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alcohol. 
Mathematics. — “Five rotations in S, in equilibrium.” By Dr. S. L. 
van Oss, Zaltbommel. (Communicated by Prof. P. H. ScHovre). 
In a previous paper (these Proceedings, Vol. IV, p. 218) the investi- 
gation of the elementary motion in S, was reduced to the consideration 
of the elementary motion in 8S, by making use of a principle to be 
read as follows: A system of rotations about planes all passing 
through one and the same point is in equilibrium when their inter- 
sections with an arbitrary S, are in equilibrium. Here we mean by 
section of a rotation with any S, the rotation of the intersecting 
space caused by its component about the plane orthogonally cutting 
this S, in the intersection of the plane of rotation. 
As an immediate result of this principle we can state the conditions 
under which three to seven planes through one point can be the 
planes of a system of rotations in equilibrium. Thus i. a. the con- 
dition for four planes, that they must belong to a hyperboloidic 
pencil, ete. ete. 
We now wish to extend this principle in order to arrive by 
investigation also at the case, that the planes do not pass any longer 
through one and the same point. 
It goes without saying, that if a system of rotations is in equilibrium, 
its section with every S, must be in equilibrium. The question 
