( 620 ) 
the above-mentioned expansions in series lost their validity; we must 
therefore investigate in a different way what becomes of the movement 
in the case mentioned. In what follows we shall investigate this 
for a mechanism with two degrees of freedom. As a base for this 
investigation a very simple mechanism is selected, namely a material 
point which moves without friction yet under the influence of gravi¬ 
tation on a given surface in the vicinity of its lowest point. Every 
time one of the cases 4 is discussed we shall pass to an arbitrary 
mechanism with two degrees of freedom. 
Movement on the bottom of a surface . 
$ 2. We shall accordingly first pass on to the treatment of the 
simple mechanism we have chosen as a base for our investigation. 
When the surface has positive curvature in the vicinity of its lowest 
point 0, when plane AT is the tangential plane in 0, and the XZ- 
and FZ-planes are the principal sections of the surface in that point, 
whilst the Z&xis is supposed positive upwards, then the equation of 
the surface in the vicinity of 0 takes the form of: 
— - ( c i** + cy + d x a r* + d^x'y -f d % xy' + dtf -f . . .) ; . (1) 
where e, and c, are positive. 
The equations of motion of the material point become: 
Availing ourselves of (1) to eliminate z we find: 
the order of greatness of /and y, then the 
omitting the terms of order h* and higher: 
* +2c,* 
y -f 2c t y = 0.\ 
(3) 
