286 
PKOFESSOK W. J. MACQUOEN KANKINE ON THE 
the solid negative. The velocity of the centre of mass of the liquid relatively to the 
solid being still taken as unity, its velocity relatively to the common centre is expressed 
as follows, L being the total mass of the liquid, 
D 
L + D 
(46 a) 
The velocity of the solid relatively to the common centre is 
L 
L + D’ 
and the respective equal and opposite momenta of the solid and liquid relatively to the 
same point are expressed by 
, LD 
— L + D' 
When the mass of liquid L becomes indefinitely great, becomes indefinitely small, 
L J 13 
— k + p approximates indefinitely to —1, and ± ^ to ±D; but notwithstanding 
these indefinitely close approximations, it is necessary to bear in mind that (as is implied 
in equation 46) the component longitudinal velocity of current u is taken relatively to 
the centre of mass of the liquid, and not relatively to the common centre of mass, the 
corresponding component relatively to the common centre being 
u 
D 
L + D’ 
If the liquid is absolutely free from stiffness and friction, the resultant pressure exerted 
between it and the solid in a horizontal direction is obviously equal to nothing, so long 
as the velocity is uniform, and only acquires a value in the event of acceleration or 
retardation ; which value is expressed by the rate of change per second in the equal and 
, LD 
opposite momenta ijTTjj* 
To adapt the formulae of this and the ensuing sections to other velocities and densities 
than those denoted by unity, let —V be the velocity of the solid, and ^ the density of 
the liquid ; then quantities denoting velocities are to be multiplied by V, those denoting 
masses by g>, those denoting momentum by Vg>, those denoting heights due to velocities 
by V 2 , those denoting energy, and those denoting intensity of pressure, by V 2 ^. 
It is to be observed that, according to the notation of this paper, motion ahead is 
treated as negative, and motion astern as positive, the latter being the direction of the 
motion of the liquid relatively to the solid. 
§ 13. Energy of Currents and of Disturbance. — The energy of the motion of the liquid 
mass contained within a given space may be taken either relatively to the disturbing- 
solid, considered as fixed, in which case it may be called the energy of current , or rela- 
tively to the undisturbed liquid, in which case it may be called the energy of disturbance. 
Assuming unity, as before, for the values of the undisturbed velocity and of the density, 
