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MATHEMATICS: A. EMCH 
x = r 0 [ cos (A + tt) v. cn (2 Kv) - n sin (4 + tt) v. sn (2 dn (2 ifr>) ], (11) 
y = r 0 [sin (4 + cn (2 fo) + M cos (4 + tt) v. sn (2 Ki>) dn (2 2&>)].(12) 
If x' = r 0 cn (2 y' = fir 0 sn (2 Kv) dn (2 z' = a cn 2 (2 (13) 
it is found that the locus of the point P'{x' , y' , z') is a quartic C 4 in space, and that 
P' is obtained by rotating the point P (x, y, z) of the pendulum curve C about the 
z-axis through an angle— (A + it) v, i.e., through an angle negatively proportional 
to the time associated with P in the motion. 
Imposing upon (11) and (12) the same condition of periodicity, as in the 
general case, the resulting curve in the xy -plane becomes a rational algebraic 
curve of order 2 k. The angle corresponding to the period 2 wi is, as before, 
$ = 2kir/mi, k and mi being relatively prime. For a given odd mi there are 
'{mi — l)/2 curves of this type, for an even mi, there are (mi — 2)1 2 such curves. 
In both cases there are m\(k — 1) real double points. 
When mi is odd there is just one curve among the set whose double-points are 
■all real. Its degree is 2 k = m\-\- 1. 
When mi is even, there is no such curve. 
In case of an odd mi the polar equation of the curve has the form 
p 2k + a lP 2k - 2 + a 2 p 2k - 4 + .... + a . p 2 <+0+ a4) 
(b lP 2Xl + b 2P 2X >+ .... +b jP 2 + b) p Wl cos mi d = 0, 
in which a ± 0, \i > A, > X 3 > ... >*1, and 2 Xi + mi ^ 2 k — 1. In 
cartesian coordinates (14) may be written in the form 
>(x 2 + y 2 ) k + a t (x'+y 2 )*- 1 + a 2 (x 2 + y 2 ) k ~ 2 + .... + a s (x 2 + y 2 ) + a + 
{bi (x 2 + y 2 )^ + b 2 (x 2 + y 2 ) Xs + • • • • +hj (x 2 +y 2 )+b\. (15) 
Transforming this curve by 
x = d= Vx''-y' /2 -2y''-l / y", y = (/' + l) / f , 
or using isotropic coordinates, transformations which do not change the 
character of the isotropic points, and "placing the curve on the analytic tri- 
angle," the result is obtained that the isotropic points absorb together 
(k- \) {2 k- mi - 1) (16) 
double points, which when added to the mi (k — 1) real double-points, gives 
(k — 1) (2 k — 1), i.e., the maximum number of double points which a curve 
of order 2 k may have. Thus we have verified directly from the equation, 
that the curve is rational, as proved before. When the curve has all double 
points real, then its degree is mi + 1 = 2 k, so that from (16) the number of 
