(TER) 
systems '). The variations of the integrals occurring in (26) and (28) 
will be 0, if the original motion is not only a possible one, but such 
that it may really take place under the influence of the acting forces. 
We shall call such a motion a real or a natural one. 
S 21. We shall conclude by briefly showing how some well 
known propositions may be presented in a form, agreeing with what 
precedes. These propositions relate to holonomic systems. Let us 
therefore assume that the connexions are expressed by 7 equations 
which must be satisfied, independently of the time, by the coordi- 
nates «,, and let us confine ourselves to possible positions, i.e. such 
as agree with these # conditions. We might determine these positions 
by 3 n—i “free” coordinates, but in what follows, it is not necessary 
to do so. 
If, in addition to the equations expressing the connexions, we 
assume still one other equation between the coordinates, we shall 
call the totality of positions satisfying that equation a surface of 
positions. 
In case one of these positions A is reached by a certain path — 
the other positions in this path not all of them belonging to the 
surface — the path may be said to cut the surface in the position A. 
For simplicity’s sake it will be supposed in such a case that the 
surface and the path have only that one position in common. 
Starting from a position A, which belongs to, or lies in asurface 
of positions S, we may give to the system infinitely small displace- 
ments, in such directions that by them the position does not cease 
to belong to the surface. 
Another infinitely small possible displacement d8 whose direction 
is perpendicular to all those displacements in the surface may be 
said to be perpendicular to the surface. It is easily shown that a 
displacement of the latter kind may always be found and that its 
direction is entirely determinate. 
Let S be a surface of positions, A a position that does not belong 
to it but is infinitely near others that do, B a position in the surface, 
such that the infinitely small displacement 4 — B is perpendicular to S, 
and C any position in S infinitely near B. Let & be the angle 
between the displacements AB and A—C, and let us denote 
by AB and AC the lengths of these displacements. Then 
A Be PIE cose 
*) See Hörper, Gott. Nachr., 1896, p. 122. 
48 
Proceedings Royal Acad. Amsterdam, Vol. IV. 
