292 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
say, let A 0 be the head, or height due to elevation and pressure, in a given elementary 
stream, at a point where the velocity is that of the undisturbed uniform current ; let V, 
as before, denote that velocity, so that uY, vY, and wY are the components of the velocity 
at any other point; then, at that other point, the head is given by the following equa- 
tion, 
V2 
■ v 2 -w 2 ); (61) 
V 2 
h=h 0+^(1- 
*9 
■u 
and the following difference may be called the disturbance of head, 
h — ^o=^(l — u ~ ~ v ' 2 - w ~) • 
Some of the general consequences of this principle have been pointed out in the paper 
“On Plane Water-lines” already referred to; and its bearing on the laws of the resist- 
ance of ships has been shown in a paper “ On the Computation of the probable Engine- 
power and Speed of proposed Ships,” published in the Transactions of the Institution 
of Naval Architects for 1864. 
In connexion with the subject of the present paper, it is sufficient to state that, when 
a current of a perfect liquid of unlimited extent in all directions flows past a solid, the 
disturbance of head takes the form of variation of pressure only, the energy of a given 
particle of an elementary stream changing its form between energy of motion and energy 
of pressure as the velocity varies — so that points of minimum velocity of current are 
points of maximum pressure, and points of maximum velocity of current are points of 
minimum pressure, — but that where the current is bounded above by a free upper surface, 
exposed to the air, that surface continues to be everywhere a surface of uniform pres- 
sure, and the disturbances of head take the form of disturbances of level, places of 
minimum velocity being marked by a swell, and those of maximum velocity by a hollow. 
For example, when a floating solid body, as a ship, moves through Stillwater, the surface 
of the water is raised at those points where the particles of water are pushed or drawn 
ahead by the ship, and depressed at those points where they run astern past her sides in 
order to fill up the space in her wake. 
The aggregate disturbance of head throughout the whole liquid mass is expressed as 
follows, 
V 2 
JI \{h~K)dx dy dz= — — • E D , 
(62) 
being obviously equal, but of contrary sign, to the total energy of disturbance per unit 
of density (see equation 47 a). 
Let the whole volume of the liquid mass be denoted by ~L=*\\\d% dy dz ; then 
JJj (h — // 0 ) dx dy dz V 2 E D 
JjJefo dy dz 2^L 
(63) 
expresses a depression of the centre of gravity of that mass relatively to the surface of 
the liquid at an indefinite distance from the disturbing solid — in other words, an eleva - 
