132 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



- 2 ^n - 1 /« - 2, n - 15 J 



(D2.) 



U = m„ (mi/i + 7W2/2 + • • • + wi„ _ i/„ _ 1) 

 + mi mg/i^a + i^\ ^3/1,3 + . . • + ^r 

 in which /i is a function of the distance of m^ from m„, and/-^;^ is a function of the 

 distance of m^ from rtij^, such that their derived functions or first differential coeffi- 

 cients, taken with respect to the distances, express the laws of mutual repulsion, being 

 negative in the case of attraction ; and then we obtain, as we desired, two separate 

 groups of equations, for the motion of the whole system of points in space, and for 

 the motions of those points among themselves ; namely, first, the group 



d x^i = x'li dt, d of II = 0, "^ 



dy„=y'„dt,d!/'„ = 0, I ........... (181.) 



d Zii = z'li dt, d z'li = 0, J 

 and secondly the group 



fi?|= {fi^i + -^^i.ma/i) dt,dj[/, = -j^dt, 



dZ, = [z'l + -^ ^i.mz'i) dt, dz\ = 



The six differential equations of the first order, (181.)? between Xnyn Zn ^'ni/'n z',, 

 and t, contain the law of rectiUnear and uniform motion of the centre of gravity of the 

 system ; and the 6n^ 6 equations of the same order, (182.), between the 6 w — 6 

 variables I ?? ^ ^r'^ 3/'^ z'l and the time, are forms for the differential equations of internal 

 or relative motion. We might eliminate the 3 n—3 auxiliary variables a?',y^ z'l between 

 these last equations, and so obtain the following other group of 3 ^^ — 3 equations of 

 the second order, involving only the relative coordinates and the time, 



> 



m 8^ 



dt. 



(182.) 



/ = 1 ^ . 



^ — w s? "*" w„ ^/ 



. ru 



su 



y 



(183.) 



but it is better for many purposes to retain them under the forms (182.), omitting, 

 however, for simplicity, the lower accents of the auxiliary variables .r'^y, z\, because it 

 is easy to prove that these auxiliary variables (180.) are the components of centrobaric 

 velocity, and because, in investigating the properties of internal or relative motion, 

 we are at liberty to suppose that the centre of gravity of the system is fixed in space, 

 at the origin oi xy z. We may also, for simplicity, omit the lower accent of 2^, un-^ 

 derstanding that the summations are to fall only on the first ^^ — 1 masses, and de- 

 noting for greater distinctness the nila. mass by a separate symbol M ; and then we 



