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I934-] POINCAR ON DIFFER EN TIAL EQUATIONS

T H E W O R K O F P O I N C A R O N D I F F E R E N T I A L E Q U A

Oeuvres de Henri Poincar publies sous les auspices de l 'Acadmie des S

par Paul Appell . Tome I publi avec la collaboration de JulPar is ,Gau thier-Vil lars , 1928. cx xix +3 82 pp .This is the second volume of the Collected Works of Henri P

appear, the f irst being volume II , which was published in 1916.* volume contained his contributions in the general f ield of automot ions ,while th e new one con tains thos e concerned p r imari ly with orpartial differential equations, and linear difference equations. The omaterial of the Collected Works is to be that contained in PoinAnalyse of his papers, and it is decidedly helpful to the reader torelevant part of theAnalyse concerning the material of these f irst two vat the beginning of volume I . A similar plan is to be followed withvo lumes .

The s tar t ing point of Poincar 's extraordinary mathematical acto be found in hisNote sur les proprits des fonctions dfinies par les quationsdiffrentielles, published in 1878 in the Journal de l 'cole Polytechniusual existence theorems for an ordinary differential system of thesuch as

dx\ dxn dtX\(X\, ' ' , Xn) Xn(xh ,Xn) 1

fai led to apply in the neighborhood of the singular points(xi, ,xn) a twhich the funct ionsXi vanish s imultaneously. Briot and Bouquet hamined such singular points in special cases by direct processes; work i t appeared that integrals of the form^e~x*= const , p layed an im po rtrole;here\j/ is an ordinary convergent power series inxi, ,xn without const an t term s, pro vide d th a t th e singula r poin t in que stion is tak en aof coordinates. But such a function satisfies the associated l inear dif-ferential equation

d d(2) Z x ~ + +Xn-~ - X^ = 0.

OX\ OXn

Poincar undertook to make this l inear part ial differential equatioof a theory of the solutions of the system (1) near such a singular p

The outcome is e laborated in hisThse of 1879,Sur les proprits des f one-tions dfinies par les quations aux diffrences partielles, which almost immedately became classic. A characterist ic result is the following: If byl inear t ransformation of the var iablesx\, ,xn we can reduce the functioX{ to the form \iXi-\~ terms of higher degree inxi, ,xn, with Xt-=^X/ forir^j (general case), and if there is a line of the complex X-plane supo in ts Xi fall on on e side of it , w hile th e origin X = 0 falls on th e othformal series are convergent, and the general integral of (1) near tpo int is furnished b y th en equa t ions\pi = ce Kit, {i l, , n).

* I reviewed the earl ier volume also; seeThe work of Poincar on automorphic functions, th is Bullet in, vol . 26, pp . 161 -172.

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364 G. D. BIRKHOFF [May,

In case not all of these restrictions are fulfilled, only partial reobtained. Indeed in the part icular case when all the Xi 's are equal tof Poincar fai l completely. However, Poincar characterist ically suexplor ing var ious i llum inat ing typical cases . A system atic t re atm eneven tod ay , a l thou gh m uch fur ther progress has been m ade .

With the local study of the solutions of (1) thus so successfullyPoincar naturally turned to apply his results to the study of the rein the large, in particular in the simplest casen = 2 when (1) may be writ ten

dx dv(3) X F> (*, y, X , F , rea

The first three parts of his great memoirSur les courbes dfinies par les qua

tions diffrentielles, published in four parts in Liouvil le 's Journal (188deal with such an equation in a masterly manner. The fourth pamemoir deals with the analogous problem forn =3. We may refer to an anasis of this work given by Hadamard in his Rice Insti tute Lectures ipublished,* where the reader will also find reference to the subsvelopments in this f ie ld . Fundamental in Poincar 's t reatment on 2\s th e geom etr ic represe ntat ion of(x, y) as a point of the plane of on a face, while the equation (3) determines the direction of the corr

integral curve. This direct ion becomes indeterminate a t the s ingXi=0, (i = l, ,n), referred to above.

The first three parts of this memoir of Poincar may be regardedmental for the theory of the simplest type of real systems of differt ions ,namely for a single equation of the first order. Indeed they arstand always as the most important works in this f ield. Lagrange, other mathematicians had fastened at tent ion upon the integrable cany rate had been content to reduce the order as far as possible byknown integrals. In the field of celestial mechanics, Laplace and theoretical astronomers had been content to use formal series as asystematic computat ion. But Poincar was the f i rs t to a t tack the getions ofnon-integrable systems of differential equations from a purely matical point of view. His principal effort in this direction forn = 2 is containedin the memoir just ci ted. Most of his later work in celest ial mechanregarded as concerned with special important cases wheren>2, in par t icular

w ith t he restr icted problem of thr ee bodies for whichn reduces to 3.Even in the casen 2, interest ing problems of analysis si tus prethemselves to Poincar. One of these of much importance was the fSup pose th at the re be given a one-to-one direct con tinuo us trans formcircumference of a circle intoitself, nam ely 0 i= /( 0 ) , whe re 0 is th e angcoordinate of a pointP, an d 0i is th e coo rdin ate of th e transfo rm ed poPoincar proved that there a lways exis ts a unique numbera such that thenth.i t e ra teP n of P by T has a coord inate 0n intermediate in value between

na 2ir and n

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1934.1 POINCAR ON DIFFER EN TIA L EQUATIONS

In other words, the constanta measures the mean rotat ion of the poiPunder indefinite iteration ofT. For this reasona may be called the "coefficiof ro tat io n" ofT. Poin car also proved t h a t ifO-/(2TT)is rati on al th er e exist peodic poin tsP such tha tP n P\ in thi s case the st ru ctu re ofT may be regardedan ent i re ly known. Furthermore, i f-\-

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366 AUREL WINTNER [May,

him for linear difference equations. The papers by Poincar in thesea long way toward providing a general theory of such l inear equaever, i t is only very recently that a general theory of such equationdeveloped.*

The last paper of the volume under review contains the second unsuccessful Prize Essay referred to above, and was only publisheously in th e A cta M ath em ati ca in 1923. H ere we find a prel imin aryof the problem as to when an ordinary l inear differential equatsecond order with polynomial coefficients and only regular singulsuch that if we writey\(x)/yz(x) =JSwheny\ andy% are two l inear ly independsolut ions , thenx is a meromorphic function ofz. This problem had been pposed and incompletely solved by Fuchs in 1880, and Poincarmainly a cri t ique of Fuchs 's work. In this art icle is to be found thePoincar 's work in the theory of automorphic functions, which is cvolume II of his Col lected Works.

The reader of volume I will also be very grateful for the careful competent revis ion which Professor Drach has provided.

G . D . B I R K H O F F

E I S E N H A R T O N C O N T I N U O U S G R O U P S

Continuous Groups of Transformations. By L. P . Eise nh art . Pr inceton, Pr incUn ivers i ty Press , an d Lond on, Oxford U nivers i ty Press , 1933. 1

This work is an up-to-date textbook on Lie 's theory of cont inuand on the recent developments of this theory. I t is , in contrast to ductions to the subject , not a book writ ten for a beginner who kcalculus. I t wil l , however, be enjoyed by every student who knowstence theorems on differential equations and is familiar with the ethe tensor symbolism. He will not be bored through a considerablebook by the almost tradit ional detai led treatment of more or less amples before reaching the first general theorem of Lie. In fact, in book many i l lustrat ions of the general theory are formulated only aso that Lie 's three Fundamental Theorems can be developed ful ly f irst chapter. Due to a similar at t i tude through the whole work, tbook, though of moderate extent , leads essential ly farther and dits predecessors, without, however, requiring too much from the re

The book s tar ts out with some elementary facts regarding totacobian differential systems and their generalizations. The necessarand uniqueness theorems of local character are used only under theof analyt ic i ty. These preparatory paragraphs are fol lowed by an eof the notion of a continuous transformation group and by a rath

* See a joint paper by myself and Trji tzinsky,Analytic theory of linear difference equations, A cta M ath em atic a , vol . 60 (1932), and a pa pe r by Trjent i t ledAnalytic theory of linear differential equations, in vol. 62 of the samjou rna l .