225 



of this position disappears oxactly lliorc^ wlioi'O fbi- specific w{>i_2;lifs 



1 

 <^ — a hexagonal section hecomes ini|)Ossii)le on account ot Akofh- 

 6 



MF.DES' Law. 



This third manner of tloatini>' was, pntbahly for the first time, 

 referred to in the "Matiiematical Gazette" of Dec. 1908, Vol. 4, 

 p. 338, Math, note N". 285, in which note, however, the second 

 one and tiie case n(nv foHowin^.^' was nol referred to at all. 



In the foiD'th pos/'fion one of the j)lanes passing throngh two 

 opi^osite parallel edges assumes the vertical direction. In this posi- 

 tion one of these edges is partially immei'sed, the other one (piite 

 outside the liquid. In consequence of this the surface section is a 

 pentagon foi- which (he intersection of the liipiid surface with (he 

 plane Just mentioned is an axis of symmeti-y. 



Such "pentagonal" tloating can only exist, however, between 

 narrow limits of density, viz. between (he densities 0,226 . . . and 

 0,24 . . . 



It should be observed that only the first and the second position 

 gradually pass into each other; further that a completely unsymme- 

 trical way of floating, in which neither one of the faces, nor one 

 of the diagonal planes, nor a space-diagonal assumes the vertical 

 position, cannot arise. 



One of the greatest difficulties connected with the problem consisted 

 in the formar exclusion of such cases. 



It further appears that between definite limits of density, several 

 positions, amounting at most to three, are possible for the same 

 cube, viz., 



Below 0,166... the first position is the only possible. 

 From 0,166 ... to 0,211 . . . the first and the third. 

 From 0,211 ... to 0,226 . . . the second and the third. 

 From 0,226 ... to 0,24 . . . (the limits ♦of pentagonal tloating) 



the second, the third and the fourth. 

 From 0,24... to 0,25 the second and the third. 

 Between 0,25 and 0,5 only the third. 



Strictly speaking one case in which one of the diagonal planes 

 coincides with the liquid-level and the specific weight therefore 

 amounts to exactly 0,5 ought to have been added to those mentioned 

 above. Dr. Brandsen has indeed proved that stability exists in this 

 case. Yet at the slightest alteration of (he specific weight the adjacent 

 Dosittons of equilibrium become unstable, e.i. those which arise by 



