( 385 ) 
lie in such a way that the centre of gravity of the cylinder lies between 
the centre of gravity of the immersed part and the centre of cur¬ 
vature which is the extremity of the smaller radius of curvature, a 
possible position of equilibrium will also be a stable one. 
On the surface ( G ) lie the points obtained by determining the 
centres of gravity of the segments of - the normal section, when 
every segment is equal to' s X the surface of the base and the 
normal section is taken halfway the height of the cylinder. 
Let PLQ be such a segment and let R be its centre of gravity. 
The locus of these centres will be a closed curve, convex in all its 
points, because the surface ( G ) is convex. The centre of gravity G 
of the cylinder will lie inside this curve. Now there will be among 
the vectors of G to the points of the locus at least one minimum: 
for, starting from a definite vector we return to the same value the 
curve being closed. Let in fig. 1 GR be this vector or one of the 
other minimum ones. Now to this minimum vector corresponds a nor¬ 
mal, pointing to a state of equilibrium stable for displacements where 
the parallelism of the generatrices with the surface of the liquid is 
retained, from which ensues that the extremity of the radius of 
curvature of the locus in R lies farther on this normal than G 
unless it coincides with G as a limiting case. 
Let PQ = p be the cord lying in the level plane. The latter is 
a rectangle with sides p and l (— length of the cylinder). The 
principal moments of inertia of this rectangle are: 
The volume of the immersed part is il when i is the surface of 
the cut-off segment. The two principal radii of curvature in the 
point R of the surface ( G ) are therefore : 
Now is the radius of curvature of the locus drawn in the 
figure. We have therefore, supposing RG to be a minimum vector, 
V, > RG. 
If now p x >p s or Z> p we are assured of the stability. The ratio 
3 > mentioned by Huygens without proof and with reserve, is there¬ 
fore unnecessarily large, it may be replaced by unity. For, the most 
unfavourable case is the one, where PQ, the level section belonging 
to the minimum value of RG, just coincides with the greatest dimension 
of the normal section. If then we take l larger than that greatest 
