( 749 ) 
The relation between £ and tp becomes: 
.<• 
Here £ again varies periodically between a greater and a smaller 
value. Now however may become equal to 0 for sin y = 0 
and for cos y> = 0. Thus barring special cases there are 3 general 
cases: 
1 st . For the extreme values of £ costp = Q. Then in the extreme 
rectangles ellipses are described with the axes along the Jf-axis 
and F-axis (fig. 13). 
2 nd . For the extreme values of £ sin <p — 0. In the extreme 
rectangles straight lines are described (fig. 14). 
3 rd . For one of the extreme values of £ sin <p = 0, for the other 
cos 7) = 0. (fig. 15). 
Special cases. 
$ 27. These we have again for the extreme values of the modulus 
x (a has the same form as in $ 16) of the elliptic functions; which 
occurs when 2 roots of the equation: 
f (£) = (/>£* + q; + r) {£ (l - p - (p£* + ?£ + r)} = 0 
have coincided. 
The special case corresponding to B of $ 9 and the second of 
$18 occurs here in two ways. We refer to the cases in which 
the same straight line is continually described (continually sin <p = 0; 
when the surface is surface of revolution, this form of motion 
is possible in every meridian) and that continually the same ellipse 
is described {cos q> — 0; this becomes for the surface of revolution 
the uniform motion in a parallel circle). 
The special case corresponding to A of $9 and to the first of 
$18 exists here too. The form of motion approaches asymptotically 
the motion in a definite ellipse. 
Envelope of the osculating curves. 
$ 28. Two cases may be indicated, in which the envelope assumes 
a simple shape. 
1. For p = — 1, 2 = 1 in (22) (the case of a surface of revolution), 
the envelope has degenerated into two concentric circles. 
2. For p = 0 and q = 0 in (22) the envelope has degenerated into 
two pairs of parallel lines, enclosing a rectangle. 
Arbitrary mechanism icith 2 degrees of freedom for which S= 2. 
$ 29. The equations of Lagrange get quite the same form here 
