52 



SCIENCE 



[N. S. Vol. LI. No. 1307 



Moon are assumed first to move in certain 

 periodic orbits, and the perturbations of tlie 

 Moon are assumed to be the same as if the 

 other bodies did move in such hypothetical 

 orbits. The principle of successive approxi- 

 mations characterizes these methods. 



The chief other advance made was based on 

 the following principle: if a function is a 

 maximum or minimum when expressed in 

 terms of one set of variables it is also a 

 maximum or minimum for any other set; 

 hence, if the difFerential equations of dy- 

 namics can be looked upon as the equations 

 for a maximum or minimum problem, this 

 property will persist whatever variables be 

 employed. This principle, developed mainly 

 by Lagrange, W. E. Hamilton, and Jacobi, 

 enables one to make the successive changes 

 of variables required in the method of suc- 

 cessive approximations by merely doing so in 

 a single function. 



Here too the results are chiefly of formal 

 and computational importance. 



The last great figure of this period is Jacobi. 

 His " Vorlesungen iiber Dynamik " published 

 in 1866 represents a highwater mark of 

 achievement in this direction. 



Nearly all fields of mathematics progress 

 from a purely formal preliminary phase to a 

 second phase in which rigorous and qualita- 

 tive methods dominate. From this more ad- 

 vanced point of view, inaugurated in the 

 domain of functions of a complex variable by 

 Riemann, we may formulate the aim of dy- 

 namics as follows: to characterize completely 

 the totality of motions of dynamical systems 

 by their qualitative properties. 



In Poincare's celebrated paper on the prob- 

 lem of three bodies, published in 1889, where 

 he develops much that is latent in Hill's work, 

 Poincare proceeds to a treatment of the sub- 

 ject from essentially this qualitative point of 

 view. 



A first notion demanding reconsideration 

 was that of integrability, which had played so 

 great a part in earlier work. In 1887 Bruns 

 had proved that there were no further al- 

 gebraic integrals in the problem of three bod- 

 ies. Poincare showed that in the so-called 

 restricted problem there were no further in- 



tegrals existing for all values of a certain 

 parameter and in the vicinity of a particular 

 periodic orbit. Later (1906) Levi-Civita has 

 pointed out that there are further integrals 

 of a similar type in the vicinity of part of 

 any orbit. 



Thus it has become clear that the question 

 as to whether a given dynamical problem is 

 integrable or not depends on the kind of 

 definition adopted. However, the most nat- 

 ural definitions have reference to the vicinity 

 of a particular periodic motion. The intro- 

 duction of a parameter by Poincare is to be 

 regarded as irrelevant to the essence of the 

 matter. 



From the standpoint of pure mathematics, 

 a just estimate of the results foimd in in- 

 tegrable problems may be obtained by refer- 

 ence to the problem of two bodies, or, more 

 simply still, of the spherical pendulum. The 

 integration by means of elliptic functions 

 shows that the pendulum bob rotates about the 

 vertical axis of the sphere through a certain 

 angle in swinging between successive highest 

 and lowest points. But the form of the differ- 

 ential equation renders this principal qualita- 

 tive result self-evident, while the most ele- 

 mentary existence theorems for differential 

 equations assure one of the possibility of ex- 

 plicit computation. Hence the essential im- 

 portance of carrying out the explicit Integra-' 

 tion lies in its advantages for purposes of 

 computation. 



The series used in the calculations of the 

 lunar theory and other similar theories were 

 given their proper setting by Poincare. He 

 showed that they were in general divergent, 

 but were suitable for calculation because they 

 represented the dynamical coordinates in an 

 asymptotic sense. 



The fact that the first order perturbations 

 of the axes in the lunar theory can be 

 formally represented by such trigonometric 

 series had led astronomers to believe that the 

 perturbations remained small for all time. 

 But the fact of divergence made the argument 

 for stability inconclusive. 



It is easy to see that this question of 

 stability, largely unsolved even to-day, is of 

 fundamental importance from the point of 



