218 
MATHEMATICS: A. EMCH 
has remarked, the theorems are thus related to the lesser Fermat theorem 
and to Wilson's theorem which are concerned respectively with the numbers 
which satisfy the difference equations u x+ i = au x , and u x+ i = xu x . 
1 Hurwitz, Zurich, Vierteljahrsch. Natf. Ges., 41, 1896, (34-64). 
2 Cotes, Phil. Trans., London, 29, 1714, (5). (Reference in Encyc. Sci. Math., Paris, 
Tome I, Vol. I, Fasc. 2, p. 169 note.) 
3 Euler, Com. Acad. Petropolitanae, 9, 1737, (98-137). 
4 Tietze, Math. Ann., Leipzig, 70, 1911, (236-265). 
ON CLOSED CURVES DESCRIBED BY A SPHERICAL PENDULUM 
By Arnold Emch 
Department of Mathematics University of Illinois 
Communicated by R. S. Woodward, April 18, 1918 
1. The geometrical aspect of the problem of closed curves in a spherical 
pendulum-motion has apparently never been fully discussed and it is the object 
of this note to present the results of an investigation of some of the geometric 
properties of these curves. 
The differential equations of the motion are 
d ^=-s y , il=-s z — gi (1) 
dt 2 I dt 2 I dt 2 I 5 y 
in which / is the length of the pendulum, S the variable reaction directed 
towards the origin (point of suspension). The velocity v of the pendulum-bob 
is, as is well known, given by v 2 = h — 2gz, in which h is a constant depend- 
ing upon the initial conditions. For z as a function of t, we may obtain 
without difficulty 
z = - + -p(t + w 2 ). (2) 
6g g 
In this Weierstrassian /^-function the one half-period W\ is real; the other w 2 
is pure imaginary. Putting u = t + w 2 , let a and b be the arguments for 
which z assumes the values — / and + h respectively, and h and / 2 the corre- 
sponding complex values of /. It is found that a = i a, where a is real and 
positive, and b = w± + z'jS, with /3 equal to a fraction of | w 2 \. The constant 
of integration may be determined such that x = r 0 , y = 0, when / = 0. 
Under these conditions d = tan -1 (y/x) is defined by 
