( 12 ) 
Mathematics. — “ The constructive determination of the velocities 
of a spacial system By Prof. J. Cardinaal. 
(Communicated in the Meeting of April 23, 1909). 
1. In Vol. LVI page 315, 1908 of the Zeitschrift fur Mathematik 
und Physik Prof. K. Th. Vahlen inserted an article on the graphic 
composition of forces in space, where he introduces the observation, 
that where such forces must be composed, we have so far occupied 
ourselves with general geometric considerations and have set aside 
the possibility of executing them. He points to the fact, that it is 
important to place oneself on the standpoint of descriptive geometry 
and to arrange the operation in such a way that the construction 
has to be made in two perpendicular planes of projection. At the 
same time he develops a method for forces in space and gives an 
example of it. The same observation can be made for an other part 
of mechanics, namely, for the theory of motion. In this paper we 
shall treat of the constructive solution of a problem out of the theory 
of motion in space. 
2. It is known that the motion of an invariable spacial system 
is determined as soon as we know that five points of the system 
must move over five indicated surfaces. The normals of the surfaces 
in these points are five director lines of a focal system, whose 
principal axis is at the same time the axis of the motion. If moreover 
we know the length of the velocity of one of these points, then the 
velocities of all the others are known in length and direction. It 
requires a great number of lines to execute this construction; we 
shall therefore somewhat simplify the form of the problem and give 
to it the following wording: 
Given are three fixed points A, B, C of a spacial system; A 
moves along the right line a, B along the line b ; C must remain 
on a definite plane. Of point A is still given the velocity AAo in a 
e nite position. Assuming this position we wish to construct: 
a. The axis of the movement. 
h. The velocities of the points B and C. 
3. Passing on to the execution (Fig. 1) we assume the plane in whi< 
thrn"T GS a be the horizontal plane of projection; we then brh 
U S e plane normal to the preceding one and we chooi 
