PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 105 



accuracy ; and since the perturbations (M.) and (O.) involve the disturbed coordi- 

 nates 9j. only as they enter into the coefficients of this small disturbing function S^, it 

 is evidently permitted to substitute for these coordinates, at first, their undisturbed 

 values, and then to correct the results by substituting more accurate expressions. 



11. The function S^ of undisturbed motion must satisfy rigorously two partial dif- 

 ferential equations of the form (C), namely, 



di "T- ^l \^ri^>" • 8^^ ^IJ . . • ^3n) = Ui (??i, . . . ;?3 J, 



and therefore, by (D.), the disturbing function S2 must satisfy rigorously the following 

 other condition : 



,,-=U,(,„. .,3.)- F,(g, ..^, ,,. . . ,3„)+F(g, . .^^,,,, ..,^„),(Q.) 



and may, on account of the homogeneity and dimension of F, be approximately ex- 

 pressed as follows : 



or thus, by (I.), 



that is, by (42.), 



^2=-f,'^2dt (T.) 



In this expression, H2 is given immediately as a function of the varying quantities 

 ??. OT^., but it may be considered in the same order of approximation as a known func- 

 tion of their initial values e. p. and of the time t, obtained by substituting for ;;. m. 



their undisturbed values (44.) (45.) as functions of those quantities; its variation 

 may therefore be expressed in either of the two following ways : 



^H2 = 2(^'S,+ ^n^), (48.) 



or 



in,= t(^-^le + '-^^p)+'-^lt.. . (49.) 



Adopting the latter view, and effecting the integration (T.) with respect to the 

 time, by treating the elements e. jo. as constant, we are afterwards to substitute for 



the quantities p. their undisturbed expressions (39.) or (I.), and then we find for the 



variation of the disturbing function S2 the expression 



MDCCCXXXV. P 



