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X. On Plane Water-Lines in two Dimensions. By William John Macquorn Rankins, 
C.E., LL.D ., F.R.SS. L.&E ., Associate of the Institution of Naval Architects , &c. 
Received July 28, — Read November 26, 1863. 
Section I. — Introduction , and Summary of known Principles. 
(1.) Plane Water-Lines in two Dimensions defined. — By the term “Plane Water-Line in 
two Dimensions ” is meant a curve which a particle of liquid describes in flowing past 
a solid body, when such flow takes place in plane layers of uniform thickness. Such 
curves are suitable in practice for the water-lines of a ship, in those cases in which the 
vertical displacements of the particles of water are small compared with the dimensions 
of the ship ; for in such cases the assumption that the flow takes place in plane layers 
of uniform thickness, though not absolutely true, is sufficiently near the truth for prac- 
tical purposes, so far as the determination of good forms of water-line is concerned. As 
water-line curves have at present no single word to designate them in mathematical 
language, it is proposed, as a convenient and significant term, to call them Neoids (from 
vrioc, the Ionic genitive of vav c). 
(2.) General Principles of the Flow of a Liquid past a Solid. — The most complete 
exposition yet published, so far as I know 7 , of the principles of the flow of a liquid past 
a solid, is contained in Professor Stokes’s paper “ On the Steady Motion of an Incom- 
pressible Fluid,” published in the Transactions of the Cambridge Philosophical Society 
for 1842. So far as those principles will be referred to in the present paper, they may 
be summed up as follows. 
When a liquid mass of indefinite extent flows past a solid body in such a manner that 
as the distance from the solid body in any direction increases without limit, the motion 
of the liquid particles approaches continually to uniformity in velocity and direction, 
the condition of perfect fluidity requires that the three components u, v, w of the velo- 
city of a liquid particle should be the three differential coefficients of one function of the 
coordinates (<p) ; viz. 
( 1 ) 
and the condition of constant density requires that the said function should fulfil the 
following condition, 
<P$ . , d s f q 
( 2 ) 
By giving to the function <p a series of different constant values, a series of surfaces are 
represented, to which each water-line curve is an orthogonal trajectory, so that if U= 
3 d 2 
