we then find 
a, =n, (C,—S)’, a, = 2n,(C_,— S$)’. 
The integral == constant then takes the form: 
C,—S) WE(C —6) cos n= 0? H 75 +. 7 
( JASE B. 
EE je = ci k, bs ais oe a= Oy i eee ae = (C= C,)a, 
2n 2n, 2n,—nN, 2n,—n, n 
where C, and C, are positive constants. 
If we put 
a on Ae Gi 
then we find 
a, =n, (C,—$)h? , a, = (Qn, — n,) (C, — 5) A’. 
The integral R= constant gets the form: 
5 V(C, - 5) (C, De 5G) cos (P = no? +- 7ö Lp, 
$ 20. In ease A we find that & lies continually between O and 
C, or between O and C, „according as C, or C, is the smaller. 
In Case B we find that & lies continually between O and C, or 
between 0 and C, Annie as Cor C, is the smaller, 
When again we represent the relations between & and @ as polar 
coordinates, we find curves of quite the same kind as in the case 
of a mechanism with two degrees of freedom for which , = 37,. 
So there are curves which do not enclose ( and which therefore 
relate to forms of motion, where p runs to and fro between two 
limits; and curves which do enclose V and which therefore relate 
to forms of motion, where g takes all values. The transition is 
formed by a curve, having a double point on the axis of the angles ; 
this points to a particular case, where the form of motion tends 
asymptotically to a movement where sin p=0 and ¢ is constant. 
To another special case the isolated point refers situated on the axis 
of the angles; it points to a form of motion, where sin p = 0 
and where ¢ is constant. 
§ 21. Osculating curves. The osculating curves of the image point 
are again Lissajous twisted curves. The vertices of the circumscribed 
parallelopiped move along a twisted curve lying on an ellipsoid, 
whose axes lying along the coordinate axes have lengths pro- 
iy) 1 | OER 
portional — to — 3 6 —; the twisted curve projects itself 
Wy. Bly IN 
in case A on the X )-plane as a hyperbola, on the YZ- and the ZY-plane 
