114 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



Analogous formulae for the motion of a Single Point. 



19. Our general method in dynamics, though intended chiefly for the study of 

 attracting and repelling systems, is not confined to such, but may be used in all 

 questions to which the law of living forces applies. And all the analysis of this 

 Essay, but especially the theory of perturbations, may usefully be illustrated by the 

 following analogous reasonings and results respecting the motion of a single point. 



Imagine then such a point, having for its three rectangular coordinates x y z, and 

 moving in an orbit determined by three ordinary differential equations of the second 

 order of forms analogous to the equations (2.), namely, 



^=i75y' = y^' ^" = 17' (78.) 



U being any given function of the coordinates not expressly involving the time : and 

 let us establish the following definition, analogous to (4.), 



T = 4-(^'^+y2 + ^'2), (79.) 



a^ i/ z' being the first, and jc" y" z" being the second differential coefficients of the 

 coordinates, considered as functions of the time t. If we express, for greater gene- 

 rality or facility, the rectangular coordinates x 7/ zas functions of three other marks of 

 position f}i f]2 ^3) ^ "^ill become a homogeneous function of the second dimension of 

 their first differential coefficients n'l ^\ ^'3 taken with respect to the time ; and if we 

 put, for abridgement, 



8T ST ST 



TO", — v^, "^o = ^r-r, ^ 



1 — g^M «'2~ Ir^^y "'3 — Irl^' 



(80.) 



T may be considered also as a function of the form 



T = F (tjTi, rzTg, t;73, ;?1, ;j2J ^j)j (81.) 



which will be homogeneous of the second dimension with respect to w^ wg ^^3. We 

 may also put, for abridgement, 



F (tjTi, tJTg, ^3, ??i, J72J ^3) — U ini, ??25 ^73) = H ; (82.) 



and then, instead of the three differential equations of the second order (78.), we may 

 employ the six following of the first order, analogous to the equations (A.), and ob- 

 tained by a similar reasoning. 



rf^_ SH ^_ , SJI ^_ I ^ "^ 



(83.) 



dt -^ "^ S^i' dt "^ "^ IvTc^' dt '^ '^ S^a' I 

 dt S>]j' dt ^1)2' dt ^»!3* J 



20. The rigorous integrals of these six differential equations may be expressed 

 under the following forms, analogous to (B.), 



__ 8S _ SS SS 



1-8,1' "2-8^, ^3-8^. I 



ss ss ss i •••(•) 



