DR. W. J. MACQUORN RANKINE ON PLANE WATER-LINES. 
387 
embrace those figures also. It may further be observed that the figure of the solitary 
wave, as investigated experimentally by Mr. Scott Bussell (Keports of the British Asso- 
ciation, 1845), and mathematically by Mr. Earnshaw (Camb. Trans. 1845), is that of a 
wave propagated in a canal of small breadth and depth as compared with the dimen- 
sions of the wave, and in which particles of water originally in a plane at right angles 
to the direction of motion continue to be very nearly in a plane at right angles to the 
direction of motion, so as to have sensibly the same longitudinal velocity. This state of 
things is so different from the circumstances of the motions of the particles in the open 
sea, that it appears desirable to investigate the subject with special reference to a mass 
of water of unlimited breadth and depth, as has been done in the previous sections of 
this paper. 
(24.) Variety of Forms of Oogenous Neoids, and their likeness to good known Forms of 
Water-Line.— The water-lines generated from ovals which have been described in the 
second section of this paper, are remarkable for the great varieties of form and propor- 
tions which they present, and for the resemblance of their figures to those of the water- 
lines of the different varieties of existing vessels. There is an endless series of ovals, 
having all proportions of length to breadth, from equality to infinity ; and each of those 
ovals generates an endless series of water-lines, with all degrees of fulness or fineness, 
from the absolute bluffiiess of the oval itself to the sharpness of the knife-edge. Fur- 
ther variations may be made by taking a greater or a less length of the curve chosen. 
The ovals are figures suitable for vessels of low speed, it being only necessary, in 
order to make them good water-lines, that the vertical disturbance (as explained in 
art. 20) should be small compared with the vessel’s draught of water. At higher speeds 
the sharper water-lines, more distant from the oval, become necessary. The water-lines 
generated by a circle, or “ cyclogenous neoids,” are the “ leanest ” for a given propor- 
tion of length to breadth ; and as the excentricity increases, the lines become “ fuller.” 
The lines generated from a very much elongated oval approximate to a straight middle 
body with more or less sharp ends. In short, there is no form of water-line that has been 
found to answer in practice which cannot be imitated by means of oogenous neoids. 
(25.) Discontinuity at the Bow and Stern. — Best limits of Water-Lines. — Amongst the 
endless variety of forms presented by oogenous water-lines, it may be well to consider 
whether there are any which there are reasons for preferring to the others. One of the 
questions which thus arise is the following : — Inasmuch as all the water-line curves of a 
series, except the primitive oval, are infinitely long and have asymptotes, there must 
necessarily be an abrupt change of motion at either end of the limited portion of a 
curve which is used as a water-line in practice, and the question of the effect of such 
abrupt change or discontinuity of motion is one which at present can be decided by 
observation and experiment only. Now it appears from observation and experiment, 
that the effect of the discontinuity of motion at the bow and stern of a vessel which has 
.an entrance and run of ordinary sharpness and not convex, extends to a very thin layer 
of water only, and that beyond a short distance from the vessel’s side the discontinuity 
