( 740 ) 
— R 0 h |/£ cos n, t 
and 
y = ~R a hVYtr iws(in ,t-<f). 
For £ and <p we must substitute the values, which these quantities 
have at the moment for which we wish to know the osculating 
curve. 
The osculating curves are Lissajous curves answering to the value 
— for the ratio of the periods of the vibrations. They are described 
I 2 
in the rectangles having as sides 2 22 0 A|/£ and — R^hVl —£. 
As £ varies between two limits the rectangles in which the curves 
are described lie between two extremes. The vertices lie on the 
2 
circumference of an ellipse having 2 R 0 h and — R 0 k as lengths of 
The shape of the curve described in a definite rectangle is still 
dependent on the value of <p, i. e. on the value of the difference in 
phase at the moment of the greatest deviation to the right. 
To an arbitrary value of ip the wellknown Lissajous curve with 
two nodes of fig. 8 answers. For or ~ the curve is sym¬ 
metrical in respect to the axes; the nodes lie in the Jf-axis on 
either side of O at distances — R 0 h (fig. 9). For <p =r 0 or it we get 
a curve, which is described in both directions alternately and which 
passes through 0 (fig. 10). 
In fig. 11 we find some of those osculating curves represented 
for a definite case of motion: two belonging to cp — nr; two for 
(p = —, and one for an arbitrary value of <p ^ • 
Out of (19) follows that = 0 for sin <p = 0. In the extreme 
at 
rectangles the curves are described which we have for <p — 0 or -t. 
Now a number of different cases are possible, of which we get a 
clear representation by representing equation (18) in polar coordinates. 
In fig. 12 some of the curves obtained in this way are represented, 
where ip is taken as polar angle, V1—C as radius vector. The different 
shapes of the curves correspond to the roots of the equation: 
£ a (! — £) — (p™ -b ?£ -f r) 9 =: 0. . „ , 
(20) 
