370 
DB. W. J. MACQUOEN EANKINE ON PLANE WATEE-LINES. 
constant be the equation of a series of surfaces each containing a continuous series of 
water-line curves (and one of which surfaces must be that of the solid body), the function 
U must satisfy the following condition, 
dV dp c?U dp dXJ dp q. 
dx ' dx' dy dy' dz dz * 
(3) 
or if ds' be an elementary arc of a water-line curve, and x', y', z' its coordinates, the 
following conditions must be satisfied, 
dad dy' dz 1 . dp dp dp 
ds' ’ ds' ds' ' ' dx ' dy ' dz’ 
(4) 
and these are the most general expressions of the geometrical properties of water-line 
curves in three dimensions. 
When the inquiry is restricted to motion in two dimensions only, x and y, the terms 
containing dz and dz' disappear from the preceding equations; and it also becomes 
possible to express the same conditions by means of equations of a kind which are more 
convenient for the purposes of the present investigation, and which are as follows. 
Conceive the plane layer of liquid under consideration of thickness unity, to be divided 
into a series of elementary streams by a series of water-line curves, one of which must 
be the outline of the solid body; let U= constant be the equation of any one of those 
curves, U being a function of such a nature that is the volume of liquid which flows 
in a second along a given elementary stream ; then the components of the velocity of a 
particle of liquid are 
JU 
dx ’ 
(5) 
the condition of continuity is satisfied ; and the condition of perflect fluidity requires 
that the function U should fulfil the following equation, 
fU fU__ 
dx 2 ' dy 9 ’ 
( 6 ) 
(When the motion of the liquid is not subject to the condition of being uniform in 
velocity and direction at an infinite distance in every direction from the solid, it is 
sufficient that 
<T-U . d * U , , TT 
■^+'a^= functlonofII: 
but cases of that kind do not occur in the present paper*.) 
(3.) Notation . — It is purely a question of convenience whether the infinitely distant 
particles of the fluid are to be regarded as fixed and the solid as moving uniformly, or 
* Professor William Thomson, in 1858, completed an investigation of the motion of a solid through a perfect 
liquid, so as to obtain expressions for the motion of the solid itself, involving twenty-one constants depending 
on the figure and mass of the solid and the density of the liquid ; but as that investigation, though on the eve 
of publication, has not yet been published, I shall not here refer to it further. 
