DE. W. J. MACQUORN RANKINE ON PLANE WATEE-LINES. 
389 
III. For the trajectory of slowest gliding, LN, there may be substituted, without 
practical error, a straight line cutting the axis OX in L at an angle of 45° ; and when 
this has been done, the excentricity OA or a is almost exactly equal to the length 
x- 634 (=i77l) ; 
and this of course is also the ratio of the area to the circumscribed rectangle. The base 
OL or l also is very nearly equal to (the sum of the length and breadth) X '634. 
IV. Hence the following approximate construction. Given, the common length QR 
of a set of water-lines of smoothest gliding which are to have a common termination at 
Q, and their breadths RP 1? RP 2 , RP 3 , &c. : required, to find their areas, bases, and foci. 
Through Q and R draw the straight lines QU and RU, making the angles RQU=45°, 
QRU=30°. Through their intersection U draw UY perpendicular to RQ. All the 
required foci will be in UY ; and RY will be the length of the rectangles equivalent to 
each of the water-line areas ; so that 
area Pj QRj=RV X RPi , 
area P 2 QR 2 =RV X RP 2 » 
&c. &c. 
Through P 1? P 2 , P 3 , &c. draw lines parallel to RU, cutting QU in Lj , L 2 , L 3 , &c. : these 
points will be the ends of the bases required, through which draw the bases L, , 
L 2 0 2 , L 3 0 3 , &c. parallel to QR, and cutting YU in Aj , A 2 , A 3 , &c. : these will be 
the required foci. 
The bases and foci and the points Pj , P 2 , P 3 , &c. being given, the water-lines are to be 
constructed by the rules given in article 11. 
(28.) Lissoneoids compared with Trochoids. — In fig. 5, Plate IX., the full line PQ is a 
lissoneoid, and the dotted line P q a trochoid of the same breadth and area. The curves 
lie very near together throughout their whole course — the only differences being, that 
the trochoid is slightly less full and more hollow than the lissoneoid, but at the same 
time the trochoid is the longer and has a greater frictional surface. Had the entrance 
of the trochoid consisted of a straight tangent from its point of contrary flexure (as in 
the bow of the ‘ Lancefield,’ mentioned in the note to article 25), the two curves 
would have lain still closer together. The same likeness to a trochoid is found in all 
lissoneoids whose length is more than about 3^ times the breadth. 
(29.) Combinations of Bow and Stern. — Although there is reason to believe that 
water-lines of equal length and similar form at the bow and stern, such as are produced 
by using one neoid curve throughout, are the best on the whole, still the naval archi- 
tect, should he think fit, can combine two different oogenous neoids for the bow and 
stern; or, according to a frequent practice, he may adapt the figure of the stern to 
motion of the particles in vertical layers instead of horizontal layers ; provided he takes 
care in every case that the midship velocity of gliding ( u 0 „, as given by equation (28 a) 
of article 13) is the same for each bow water-line and stern water-line at their point of 
junction. 
