PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 113 



Hence, in the same order of approximation, if the part Ej, like the whole function E, 

 be chosen so as to vanish with the time, we shall have 



E. = -X'{4!^+H,(,«„....3.|A,...^)}= (R., 



^ ^ 3n J 



and thus a first approximate expression Ej can be successively and indefinitely cor- 

 rected. 

 Again, by (L^) and (GK), and by the definition (77.), 



the function C must therefore satisfy rigorously the partial differential equation, 



17 = H2(^^,g^,...^,Xi,...X3„j: (T^) 



and if we put 



C = Ci + C2, (Ui.) 



and suppose that the part Cg is small, then the rigorous equation 



* * an 3m 



becomes approximately, by (G^) and (L^), 



and gives by integration, 



C2=X'{-'# + H,0,g,...|g.,X„...O}'''. • • • • (X--) 



the parts C^ and Cg being supposed to vanish separately when ^ = 0, like the whole 

 function of elements C. 



And to obtain such a first approximation, Ej or C^, to either of these two functions 

 of elements E, C, we may change, in the definitions {7Q.) (77-), the varying elements 

 X, X, to their initial values e, p, and then eliminate one set of these initial values by 

 the corresponding set of the following approximate equations, deduced from the for- 

 mulae (G^) : 



»<8H 

 fC; = e. 



■i+rit'^*-' (Y.) 



and 



A 



i=Pi-f!-JTd' (Z'-) 



It is easy also to see that these two functions of elements C and E are connected 

 with each other, and with the disturbing function Sg, so that the form of any one 

 may be deduced from that of any other, when the function Si of undisturbed motion 

 is known. 



MDCCCXXXV. Q 



