669 
of Edinburgh, Session 1871-72. 
it. Some or all of these solids being perforated, let x, x) x, &c., 
be the quantities of liquid which from any era of reckoning, up to 
the time t , have traversed the several apertures. According to an 
extension of Lagrange’s general equations of motion, used in Yol. I. 
of Thomson and Tait’s “ Natural Philosophy,” §§ 331... 336, proved 
in §§ 329, 331 of the German translation of that volume, and to 
be farther developed in the second English edition now in the press, 
we may use these quantities x, x) ••• as if they were co-ordinates 
so far as concerns the equations of motion. Thus, although the 
position of any part of the fluid is not only not explicitly specified, 
but is actually indeterminate, wheni/f, <p , ... x, x) ••• are all given, we 
may regard x, X as specifying all that it is necessary for us to 
take into account regarding the motion of the liquid, in forming 
the equations of motion of the solids; so that if and 'k, 
<f> ... denote the generalised components of momentum and of force 
[Thomson and Taifc, § 313 (a) (5)] relatively to if/, and if 
k, k, ... K, K' . . . denote corresponding elements relatively to x? 
X'..., we have (Hamiltonian form of Lagrange’s general equations) 
dt dxf/ 
dK frT 
dt dx 
? dt dp 
d K' bT 
’ dt + df 
= . 
= K'. 
( 1 ), 
where T denotes the whole kinetic energy of the system, and b dif- 
ferentiation on the hypothesis of rj, ••• k , k ... constant. 
2. To illustrate the meaning of x, K, k, x) let B be one of the 
perforated solids, to be regarded generally as movable, draw an 
immaterial barrier surface O across the aperture to which they 
are related, and consider this barrier as fixed relatively to B. Let 
N denote the normal component velocity, relatively to B and O of 
the fluid at any point of O; and let ffdcr denote integration over 
the whole area of 12 : then 
ff NAr = X 
■ ■ (2); 
X^fdtffKdo- . 
• • (3), 
which is a symbolical expression of the definition of x* To the 
