257 
the Stability of Ships. 
Vessel coincide with the arcs of the conic parabola, and the 
sides of another vessel coincide with the arcs of another conic 
parabola, whatever be the form thereof, varying according to 
the parameter, the weights of the vessels, breadths at the 
water-line, and the other conditions being the same in both 
cases, the stabilities of the two vessels, at all equal angles of 
inclination, will be equal. If, for instance, the forms of two 
vessels should be such as are represented in Tab. XII. fig. 21. 
and fig. 22. the weights and other conditions being the same, 
the stabilities of each of these vessels will be equal to that of a 
vessel PBQFAK, the sides of which are plane surfaces, paralleL 
to the masts. 
The propositions immediately preceding, relate to the conic 
or Apollonian parabola : they have been inserted, with a view 
of establishing and extending the theory of stability. It may 
also be remarked, that the sides of vessels are in some instances 
constructed nearly of these forms; for the same reasons, it 
may be not altogether useless, to examine on what principle the 
stability of vessels is to be investigated, when the forms of the 
sections are parabolic curves of the higher orders, such as are 
represented in fig. 23. The line cBCO is a conic or Apollonian 
parabola, dBDO is a cubic, and <?BEO a biquadratic parabola. 
/BFO (fig. 23.) is a parabola of 8 dimensions, and g*BGO 
a parabola of 50 dimensions, which are drawn from a geome- 
trical scale, in order to give a true representation of the forms 
of these curves. 
The general equation, determining the relation between the 
abscissas and ordinates of any parabola, of the dimensions n y is 
y n =zp n ~ 1 x x, if the ordinates are drawn perpendicular to the 
axis of the curve; and y = a -f px + qx* + rx\ &c. -f vx\ 
MDCCXCVIII. L 1 
