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X. On the Mathematical Theory of Stream-lines , especially those with four Foci and 
upwards. By William John Macquorn Rankine, C.F., LL.l)., F.B.SS. Bond. 
& Edin., &c. 
Received January 1, — Read February 10, 1870. 
Introduction. 
§ 1 . Object and Occasion of this Investigation. — A Stream-line is the line that is traced 
by a particle in a steady current of fluid. Each individual stream-line preserves its figure 
and position unchanged, and marks the track of a filament or continuous series of par- 
ticles that follow each other. The motions in different parts of a steady current may be 
represented to the eye and to the mind by means of a group of stream-lines ; for the 
direction of motion of a particle at a given point is that of a tangent to the stream-line 
which traverses that point ; and when the fluid is of constant density, as is sensibly the 
case with liquids, the comparative velocities at different points are indicated by the com- 
parative closeness of the stream-lines to each other. Even when the fluid is gaseous, 
the comparative mass-velocities are indicated by the closeness of the stream-lines — the 
term mass-velocity meaning the mass which traverses a unit of area in a unit of time. 
Gaseous fluids, however, will not be considered in the present paper. 
Stream-lines are important in connexion with naval architecture ; for the curves 
which the particles of water describe relatively to a ship, in moving past her, are stream- 
lines ; and if the figure of a ship is such that the particles of water glide smoothlv over 
her skin, that figure is a stream-line surface *, being a surface which contains an indefi- 
nite number of stream-lines. The stream-lines of a current gliding past a circular 
cylinder in a direction transverse to its axis, and also those of a current gliding past a 
sphere, have long been known. 
In a paper entitled “ On Plane Water-lines in two Dimensions,” read to the Royal 
Society in I860, and published in the Philosophical Transactions, I have given a detailed 
*' Note added December 1870. — This limitation is necessary in speaking of the figures of ships ; for although 
every surface is a possible stream-line surface, the surface of a ship is not even approximately an actual stream- 
line surface unless it is such that she does not drag along with her a mass of eddies of such volume and shape 
as to cause the actual tracks of the particles of water to differ materially in form from those which would be 
described in the absence of eddies. The surfaces which fulfil this condition are what are called by shipbuilders 
“fair ” surfaces ; and their forms have in a great many cases been determined by practical experience. In 
order to determine, at all events approximately, the actions of such surfaces on the water, it is necessary to be 
able to construct them by geometrical rules based on the principles of the motion of fluids ; and the methods 
described in this paper afford the means of doing so. — W. J. M. R. 
MDCCCLXXI. V2 P 
