554 



NATURE 



[November 28, 1912 



cessfuUy expounding the nature of the Fuchsian 

 functions. 



Many must be able to recall the delight with 

 which they read those famous memoirs in the 

 early volumes of the Ada Mathematica, and the 

 eagerness with which they turned to each new 

 part, in the hope of finding more of this enchant- 

 ing causerie. Few formulae, and short ones at 

 that ; just a succession of brief, almost conversa- 

 tional, sentences opening up a new and vast domain 

 in which even such a subject as elliptic modular 

 functions took a place like that of reciprocants in 

 the general theory of differential invariants; new 

 vistas and new problems presenting themselves on 

 every side. It is easy enough to trace the lineage 

 of the automorphic functions. Immediately sug- 

 gested by Fuchs's work on differential equations, 

 and actually a generalisation of modular functions, 

 they are historically the outcome of Gauss's memoir 

 on the hypergeometric series, and Riemann's 

 paper on the P-function. To say this is no 

 detraction from Poincare's merits : the fact is that, 

 like Lejeune-Dirichlet, he won many of his highest 

 triumphs by his extraordinary power of seizing 

 the main points of an existent theory, simplifying 

 it by an appropriate analysis, and then extending 

 it beyond all expectation. Compare, for instance, 

 the present positions of the theories of modular 

 functions and of Fiichsinn functions. In the 

 former, apart from further application to arith- 

 metic and the like, the one main problem that still 

 remains is to find out, if possible, the arithmetical 

 characters of all the sub-groups of the modular 

 group; in the latter there are difficulties at the 

 outset, arising from the fact that in certain families 

 of Fuchsian groups there are conditions of in- 

 equality which involve troublesome relations 

 connecting the constants of the generating sub- 

 stitutions. In this and in other matters Poincar^ 

 (lid not go into detail : but he pointed out the 

 way for others by his distribution of the functions 

 into families, and by his geometrical method with 

 its non-Euclidean interpretation. Perhaps the 

 crowning result of his work in this direction is 

 his theorem that the coordinates of any point on 

 an algebraic curve can be expressed as one-valued 

 I'\ichsian functions of a parameter. This is analo- 

 gous to the representation of a point on a circle 

 by (sin 6, cos 6), and is to be distinguished from 

 the Puiseux-Weierstrass representation of an 

 clement of the curve. 



A more definite example of Poincare's power of 

 dealing- with a classical problem is afforded by 

 his work on rotating fluid masses. Long ago it 

 was shown by Jacobi that an ellipsoid of three un- 



No. 2248, VOL. go] 



equal axes was a possible figure of relative equi- 

 librium : but it was reserved for Poincare to take 

 up the problem afresh, and develop the solution 

 into what may fairly be called (apart from details) 

 its final and definite form. He shows the exist- 

 ence of whole families of figures of equilibrium, 

 including as particular cases those already known ; 

 gives analytical criteria for stability ; and proves 

 that when, by varying the parameter that gener- 

 ates a particular family, we pass from stability 

 to instability, the critical surface is one of "bifur- 

 cation," that is, it simultaneously belongs to two 

 distinct families. In some respects this is analo- 

 gous to the way in which a curve /(.v, y, /i) = o, by 

 variation oi ;x, acquires a double point, and then 

 alters what may be called its connectivity; and 

 in any case, without pressing the analogy, 

 Poincare's results here seem typical of what 

 happens, with regard to stability, in the variation 

 of dynamical systems. The value and originality 

 of these researches was recognised by .Sir G. H. 

 Darwin in his address to the Royal Astronomical 

 Society, when its gold medal was presented to 

 Poincare (Feb. 9, igoo). 



The contributions of Poincare to celestial 

 mechanics not only brought new life to a 

 subject which showed signs of becoming stale, 

 but undoubtedly opened up a fresh line of in- 

 vestigation. Starting with an idea due to G. VV. 

 Hill, who, in his turn, was indebted to Euler, 

 he brought the whole range of his great knowledge 

 and power of analysis to bear on a problem which 

 has baffled the ingenuity of mathematicians for 

 more than two hundred years. That he did not 

 succeed in solving it, either in the old or the 

 modern sense, is no criticism on his achievements ; 

 it is sufficient to say that he opened the way and 

 explored a new region by routes which may ulti- 

 mately lead to the final goal — a demonstration of 

 the stability or instability of the solar system. 



His investigations on the general problem of 

 three bodies ,ue principally contained in the three 

 volumes entitled " Les Methodes Nouvelles de la 

 Mecanique Celeste," which form a natural 

 sequence to the earlier prize essay of i88g. The 

 foundation of the work is the now well-known 

 periodic solution of a set of differential equations. 

 Hill had developed one such solution arising in 

 the motion of the moon round the earth ; Poincar6 

 considers periodic solutions of any class of differ- 

 ential equations, examining their general pro- 

 perties and the conditions for their existence. He 

 then takes up the special properties of the equa- 

 tions of dynamics and, descending still further 



