380 
DR. W. J. MACQITORN RANKINE ON PLANE WATER-LINES, 
where ip denotes a function of b and q, such that 
x d\ n . 
’ ‘ ‘ ’ 
and consequently that 
c 2 ( (Pty \ 2 /d q ^\ 2 
u^ + v 2 \dbdq) ~^~\db' 2 ) ' 
(51) 
(52) 
The curves to which this method of investigation leads are inferior to oogenous neoids 
as water-lines for ships, because they have comparatively sharp curvature amidships, 
which causes them to have small capacity for their length and breadth, and would give 
rise to comparatively sudden changes in the speed of gliding. They will therefore not 
be further discussed in the present paper, except to state that the simplest of them is 
the well-known cissoid. 
Section III. — Remarks on the Practical Use of Oogenous Water-Lines. 
(23.) Previous Systems of Water-Lines. — Owing principally to the great antiquity of 
the art of shipbuilding, and the immense number of practical experiments of which it 
has been the subject, that part of it which relates to the forms of water-lines has in many 
cases attained a high: degree of excellence through purely empirical means. Excellence 
attained in that manner is of an uncertain and unstable kind ; for as it does not spring 
from a knowledge of general principles, it can be perpetuated by mere imitation only. 
The existing forms of water-lines, whose merits are known through their practical 
success, constitute one of the best tests of a mathematical theory of the subject; for if 
that theory is a sound one, it will reproduce known good forms of water-line ; and if it is 
a comprehensive one, it will reproduce their numerous varieties, which differ very much 
from each other. 
The geometrical system of Chapman for constructing water-lines is wholly empirical ; it 
consists in the use of parabolas of various orders, chosen so as to approximate to figures 
that have been found to answer in practice, and it has no connexion with any mechanical 
theory of the motion of the particles of water. 
The first theory of ships’ water-lines which was at once practically useful, and based 
on mechanical principles, was that of Mr. Scott Russeel, explained in the first and 
second volumes of the Transactions of the Institution of Naval Architects. It consists 
of two parts ; the first has reference to the dimensions of water-lines intended for a given 
maximum speed, and prescribes a certain relation between the length of those lines and 
the length of a natural wave which travels with that speed ; the second part relates to 
the form of those lines, and prescribes for imitation the figures of certain natural waves, 
as being lines along which water is more easily displaced than along other lines. The 
figures thus obtained are known to be successful in practice but it is also well known 
that there are other figures which answer well in practice, differing considerably from 
those wave-lines; and it is desirable that the mathematical theory of the subject should 
