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PROFESSOR W. J. MACQUORN RANKINE ON THE 
investigation of the mathematical properties of a very extensive class of stream-lines, 
representing the motions of particles of water in layers of uniform thickness. Those 
stream-lines closely resemble the water-lines, riband-lines, and other longitudinal sections 
of ships of a great variety of forms and proportions ; and there is scarcely any known 
figure of a fair longitudinal line on a ship’s skin to which an approximation may not be 
found amongst them; hence I have proposed to call them Neoids ; that is, ship-shape 
lines. 
In the Philosophical Magazine for October 1864, was published a paper which had 
been read by me to the British Association, containing a summary of the properties of 
some additional kinds of stream-lines, some in two, and others in three dimensions, and 
of those stream-lines in particular which generate stream-line surfaces of revolution. 
All these stream-lines also are neoids, or ship-shape curves. 
All the neoid stream-lines before mentioned are either unifocal or bifocal ; that is to 
say, they may be conceived to be generated by the combination of a uniform progres- 
sive motion with another motion consisting in a divergence of the particles from a cer- 
tain point or focus, followed by a convergence either towards the same point or towards 
a second point. Those which are continuous closed curves, when unifocal are circular, 
and when bifocal are blunt-ended ovals, in which the length may exceed the breadth 
in any given proportion — for example, the curves marked L B in figs. 2, 3 & 4, Plate 
XV. To obtain a unifocal or bifocal neoid resembling a longitudinal line of a ship 
with sharp ends, such as A, fig. 1, it is necessary to take a part only of a stream-line, 
and then there is discontinuity of form and of motion at each of the two ends of that 
line. 
The occasion of the investigation described in the present paper was the communi- 
cation to me by Mr. William Froude of some results of experiments of his on the re- 
sistance of model boats, of lengths ranging from 3 to 12 feet. A summary of those re- 
sults is published at the end of a Report to the British Association, “ On the State of 
Existing Knowledge of the Qualities of Ships.” In each case two models were com- 
pared together of equal displacement and equal length ; the water-line of one was a 
wave-line, as at A (Plate XV. fig. 1), with fine sharp ends ; that of the other had blunt 
rounded ends, as at B — suggested, Mr. Froude states, by the appearance of water-birds 
when swimming. At low velocities, the resistance of the sharp-ended boat was the 
smaller ; at a certain velocity, bearing a definite relation to the length of the model, the 
resistances became equal ; and at higher velocities the round-ended model had a rapidly 
increasing advantage over the sharp-ended model. 
Hence it appeared to me to be desirable to investigate the mathematical properties of 
stream-lines resembling the water-lines of Mr. Froude’s bird-like models ; and I have 
found that endless varieties of such forms, all closed curves free from discontinuity of 
form and of motion, may be obtained by using four foci instead of two. They may be 
called, from this property, quadrifocal stream-lines , or, from the idea that suggested 
