158 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 8 
A special case arises when the condition (36) can be satisfied indefi- 
nitely for all the polynomials ¢;, the function g’ being, in this case, 
an infinite series. The integral curves of (34) are then no longer 
spirals but a family of closed curves enveloping each other and 
enclosing the origin. The oscillations of the system are in this case 
undamped and continue indefinitely.’ 
There may also be another type of periodic oscillations, in which 
the integral curves of (34) are spirals winding, not into the origin, but 
asymptotically about a limiting cycle (Fig. 2). The process, in such 
case, is not at first exactly periodic, but becomes more and more nearly 
so as time goes on. 
This case bears a certain analogy to conditions that sometimes 
arise in cyclones and water spouts.’ In point of fact Fig. 2 is repro- 
duced from Bjernke’s Dynamic Meteorology® and illustrates the so- 
called eye of a cyclone. I presume that the sleeve of a waterspout is 
a concrete and material visualisation of the limiting cycle in a vortex 
of this type. 
7 The purely periodic type of process was considered by the writer on a former occa- 
sion (Proc. Nat’l. Acad. Vol. 6, 1920 p. 410; Jl. Am. Chem. Soc. Vol. 42, 1920 p. 1595.) 
It may be remarked that the expression then given for the period of oscillation holds 
true only near the origin. Thestatement that this period is independent of the amplitude 
requires correction. 
8’ Monthly Weather Review 1915 Vol. 43, p. 550. 
* Carnegie Institution No. 88 Part 2, p. 52. 
