MATHEMATICS: G. D. BIRKHOFF 
315 
the equations of motion can be directly integrated. As a second appli- 
cation of the transformation theory the equations of displacement are 
derived. 
For reversible problems it is known that the equations of motion can 
always be interpreted as those of a particle constrained to move on a 
fixed smooth surface. In the irreversible case I have been able to show 
that Xj y may be regarded as the coordinates of a particle constrained 
to move in a smooth surface which rotates uniformly about a fixed axis 
and carries with it a conservative field of force. 
Among all types of orbits the most fundamentally important are those 
which are periodic. 
The existence of periodic orbits is intuitively evident in the reversible 
case, when the orbits may be looked upon as geodesies. If proper re- 
strictions are imposed a closed geodesic will exist along which the arc 
length is a minimum, and this geodesic will correspond to a periodic 
orbit of minimum type. Orbits of minimum type may also be derived 
by means of an interesting criterion for the reversible case due to Whit- 
taker^ first rigorously estabKshed by Signorini.^ Notwithstanding the 
fact that the integral analogous to arc length may change sign in the 
irreversible case Whittaker stated the direct formal extension of his 
criterion for this case. I have estabhshed that such an extension is 
legitimate if X is of one sign, but not otherwise. This restriction on X 
is fulfilled in the restricted problem of three bodies. 
Unfortunately, as Poincare pointed out, only unstable periodic orbits 
can be of minimum type. 
Another method may be employed to find a large class of stable pe 
riodic orbits which I call of minimax type. This entirely new method 
may be stated in a special case as follows : There is a minimum length 
of string, constrained to lie in a given surface of genus 0, which may be 
slipped over that surface. In some intermediate position the string 
will be taut and will then coincide with a closed geodesic. 
Poincare has proved that a closed geodesic exists on any convex sur- 
face by an entirely different method. 
A third method for the discovery of periodic orbits is that of analytic 
continuation. Hitherto the application of this method has been limited 
by the restriction that the variation of the parameter involved be 'suf- 
ficiently small.' This restriction turns out to be unnecessary in the 
reversible case if the orbits near any orbit always intersect it. If the 
conditions 
X>0, 7>0, A (log 7) >0 
hold in the irreversible case the restriction is also unnecessary. 
