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DE. W. J. MACQUOEN BANKINE ON PLANE WATEE-LINES. 
ceases, through some slight modification of the water-lines of which the mathematical 
theory is not yet adequate to give an exact account*. 
Still, although the effect of the discontinuity in increasing resistance may not yet 
have been reduced to a mathematical expression, and although it may be so small that 
our present methods of experimenting have not yet detected it, it must have some 
value ; and it is desirable so to select the limits of the water-line as to make that value 
as small as possible. In order that the abrupt change of motion may take place in as 
small a mass of water as possible, it would seem that the limits of the water-line 
employed in practice should be at or near the point of slowest gliding ; that is, where 
the water-line curve is cut by the trajectory of slowest gliding LN, in Plate VIII. fig. 1, 
and Plate IX. fig. 3, as explained in articles 14, 15, and 16 ; and that conclusion is borne 
out by the figures of many vessels remarkable for economy of power. 
(26.) Preferable Figures of Water-Lines. — In forming a probable opinion as to which, 
out of all the water-lines generated by a given oval, is to be preferred to the others, 
regard is to be had to the fact, that every point of maximum disturbance of the level 
of the water, whether upwards or downwards, that is to say, every point of maximum 
or minimum speed of gliding (see article 20), forms the origin of a wave, which spreads 
out obliquely from the vessel (as may easily be observed in smooth water), and so 
transfers mechanical energy to distant particles of water, which energy is lost. Hence 
such points should be as few as possible ; and the changes of motion at them should be 
as gradual as possible; and these conditions are fulfilled by the curves described in 
article 17, by the name of “ Lissoneoids,” being those which traverse the point P in 
the figures, and which may have any proportion of length to breadth, from y/3 to 
infinity. 
(27.) Approximate Pules for Construction and Calculation. — The description of those 
curves, already given in article 17, has been confined to those properties which are 
exactly true. The following rules are convenient approximations for practical purposes, 
when the proportion of length to breadth is not less than 4 : 1 (see Plate IX. figs. 3 & 4). 
I. A tangent to the curve at Q, the point of slowest gliding, passes very nearly through 
the point P of greatest breadth. 
II. The area PQR enclosed within the water-line is very nearly equal to the rectangle 
of the breadth PR, and excentricity a. (When the length is not less than six times the 
breadth, this rule is almost perfectly exact.) 
* In confirmation of this, experiments made on the steamers ‘ Admiral ’ and ‘ Lancefield,’ by Mr. J. E. Na- 
pier and the author, may be specially referred to. The water-lines of the ‘ Admiral 5 are complete trochoids, 
and tangents to the longitudinal axis at the how and stem. The engine-power required to drive her at her 
intended speed was computed from the frictional resistance, according to principles explained in publications 
already referred to in the note to article 21 ; and the result of the calculation was closely verified by experiment. 
The water-lines of the ‘ Lancefield ’ are only partly trochoidal, being straight from the point of contrary flexure 
to the how, so that, instead of being tangents there to the longitudinal axis, they form with it angles of about 
13§°. Yet the same formula which gave the resistance of the ‘Admiral ’ has been found to give also the resist* 
ance of the ‘ Lancefield ’ without any addition on account of the discontinuity of motion at the how. 
