Januait 16, 1920] 



SCIENCE 



55 



at least once in every set of N successive in- 

 tegers of the sequence. 



In any domain of transitivity the two ex- 

 treme types of motion are the recurrent 

 motions on the one hand and the motions 

 which pass arbitrarily near every state of 

 motion in the domain on the other. Both 

 types necessarily exist, as well as other inter- 

 mediate types. 



The precise nature of such recurrent mo- 

 tions has yet to be determined, but Dr. H. C. 

 M. Morse in his 1918 dissertation at Harvard 

 has shown that there exist," non-periodic 

 recurrent motions of entirely new type in 

 simple dynamical problems. 



Such are a few of the steps in advance that 

 theoretical dynamics has taken in recent 

 years. I wish in conclusion to illustrate by 

 a very simple example the type of x)owerful 

 and general geometric method of attack first 

 used by Poincare. 



Consider a particle P of given mass in 

 rectilinear motion through a medium and 

 in a field of force such that the force act- 

 ing upon P is a function of its displace- 

 ment and velocity. In order to achieve sim- 

 plicity I will assume further that the law of 

 force is of such a nature that, whatever be the 

 initial conditions, the particle P will pass 

 through a fixed point infinitely often. 



If P passes with velocity v it passes 

 at a first later time with a velocity v^ of 

 opposite sign. We have then a continuous 

 one-to-one functional relation v^ = f (v). If 

 V is taken as a one-dimensional coordinate in 

 a line, then the effect of the transformation 

 v^ = f («) is a species of qualitative "reflec- 

 tion " of the line about the point 0. 



If this " reflection " is repeated the result- 

 ant operation gives the velocity of P at the 

 second passage of 0, and so on. But the 

 most elementary considerations show that 

 either (1) the reflection thus repeated brings 

 each point to its initial position, or (2) the 

 line is broken up into an infinite set of pairs 

 of intervals, one on each side of 0, which are 

 reflected into themselves, or (3) there is a 

 finite set of sucih pairs of intervals, or (4) every 

 point tends toward (or away from it) under 

 the double reflection. 



Hence there are four corresjKinding types of 



systems that may arise. Either (1) every 

 motion is periodic and is a position of 

 equilibrium, or (S) there is an infinite 

 discrete set of periodic motions of increas- 

 ing velocity and amplitude (counting the 

 equilibrium position at as the first) such 

 that, in any other motion, P tends toward 

 one of these periodic motions as time in- 

 creases and toward an adjacent periodic mo- 

 tion in past time, or (3) there is a finite 

 set of i)eriodic motions of similar type such 

 that, in any other motion, P behaves as just 

 stated, if there be added a last i)eriodic mo- 

 tion with " infinite velocity and amplitude " 

 as a matter of convention, or (4) in every mo- 

 tion P oscillates with diminishing velocity and 

 amplitude about as time changes in one 

 sense and with ever increasing velocity and 

 amplitude as time changes in the opposite 

 sense. 



Here we have used the obvious fact that 

 there is a one-to-one correspondence between 

 velocity at and maximum amplitude in the 

 immediately following quarter swing. 



This example illustrates the central role of 

 periodic motions in dynamical problems. It is 

 also easy to see in this particular example that 

 the totality of motions has been completely 

 characterized by these qualitative properties 

 in a certain sense which we shall not attempt 

 to elaborate. 



What is the place of the developments re- 

 viewed above in theoretical dynamics? 



The recent advances supplement in an im- 

 portant way the more physical, formal, and 

 computational aspects of the science by pro- 

 viding a rigorous and qualitative background. 



To deny a position of great importance to 

 these results, because of a lack of emphasis 

 upon the older aspects of the science would be 

 as illogical as to deny the importance of the 

 concept of the continuous number system 

 merely because of the fact that in computa- 

 tion attention is confined to rational numbers. 

 George D. Birkhoff 



SIR WILLIAM OSLER (1849-1919) 



After a tedious and painful illness. Sir 

 William Osier, Regius professor of medicine 

 at Oxford, died at his home in Norham Gar- 

 dens on December 9, 1919. In spite of in- 



