259 
the Stability of Ships. 
ultimately coincides, when the dimensions are increased sine 
limite. This extreme case has relation to the subject of stabi- 
lity : for, whatever may be the effect of giving to the sides of 
ships the forms of the several higher orders of parabolas, it is 
certain, that as the dimensions of these curves are increased, 
the stability will approach to that which is the consequence of 
making the sides parallel to the masts ; but it has been shewn, 
that when the sides coincide with the form of a conical para- 
considered as a parabolic curve of infinite dimensions, the two portions of the curve 
BH, HO, (fig. 23.) being inclined at a right angle, when coincident with the sides of a 
rectangular parallelogram : but, since the curvature is nothing at the vertex O, the ab- 
scissa being then — o, and before the abscissa has increased to any finite line, the cur- 
vature at the extremity of the corresponding ordinate OH is infinite ; and since the cur- 
vature between the points O and H must necessarily pass through all the intermediate 
gradations of magnitude, it becomes a question to define the abscissa and correspond- 
ing ordinate, when the radius of curvature is a finite line : 2dly, when it becomes eva- 
nescent ; and, lastly, when it is again infinitely great. By referring to the preceding 
expressions for the abscissa and corresponding radius of curvature, it is found, that if 
p represents the parameter, and x is made — — , (the number n denoting the dimen- 
sions of the curve,) when n is increased sine limite, the radius of curvature will be 
greater than any line that can be assigned : and such is the curvature of any portion of 
the line OH, between the points O and H. 2dly, if x is = the radius of cur- 
vr 
vature will be = p, the ordinate approximating to equality with the line OH. 3dly, 
if u: — — - — , the radius of curvature will be smaller than any finite line : and, lastly, 
if x — p, or any finite line, the radius of curvature will be greater than any assignable 
line : which conclusions are immediately inferred from the equation expressing the 
2 n — 2 2 n — 2 ") 
radius of curvature, or r -p X — < * when the number of di- 
n x «— 1 x p n x x n 
mensions n is increased sine limite, these successive changes in the radius of curvature 
taking place while the abscissa x is increased from o to any finite magnitude. 
LI 2 
