( 588 ) 
If we put in the four general formulae y=1 and if we remark 
that 
i sin(n + 1) ed 
C $ (oe ee 
4 (cos iF) os 
there appear besides two relations really alike to the formulae (2) 
and (4) the two new ones : 
en 2r +1 
sin (tcos p cos yy) sin (t sin p sin w) 1 (t) dt __ 
, | : 
0 
(pele sin (QA + 2) p sin (2A + 2)y 
En 7 st (Ar 1 — (27-7 op ; 
wo 2r+2 
fz (t cos p cos Y)) sin (t sin p sin w) L (t) dt 
EE oS ee 
0 
2 eae sin (2 r -+ 2) psin(2r + 2)p 
i Zr 2 
Kinematics. — “The motion of variable systems” (24 part). by 
Prof. J. CARDINAAL. 
1. In the previous communication the construction of the direction 
of velocity was our subject, no mention being made of the length of 
velocity ; in the following some relations will be deduced in which 
these lengths appear. 
As is known, the rule that the extremities of the velocities form 
a system similar to the original one holds good for the motion of 
a plane system remaining congruent or similar to itself. If the 
system remains affine to itself, the extremities of the velocities 
form an affine system. 
In each of these cases the points of coincidence of the original 
system (2) and the system formed bij the extremities of the velo- 
cities (2) are the same as those of the system = and the one lying 
at infinitesimal distance. We shall now consider, which theorems 
hold good for the motion of spacial systems.') 
1) Whilst writing this paper I got to hand number 1—2 of the 47th. vol. of the 
vZeitschrift der Mathematik und Physik”, in which L. Burmester treats of a subject 
closely allied to the paper, cited in the previous communication. As however the proofs 
given there differ from those given here, though the results agree, I have made no 
change in this text, 
