminating, after the integration, the 3 n quantities p^, by the same 

 approximate forms. And when an improved expression, or second approximate value 

 Si + A Si, for the principal function S, has been thus obtained, it may be substituted 

 in like manner for the first approximate value S^, so as to obtain a still closer ap- 

 proximation, and the process may be repeated indefinitely. 



An analogous process applies to the indefinite improvement of a first approximate 

 expression for the function Q or V. 



Rigorous Theory of Perturbations, founded on the Properties of the Disturhing Part 



of the whole Principal Function. 

 8. If we separate the expression H (17.) into any two parts of the same kind, 



Hi + H2=H, (40.) 



