DR. W. J. MACQTTORN RANKINE ON PLANE WATER-LINES. 
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following equations : — 
u j l 2 (y i —x‘ 2 ) . v 2 l 2 xy. \/{(u— c) 2 + fl 2 } If 
c r A c r 4 c r* ' ■ ' 
As a convenient name for water-line curves of this sort, it is proposed to call them 
Cyclogenous Neoids, that is, sliijg-shape curves generated from a circle. 
The water-line surfaces generated by a sphere are known ; but no use will be made 
of them in this paper. (See paper by Dr. Hoppe, Quart. Journ. Math., March 1856.) 
Section II. — Properties of Water-line Curves generated from Ovals , or Oogenous Neoids. 
(6.) Derivation of other Water-Line Curves from Cyclogenous Neoids. — When a form 
of the function F has been found which satisfies equation (12) of art. 4 (that is to say, 
which fulfils the condition of liquidity), an endless variety of other forms of that function 
possessing the same property may be derived from the original form by differentiation 
and integration. 
The original form, and also the derived forms, must possess the properties of vanish- 
ing for x=co and for y= oo , and of becoming =0 or a constant for y= 0. The first of 
those properties excludes trigonometrical functions, and consequently exponential func- 
tions also, which are always accompanied by trigonometrical functions, and leaves avail- 
able functions of the nature of potentials. The second property excludes derivation by 
means of differentiation and integration with respect to y, and leaves available differen- 
tiation and integration with respect to x. 
The original form of the function F which will be used in this paper is that appro- 
priate to cyclogenous neoids, or water-line curves generated from a circle, as given in 
equation (14) of art. 5, viz. — 
F=-^X constant. 
When one or more differentiations with respect to x are performed on this function, and 
the results substituted for F in equation (10), there are obtained curves which are real 
water-lines, but which are not suitable for the figures of ships, some of them being lemnis- 
cates, others shaped like an hour-glass, and others looped and foliated in various ways. 
It is otherwise as regards integration with respect to x; for that operation, being performed 
once, gives the expression for the ordinate in a class of curves all of which resemble 
possible forms of ships, and which are so various in their proportions, that every form of 
ships’ water-lines which has been found to succeed in practice may be closely imitated by 
means of them. As that class of curves consists of certain ovals, and of other water-lines 
generated from those ovals, it is proposed to call them Oogenous Neoids (from ’Qoyevvc). 
(7.) General Equation of Oogenous Neoids. — The integration with respect to x, already 
referred to, is performed as follows : — The coordinates of a particle of water being x and 
y, let x’ denote the position of a moveable point in the axis of x : then the function to 
be integrated is 
y . . 
(x—x’f+y' 2 
