( 363 ) 
here, however, is to find out how many of those intersections will 
have to be examined before we can conclude about the system being 
in equilibrium or not. 
To this end we direct our attention in the first place to the case, 
that a system @ has two sections in equilibrium, namely with the 
spaces A and J. 
If the section Q/A is in equilibrium, then 2 must necessarily be 
reducible to a single rotation about a plane in A; likewise, if 2/5 
is in equilibrium, then 2 can be reduced to a single rotation about 
a plane in the space D. 
So from the equilibrium of the sections it does not yet follow 
that the system itself is in equilibrium, for the possibility remains 
that it may be reducible to a rotation about the plane common to 
the two spaces of intersection. 
If, however, we can point out three spaces S, not passing through 
the same point, their sections being in equilibrium, then the equilibrium 
of the system itself is guaranteed. Let us now apply this result to 
determine five planes which can be the bearers of a system of 
rotations in equilibrium. 
The neccessary condition which these planes must satisfy is that 
they be intersected by three spaces S,, not passing through one and 
the same point, in rays of a linear congruence. In other words: They 
must intersect three pairs of straight lines, the director lines of these 
congruences. 
Now we know that in S, there are just 5 planes intersecting 6 
given lines. They are the five “associated planes” of Srerr (Rend. 
di circ. math. di Palermo, t. IL, 1888). 
Now we have the necessary condition; we shall show, that it is 
also sufficient. 
Let 2 be a system of rotations about 5 associated planes, A an 
S, so that 2/A is in equilibrium. If 2 were not in equilibrium itself, 
this system would have to be equivalent to a rotation w about a 
plane @ in A. If we reverse the direction of the rotation about this 
plane, then the combination (€2—w) is in equilibrium. If we now 
consider a second intersecting space B, not through e, then the planes 
of @ are intersected in 5 rays of a congruence and the plane of @ 
in a line not belonging to this congruence. The section of B with 
the combined system {£2—w would, however, have to be in equili- 
brium. This is impossible, unless w is equal to naught, i.e. unless 
2 is in equilibrium. 
Nothing remains but to determine the ratios of the intensities of 
the rotations of 2. This should be done as follows: 
