PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 273 



although the consideration of the point, called usually the centre of gravity, is very 



simply suggested by the process of the tenth number, yet this internal centre is even 



more simply indicated by our early corollaries from the law of varying action ; which 



show that the components of relative final velocities, in any system of attracting or 



1 g V 

 repelling points, may be expressed by the differences of quantities of the form — g-7, 



1 8 V 1 S V 

 — ^— , — J— : and therefore that in calculating these relative velocities, it is advan- 



m oy' m oz ° ' 



tageous to introduce the final sums 2 mx, ^mi/, ^mz, and, for an analogous reason, 

 the initial sums 2 m a, ^mb, tmc, among the marks of the extreme positions of the 

 system, in the expression of the characteristic function V ; because, in differentiating 

 that expression for the calculation of relative velocities, those sums may be treated as 

 constant. 



On Systems of two Points, in general ; Characteristic Function of the motion of any 



Binary System. 



13. To illustrate the foregoing principles, which extend to any free system of points, 

 however numerous, attracting or repelling one another, let us now consider, in parti- 

 cular, a system of two such points. For such a system, the known force-function U 



becomes, by (2.), 



\] = m^m^f{r), (71.) 



r being the mutual distance 



r = ^'('^i - ^2? + (3/1 - ViY + (^1 - ^2?, (72.) 



between the two points m^, ^2, and f (r) being a function of this distance such that 

 its derivative or differential coefficient /' (r) expresses the law of their repulsion or 

 attraction, according as it is positive or negative. The known differential equations 

 of motion, of the second order, are now, by (1.), comprised in the following formula : 



m, {x\ h a?, +y\ ^y^ + z\ I z,) + m., {x\ I oc^ -\-y\ ly, + A ^ z^) = m^ m^ If if) ; . (73.) 

 they are therefore, separately. 



•^ 1 — ^2 8^^ 5 3/ 1 — ^2 s^/j J z I — m^ §^^ , j 



' . . . (74.) 



The problem of integrating these equations consists in proposing to assign, by 

 their means, six relations between the time t, the masses m^ m^, the six varying 

 coordinates x^ y^ z^ x^ y^ Z2, and their initial values and initial rates of increase 

 «i &! Ci ^2 ^2 ^2 ^'1 ^'1 ^1 ^'2 ^'2 ^2- If we knew these six final integrals, and combined 

 them with the initial form of the law of living force, or of the known intermediate 

 integral 



lm^{x\^+y\^^z\^)-\-\m.,{x',,^-\-yK,^^z'^^)=m,mJ{r)-\-Ui . . (75.) 



