( 390 ) 
This determinant proves to be of* the following form 
S* - s 4- X = 0.. (4) 
Out of the properties of the (G) and {M) surfaces we know that 
(4) will be sure to have two positive roots corresponding to the two 
points of surface (M ). Now the condition of stability is, that the centre 
of gravity of the parallelopipedon lies between x, y, z of the surface 
(G) and the most adjacent point §, rj, S on the normal in x,y,z. 
Out of (2) and (3) then follows, that a must be smaller than each 
of the roots of (4), which comes to: 
.(5) 
o s <* + X > 0.(6) 
The calculations of the positions of equilibrium soon become 
impossible, but the shape of the surface (M) often makes it possible 
that much is said of the positions of equilibrium. This surface (M) is 
the locus of the points where 2 adjacent normals of surface (G) coincide. 
When passing the surface (M) we shall have to get 2 real normals 
more or less, whilst on the contrary the number of normals we can 
construct, out of 2 given points, to the surface ( G ) is the same if starting 
from one point we can reach the other without passing the surface 
(3/), whilst their properties of being minimum distances or not remain; 
as these too change only after coincidence of two normals. We must 
thus try to determine the number and the shape of the compartments 
in which the surface (M) divides the space. The situation of the 
parts will inform us how the point out of which the normals are 
constructed lies with respect to the two sheets of the surface {M), 
which position is again the same for all points in the same 
compartment. To obtain then a graphic solution, the results found 
must be applied in the cube discussed in § 4. Each compartment in 
that cube points to a definite number of positions of equilibrium, 
amongst which again a definite number of stable positions appear. 
After this we must still investigate whether the normal indicating a 
stable position points to a serviceable level plane, i.e. to an immersed 
part of the form for which the surface {G) is determined. 
In this or suchlike manner, the following results have been obtained. 
7. Second case. 
This is not to be realized: no parallelepipedon can float in a 
stable way with but one point above or below the surface of the 
liquid; yet the latter case, where s < is submitted to a closer 
investigation on the supposition, that the faces of the parallelepipedon, 
concurring in the immersed point, are indefinitely produced without 
