110 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



It is evident also, that these coefficients «,^ , have the property 



«,,£=-«,,5. (^^) 



and 



«,,, = 0; (68.) 



S H d X.- 



the term proportional to -~ disappears therefore from the expression (Bi.) for-—; 



and the term 



8 H2 SJIg . 8JI2 ^ 



destroys the term 



8H2 8H2 . 8H2 ^«, 



8 Xj * *> * * 8 x^- 8 X, dt ' 



when these terms are added together ; we have, therefore, 



i Hg </ X 



8 X ^^ 



2.'-^^r = 0, (E.) 



or 



' dt ~~ dt 



tZ H.2 _ 8 H2 



that is, in taking the first total differential coefficient of the disturbing expression H2 

 with respect to the time, the elements may be treated as constant. 



Simplification of the differential equations which determine these gradually varying 

 elements f in any problem of Perturbation ; and Integration of the simplified equations 

 by means of certain Functions of Elements. 



16. The most natural choice of these elements is that which makes them corre- 

 spond, in undisturbed motion, to the initial quantities e^ p^. These quantities, by the 

 differential equations (H.), may be expressed in undisturbed motion as follows, 



and if we suppose them found, by elimination, under the forms 



^i = ^i + ^i {t, ^1, yi2, . . . rjs^, T^i, STg, . . . Tffsn), 1 



\ (70.) 



Pi =^i-^% (t, yii, >l2, ' ■ - %„J ^1, ^25 • • • ^3n)j J 



it is easy to see that the following equations must be rigorously and identically true, 

 for all values of ;?, rff-, 



= '^. (0, ??j, ;?2J • • • ^3nJ ^1> ^2> • • • '^3 J- J 



When, therefore, in passing to disturbed motion, we establish the equations of defi- 

 nition^ 



