November 28, 1912] 



NATURE 



355 



into details, the applications to the problem of 

 three bodies and to restricted cases of this problem. 

 No g-eneral method for finding the solutions, nor 

 for discovering the full number of them, is obtained, 

 but these needs are being supplied by the 

 researches of Darwin, Moulton and others into 

 the possible orbits which may be described in 

 various circumstances. 



The periodic orbit only represents a particular 

 solution of the equations of motion. Poincare 

 obtains a general solution within a limited range 

 of the arbitrary constants by considering those 

 differing slightly from the periodic solution. In 

 this connection arise the "characteristic expon- 

 ents " which may be somewhat loosely taken to 

 give the various periods present in the general 

 solution. These exponents form the bridge which 

 enables him to enter into such questions as the 

 existence of integrals, the anal3'tic forms of 

 possible solutions and the convergence or diver- 

 gence of the series thus formed. His proof that 

 there cannot exist any algebraic or transcendental 

 integral of the problem of three bodies (under a 

 restriction as to the magnitude of the masses) 

 beyond those known is an important advance on 

 Bruns' result — that no new algebraic integral 

 exists, although the latter is true for any values 

 of the masses. 



Not less important is his examination of the 

 older methods from the logical point of view. His 

 presentation of these is nearly always fresh and 

 novel ; he is rarely content with previous methods 

 of arriving at the results. This change is perhaps 

 necessary, for he has a different object in view ; 

 nevertheless, the reading of them frequently gives 

 the impression that Poincare simply took the 

 premises and the conclusions and found it less 

 difiicult to work out the latter from the former 

 in his own way than to go fully into the author's 

 work. Perhaps the most startling result was his 

 discovery that the majority of the series which have 

 been used to calculate the positions of the bodies 

 of the solar system are divergent. This fact, of 

 course, required an examination into the reasons 

 why the divergent series gave sufficiently accurate 

 results : hence arose the theory of asymptotic 

 scries now applied to the representation of many 

 functions. 



The crux of the problem is the divergent series. 

 The functions are only represented in the numerical 

 sense by series, and we do not know their limits. 

 Can we argue one way or the other as to the 

 stability of the system? In other words, is the 

 ultimate divergence peculiar to the functions, or 



NO. 2248, VOL. go] 



is it merely due to our inability to obtain expres- 

 sions from which a conclusion can be deduced? 

 The question remains unanswered. Gylden be- 

 lieved that he had overcome the difficulty, but 

 Poincare has shown that it still exists. 



Whilst the greater part of Poincare's researches 

 are thus confined to the logical side of the prob- 

 lems in celestial mechanics, we have occasional 

 papers in which he developed methods useful for 

 actual calculation, in addition to those chapters 

 of the " Methodes Nouvelles " which are devoted 

 to this part of the subject. Amongst them may 

 be mentioned one on the lunar theory, in which 

 he developed a method with rectangular coordin- 

 ates which appears to be of value for obtaining 

 algebraic expressions for the coordinates of the 

 moon. There are also two papers dealing with 

 librations in planetary systems which open a way 

 to the more extensive treatment of this complex 

 subject. They have received less notice on account 

 of their narrower range of application ; they are 

 incorporated with other matter in his " Lecons de 

 Mecanique Celeste." The recently published 

 volume on cosmogony is of a different nature. 

 It is chiefly a presentation, given originally in a 

 course of lectures, of the works and theories of 

 others, but he does not hesitate to express his own 

 opinions as to their importance in a discussion of 

 the evolution of solar and stellar systems. 



A pure mathematician might be pardoned for 

 doubting whether the world, as a whole, bene- 

 fited by Poincare's appointment to a chair of 

 mathematical physics. The redactions of his early 

 lectures on electricity and optics have to be read 

 with a certain amount of reserve; he is not yet 

 sure of his ground, and is assimilating the ideas 

 of others. It is difficult to conjecture what he 

 might have done if he had been able to follow 

 up his original bent, which was undoubtedly pure 

 analysis ; it would certainly have been something 

 very great. On the other hand, he popularised 

 the Maxwellian theory of electricity, and ultimately 

 mastered it, as well as more recent developments, 

 so that he was able to make contributions to the 

 theory of electrons and that of diffraction. .-Vnd 

 even in a bare outline, such as this, of his best 

 work, we ought not to pass over his masterly 

 papers on potential and similar subjects, which 

 form the bridge, so to speak, between Neumann 

 and Fredholm. 



Poincare did not disdain to write for a popular 

 audience. "La Science et I'Hypothfese" has 

 deservedly had a wide circulation, and affords a 



