104 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



Comparing the forms (44.) with the second set of equations (I.) for tlie integrals of 

 undisturbed motion, we find that the following relations between the functions ^, Sj 

 must be rigorously and identically true : 



ni = (Pi\t,e^,e^,..e^^,-jj-^,-fj^,..-Y^J\ ....... (47.) 



and therefore, by (K.), that the integrals of disturbed motion may be put under the 

 following forms, 



^. ^u^2.--^3«'i'i + y;;'^2+ g^.-'i'sn + iT^/ (L.) 



We may therefore calculate rigorously the disturbed variables r^ by the rules of un- 

 disturbed motion (44.), if without altering the time t, or the initial values e^ of those 

 variables, which determine the initial configuration, we alter (in general) the initial 

 velocities and directions, by adding to the elements p^ the following perturbational 

 terms, 



A;>. = ||,Ap, = g,.. A;'3„ = ^„: (M.) 



a remarkable result, which includes the whole theory of perturbation. We might 

 deduce from it the differential coefficients ??'i, or the connected quantities rar^, which 

 determine the disturbed directions and velocities of motion at any time t ; but a 

 similar reasoning gives at once the general expression, 



^^£ = 87-+ "4^^ [ty ^1, ^2. • • ^3«./^l + 8^'-?^2 + 8^, . . • ;?3n + fj-)^ ' W 



implying, that after altering the initial velocities and directions or the elements j9. as 



before, by the perturbational terms (M.), we may then employ the rules of undisturbed 

 motion (45.) to calculate the velocities and directions at the time t, or the varying 

 quantities ts., if we finally apply to these quantities thus calculated the following new 



corrections for perturbation : 



A ^ ^2 A ^ ^2 A ^ ^2 /r\\ 



^^i = H' "^"^2 = 8^, ••^^3n = 8;5;; (O.) 



Approximate expressions deduced from the foregoing rigorous Theory. 



10. The foregoing theory gives indeed rigorous expressions for the perturbations, 

 in passing from the simpler motion (H.) or (I.) to the more complex motion (G.) or 

 (K.) : but it may seem that these expressions are of little use, because they involve an 

 unknown disturbing function S2, (namely, the perturbational part of the whole princi- 

 pal function S,) and also unknown or disturbed coordinates or marks of position ;?.. 

 However, it was lately shown that whenever a first approximate form for the princi- 

 pal function S, such as here the principal function S^ of undisturbed motion, has been 

 found, the correction Sg can in general be assigned, with an indefinitely increasing 



