( 749 ) 
C,. If C,=C;, then of necessity k=0; so this belongs to the 
second special case. If C,=0, then by putting $= C,5/ the relation 
(6) passes into the one which we had with the mechanism with two 
degrees of freedom, for which n, = 7,. 
The special cases will be discussed in § 14. 
$ 9. Osculating curves. In order to illustrate the motion of the 
mechanism somewhat better, we use an image point. 
To this end we choose the point whose rectangular coordinates 
xv, y, and z are at an arbitrary moment equal to the values of the 
principal coordinates q,,g,, and q, at that moment. The motion of 
this point is then given by 
Va 
B = —— cos(n,t + 2n,8,), 
nN, 
Va 
e= = cos (n,t -+ 2n,8,), 
Ns 
pe Ve COS (Or, Fn.) t+ 2 (n, +7,) B;}. 
» 
By eliminating ¢ between these equations two by two and by 
ascribing to the es and 9'’s, the values at a definite moment, we 
find the projections of the osculating curves on the planes of coor- 
dinates. 
These projections are Lissasous curves; the osculating curves them- 
selves we can call Lissasous twisted curves. 
§ 10. Such a twisted curve remains enclosed inside a rectangular 
parallelopiped bounded by the planes: 
1 « 
oe ey 55 eS ee = Berne 
n, Ts n__n; 
In consequence of the variability of the @’s this enclosed parallelo- 
piped varies continually. The vertices move along a twisted curve, 
which according to (8) projects itself on the XY-plane as a hyper- 
bola, on the XZ and on the YZ-plane as an ellipse. Out of (5) 
follows that this curve is situated on an ellipsoid, whose axes lying 
on the axes of coordinates are in the ratio: 
Lea 1 
IN Ny n,-- Ms 
As the a's change periodically between definite limits, the vertices 
will move to and fro along the above mentioned twisted curve 
hetween two extreme positions. (fig. 2). 
