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MATHEMATICS: G. D. BIRKHOFF 
An application of periodic orbits which is fundamental lies in the 
construction of surfaces of section. The existence of a ring-shaped 
surface of this sort was noted by Poincare in the restricted problem 
of three bodies and allied dynamical problems. The results of my 
paper show that such surfaces exist in a wide variety of cases and may 
be of any genus and possess any number of boundaries. By means of 
a surface of section the dynamical problem reduces to a one-to-one 
analytic area-preserving transformation of the surface of section into 
itself. 
Periodic orbits correspond to invariant points of this transformation. 
Consequently the discovery of further periodic orbits hinges upon the 
proof of the existence of such invariant points. Two theorems con- 
cerning such points are proved by me. 
The first of these theorems is the following: If a surface of genus p 
admits of a one-to-one analytic transformation into itself which may 
be looked upon as obtained by a deformation then the excess of com- 
pletely unstable invariant points over all other types of invariant points 
is precisely 2p — 2. 
If the transformation is merely restricted to be continuous the same 
argument shows that there exists at least one invariant point for ^ =1= 1 . 
For the case p = 0 this last result constitutes a well-known theorem 
due to Brouwer.^ 
The dynamical application of the result for the analytic case is to a 
fundamental equality obtaining between the various types of periodic 
orbits. 
The second theorem is the following: If the region outside of a circle 
in the plane is transformed into itself in such a way that points on the 
circumference are advanced in one sense, so that points distant from the 
center are regressed by more than a fixed positive angle about the center, 
and so that areas are preserved, then at least two points are left in- 
variant by the transformation. This theorem is to be regarded as the 
extension of a theorem stated by Poincare and proved by me.^ 
These two related theorems are used to prove that infinitely many 
periodic orbits exist whenever a surface of section is at hand. 
^Whittaker, Analytical Dynamics, pp. 376-384, Cambridge, 1904. 
2Signorini, Palermo, Rend. Circ. Mat., 33, 1912, (187-193). 
3Brouwer, Math. Ann., Leipzig, 69, 1910, (176-180). 
^Birkhoff, Trans. Amer. Math. Soc, New York, 14, 1913, (14-22). 
