On the Pfaff problem

Transcription

1 Sur le problème e Pfaff Bull sc math astro (2) 6 (882) 4-36 O the Pfaff problem By G DARBOUX Traslate by D H Delphech The metho for the tegrato of partal fferetal equatos a arbtrary umber of epeet varables that Pfaff mae ow 84 the Mémores e l Acaéme e Berl has bee eglecte for qute some tme The beautful scoveres of Jacob a Cauchy have oly attracte the atteto of geometers that were occupe wth that theory Meawhle the Pfaff metho whch s moreover a geeralzato of oe that s ue to Lagrage for the case of two epeet varables offers some coserable avatages It replaces calculatos that are ofte complcate wth the use of certa fferetal ettes that gve the ey to the tutve soluto of ffcultes that preset themselves the other methos The beautful results that were obtae by Le varous memors that were serte to the Mathematsche Aale show all of what oe ca fer from these ettes for example f oe woul le to reuce to the smallest possble umber the tegratos that oe must successvely perform before arrvg at the complete soluto of a partal fferetal equato I the wor that oe s presetly reag I propose to expla the soluto to the Pfaff problem wthout ay recourse to the theory of partal fferetal equatos a above all I am oblge to exhbt the varace propertes that play a fuametal role that soluto I am ot at all cocere wth the tegratos that are ecessary orer to brg a fferetal expresso to ts reuce form a moreover from some formulas that I wll gve the operatos that oe must o orer to obta the soluto to that problem ca be cope some fasho oto the oes that refer to the tegrato of a partal fferetal equato I the frst part I stuy the reuce forms a I show that the tegrato of the frst Pfaff system suffces a mmeately gves the reuce form whe oe s ealg wth the fferetal expresso that correspos to a partal fferetal equato I the seco part I stuy the relatos betwee the reuce forms a I prove partcular three propostos that serve as the bass for the theory of Le groups ( ) ( ) The frst part of ths paper was wrtte 876 a commucate by Bertra who taught the theory of partal fferetal equatos at the Collège e Frace at the tme Bertra has ly explae the metho that I have propoue here hs frst lecture Jauary 877 Some tme later a beautful memor of Frobeus appeare the Joural e Borchart that carre a ate that was prevous to Jauary 877 moreover (vz September 876) a whch that geometrc scholar followe a path that s very closely aalogous to the oe that I commucate to Bertra the sese that t reste upo the use of varats a the blear covarats of Lpschtz Upo returg to my wor at that pot tme t seeme to me that my exposto was more free from calculatos a as a cosequece from the mportace that Pfaff metho s sa to tae o so there woul be some terest mag t ow

2 Darboux O the Pfaff problem 2 Coser the fferetal expresso: PART ONE I X x + + X x where X X are gve fuctos of x x We eote them by the otato Θ where the ex refers to the system of fferetals that s aopte Oe wll thus have: () Θ = X x + + X x a f oe employs other fferetals that are eote by the character δ the oe has: (2) Θ δ = X δx + + X δx From the two preceg equaltes oe euces that: δθ = δx x + + X δx a cosequetly: Θ δ = X δx + + X δx δθ Θ δ = (δx x X δx ) X X = ( xδ x xδ x ) the summato beg extee over all combatos of ces 2 a cosequetly cossts of ( )/2 terms To abbrevate we set: X X (3) a = a the preceg equalty wll become: (4) δθ Θ δ = a ( x δ x x δ x ) By vrtue of the ettes: I the same year vz 877 a mportat memor by Le o the same subject also appeare the Archv for Mathemat Chrstaa (t II pp 338) However ths paper rests upo methos that are completely fferet from the oes that I wll preset

3 Darboux O the Pfaff problem 3 a + a = a = whch follow from formula (3) oe ca further wrte equato (4) the form: (4 cot) δθ Θ δ = ax δ x = = Now suppose that oe replaces the varables x the fferetal expresso () wth some other varables y Upo performg the substtuto that s efe by the formulas: (5) x = ψ (y y ) whch gves: ψ x = y the expresso Θ wll tae the form: (6) Θ = Y y I all of what follows we shall assume that the fuctos y are epeet As a result the ew varables y ca be regare as epeet fuctos of the ol oes x As for the coeffcets Y oe ca always trasform them by the use of formulas (5) to fuctos of the varables y Havg sa that apply formula (4) to the ew expresso for Θ If we set: Y Y (7) b = the we wll have: a cosequetly: (8) δθ Θ δ = ax δ x = = = b y δ y = = b y δ y = = Ths formula s fuametal to our theory Furthermore before cotug we shall gve a rect proof of t wthout appealg to the property that s expresse by the equato: δx = δx that we mae use of From a comparso of expressos () a (6) for Θ oe euces the equaltes: X + + X = Y

4 Darboux O the Pfaff problem 4 whch serve to efe the quattes Y From them oe euces that: a cosequetly: Y Y Y = = x X x x 2 α α α α Xα + α α α α X α X α α α α α α α α α where the sum o the rght-ha se s tae over all systems of fferet values of α α a cosequetly t cossts of ( )/2 terms If oe multples the preceg equato by y δy y δy a oe the taes the sum of the ( )/2 equato thus obtae the the coeffcet of: the rght-ha se wll be: X α α X α α e: Oe wll the have: α α α α ( yδ y yδ y ) x α δx α x α δx α (9) Y Y ( yδ y yδ y ) X α X α = ( xαδ xα xα δ xα ) α α whch s the same thg as equato (8) II Havg sa ths coser the varables x to be fuctos of oe auxlary varable t that are efe by the fferetal equatos: () ax + + a x = λ Xt a2 x + + a2x = λ X 2t a x + + ax = λ X t

5 Darboux O the Pfaff problem 5 where λ wll be a quatty that oe ca choose arbtrarly to be a costat or a fucto of t epeg upo the stuato We remar that equatos () ca be replace wth the sgle equato: () a = = a x δ x = λ t X δx that oe obtas by ag them after havg multple them by δx δx respectvely prove that oe requres that ths equato s verfe for all of the values that are attrbute to the auxlary varables δx Therefore the system () ca be replace wth the sgle equato: () b δθ Θ δ = λ Θ δ t whch must be true for ay fferetals δ I the applcatos t wll always be preferable to rectly form the two ses of the latter equato stea of successvely calculatg the quattes a that appear system () From ow o the preceg remars wll lea us to a fuametal property of system () Suppose that oe performs a chage of varables a oe replaces the varables x wth some other varables y that are equal umber a whch are epeet fuctos of the former It s easy to see that the system () s trasforme to the oe that oe forms the same maer by tag ew epeet varables Ths results mmeately from the fact that ths system whe wrtte the form () b s obvously epeet of ay choce of epeet varables However for the sae of eatess coser equato () a Oe ows by vrtue of equalty (8) that ts left-ha se wll become: b y δy As for the rght-ha se t wll obvously trasform to the followg oe: λ t Y δy Therefore equato () a wll tae the form: b yδ y = λ t Yδ y Sce the fuctos y are epeet ther fferetals δy are arbtrary le the fferetals δx Oe ca the equate the coeffcets of the fferetals the two ses a oe wll obta the equatos:

6 Darboux O the Pfaff problem 6 () b y + b2y + + b y = λy t b y2 + + b 2y = λy2t b y + + by = λy t Therefore wheever the fuctos x satsfy equatos () the fuctos y wll satsfy equatos () The coverse s obvously prove the same maer Oe ca thus say that systems () a () are absolutely equvalet sce they are two forms of the same system of fferetal equatos whe wrtte fferet varables As they are compose the same maer by meas of varables that eter to them we express ths property a abbrevate maer by sayg that t amouts to sayg that system () s varat We shall mae use of ths proposto orer to cate the reuce forms to whch oe ca covert the fferetal expresso Θ III Frst suppose that s eve The sew etermat: ± a a 22 a wll be a perfect square We beg by assumg that ths etermat s o-zero Oe ca the solve equatos () for x x a oe wll obta a system of the form: x H = = x H = λ t that amts epeet tegrals of t Tae these tegrals to be ew varables that we eote by y y a a fucto y that s subject to the sgle coto that t ot be a tegral of the system y y the efe a system of epeet fuctos a the system () whe wrtte the ew varables wll tae the form () Oe must the express the ea that equatos () are verfe whe oe assumes that the fuctos y y them are costats Oe must the have: Y Y y = λ Y t Y From ths oe euces that: 2 Y y = λ Y 2 t = λ Y t 2

7 Darboux O the Pfaff problem 7 Y = logy logy = 2 logy = = = λ t y The latter equatos show that the fuctos Y Y epe effectvely upo the y but that ther mutual ratos are epeet Oe ca thus assume that for < oe has: Y = KY Y beg epeet of the varable y whle o the cotrary K ecessarly cotas t Oe thus comes ow to a fferetal expresso of the form: Θ = K( Y y + + Y y ) whch has at least oe term but whch aga ejoys the property of cotag the varable y oly the factor K Oe ca further wrte: (2) Θ = y ( Y y + + Y y ) upo ow eotg the coeffcet K by y Now suppose that s o The etermat: = ± a a wll the be zero sce t s sew-symmetrc of o orer a cosequetly equatos () wll ever be mpossble f oe sets λ = them We frst suppose that all of the mors of frst orer are o-zero I ths case equatos () where oe maes λ = eterme the ratos of the fferetals completely They therefore amt epeet tegrals that we aga eote by y y a that we tae for the ew varables whe we a a fucto y to them that wll ot be a tegral a wll cosequetly form a system of epeet fuctos wth them Equatos () must the be verfe by the substtuto of the equatos: whch wll gve: λ = y = y = Y Y = Y2 Y = 2

8 Darboux O the Pfaff problem 8 Y Y = It s easy to f the most geeral form for the fuctos that satsfy these equatos Iee set: Ψ Ψ Y = Y = + Y The equatos express the ea that the ervatves of the fuctos y are zero Oe ca thus set: Θ = Ψ + Y y + + Y y Y wth respect to whch the fuctos Y o ot epe upo y However two fferet cases ca preset themselves here I geeral Ψ wll cota y a cosequetly Ψ y y wll be epeet fuctos Upo chagg the otato a eotg Ψ by y oe wll get the frst reuce form: (3) Θ = y + Y y + + Y y However t ca also happe that Ψ oes ot cota y Oe wll the have: or more smply: (4) Θ = Ψ Ψ Θ = + Y y Y y Y y + + Y y It s moreover very easy to stgush these forms from each other a pror Iee the latter s characterze by the property that Θ s aulle whe oe has: y = y = Oe thus sees that oe wll obta ths form wheever the equato: X x + + X x = s a cosequece a smple lear combato of equatos () whch oe has set λ = For example coser the form three varables: Here system () becomes: F = X x + Y y + Z z =

9 Darboux O the Pfaff problem 9 (5) x Y Z z = y Z X z = z X Y If oe replaces x y z the form wth quattes that are proportoal to them the oe obtas the well-ow expresso: (6) Y Z Z X X Y X + Y + Z z z If ths expresso s o-zero the oe ca covert F to the form: γ + M α + N β where α β are tegrals of the system (5) M a N are fuctos of α a β a γ s a fucto that s epeet of α β O the cotrary f the expresso (6) s zero the the term γ wll sappear a what wll rema s: whch s agreemet wth ow results F = M α + N β = µ u IV Up to ow we have assume that the system () s etermate Now mage that t s ot Thus f s eve the the etermat: ± a a wll be zero a cosequetly the same wll be true for all of ts frst-orer mors by vrtue of a ow property of sew-symmetrc etermats If s o the the frstorer mors of the same etermat wll all be zero Equatos () the reuce to at least stct oes a t o loger suffces to eterme the mutual ratos of x x t However I remar that they always form a system that s equvalet to system () sce the argumet that we mae orer to establsh that equvalece suffers o excepto To smplfy suppose that oe has mae λ = Equatos () wll be etermate Suppose that they reuce to p stct equatos where p ca be equal to zero I arbtrarly appe p fferetal equatos for example the followg oes: ϕ = ϕ 2 = ϕ p =

10 Darboux O the Pfaff problem where ϕ ϕ p are arbtrary fuctos a I thus obta a perfectly eterme system I further call the tegrals of the complete system y y a upo ajog to them a fucto y that s ot a tegral I aga obta epeet fuctos y that I substtute for the varables x The system () whch oe sets λ = wll be verfe le the frst oe whe oe sets: y = y = By a argumet le the oe that we mae the case where s o we are le to the same coclusos a we f oe of the forms (3) or (4) I summary we ca state the followg theorem: A form Θ varables ca always be coverte by the tegrato of the system () to oe of the three forms: (A) y ( Yy + + Y y ) Y y + + Y y y + Y y + + Y y where the varables y y are epeet a where the fuctos Y epe oly upo y y Some of the fuctos Y ca be zero moreover The frst of these three forms presets tself oly whe s eve a the etermat: s o-zero ± a a Oe ca further state the preceg result the followg maer: Let Θ eote a fferetal form varables Oe ca always covert Θ to oe of the three forms: y Θ Θ y + Θ where y s a varable that s completely epeet of the oes that fgure the ew fferetal expresso Θ V We may ow prove the followg theorem: A form Θ ca always be coverte to oe of the followg two types: (7) y zy z2y2 z py zy + z2y2 + + z py p p

11 Darboux O the Pfaff problem where the fuctos y y z costtute a system of epeet varables; e they are fuctos that are epeet of all the varables that eter to the form Θ The frst of these two preceg types wll be sa to be of etermate type whle the other oe wll be sa to be of etermate type We shall prove that ths proposto s a almost mmeate cosequece of the preceg oe Iee t s obvous for forms oe a two varables It wll the suffce to show that f t s true for a form varables the t s also true for a form that cotas oe more varable I orer to o ths we remar that a form varables ca be coverte to oe of the three types A Neglectg the seco oe whch epes upo oly varables a for whch cosequetly the theorem s allowe we remar that the other two are compose a very smple maer wth the fucto varables Y y + + Y y Replacg that form varables wth oe of the two types (7) we obta oe of the followg expressos for the form varables: y (u v u v 2 u 2 v p u p ) y (v u + v 2 u 2 + v p u p ) (y + u) v u v 2 u 2 v p u p y + v u + v 2 u v p u p where u u v are epeet fuctos of y y a where cosequetly y u u v are epeet fuctos of the orgal varables The last two expressos obvously fall to the etermate type As for the frst two oe coverts them to the seco type by substtutg the followg fuctos for the fuctos v v p : v y = ± w v p y = ± w p The theorem s thus establshe The followg cosequece s a obvous result: If the reuce form for the expresso varables Θ s: z y + + z p y p the the 2p fuctos z y of the varables x are epeet so oe ecessarly has 2p If the reuce form s: y z y z p y p the oe must lewse have 2p +

12 Darboux O the Pfaff problem 2 VI We shall ow solve the followg problem: If oe s gve a form Θ varables the whch of the two types (7) ca t be coverte to a what s the value of the umber p the? Ths problem s susceptble to a extremely smple soluto Iee suppose that oe trasforms the expresso Θ by tag the ew varables to be the oes that fgure the reuce form a choosg the other oes a arbtrary maer so that they woul complete the umber of epeet fuctos Observe that ths must become the system () Ths system ca be replace wth the sgle equato: (8) δθ Θ δ = λ Θ t whch must be val for ay fferetal δ Suppose to beg wth that the reuce form of Θ s: Θ = y z y z 2 y 2 z p y p Oe wll have: δθ Θ δ = z δy y δz + + z p δy p y p δz p a the system () or equato (8) whch s equvalet to t gves us: (9) y = z = λ zt y2 z2 λ = = z2t y p = z p = λ z pt = λ t Oe sees that oe wll ecessarly have λ = a that equatos () reuce to 2p whch wll be completely tegrable O the cotrary f the reuce form s: Θ = z y + + z p y p the system () wll be equvalet to the followg oe: (2) y = z = λ zt y2 = z2 = λ z2t y p = zp = λ z pt

13 Darboux O the Pfaff problem 3 Here t wll ot be ecessary to mae λ = whch stgushes ths case from the frst oe Moreover the equatos amt 2p epeet tegrals of t: z2 y = C z = C y p = C p z p z = C p We ca therefore state the followg theorems: If equatos () whe regare as etermg the fferetals x are mpossble as log as λ s o-zero the the form Θ s reucble to the etermate type: y z y z 2 y 2 z p y p The umber 2p s equal to the umber of stct equatos to whch equatos () reuce whe oe sets λ = a cosequetly t wll be easy to eterme a pror Moreover the 2p equatos to whch equatos () the reuce are completely tegrable a the varables y z of the reuce form are fuctos of ther 2p tegrals If equatos () ca be verfe by supposg that λ s o-zero the the form s reucble to the etermate type: z y + + z p y p The umber 2p s equal to the umber of stct equatos to whch equatos () the reuce Moreover these equatos are always completely tegrable a oe wll have a system of tegrals of these equatos terms of the varables of the reuce form that are gve by the formulas: y = α z e λ t = β y p = α p z p e λ t = β p I other wors these fferetal equatos amt the fuctos y y p a the quotets z 2 / z z p / z for epeet tegrals of t As a applcato we stuy the reuce form Θ the most geeral case If s o the the etermat: ± a a s o-zero a oe ca solve equatos () for the fferetals x ; λ s o-zero a equatos () are all stct Here oe the has the seco type (7) a the reuce form s: z y + z 2 y z /2 y /2

14 Darboux O the Pfaff problem 4 O the cotrary f s o the the etermat: ± a a s zero; however ts frst-orer mors are o-zero geeral As we have see oe must the have λ = apart from a exceptoal case a the equatos the reuce to stct oes The reuce form s: y z y z ()/2 y ()/2 VII We have see how oe recogzes whch type s attache to a fferetal form a how oe etermes the umber p It remas for us to show the tegratos that are ecessary orer to covert a gve fferetal expresso to ts caocal form The beautful scoveres of Mayer a Le greatly msh the ffculty ths subject However ths paper I wll occupy myself oly wth the varace propertes that relate to a fferetal form I wll thus cotet myself wth explag the geeral process of tegratos my sole objectve beg to show that the Pfaff metho whe apple to a partal fferetal equato leas to the same results as those of Cauchy Frst coser a fferetal expresso: Θ = X x + + X x whose caocal form s: (2) z y + + z p y p We ow that the Pfaff system: δθ Θ δ = λ Θ δ t s the completely tegrable f 2p < a cosequetly amt 2p epeet tegrals of t ay case There wll thus be at least 2p varables x that are ot tegrals Suppose to fx eas that the latter are: Whe oe sets: x 2p = x 2 p x 2p+ = x 2p x 2p+ x x + x = 2 p x x 2 p x beg umercal costats the 2p tegrals of the Pfaff system reuce to fuctos of x x 2p There wll the be oe tegral that reuces to x aother that reuces to x 2 a so o ( ) We let [x ] or u eote those of these tegrals that reuce to ( ) Ths classfcato of tegrals of a system of equatos s as oe ows ue to Cauchy the case where there s just oe epeet varable As far as completely tegrable systems are cocere t has alreay bee utlze by Le the paper that we alreay cte o the Pfaff problem

15 Darboux O the Pfaff problem 5 x We ow that the fuctos u epe solely upo the varables y y p that appear the caocal form (2) a the quotets z 2 / z z p / z Havg sa ths we tae the ew varables to be: u u 2p x 2p x whch are obvously epeet fuctos of the frst oes The form Θ becomes: (22) K(U u + + U 2p u 2p ) where U U 2p epe upo oly the u u 2p whle K by cotrast cotas oe or more varables x 2p x Ths s smple to prove several ways For example f oe starts wth the caocal form (2): z z 2 p z y + y2 + + y z z the oe ows that the z / z are fuctos of the varables u Therefore f oe replaces the y z / z wth ther expressos as fuctos of the tegrals u a f oe remars that z s a epeet fucto of the preceg oes the oe ee fs the expresso (22) I remar that the fucto K that appears that expresso s ot efe completely Nothg prevets oe from vg t by a arbtrary fucto ϕ(u u 2p ) o the coto that oe multples the quattes u by the same fucto ϕ However oe ca eterme K completely by the followg coto: Suppose that K reuces to a fucto: ψ(x x 2 x 2p ) for x 2p = x 2 p x = x We ve K by ψ(u u 2 u 2p ) a the the ew value of K wll be efe completely a wll ejoy the property of reucg to whe oe sets x 2p = x Havg sa ths we wrte ow the etty: X x + + X x = K(U u + + U 2p u 2p ) p x x = a set x 2p = x 2 p x = x o both ses Let X p eote what X p becomes Sce K wll the become equal to u wll become equal to x a oe wll have: X x + + X x 2p = U x + + U 2p x 2p 2 p 2 p a cosequetly oe ca wrte: U = X

16 Darboux O the Pfaff problem 6 whch leas us to the followg theorem: Suppose that the caocal form of a fferetal expresso: s Θ = X x + + X x z y + + z p y p The former Pfaff system wll be completely tegrable f 2p < a wll amt 2p epeet tegrals ay case Therefore there wll always be at least 2p + of the varables x that are ot tegrals of that system Let x 2p x be 2p + varables that ejoy that property Coser the 2p tegrals of the Pfaff system that reuces to x x 2p whe oe sets x 2p = x x = 2 p x a let u eote the oes that reuce to x If oe chooses these tegrals to be the ew varables the the expresso Θ taes the followg form: K(U u + + U 2p u 2p ) where oe euces U h from X h by replacg x x 2p wth u u 2p respectvely a x 2p x wth the costats x x Now coser the case whch the form (23) y z y z p y p 2 p Θ s reucble to the type: Oe ows that the Pfaff system wll be possble oly f oe sets λ = t a that all cases t wll amt 2p tegrals that wll be z z p y y p Here we may argue as the preceg Amog the varables x there wll be at least 2p of them that wll ot be tegrable Let: x 2p+ x be 2p varables that ejoy ths property Let u eote those of the tegrals that reuce to x whe oe replaces x 2p+ x wth umercal costats x + x Fally perform a chage of varables that substtutes the followg varables: 2 p for the orgal oes Oe wll have: u u 2p u 2p+ x (24) H + U u + + U 2p u 2p for the ew form of the fferetal expresso Iee the caocal form (23) the varables z y that are the tegrals of the Pfaff system ca be regare as fuctos of u

17 Darboux O the Pfaff problem 7 u 2p Therefore f oe supposes that they are expresse as fuctos of u u 2p the oe wll ee obta a result of the preceg form I the expresso (24) the fucto H s ot efe a t s clear that the expresso oes ot chage f oe replaces H wth: H ϕ(u u 2p ) o the coto that oe must a ϕ / u to U If H reuces to ψ(x x 2p ) for x 2p+ = x 2 p + x = x the we agree to subtract: ψ(u u 2p ); the ew value of H wll the reuce to zero for x 2p+ = Now wrte ow the etty: x 2 p + x = x X x + + X x = H + U u + + U 2p u 2p a set x 2p+ = x 2 p + x = x t Oce more let X be what X becomes uer that substtuto Sce u the becomes equal to x a H becomes equal to zero oe wll have: a cosequetly: X x + + X 2 p x 2p = U u + + U 2p u 2p U = X We may thus state the ew proposto as follows: Suppose that the caocal form for a fferetal expresso: s Θ = X x + + X x y z y z p y p The frst Pfaff system wll be possble oly f oe set λ = t a wll amt 2p tegrals Let x 2p+ x be a system of varables that o ot tae part these tegrals a let u eote the tegral of the Pfaff system that reuces to x for x 2p+ = x + x = 2 p x The expresso Θ ca be coverte to the form: H + U u + + U 2p u 2p where oe euces U from X by replacg x x 2p wth u u 2p a x 2p+ x by costats x + x H s a fucto that reuces to zero for x 2p+ = x + x = x 2 p 2 p

18 Darboux O the Pfaff problem 8 It s goo to remar that H wll be eterme wth o ffculty by a quarature whe u u 2p are ow Because oe has: H = Θ U u U 2p u 2p a everythg wll be ow the rght-ha se The two preceg theorems lea to several cosequeces Oe sees mmeately that the varous systems of fferetal equatos to whch the applcato of the metho leas acqure a epeet exstece a sese Oe may wrte each of them after havg tegrate the preceg oe Mayer alreay mae aalogous remars relatg to the completely tegrable systems Moreover oe sees that by startg wth the seco system oe o loger has etermacy a oe o loger fs that the forms belog to the two geeral types Oe ca mae a mportat applcato of the preceg results to the partcular form that oe ecouters the theory of partal fferetal equatos Let: (25) p = f(z x x p p 2 p ) be a partal fferetal equato where p eotes z / It s clear that the tegrato of that equato s equvalet to the followg problem: Aul the form: Θ = z f x p 2 x 2 p x 2 varables z x x p 2 p by establshg relatos betwee these varables Oe ows that the soluto to ths problem offers o ffculty as log as Θ s reuce to the caocal form Now I say that orer to covert Θ to the caocal form t wll suffce to tegrate the frst Pfaff system relatve to the form Iee wrte ow that system: δθ Θ δ = λ Θ δ t or f δx δf x + p 2 δx 2 δp 2 x p δx x δp = λ t(δz f δr p δx ) whch gves the equatos: f f x = λ f t f x + p 2 = λ p 2 t 2 f x + p = λ p t

19 Darboux O the Pfaff problem 9 f z x = λ t f x p 2 = p2 f x x = p whch oe easly puts to the followg form: (26) x x2 x p2 p = = = = = = f f f f f f + p2 + p p2 p 2 z z z = px + + px Oe recogzes the fferetal equatos of the characterstc Here we see that x s ever a tegral Let [z] [p ] [x ] eote the tegrals of that system that z p x reuce to for x = x x beg a arbtrary costat There wll be o ffculty etermg these tegrals as log as the system (26) s completely tegrable If we ow apply the frst of the two theorems that we prove the we see that oe wll have: z f x p2x2 px (27) = L{ [ z] [ p2] [ x2 ] [ p3] [ x3] [ p ] [ x ]} whch L epes upo x We thus obta the reuce form that must be the cocluso of our calculatos o the frst try The preceg metho s ecoutere the Cauchy metho a t plays a fuametal role there It s potless to retur to the well-ow propostos a to show how they lea to the tegrato of the propose partal fferetal equato For us t suffces that we have establshe that by meas of a smple supplemet the Pfaff metho becomes as perfect as the others However t s also justfe for us to a that ths classfcato of tegrals that allowe us to arrve at our objectve costtutes a very essetal avace that s oce more ue to Cauchy (to be cotue)