PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 131 



dt^ll = ^{d^'^nT^ dvT^f}); (A2.) 



and in order to separate the absolute motion of the whole system in space from 

 the motions of its points among themselves, let us choose the following marks of 



position : 



X .mx 'X.my X.mz ^ 



^ii~ ^m ' y^i ~ Xm ' ^11 ~ Xm ' (1/6.) 



and 



|,.=:a?,--^„, ;?. =3/. --3/„, ^. = ;5. -2;„; (177.) 



that is, the 3 rectangular coordinates of the centre of gravity of the system, referred 

 to an origin fixed in space, and the 3 /i— 3 rectangular coordinates of the n— 1 masses 

 Wi m2 . . . m„ _ 1, referred to the nth. mass m„, as an internal and moveable origin, but 

 to axes parallel to the former. We then find, as in the former Essay, 



T = i(a/,2-|-y7 + ;,7)2m 



the sign of summation 2^ referring to the first w — 1 masses only ; and therefore. 



. (178.) 



^.Xm 



+ 4 ^-i{(.W + 67)^ + 07 > 



+^,{(^47+(^-w)^+(^'S)7-. 



(179.) 



If then we put for abridgement. 



^1— m 8^' 

 , _ J_ ST 



, _ 1 8T 



Xm ' 



, X^.mrl 

 n — 



Xm ' 

 Xm ' 



> 



(180.) 



(B«.) 



m 8 ^' "~ ^ 

 we shall have the expression 



H = 4 W + y / + «?) 2 m + i 2, . m (^ 2 + y,2 + a'/) - 



+ 2^ «2, . m ^,)2 + (2, . my,)2 + (2, . m ^7} - U, 



of which the variation is to be compared with the following form of (A^.), 

 dt^K = (da^i,lj[/„ — da/iilx^, + di/,fy',i — dy\^y^^ + c^z^^Sz'^, — dz'„hz,) 2 w 

 + ^,.m(id^^a/, - dx',^^ + dTjli/, - dy\^7}-\- dZ,^z\ - dz',m 

 in order to form, by our general process, 6n diiferential equations of motion of the 

 first order, between the 6 n quantities x,^i/i, %,, j/^^ y,l z\, InZ, x' ,y\ z\ and the time t. In 

 thus taking the variation of H, we are to remember that the force- function U de- 

 pends only on the 3 w — 3 internal coordinates In^ being of the form 



s 2 



} . (C2.) 



