MATHEMATICAL THEORY OE STREAM-LINES. 
269 
such shapes to Mr. Froude, Cycnoids, or swan-like lines ; while the stream-lines in 
which particles of liquid flow past them may be said to be Cycnogenous * . 
Chapter I. Summary of Cinematical Principles. 
§ 2. Normal Surfaces to Stream-lines in a Liquid . (For details on this part of the 
subject, see Stokes “ On the steady Motion of an Incompressible Fluid,” Cambridge 
Transactions, 1842; also Rankine “On Plane Water-lines in Two Dimensions,” Philo- 
sophical Transactions, 1863.) — Let a perfectly liquid mass of indefinite extent flow 
past a solid body in such a manner that, as the distance from the solid body in any di- 
rection increases without limit, the motion of the liquid particles approaches indefinitely 
to uniformity in velocity and direction. Let u, v, and w be the rectangular components 
of the velocity of any particle ; then the condition of constant density requires that the 
following equation should be fulfilled, 
du dv dw „ 
dx~^~ dy~^~ dz ’ 
( 1 ) 
and the condition of perfect fluidity being combined with that of the approximation to 
uniformity of motion at an indefinite distance requires that the three following equations 
should be fulfilled : 
dv dw _ dw du _ du dv _ , . 
dz dy ’ dx dz ’ dy dx ' ' 
These four conditions are fulfilled by making 
d<p dp dp 
U = Tx’ V =Ty ’ W ~dz' 
(3) 
the velocity-function , <p, being a function which fulfils the condition 
The equation 
df df df\ 
dx 9 dip dz 9 J ^ 
<p=a (a constant) 
(4) 
( 5 ) 
is that of a surface of equal action, which is normal to the direction of motion of every 
particle that it traverses ; in other words, it is normal to all the stream-lines that it cuts. 
If a series of different values be given to the constant a, the equation (5) represents a 
series of such normal surfaces; and every stream-line is a normal trajectory to that series 
of surfaces. In symbols, let ds' denote an elementary arc of a stream-line, and x 1 , y 
and z! the coordinates of a fixed point in it, those coordinates being regarded as functions 
* K vKvoedris, KVKi'oyevr/s. It is to be observed that the swan-like curves here described are different from 
the lines of the vessel which some years ago was built from the designs of Mr. Peacock, and described in the 
Mechanics’ Magazine ; for the lines of that vessel are oval, and approximate to bifocal neoids, and are wholly 
without the peculiarly shaped ends that characterize Mr. Feoude’s cycnoi'd models. 
2 p 2 
