255 
the Stability of Ships. 
the vessel’s stability, when inclined from the perpendicular 
through the given angle MOI. 
From the preceding construction and demonstration, a pro- 
perty of stability is inferred, which may be expressed in the 
following proposition. 
If the vertical sections of a vessel are terminated by the arcs 
of a conic parabola, and the sides of another vessel are parallel to 
the plane of the masts, both above and beneath the water-line, 
the stabilities of the two vessels will be equal at all equal in- 
clinations from the upright, if the breadths at the water-line 
BA, and all the other conditions, are the same in both cases. 
It is thus demonstrated : 
For brevity, let the angle of inclination from the upright, 
or the angle ASH, be denoted by the letter S ; let BA = t } 
and LD = a : rad. = 1 . 
From the preceding construction and demonstration, it ap- 
pears that XV : XN : : 1 : tang. S, and by the properties 
of the figure -§-XN : XV : : XV : p , joining these 
ratios - 1:2 : : XV : p x tang. S. 
Wherefore XV = 'tl 
2 5 
and XL = — = £211221^ — LN • 
P 4 
also, since XV : NV : : cos. S : 1 , 
NV = xy P * tan g- s . 
cos. S 2 x cos. S 5 
and because LD = VK = a, and LE = VQ = JZ-, 
and the angle VQP = ASH = MOI, it follows that 
VP 
3 a X sin. S 
; and therefore 
NP 
NV + VP == p * tang ’ s ±1 
1 2 x cos. S * 
X sin. S 
I 
