of Edinburgh, Session 1871—72* 
669 
it. Some or all of these solids being perforated, let 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 Vol. 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//, (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 
<£>... denote the generalised components of momentum and of force 
[Thomson and Tait, § 313 (a) (6)] relatively to i{/, <p..., and if 
k, k , ... X, K'... denote corresponding elements relatively to x> 
Xh.., we have (Hamiltonian form of Lagrange’s general equations) 
d£ bT drj bT 
dt d\f/ ’ dt dp 
d\< bT .p. dK.' bT 
It + Hx = ~di + ~<hl = 
( 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, X, 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 JJda- denote integration over 
the whole area of O : then 
f/Kda = x . . . ( 2 ); 
and 
• • • ( 3 )> 
which is a symbolical expression of the definition of x- To the 
