ODE History

8/4/2019 ODE History

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18 9 3] ORDINARY DIFFERENTIAL EQUATIONS. 119

S OM E O F T H E D E V E L O P M E N T S I N T H E T H E O R YO P O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N SB E T W E E N 1878 A N D 1893.*BY PRO F. T. CRAIG.

S I N C E th e princip les of the infinitesimal calculus were established, the analyst has been confronted by three problems,to wit :The solution of algebraic equations;The integration of algebraic differentials;The integration of differential equations.T he history of these thr ee prob lems is th e same. After longan d ineffectual efforts to con duct the m to simpler problem s,m athe m atician s have reconciled themselves to study thesethree great problems for themselves, and have been rewardedby abund ant success.For a long time the algebraist hoped and strove to solve allalgebraic equations by aid of radicals. T ha t hope has, however, been aban done d, and to-day th e algebraic functions areas well known as the radicals to which it was hoped tocon duc t the m . In th e same way the integra ls of algebraicdifferentials, which were long studied with the aim of redu cingthem to th e elementary functions of algebra, to the loga rithm icor trigonometric functions, are to-day expressed by the aid ofnew transcendants; and the ell iptic and Abelian functionshave as well defined a place in analysis as the logarithmic andtrigonometric functions had less than a century ago.M uch th e same th in g is tru e of differential equ ations. Th enum ber of equations integrable by quad ratures is extremelylimited, and before the mathematician had decided to studythe integrals for themselves, to study them as functions defined by a differential equation, all this analytical domain wasbut a vast terra incognita, wh ich seemed to b e forever inte rdicted to the explorer. Cauchy was the first to pe netra te tothe interior of this unk now n region, which he did by aid ofthe very ingenious method which he called the calculus oflimits. M any othe rs followed him , am ong whom it suffices forthe present to men tion Fu chs , Briot and Bouq uet, and Madame Kowalevski, all of whom employed his method withsuccess.Before taking up the class of differential equations withwhich these and other illustrious names have been particularlyidentified, it will be desirable to speak of t he class of equations wh ich, in simplicity at least, ran ks first. The se are thelinear differential equations.

* Read before the New York Mathematical Society, February 4,1893.


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12 0 ORDINARY DIFFERENTIAL EQUATIONS. [M ar chFrom the t ime of Euler unti l within recent years , the onlyclass of ordinary linear differential equation for which a general me thod of integration was know n, was th at of equationswith con stant coefficients. Th ese are still th e only equa tionswhose purely external form shows th at they are integrable.Among these is , of course, included Legendre's equation,

+ . . . + L(ax + b)(^ + My = 0;which by the transformation ax + b = e t is at once conductedto an equation with constant coefficients.Since the epoch-m aking researches of Fu ch s, published inGrelle in volumes 66 and 68, th e system atic and well-directedefforts of the most illustrious mathematicians of the age havesucceeded in increasing the number of linear differential equat ions, for which there exist methods of integration both surean d general . T he dist inctive characteris tics of sucn equationsare not of a nature to be perceived at a first glance as in thecases just m entioned , bu t can nevertheless be recognized byaid of purely algebraic operations.T he c hara cteristic prop erty of a differential equation of thistype is th at i ts gen eral integral is a uniform function of theindepen dent var iable.I t is by the stud y of the singular p oints of th e equation tha twe recognize w heth er or not this cha racteristic exists. If itexists , th e equation can be integrated . I ts general integra l is ,in fact, the quotient of two synectic functions, of which one,the denom inator, can be writ ten down almost at once. Th isis a polynomial if the singular points are l imited in num ber,an d a holom orphic function which can be constructed byWeierstrass's method if there are an infinite number of singula r points . T he nu m erato r function must be obtained bym ean s of the differential equation. T he equation is thu s integrated, since its general integral is represented by one andthe same function, the quotient of those just mentioned, forall values of th e variab le. In general th is integ ral is a newfun ction , bu t in special cases we can express i t by aid of th efunc tions already intro du ce d into analysis. The se cases aretwo in numb er. Th e f irst case, s tudied by Ha lphe n, is whenth e gen eral integ ral is un iform no t only for all finite values ofth e variable, bu t for infinite values. W hen th is is th e case,th e integ ral is a rat ional function. Th is result can only holdfor eq uatio ns w ith ratio na l coefficients. T he second case wassuggested by investigations of Hermite and discovered by


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1 8 9 3 ] OKDINABY DIFFERENTIAL EQUATIONS. 12 1Pi ca rd . H ere th e coefficients of th e equation, in add ition tobein g uniform functions of th e variable, are doubly periodicfunctions having the same period, and the general integral isunifo rm . W hen this is th e case th e integral can be expressedin finite fe m by mean s of entire polynomials, th e exp onentialfunction, and Jacobi's H-function.Starting with these three categories of linear differentialequations as a basis, H alp he n, in his crowned " Mm oire surla rduction des quations diffrentielles linaires aux formesintgrbles" proposed the following double problem :(1) Having given a linear differential equation in the variable X and the unknown Y, to determine if there exists a substi tutionsuch that , taking x for the new variable and y for the newunk now n function, the transformed equation shall belong toone of the three categories :I. Equations with constant coefficients;I I . Equations whose general integral is rational;I I I . Equations whose general integral is uniform, and whosecoefficients are doubly periodic functions having the sameperiods.(2) Having found such a substitution, to effect the integrat ion.In solving a new problem the att em pt is always m ade tosimplify it by a series of suitable transfo rma tions; b ut the re isa limit to these transformations, for in any problem there is,so to speak, something essential, which it is impossible for anytransform ation to alter . Fro m this arises the im portance ofthe general notion of invariants, which presents itself in everyma thematical question. Lagu erre was th e first to introducethis conception of invariant into the theory of linear differential equations, which can thu s be changed into their simplestpossible forms. Brioschi has also made im po rta nt con tribution s to the theo ry of the in varia nts of linear differentialequations; but i t was Halphen who, start ing from his alreadyestablished the ory of differential inva riants, has, in the mem oircited, made the most important contribution to the theory ofthese invariants, and employed it to solve the above problems.It is impossible to give here the results of Halphen's investigations, but two or three of his conclusions may be mentioned.First, he says, in order that there may exist a substitutionof the above form which will transform a given equation intoone with constant coefficients, it is necessary and sufficientthat i ts absolute invariants be constants.Second, in order th a t there may exist a transform ation ofthe above form which will transform a given equation into an


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122 ORDINARY DIFFERENTIAL EQUATIONS. [M arc hequation whose general integral is rational, or into an equationwith doubly periodic coefficients whose general integral isuniform, it is necessary but not sufficient that the invar iantsbe connected by algebraic relations of deficiency zero or one.Fo r th e furthe r stud y of this second proposit ion Ha lphen 'smemoir must be consulted.Tw o inte restin g resu lts in th e case of th e second problemmay also be mentioned here.(1) Ha vin g given a differential equation in which t he independent variable and the coefficients are rationally expressible in terms of a parameter a:If th e ratios of the integra ls rega rded as functions of a haveonly algebraical critical po ints in nu m be r th ree at m ost (infinity included), and, when these points are three in number,if their orders m, n, p satisfy the condition

I + 1 -+ 1 - > 1 ;m n p

we can make a change of the variable and the unknown function w hich will transform the equation in to one whose generalintegral is rational,(2) Having given a linear differential equation in which thevariable and the coefficients are rationally expressible in termsof a parameter a :If the ratios of the integrals considered as functions of a haveonly algebraical critical points (infinity included); and if thesepoints are three in number and their orders satisfy the relation i +L+Ui,

m n por if th ey are four in. nu m be r and all of th e second or de r;we can find a substitution which will change the equation intoone whose coefficients are dou bly periodic an d whose gen eralintegral is uniform.This last proposition is of particular interest as it givesnew cases of integ rability of Gauss's equ ation, an equ ationwhose importance has been recognized by the ablest analysts,and u pon which an enormous amo unt has been writ ten. I tis sufficient here to mention the names of Schwarz, Fuchs, Bri-oschi, an d K lein. T he new cases of integ rability recognizedby Halphen's theorem are when the equation can be integrated by aid of the ell iptic functions. H alph en concludesthe t hi rd chapter of his mem oir, which deals with the generaltheo ry of in va rian ts of line ar equa tions of all ord ers, by indicating th e m ethod of proced ure which i t is desirable to


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1 8 9 3 ] ORDINARY DIFFERENTIAL EQUATIONS. 123follow when in the study of the critical points we substituteth e consideration of th e invariants for the consideration of theequation itself. In closing his introductory chapter Halphensays, " En rsum, I have treated in th is m emoir a theoreticalquestion relative to the transformation of linear equations intoother equations belonging to those types for which we knowthe integration to be possible. Th e developments indispensable to th e solution of th is question co nstitu te a part, lessthan half, of thi s work. All th e rest is devoted to applications which I have tho ug ht i t best to m ultiply, even at therisk of wearying the rea der. Bu t i t seems to me th at insuch a matter as this applications treated completely oughtto dom inate, and that th e aim to be pursued is this : to in-integ rate effectively equa tions wh ich we know to be inte g r a t e . "The notion of invariants arrived at by a substi tution in thecase of differential equ ation s, is only pa rt of a m uc h mo regeneral theory which can only be briefly alluded to here.All branches of m athem atics, wh ether pu re or applied, areintimately connected th e one with the other, and notions a tfirst restricted to a special field of research are susceptible ofreceiving unforeseen extensions. Such is th e notion of group,m et with to-day in every branc h of mathem atical research.Poincar in his "Notice sur Halphen99 says : " T h e mode ofproced ure of mathem atical science is always th e same. I tstudies transformations of different n at u re s; a nd, to th atend, i t must search for that which remains constant and unaltered du rin g these transform ations. Above all , i t has foraim th e study of grou ps and for means the search for invariants.T his does no t ap pear in every case with th e same distinctness,bu t it is always tru e. If we can see at first sigh t th at projective geom etry is no thin g else tha n the theory of l inear substi tution s, we do no t perceive so quickly th at elementarygeom etry is conducted to the theory of orthogonal substi tut i ons . "Since the time of Galois the theory of groups of substitutions has played a rle of the highest importance in algebra.An analytical theory presenting close analogy to Galois'stheory has been developed by Lie in his " Theorie der Trans-format ions fr tippen /9 a most adm irable acc ount of wh ich byDr. Chapman has recently appeared in the B U L L E T I N of thisSociety. Lie has m ade th e discovery of prim e im porta nce ,th at tlie search for all these gro ups for a given n um ber ofvariables and param eters is conducted to the in tegration oford inar y differential equa tions. Lie's the ory is of th ehighest im portanc e in th e in tegral calculus, the real aim ofwhich is to integrate differential equations; it is not confinedto the transformations of points, but concerns i tself with the


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1 24 ORDINARY DIFFERENTIAL EQUATIONS. [M ar chcon tact t ransformations, so im po rtan t in the theory of part ialdifferential equa tions. Lie has also considered th e subjectof continuous groups of infini te order and developed thegeneral principles involved in the research for the invariantsof differential equations.The detai led investigations of Laguerre, Brioschi , andH alp he n in th e case of linear differential equ ations h avealread y been m entione d. To these nam es m ust also be addedth at of F orsy th, who in h is mem oir " Invariants, covariants,and quotient-derivatives associated with linear differentialequations " and in other papers has added imp or tant and in

terest ing results to th e g enera l theory of the l inear differentialequations.I t m us t suffice here to me rely m entio n a m ost inte res tin gmem oir by App el l on t he s tudy of th e inv ar iants and thecases of integrabil i ty:(1) of equations of the formdl -


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1 8 9 3 ] ORDIHARY DIFFERENTIAL EQUATIONS. 125m ode rn times . H is theo ry of fuchsian functions is connectedat o nce with th e theo ry of differential equations, since itenables us to integrate linear differential equations withalgebraic coefficients, and w ith th e gen eral theory of functions, since these transcendents present certain remarkablef)eculiarities which are of such a nature as to cast considerableigh t on th e m ann er of being of analytical functions. The sefuchsian functions are generalizations of the mo dular func tions, studied by He rm ite in his researches on th e ell ipticfunctions, which possess an infinite number of singular pointsdis trib ute d along a circle. I t will be sufficient to m entio none sing le p rop erty , discovered by Poin car , of these functions, for th e geometer to recognize the ir imp ortance. Th efuchsian function s are of two kin ds : one existing in th e entire plane, th e other existing only in th e interior of th e funda m en tal circle. I n bot h cases the re exists an algebraicrelation between two fuchsian functions which have the samegro up . T he determ ination of the deficiency of this relationis of the highest importance, and has been obtained by Poincar both by analy tical processes an d by aid of t h e geom etryof position. T he ex istence of these relations mak es it possibleto utilize these function s in the stud y of algebraic func tionsan d algebraic curves. Th us , we can express the co-ordinates ofa point of any algebraic curve whatever as fuchsian functionsythat is as tiniform f unctions o a single parameter. '

So pro fou nd a res ult is sufficient in itself to show th e interest and importance attaching to these new functions ; but,for the present purpose, it is desirable to give at least a faintidea of th eir relations to th e linear differential e quation s. Toat te m pt to give a full idea of t he imp ortan ce of the se func tions of Poin car 's , would be, for the present w riter, to attem ptth e impossible ; indeed , a mere statem ent of Poinca r 's r esults produces an impression of exaggeration.Since the t ime of Cauchy, mathematicians, recognizing theenormous difficulties and complications of the problem, haveno t atte m pted to study the na ture of the integrals of differential equations, ordinary or partial, for all values of thevariable, that is , thro ug ho ut th e plane,b ut have confinedthemselves to the investigation of the properties of theseintegrals in the neighborhood of certain given points. The yth us perceived th at these properties are very different according as we are concerned with an ordinary point or a singularpoin t. T he researches of Brio t and B ouque t are too wellknown to need more than mention, and since their appearancemo st im po rtan t addition s to the theory of non-linear equationshave been made, notably by F uch s, Poincar , Picard, andmo re recently by Painlev. Bu t th e study of the integralsof differential equations in the neighborhood of a given point,


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12 6 ORDINARY DIFFERENTIAL EQUATIONS. [M arc hwh atever may be its utility from th e point of view of nu m erical calculation, can only be regard ed as a first s tep in thi sstu dy . The se deve lopm ents, which hold only in a verylimited region, do not teach us concerning the integrals ofdifferential equations t ha t which th e ^-functions teach usconcerning the integrals of algebraic differentials : they cannot be considered as true integrations.They m ust the n be taken as a poin t of d epartu re for amore profound study of the integrals of differential equations,in w hich it is proposed to free ourselves from the restrictionsof these "lim ited region s," where, to use a phrase of Poin-car 's , "on s'tait systmatiquement cantonne'9 and to followth e in tegral as the variable moves throug ho ut the entirep lane .Th is generalized study can be entered upon from two pointsof view.(1) We may propose to express the integrals by developmentswhich alioays hold, and are no longer limited to a particulardom ain. Th is leads to the introduction of new transcendantsinto the theory ; but this introduction is necessary in any case,for the functions which have previously been introduced intoanalysis will only permit us to integrate a very small numberof differential equations.(2) This method of integration, though affording us a knowledge of th e prop erties of th e differential equations from th epo in t of view of th e theo ry of function s, is no t in itself sufficient if, for exam ple, we wish to apply th e equations toque stions of mech anics or physics. These developm entswould not readily teach us, for exam ple, if th e function wasone which continually increased, if it oscillated between certain l imits, or if i t ' increa sed beyond all l imit . In otherwords, if the function is considered as defining a plane curvewe would not know th e general form of the curve. In certainapplications all these questions are of as much importance asthe num erical calculation. We have here the n a new problem for solution. These are th e two problem s tha t Poinc arproposed to himself and undertook and succeeded in the taskof solving,truly a task for a giant of intellect." Desiring," to quote Poincar, " to express the integrals ofdifferential equations by the aid of series always convergent, Iwas na tur ally led first to atta ck th e linea r differential equat ions . These equations, in fact, which have been duringrecent years the object of the investigations of Fuchs, Thome,Probenius , Schwarz, Klein , and Halphen, were the bestkn ow n of all. W e had for a long tim e possessed th e developments of their integrals in the neighborhood of a given point,and in a quite large number of cases had succeeded in completely int eg rati ng the m by aid of func tions already know n.


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1 8 9 3 ] ORDINARY DIFFERENTIAL EQUATIONS. 12?It was the n in comm encing m y studies at this po int t ha t Ihad the most chances of arriving at a resuit ." B ut i t was further necessary to m ake some hypothesisconcerning the coefficients of the equations which I wished tostu dy . If, in fact, I ha d ta ke n for coefficients any functionswhatever, I would equally have obtained for integrals anyfunctions whatever, and consequently would not have beenable to say an yth ing precise on th e subject of t he na tu re ofthes e integ rals, which was my aim. I was th us led to exam ine the l inear equations with rational and with algebraiccoefficients."

The following is Poincar's classification of these linearequ ations, th e one wh ich, from th e po int of view of the pro posed problem, is the most natural .L e t y be an integral of a linear equation of order n withrat ion al coefficients. W rite

where X and the F's are rational functions of x. It is cleart h a t z, l ike y, will satisfy a lin ear equ ation of o rde r n withratio na l coefficients. Po inca r says th at these equations belongto the same family. In fact, it is easily seen that the knowledge of the properties of the function y involves that of theprope rties of the function z. In each family the re are an infinityof different equations, but certain functions of the coefficientshave the same value for equ ations of th e same family.. In othe rword s, the re are invariants which remain unaltered by theabove substi tution . These invariants are not the same as thoseof H alp he n for linear equation s. The se latter arise from atransformation which consists in replacing x by any functionwhatever of x' and mult iplying y by any other f unction whatever of x'. On the contrary, the functions entering intoPoincar 's substi tution are not any whatever, but are rational.T h is shows well th e difference between H alp he n's investigations and those of Poincar. Ha lph en sought above all tofind the relations between different integrals, and could thuswith impunity introduce any functions whatever into his calculations. Poinc ar, on th e o ther han d, sought to study thenature of the integral itself ; this nature would manifestly bealtered if the integral were multiplied by an arbitrary function, as is the case in Halphen's work.This intimate study of the nature of the integrals can onlybe m ade by the introduction of new transc end ents. Thesenew transce nde nts have a close analogy to th e ell iptic functions, which is not surprising, as Po incar was atte m ptin g todo for differential equ ation s w ith algebraic' coefficients w ha t


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12 8 ORDINARY DIFFERENTIAL EQUATIONS. [M arc hhad already been done by aid of the elliptic and Abeliantheta-functions for the integration of algebraic differentials.This analogy with the elliptic functions served Poincar asa guid e in all his researches. T he elliptic functions are un i-form functions which are u naltered when the variable isincreased by certa in periods. Th is notion is so useful inmathematical analysis that mathematicians had long seen thedesirability of generalizing it by seeking uniform functions ofa single variable x, which would remain unaltered when thisvariable was subjected to certain transformations. Thesetransform ations cann ot be chosen arbitrari ly, bu t m ust evident ly form a g ro up : and fur ther , th is group m ust containno infinitesimal transf orm ation ; th at is , # m ust not vary byan infinitesimal a m ou nt. Fo r, if this were so, on rep eatin gthis transformation indefinitely, x would vary continuously ;and our uniform funct ion, remaining unal tered when thevariable changed continuously, would necessarily reduce to aconstan t. In othe r words, the gro up of transformations m ustbe discontinuous.In ell iptic functions the transformations of the group consist in ad din g constants. Th e functions are studied bydividing the plane into an infinite nu m ber of parallelogramsof periods. All the parallelograms are obtained by transforming one of them by th e different s ubsti tutions of th egr ou p; so th at a knowledge of th e function in one of theseparallelograms involves a knowledge of i t throughout theent i re plane.So if we consider a more comp licated discontinuous gro upgene rated by a transc end ent of h igher order, we can divideth e p lane (or the pa rt of th e plane in which the functionexists) into an infinity of regions, or curvilinear polygons, insuch a way that we can obtain all these regions in applying toone of the m the different transform ations of th e grou p. Th eknowledge of the function in the interior of one of thesecurv ilinear polygons will involve th e know ledge of th e function for all possible values of the variable.In the ell iptic functions, considering the integrals of " th efirst kind," we regard the variable x by th e process of inv ersion as a func tion of th e inte gra l. T he function th us definedis unifo rm a nd doubly periodic. So, considering a l inearequ ation of the second ord er, and by a species of inversion,Poincar regards the variable x not as a function of anintegral but as a function of the ratio of the two integrals ofth e equa tion. In ce rtain cases th e function th us defined isuniform , and the n rem ains unchang ed by an infinity of l inearsubst i tut ions changing z into

yz + '


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1 893 ] ORDINARY DIFFERENTIAL EQUATIONS. 12 9W hen this is th e case, the group formed by these sub stitutions must be discontinuous, and the curvilinear polygonsreferred to m ust be lim ited by arcs of circles. Poincar firstsupposes the coefficients of the substitutions

\*> yz+)to be re al, or, wh at comes to the same thin g, supposes tha tthese substitutions do not alter a certain circle which he callsth e "fundamental circle." In this case the arcs of circleswh ich serve as sides of the polygons c ut th e fun dam entalcircle orthogonally.W hat is the n the condition tha t a group generated by af lven curvilinear polygon shall be disco ntinuo us? Poin carnds this condition, and the n constructs the discontinuousgroup s formed by subs ti tutions which leave the fundam entalcircle una ltered, and which h e calls fuchsian groups. He re animp ortant problem now presents itself: having given a fuchsian gro up , do th ere exist functions which are una ltered bythe substi tutions of this group ? In answering this questionPoincar appeals again to the analogy with the elliptic funct ions. These functions can be regarded as the quotients oftwo 0-series. Th ese aux iliary transce nd ents are no t onlyuniform , bu t are also entire functio ns; they are not doublyperiodic, bu t reproduce themselves mu ltiplied by an exponential wh en th e variable is increased by a period . So in th enew case the fuchsian functions are expressed as the quotientof two "thetafuchsian" series, which are finite and uniform,perfectly analogous to th e ^-functions, and reproduce them selves m ultiplie d by a sim ple factor when we apply to th evariable one of the substi tutions of the group.In order to complete the analogy with the elliptic functions,it was necessary th at the oth er prop erties of thes e functions,such as addition, mu ltiplication, and transformation, shouldbe extended to the new transce nde nts. Th e theory of transform ation is imm ediately generalizedalways with this dif-ference, th a t th e grou p of fuchsian functions being very muchmore complicated than that of the ell iptic functions, thecases to be considered are much more numerous and varied.A po int of great inte rest in th e extension of this theory oftransformation is the new light thrown upon the reduction ofthe Abelian integrals. T he theory of addition cannot beexte nd ed to all th e fuchsian function s ; it is only possible inone pa rticu lar case an d for one special class of these tra ns cend ents. Th e question need not, however, be entered intohere.In extending the l inear substi tutions to the case where the


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130 ORDINARY DIFFERENTIAL EQUATIONS. [M ar chcoefficients are no lon ger restricte d to be rea l, b u t are a rbitrary, Poincar arr ives at the discontinuous groups which hecalls kleinean, and so to a new class of functions, the kleineanfunctions, wh ich a re perfectly analogous to the f uchsian funct ions . T h e on ly difference which it is necessary to m en tionis tha t wh ich arises from th e form of th e region inside ofwh ich these new functions exist. Th is region instead ofbeing a circle is l imited by a non-analytical curve which hasno determ inate radius of curv atu re; in other cases the regionis lim ited by an infinity of circum ference s. A no the r class offunc tions can only be m ention ed. The se are the zeta-fuch-sian functions, which play the same part in the integration ofth e fuchsian functions th at the zeta-functions play in theintegration of the el l ipt ic functions.In con cluding this very impe rfect sketch of wha t Poincarhas done for th e in tegra tion of l inear differential equations,we can say that he has shown how to express the integrals oflin ea r e qua tions w ith a lgeb raic coefficients by aid of the senew t ransc en den ts in t h e same way as the integrals of algebraic d ifferentials were previously expressed by aid of th eAb el ian funct ions . F ur th er , these lat ter integrals are them selves suscep tible of be ing ob tained by aid of th e fuchsianfun ction s; and we thu s have for th em expressions entirelydifferent from thos e g iven by th e ^-functions of severalvariables.Poincar's further investigations on the integration of l ineardifferential equ ation s by aid of algebra ic and Ab elian funct ion s, his work on no n-l inear equations (with one exceptionto b e referred to lat er) , on curve s defined by differential equat ions, on irregular integ rals , on pa rt ial differential equations,and on the differential equations of celest ial mechanics, mustbe passed over. En ou gh ha s been said, however, to show thegre at value of his work. In leaving the subject of l inea rdifferential equ ation s a m ere m en tion w ill have to suffice oftwo interes t ing ca pers w hich have recent ly appeared : oneby A ppe ll , on linear differential equa tions transformab leinto themselves by chan ge of the function and the variable;an d one by Helge von Ko ch, on infinite determ inants andlinear differential equations.

The l inear differential equations possess one remarkablepro per ty , the s ingular points are the same for al l the integralsI n th e case of equ ations whose coefficients are entire polyno mials in x9 these singular points are the values of x whichan n ul th e f irst coefficient. I t is up on th is circumstanc e th atthe method of integration of these equations by the zeta-fuchsian functions is founded . No n-linear equations do no t ingen eral possess this prop erty. T his has led to im po rtan t in-


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1 8 9 3 ] ORDINARY DIFFERENTIAL EQUATIONS. 131vestigations by Fuchs and Poincar with the end of ascertain*ing w he the r or no t th er e existed oth er classes of differentialequations for which al l the part icular integrals had the samesingu lar points . T he researches of these ma them aticianswere confined to equations of the first order.Fu ch s found the necessary and sufficient cond it ions th at anintegral of the equation

where y, y' enter algebraically, shall have only fixed criticalpoints x t. These conditions once satisfied, if we wish the general integral to be un iform, i t rem ains to express th at thesepoints x i9 which are known, are not cr i t ical points of the integrals . Poincar resuming the question, arr ived at conclusionsas interes t ing as unexp ected. H e found tha t when the cr i ticalpoints of the equation

are f ixed this equation can ei ther be integ rated algebraically,or by a quad ratu re, or can be condu cted to a Riccati's e quation.Pica rd has un de rtak en the generalizat ion of the work ofFu ch s an d Po inca r to th e case of differential e qua tions ofth e second ord er. H e says : " I t would seem at a first glanc eth at the extension of t he reasonings employed in the case ofequ ations of the first ord er would be easy, b ut this is no t th ecase. We may indeed com mence by following the m ethod sof reasoning employed by Poincar ; the end of his reasoningis un fortu na tely no t applicable. W e find ourselves always inpresence of th e same fact : A certain bi-uniform transform ation is no t necessarily bi-ration al, a nd it is th is fact w hichchanges thro ug ho ut th e ent i re character of this theo ry."Pica rd confines himself to th e case of bi-ration al transfo rm ations. T he difficulties in th e way of a stu dy of non-lineardifferential equ ations of th e second o rder have caused th issubject to remain as yet almost untouched ; indeed, thoughin rece nt years m uc h has been don e, the non-l inear equ ationsof the first order still afford a vast field for research.A problem of the high est imp ortanc e connected with thedifferential equ ations of the first ord er and degree is to d eter m ine in any case wh ether or no t the equation is integrab lealgeb raically. T o solve th is problem i t is ma nifestly sufficientto f ind a superior l imit for the degree of the integral ; af terthat the only operations remaining to be performed would bepure ly algebraical . T his is a problem on th e solution ofwhich, i t would seem, m athema t icians mig ht have beentempted to bestow much labor ; nevertheless, they have concerned themselves very l i t t le w ith i t . Ind eed , from the t ime


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1 32 ORDINARY DIFFERENTIAL EQUATIONS, [M ar ch.of Darboux's masterly memoir in 1878, the problem was en-t irely neglected, until the Acadmie des Sciences, recognizingthe importance of a more profound study of differential equat ions of the first order and degree, proposed as a subject ofcompetit ion for the Or and Prix des Sciences Mathmatiquesfor 1800 the following :" To perfect in some important point the theory of the dif-ferential equations of the first order and first degree."The outcome of the compet i t ion has been two impor tan tmemoirs : one by M. Painlev, to which was awarded theGrand Pr ize ; and the other by M. Autonne , who receivedhonorable mention.M. Autonne takes for poin t of depar ture a geometrical in-terpreta t ion of w hich every differential equa tion of the firstorder is susceptible. He shows that such an equation can beconsidered as giving curves situated on a certain algebraicsurface, and of which the tangents belong to a certain l inearcomplex. The surface is unicursal if the equation of the firsto rder is at the same t ime of the first degree. Ta kin g the n aunicursal surface, the author forms the equation which corresponds to it, and which he calls "rglementaire." His workthen consists of a s tudy of the equations of the first order anddegree considered as "rglementaires" Autonne makes aclassification of the crit ical points of the equation. It willbe sufficient to ment ion , in addition to the ordinary crit icalpoints forming the genera l case, those which he calls " di-criiicaV9 These are points throu gh which pass an infiniten u m b e r of simple branches of integrals having an arbi t rarytangent .Painlev's researches consti tute one of the most importantcontr ibut ions ever made to the theory of the non-linear dif-ferential equations of the first order and degree. He considers any differential equation whatever of the first orderwhere the function and its derivative figure algebraically, andmakes at first an important dis t inct ion between the fixed andthe movable cri t ical points of the integrals,by movable cri t ical points meaning those which are susceptible of changes ofposition as the constant of integr ation changes. Such pointscannot be points of indterminat ion for the integral . Theut i l i ty of making this remark explicit ly is seen if it is noticedthat equat ions of an order higher than the first can havemovable singular points. As an i l lustration, take the equat ion W-y/y +W^o,whose general integral is

-L


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1 8 9 3 ] ORDINARY DIFFERENTIAL EQUATIONS. 133the essential singularities of the integrals depend evidently onthe constant G 9.Painlev draws a system of cuts in the plane of the independent variable, which prevent this variable from turningroun d the fixed critical points, and then studies the equationsfor which the integrals take only a limited number of valuesaroun d the movable cri t ical points. U nd er this hypothesiswe can conceive the general integ ral p ut into a form whichwill bring into evidence a class c of algebraic curves associated with the given equation.Every curve of this class is a rational transform of thecurve re prese nted by th e differential equa tion wh en for an yxe value wh atever of the variable we regard the functionan d its deriv ative as th e co-ordinates. If the deficiency of th ecurves c is greater than unity, we can by algebraic operationsdetermine the class, and the integral itself can be obtainedalgebraically. If the curves c are of deficiency one, it may benecessary, in order to find the integral, to obtain a solution ofa linear equation, and to ascertain if a certain Abelian integral has only two perio ds. Only th e case of deficiency zeroescapes thi s m ethod ; th is circu ms tance necessarily prese ntsitself when the equation is of the first degree with respect tothe derivative.Painlev's methods can be applied in the attempt to ascertain whether or not the integrals of a given differentialequation are algebraic, or can take only a limited number ofvalues in th e plane. T he distinction between fixed andmovable critical points permits the separation of this questio n in to two pa rts, and in some cases th e solution of th isproblem can be arrived at,a solution which in all its generality will doubtless not soon be found.T he equalit ies and inequalit ies added by. Au tonne andPainlev to those of Darboux c onsti tute a very im por tantprogress in the solution of the problem of finding out whetheror n ot a differential equ ation of the -first order an d degree isalgebraically integrable; but, as Poincar has pointed out ina recent paper in the Rendiconti del Gircolo Mat&matico diPalermo, m uch rem ains to be done. Suppose, in fact, tha tthe general integral is writ ten

F = const.,where J77 is a rationa l function . A no ier form of the generalintegral will be given by equating to a constant any entirepolynomial whatever in F. It follows as a consequence ofthis that the superior limit of the degree of the general algebraic integral cannot be found, at least unless some means


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1 3 4 ORDINARY DIFFERENTIAL EQUATIONS. [M ar chcan be found of expressing in th e inequalities th at this integral is irreducible.Painlev himself clearly perceived this difficulty, but wasun ab le to overcome it. H e was unab le to solve th e problemin all its generality, but only to show that in a certain number of cases th e integ ral could n ot be algebraic. As alre adystated, Painlev's problem was to determ ine whether a givendifferential equ ation ad m itted a n algebraic integ ral of agiven deficiency. In th e pape r referred to, Poin car gives aformula which contains the complete solution of Painlev'sproblem whenever the dimension of the equation exceeds 4.He has also demonstrated some properties of equations whichare integra ble algebraically. Suc h resu lts, he says, may notfor the mo m ent possess any g reat value, bu t they may ac quireone the moment i t is known whether these properties can beextended to non-integrable equations, or if they are notalways true for these equations. In th e first case we wouldhave a general theorem applicable to all differential equations,and in t he second case we would have a criterion w hich wouldenable us to dem onstrate th at the equations of certain categories were non-integrable.In the subject of partial differential equations most important researches have been made during the past ten years,notably by Gou rsat, Poincar, Picard , and Darboux. Inde ed,th e lat ter 's treatis e on th e the ory of surfaces m igh t, from acertain point of view, be regarded as a treatise on these equations, particu larly the equa tions of the second order. Pica rdhas incorporated some of his more important results in thefirst fascicule of the second volume of his Traite iVAnalyse.There also, among other methods for the proof of the existence of an integral of a differential equation, he gives hisown elegant metho d of proof by successive ap pro xim ation s,an English translation of which by Dr. Fiske has been pu blished in the B U L L E T I N of this Society. Picard has ma desome interesting extensions of this method, particularly topar tial differential equations. T he whole subject, howev er,of th e re cen t develo pm ents of the theory of these equationsand the ir application s to geom etry m ust be reserved foranother paper .

JOHNS HOPKINS UNIVERSITY,BALTIMORE, January 15, 1893.

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