( 741 ) 
The cases are : 
1. The curve indicated in the figure by - - - keeps to the right or 
to the left of O x ; <p changes between two limits; the limits are 
equal and opposite; the positive is smaller than For the extreme 
values of £ we find <p either both times 0 or both times n. 
2. The curve-intersects the straight line <p = - at two points 
above O x and at 2 points below O v For the extreme values of £ 
we again find <p either both times D or both times n. 
3. The curve consists of two closed parts (a continuous line in the 
figure), which surround 0 Now <p assumes all values. For the 
extreme values of £ (p =■ 0 one time <p = 0 and tp = ji the other. 
The transition case between 2 and 3 is represented by_ 
Fig. 11 relates to the 2 nd case; for the two extreme values of £ 
we find <p = jt. 
§ 18. These occur for the extreme values of the modulus x of the 
elliptic functions; two roots of equation (20) have coincided. 
1. * = 1. The elliptic functions pass into hyperbolic ones. The 
geometrical representation just now discussed of the relation between 
£ and <p and already mentioned as transition case between the 
second and third cases has a node situated on the axis of the angles. 
The form of motion approaches asymptotically to a form of motion 
belonging to = 0 or <p =: r. 
2. x — 0. The elliptic functions pass into goniometrical ones. The 
curve of fig. 12 becomes an isolated point C (special case belonging 
to the 1 st case of § 17 as limiting case) or it consists of an isolated 
point and a closed curve (special case belonging to the 3 rd case of 
$ 17 as limiting case). If the initial value of £ coincides with the 
twofold root of (20) we find that £ remains constant; ip is conti¬ 
nually 0 or 7 t. Thus the same curve is continually described. 
Arbitrary mechanism with 2 degrees of freedom for which S= 4. 
$ 19. In the case that n, = 3 n 1 4- o the terms of order h* can 
give no disturbing terms in the equations of motion. 
So we may write: 
