The Quarterly of Applied Mathematics, Vol. 1, Nr 4

Q U A R T E R L Y

OF

A P P L I E D M A T H E M A T I C S

H. L. DRYDEN J. M. LESSELLS

H, BATEMAN J, P. DEN HARTOG j. N. GOODIER R. V. SOUTHWELL

E D IT E D BY

T. C. FRY TH.

W. PRAGER L S.

J. L. SYNGE

w i t h t h e c o l l a b o r a t i o n o f

M, A. BIOT L. I

H. W. EMMONS K. (

F. D. MURNAGHAN W.

G. I. TAYLOR S. P

,-U? k

if ’

v. KARMAN SOKOLNIKOFF

. BRILLOUiN L FRIEDRICHS I, SEARS . TIMOSHENKO

Vo l u m e I J A N U A R Y ; 1944 / Nu m b e r 4

■ J Q 4 3

}

Q U A R T E R L Y O F

A P P L I E D M A T H E M A T I C S

T h is periodical is published u n d er th e sponsorship of B row n U n iv ersity . F o r its su p p o rt, an op eratio n al fu n d is being s e t u p to w hich in d u strial o rg a n iz atio n s m a y co n trib u te. T o d ate, co n trib u tio n s of the following in d u stria l com pan ies are g ratefu lly acknow ledged:

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Pr i n t e d b yt h e

Ge o r g e Ba n t a Pu b l is h i n g Co m p a n y Me n a s h a, Wis c o n s in

Q U A R T E R L Y

OF

A P P L I E D M A T H E M A T I C S

E D I T E D B Y

H. L. DRYDEN J. M. LESSELLS

T. C. FRY W. PRAGER J. L. SYNGE

TH. v. KARMAN I. S. SOKOLNIKOFF

H. BATEMAN J. P. DEN HARTOG J. N. GOODIER R. V. SOUTHWELL

W I T H T H E C O L L A B O R A T I O N O F

M. A. BIOT H. W. EMMONS F. D. MURNAGHAN G. I. TAYLOR

L. N. BRILLOUIN K. O. FRIEDRICHS W. R. SEARS S. P. TIMOSHENKO

V o l u m e

I /V*- ^ • *943

Printed by the

Ge o r g e Ba n t a Pu b l is h i n g Co m p a n y

Menasha, Wisconsin

CONTENTS

W . S. A m en t: T h e lines of principle stress in th e plane problem of p la stic ity . 278 H . B a te m a n : T h e tra n sfo rm a tio n of p a rtia l differential e q u a tio n s . . . . 2 8 1 L. B e rs : C oncerning th e acceleration p o t e n t i a l ... 93

an d A. G e lb a rt: On a class of differential eq u a tio n s in m echanics of c o n t i n u a ... 168 L. B rillo u in : T h e a n te n n a p r o b l e m ... 201 C. L. B row n : T h e tre a tm e n t of d isco n tin u ities in beam deflection prob lem s . 349 W . Z. C h ien : T h e in trin sic th e o ry of th in shells a n d p l a t e s ...297

■ an d A. W e in ste in : On th e v ib ra tio n s of a clam ped p la te u n d e r tension 61 L. H . D o n n e ll: A c h a rt for p lo ttin g re la tio n s betw een v aria b les over th e ir e n ­

tire real r a n g e ... 276 H . L. D ry d e n : A review of th e sta tis tic a l th eo ry of t u r b u l e n c e ... 7 W . F e lle r: On A. C. A itk e n ’s m eth o d of i n t e r p o l a t i o n ...86 K. O. F rie d ric h s an d J . J . S to k e r: F orced v ib ra tio n s of sy stem s w ith n on­

lin ear re sto rin g f o r c e ... 97 R . E. G a sk e ll: On m o m en t b alancing in s tru c tu ra l d y n a m i c s ...237 A. G e lb a rt: (See L . B ers)

G. H . H a n d e lm a n : A v a ria tio n a l principle for a s ta te of com bined p lastic stress 351 G. E. H a y an d W . P ra g e r: On plane rigid fram es loaded p erp en d icu larly to

th e ir p la n e ... 49 M . H e rz b e rg e r: A d ire c t im age erro r t h e o r y ...69 N. J . H o ff: A stra in energy d e riv a tio n of th e torsional-flexural bu ckling loads

of s tra ig h t colum ns of th in -w alled open s e c t i o n s ...341 T h . v. K a rm d n : T ooling up m a th e m a tic s for en g in e e rin g ... 2 P . W . K e tc h u m : On th e d isco n tin u o u s flow aro u n d an airfoil w ith flap . . . 149 W . M . K incaid an d V. M o rk o v in : A n ap p lica tio n of o rtho g o n al m o m en ts to

problem s in sta tic a lly in d e te rm in a te s t r u c t u r e s ... 334 Y. H . K u o : On th e force a n d m o m en t a c tin g on a b o d y in sh ea r flow . . . 273 J . L e h n e r an d C. M a r k : A n ap p lica tio n of th e m eth o d of th e acceleration

p o t e n t i a l ... 250 C. C. L in : On th e m otion of a pen dulu m in a tu rb u le n t f l u i d ...43 S. L u b k in an d J . J . S to k e r: S ta b ility of colum ns an d strin g s u n d e r periodically

v a ry in g f o r c e s ...215 C. M a r k : (See J . Lehner)

V. M o rk o v in : On th e deflection of an iso tro p ic th in p l a t e s ...116 ---(See W . M . K i n c a i d )

W . P r a g e r : (See G. E . H a y )

W . C. R a n d e ls : A new d e riv a tio n of M u n k ’s f o r m u l a e ... 88 K . R ie s s : E lec tro m a g n etic w aves in a b e n t pipe of re c ta n g u la r cross section . 328 S. A. S ch elk unoff: T h e im pedance of a tra n sv e rse w ire in a re c ta n g u la r w ave

g u i d e ... 78 On th e a n te n n a p r o b l e m ... 354 H . J . S te w a rt: P eriodic p ro p e rtie s of th e sem i-p erm an en t atm o sp h eric pressure

s y s t e m s ... 262

J . J. S to k e r: (See K . 0 . Friedrichs)

;----(See S . L u b k i n )

A. G. S tra n d h a g e n : Use of sine tran sfo rm for non-sim ply su p p o rte d beam s . 346 J . L. S y n g e: O n H e rzb erg er’s d irec t m ethod in geom etrical op tics . . . . 268 H . S. T s ie n : S ym m etrical Jouk o w sk y airfoils in sh ear f l o w ...130 A. W e in ste in : (See IV. Z . Chien)

Book re v ie w s ... 189

281

Q U A R T E R L Y O F A P P L I E D M A T H E M A T I C S

V o l. I J A N U A R Y , 1 9 4 4 N o . 4

THE TRANSFORMATION OF PARTIAL DIFFERENTIAL EQUATIONS*

B Y

H. BATEMAN

California Institute of Technology

1. In tro d u c tio n . In th e early stages of th e use of p a rtia l differential eq u a tio n s for the solution of problem s of m echanics an d physics th e sep a ratio n of v ariab les and constru ctio n of sim ple solutions was the p rim a ry aim . T h e in tro d u ctio n of the idea of an exact differential b y F o n ta in e an d E u ler led to th e idea of associated differential eq u atio n s such as those for the velocity p o ten tial and strea m fun ctio n in h y d ro ­ dynam ics, th e a d jo in t eq u a tio n s of L agrange a n d R iem an n, th e c o n ta c t tra n sfo rm a ­ tions of L egendre an d A m pere, th e tran sfo rm atio n s of E u ler an d L aplace for the solution of differential eq u a tio n s by definite in teg rals an d o th e r tran sfo rm atio n s too num erous to m ention. A n o th er aim w hich led to th e s tu d y of tran sfo rm atio n s was t h a t of reducing an eq u a tio n to a canonical form . L ap la ce’s red u ctio n of a lin ear p a r­

tial differential eq u a tio n of th e second o rd e r to a form in w hich only one p artia l d eriv a tiv e of th e second o rd e r occurs led to th e s tu d y of tran sfo rm atio n s w hich p re­

serve this form an d of q u a n titie s w hich have a p ro p e rty of invariance. C onditions were th en found t h a t an eq u a tio n m ay be reducible b y m eans of a specified ty p e of change of v ariables to some p a rtic u la r eq u a tio n s which h ad been fully stu d ied . T h e conditions found by C am pbell [l ] t (constan cy of his tw o in v a ria n ts) t h a t L a p la c e ’s canonical eq u a tio n m ay be reducible to th e eq u a tio n of E u ler and Poisson, m ay be cited as an exam ple.

A classification of tran sfo rm atio n s m ay be m ade by including in group A all tran sfo rm atio n s w hich arise from th e condition o r conditions t h a t a lin ear differential form m ay be of a specified ty p e (for exam ple an ex a c t differential). T ra n sfo rm a tio n s arising from th e s tu d y of a n u m b e r of lin ear differential form s m ay be included in this group. T ra n sfo rm atio n s associated w ith th e C alculus of V a riatio n s are also included because th e eq u a tio n s of E u ler and L agrange are closely asso ciated w ith th e condi­

tions for an ex a ct differential. T h e extension of L egen d re’s tra n sfo rm a tio n found b y C ara th e o d o ry [2] m ay be m entioned here. In this article a tte n tio n will be d evo ted a l­

m ost en tire ly to tran sfo rm atio n s of group A.

T ra n sfo rm atio n s of g roup B include all those w hich arise from th e con ditio ns t h a t a q u a d ra tic differentia] form m ay be of a specified typ e. T h e tra n sfo rm a tio n of a

* Received March 2, 1943.

f Numbers in square brackets refer to the list of references at the end of the paper.

282 H . B A T E M A N [V ol. I , N o . 4

linear differential eq u atio n to a form in w hich the v ariables are se p a rated is th u s a B- tran sfo rm atio n . T h e tran sfo rm atio n s of g roup B are n o t necessarily p o in t tra n sfo rm a ­ tions, for instance, if Qia, b, c) is a non-negative q u a d ra tic form in th e real variables a, b, c tran sfo rm atio n s from (x, y , z, t, u, v, w) to (X , Y, Z , T , U, V, W ) m ay be con­

sidered in w hich Q{dx — udt, d y —vdt, dz — wdt) goes over into Q ( d X — Udt, d Y — Vdt, d Z — W d t) th e coefficients of Q in the first case being fun ction s of y , z, t, u, v, w and in the second case functions of X , Y, Z , T , U, V, W . Since the eq u a tio n <3 = 0 im plies th a t d x = udt, d y = v d t , d z = wdt it also im plies t h a t d X = U d T , d Y = V d T , d Z = W d T . In o th e r w ords if u, v, w can be regarded as th e co m p o n en t velocities of a recognizable m oving p article of fluid th en U, V, W can be reg ard ed as co m p o n en t velocities of a recognizable p article of a corresponding fluid. Such a tran sfo rm atio n is of in te re st because the d en sity of each fluid can be defined in such a w ay t h a t th e eq u a tio n of c o n tin u ity is in v a ria n t u n d er th e tran sfo rm atio n .

G roup C m ay be regarded as including all o th e r tran sfo rm atio n s an d som e tra n s ­ form ation of the o th e r g roup w hich arise in th e reduction of an e q u a tio n to a canonical form .

2. A ssociated equations of th e typ es of M onge an d L eg en d re. In his w ork on p a r­

tial differential eq u atio n s of the second o rd e r in tw o variables x , y w hich can be re­

g arded as indep en d en t, M onge [3] used z as d e p e n d e n t variable, p an d q as th e first d eriv a tiv e s zx, z„ respectively an d r, s, t as th e second d eriv ativ es z zx, zxy, z vy. As we shall have app licatio n s to fluid dynam ics in m ind, we shall d ev iate sligh tly from the n o tatio n of M onge an d use u, v in place of p and q so t h a t when z is th e velocity po­

te n tia l u an d v rep resen t th e co m p o n en t velocities as usual. T h is plan also allows us to use th e sym bol p to d en o te th e pressure a n d q to d en o te th e re s u lta n t velocity.

T h e eq u atio n s of ste a d y m otion of a com pressible fluid u n d er no body forces when the flow is irro ta tio n a l an d th e fluid b aro tro p ic (d en sity a function of pressure only) can, as we know, be deriv ed from a v aria tio n a l principle

') d x d y = 0, u — z x, v = s„, (1)

in w hich p is a specified function of q. F o r g re a te r g en e rality a t th e o u tse t we shall suppose, how ever, t h a t p is a specified function of u an d v. L a g ran g e’s p a rtia l differ­

ential eq u atio n for this v a ria tio n a l problem is th en d eriv ab le from H a a r's condi­

tion [4] th a t p v d x — pudy should be an exact differential. W hen th e differen tiatio ns are m ade, th e eq u atio n has the form

puuf + 2 p uvs + p vvt = 0. (2)

W hen L egend re’s tran sfo rm atio n is applied to th is differential eq u a tio n th e new d e­

p e n d e n t v ariable is

w = u x + vy — z (3)

and since d z = u d x + v d y ,

du> = x d u + ydv. (4)

W hen u an d v can be regarded as in d ep en d e n t th is eq u atio n gives th e relation s

x = y = wv, (5)

an d th e e q u a tio n for w is

p u u ^ w 2.p Ul'Wuv T pvv^uil ~ 0. (6)

W hen, how ever, u an d v are related so th a t th ey can be regarded as fu nctio ns of a single v ariab le r th e eq u atio n (4) indicates t h a t w is then also a fun ctio n of r an d we have th e equ atio n s

1944] T R A N S F O R M A T I O N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S 283

w { t ) = x u(t) + yv(r) — s, w ' ( t ) = x u ' ( t ) + yv'ir), which furnish a solution of (2) if

PuuU,2( t) + 2p Uvu ' ( t ) v ' \ t ) + Pvx-v'-(t) = 0.

L e t us now seek the conditions th a t 3 q u a n titie s

R = R ( u , v), S = S ( u , v), T — T (u , v),

(7)

(8)

(9) m ay be the second d eriv ativ es Z zx, Z xy, Z21 of a single function Z ( x , y ) . T h e re q u ired conditions R V = S X, S y = T x m ay be w ritten in the form

R,,s T R vt — S ur + S t.s, S„s + S rt = T ur + T rs. (10) W e now seek the conditions th a t these tw o eq u atio n s arc b o th satisfied in v irtu e of eq u a tio n (2). T h is will be th e case when

R v 1L‘p vr) R u wp ltv ~i~ by Su w p„ u, Sv — b IVp y, , Tu = w'puu, T t = w ' p uv + li'y S„ = h' — w ' p uv, S r = — w ' p vv.

E q u a tin g th e different expressions for R uv, 5 UC, T uv we o b tain the e q u a tio n s llyi — Wyypyyy. lVVp Utly IlV ™ WyiPw >

bu 1C r p uu “ puvy b v — W i; p u v Wu pvt"

T h e elim ination of h and h ' yields the tw o eq u atio n s IVuuprv 2.wurp ur T w vvp uu — 0, IVuUpW 2 Wyirpny “(~ Wvrpytu — 0

(11)

(12)

(13) which show th a t w and w ' are solutions of eq u a tio n (6). T h e case of chief h y d ro d y - nam ical in te re st is t h a t in which th e second d eriv a tiv e s of w an d w ' are all zero. W e shall, how ever, look first a t a possible a lte rn a tiv e case.

E q u a tin g the different expressions for S u an d S v we o b tain th e eq u a tio n s

H ence

Wpuu = w'pur +

J

w'pvv = Wpuv +

J

Wv prv + w'purv = 2 w vpuv + Wpuvr ~ Wup,.v, 2wû puv — Wv pu u + w ' p uuv = Wupuu + Wpuuu,

WÛ prv + w'Purv = Wvpuu + Wpuuv

(14)

(15)

T h ese e q u a tio n s lead to th e relation

w 'D v = w D u, where D = p uup xv — p l v. (16) T his eq u atio n is satisfied iden tically w hen D is c o n sta n t b u t it m ay also be satisfied if w = E„, w ' = E U w here D an d E are related. An im p o rta n t case of this second ty p e occurs when D is a function of q only a n d w = v , w ' = u. In th is case

h — — p u, h' = — p v, R = v p v — p, T = u p u — p, S — — u p v — — vp„, (17) an d p is a function of q only. If p u = — up, p v = —vp, w here p is th e d en sity of th e fluid we have

R = — p — pv2, S = puv, T — — p — pu2, (18) and so Z is a kind of stress function w hich satisfies th e eq u atio n

R T - S 2 = p ( p + pu2 + pv2) = p (p + pq2) = F ( R + D (19) since R - \ - T = — 2p —pq2. T h e p a rtia l differential eq u atio n for Z is th u s of L egen d re’s ty p e [5]

3C(R, 5, T ) = 0. (20)

In the special case in which — F ( R + T ) = K Z — |( i ? + 7’) 2 th e eq u atio n reduces to one w hich occurs in S a in t V e n a n t’s th eo ry of plastic bodies. T h is e q u a tio n has been discussed by H en ck y [6], P ra n d tl [7] and C ara th e o d o ry [8]. Oseen [9] uses th e m e th ­ od of L egendre in which th e eq u a tio n is first solved for R , d ifferen tiated w ith resp ect to y an d so reduced to an eq u atio n

(.K2 - V % y \ V XX - Vyy) + 2 V XV Xy = 0 (21) in which

S = V X, T = Vy.

I t should be rem arked t h a t if p x — u = z x, p u = ~ v — —z y,

p + pu2 + pu2 = p, 1 /p — — p, (22) we m ay w rite

R — — p — pi'2, S = puv, T = — p — pii2, (23) an d the eq u atio n s R V = S Z, S y — T x lead to th e p a rtia l differential e q u a tio n for th e strea m -fu n ctio n z.

In th e th eo ry of plane w aves of finite am p litu d e eq u a tio n s of L eg en d re’s ty p e oc­

cu r in a t least tw o w ays one of w hich is discussed b y J. R. W ilton [10]. In th e o th e r w ay use is m ade of th e eq u a tio n s

R = p, S = — pu, T — p + p u 1, (24)

where now y denotes the tim e an d x- a co-o rdinate in th e direction in w hich th e w aves are travelling. T h e q u a n titie s u, v are again th e d eriv a tiv e s of a velo city p o ten tial z, p is th e d en sity , p th e pressure an d u th e velo city of th e fluid. T h e q u a n titie s R , S , T are th e second d eriv a tiv e s of a stress-function Z . T h e ad d itio n al eq u atio n s from w hich the relation betw een R , S an d T m ay be derived, are

v + h i2 = f'(p), P = /(p) — pf'(p)- (25)

2 8 4 H . B A T E M A N [V o l. I , N o . 4

T h e desired relatio n is th u s

T - R ~ lS2 = /(R) - R f ' ( R ) . (26) T h is eq u atio n , like t h a t considered by W ilton, m ay be solved by th e m eth od of L egendre in w hich th e eq u a tio n is differen tiated w ith re sp ect to one of th e in d ep en d ­ e n t variables (in th is case a;) so as to reduce it to an eq u a tio n of th e M onge-A m pere type. T h e tran sfo rm atio n to the new e q u a tio n can be regarded as a special B acklund tran sfo rm atio n [ l l ] as Oseen [9] observes. If U = Z X, th e new eq u a tio n is

U yy - 2{ U y/ U z) U zy + ( U l / U l ) U zz = - R f " { R ) U zx = c2U zz • (27) or

I I U ZZ + 2 K U zy + L U yy = 0,

w here I l = { U2v/ Z f x) - c 2, K = - ( U y/ U z), Z = 1. T h e in v a ria n t G is

G = K - — H L — c~ (28)

an d th e condition Gt^O is satisfied so long as c - ^ O . In the p re sen t case

1944] T R A N S F O R M A T I O N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S 285

(29) R u = — p u / c2, R v = — p / c2, S u = p(u2/ c2 — 1)S„ = pii/c2,

r „ = Pu( 1 - u2/ c2), T v = - p( 1 + m2/ c 2) . T h e tw o eq u a tio n s (10) are b o th e q u iv a le n t to

(u2 — c2)r + 2 us + 1 = 0, (30)

an d it is readily seen th a t w — — 1, w ' = — u, g = k <l — hl = c!-7i §.

I t should be noticed t h a t if we solve eq u atio n s (9) for u an d v in the form

u = F ( R , S ) , v = G(S, T ), (31)

the eq u a tio n u y = vz is satisfied on ac co u n t of R y = S x, S y = T x if

Fr = Gsi F s — Gt. (32)

T hese tw o eq u atio n s th en are consequences of th e single e q u a tio n 3C(R, S, T ) = 0 . T h e expression of such an e q u a tio n in th e tw o form s (32) m ay be reg ard ed as a p ro blem of som e in te re st.

In th e case w hen D is a c o n s ta n t an d p is a function of q only

D = p qp j q = P2[l ~ (? A )2]- (33)

an d th e flow is e ith e r en tire ly subsonic ( D > 0 ) o r en tire ly supersonic (D <0). In m an y cases in w hich p is a function of q only, D can have e ith e r sign an d so th e flow is p a rtly subsonic an d p a rtly supersonic. I t is then of som e in te re st to seek th e co n d i­

tion satisfied by th e function 3C(R , S, T) w hen D> 0 . F o r this purpose we w rite the e q u a tio n in th e form

0 = 3C{R, S , T) = [(£ - T )2 + 4 5 2] I/2 - J { R + T ) , (34) w here / is a function w hich is such th a t

-pf'(p ) — ~ J [ ~ 2/(p)]. (35)

H ere p = f ( p ) —pf'(p) is the relation betw een th e pressure p an d d en sity p. Now we find by differentiation th a t

3C„ + X T - 2 = - 2 - 2J ’(R + T ) = - 4 - 2 p /"(p )//'(p ) = 4(cV ?2 ~ D- (36) H ence 3C/i + 3Cj- — 2 > 0 when c2> g 2and 3C« + 3Cr — 2 < 0 when c2< q 2. In th e case of th e p lastic e q u a tio n J is a c o n s ta n t an d so

3C« 3Cr = 0. (37)

T h e corresponding flow is ch aracterized b y the relation q- = 2c- an d is co n seq u en tly supersonic.

A sim ple case in w hich D is c o n sta n t is o b tain ed b y w riting

p = a n2 + 2 cuv + fa.’2, (38)

w here a, b and c are c o n stan ts. T h e functions iv, w ' b oth satisfy

bw„u + a w vv — 2cwUv = 0, (39)

and we m ay w rite w — b V u, w ’ = a V v, where V is a solution of this eq u a tio n . If a fu nc­

tion W is defined b y th e eq u atio n s

W u = c V u -

avv,

i? = 2b(lV + cV), S = - 2abV, T = 2 a ( W + cV), (41) w here V an d W are connected b y th e foregoing eq uatio ns. In th is case the relatio n betw een R , S an d T is sim ply

a3C = a R - b T = 0. (42)

T h e q u a n tity 3Cb + 3Ct —2 is now sim ply — (a+Z>)/a, a c o n sta n t. T h ere is no change in sign of th e expression. I t will be noticed t h a t th e eq u atio n s a R —b T = 0 an d

ar + 2 cs + bt = 0 (43)

sa tisfy the cond ition of a p o la rity

A b + B a - 2Cc = 0, (44)

w hen the first eq u a tio n is w ritten in th e form A R - \ - 2 C S - \ ~ B T = 0.

3. T h e tran sfo rm atio n of th e M onge-A m pere equation. If for th e e q u a tio n hr + 2 k s;'+ It -T m + n{rt — s2) = 0, (45) th e expression

g = k2 — III -f- m n (46)

is n o t zero a n d so th e tw o system s in the m eth o d s of M onge an d Boole are d istin c t, th e eq u atio n is tran sfo rm ed b y a c o n ta c t tran sfo rm atio n

2 8 6 . H . B A T E M A N [V o l. I , N o . 4

X = X ( x , y, z, u, v), Y = Y ( x , y, z, u, v), Z - Z { x , y, z, u, v), \ U = U{x, y, z, 11, v), V = V ( x , y, z, it, v), dZ = U d X — V d Y = a{dz — u d x — vdy) j (47) in to an eq u atio n

H R + 2 K S + L T + M + X ( R T - S 2) = 0 (48)

1944] T R A N S F O R M A T I O N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S 287

for w hich th e q u a n tity G = K i - I i L - { - M N = 0.

In a p ap e r published in 1904 S ophus L ic -[l2 ] rem ark ed t h a t it would be d esirab le to have a d irec t proof of this th eorem an d K iirsch ak [13] gave one based upon a re p resen tatio n of the eq u atio n in th e form of a Jacob ian

d{a, b )/d { x , y) (49)

w here a an d b are functions of x , y , z, u, v a n d d / d x = d / d x - \ - u { d / d z ) -\-r{d /d u ) -\-s{d/dv), d / d y = d / d y + v ( d / d z ) -\-s{d /d u ) -)rt{d/dv). W hen th is re p re se n ta tio n is n o t used th e proof is algebraically m ore difficult b u t th e an aly sis is w o rth giving on ac­

co u n t of th e n um erous re la tio n s to w hich it leads. R eference for th is ty p e of proof m ay be m ade to a p ap e r b y R. G arn ier, S u r la transform ation des dérivées secondes d a n s les transform ations de contact et les transform ations ponctuelles, B ull, des Sci. M a th . (2),

64, 13-32 (1940).

W e shall suppose th a t dz = udx-\-vd y an d th a t co n seq u en tly dZ = U d X - \- V d Y . T o m ake d U = R d X + S d Y , d V = S d X + T d Y consequences of d u = r d x + s d y , d v = s d x f - t d y we shall require th a t

d U - R d X - S d Y = ( U u - R X U - S Y f ) { d u - r d x - sdy) + ( U v — R X V — S Y u)(dv — s d x — tdy), d V - S d X - T d Y = (Vu - S X U - T Y u){dn - r d x - sd y)

+ ( F , - SX„ - T Y v)(dv - s d x - tdy).

W ith the n o ta tio n

(50)

Z y + v Z z, etc., {uv) = U v

(vv) = V v

R X V s x v -

■ S Y V, T Y V,

r{uv) + s{vv) + Vi - S Xi - T Y i = 0, s{vu) + t(vv) + F 2 — S X2 — T Y 2= 0.

Z i — Z i + uZz,

{uu) = Uu - R X U - S Y U {vu) = V u - S X u - T Y u th e eq u atio n s to be satisfied are

r{uu) + s(uv) + U i - R X i - S Y i = 0,- s{uu) + t{uv) + Ui - R X2 ~ T Y i = 0,

H ence

rA = U vV x- U1V v+ R { X1V v- FiA%) + 5 ( 7 ^ , + U xX v- X xU v- F i F ,) + T{ U \Yv - IT U v) + ( R T - S > ) { Y xX v - ATK.),

tA = U 2Vu - V i U u+ R { X UV2 - X2V U) + S { X2U u + Y u V2- X v U2- Y 2V u) + T { Y2U u- Y u Ut) + { R T - S * ) { X2Yu - X u Y 2) ,

s A ^ U xV u - V1U u + R { X u V1- X 1V u ) + S { X1U u + Y u V1- X u U1- Y 1Vu) + T { U u Y1- U 1Y u) + { R T - S * ) { X1Y u- X u Y 1),

sA = V2U v- U2V v+ R { X2Vv- X z V 2) + S { U2X V+ Y i V , - X t U , - V t Y r) + T{ U2Y v - Y2 U v) + { R T —S2) { Y2X V - X 2Y v) ,

A { r t - s 2) = U xV 2- ¿72F i + R ( F 1A 2- F 2AT) + T { U2Y y- U xY 2)

+ 5 (U2X x- U xX2+ V xY2- V2Y x) + (R T - S2)(A iK *- X2Y x) , A - U uV v- U v V u + R { X vV u - X u V v) + T { U vY u - U u Y v)

+ S { Y vV u- Y u V v+ U vX u - U u X l) + { R T - S ' - ) { X u Y v- X vYu).

(5 1)

(52)

(53)

T h e two expressions for s are eq u iv ale n t on ac co u n t of th e relatio ns

[.U V] = [ X V ] = [ Y U ] = [X V] = 0, [AT£/] = [ Y V ] = a (54) w hich, in ad d itio n to th e relation s [F Z ] = [ Z X ] = 0, \ U Z ] =crU‘[ V Z ] = a V are s a tis ­ fied because th e tran sfo rm atio n is a c o n ta c t tran sfo rm atio n . In these re la tio n s [.A B] is th e Poisson b ra c k e t

[A B ] = A uBx - A t . B u + A vB2 - A2B V. (55) T h e relations arc derived b y Lie [12] b y a clever device. In th e book of C erf [14] th e relations are derived from the eq u atio n

<t\ A B ] = [a i] (56)

w here A , B are the expressions for a (x, y , z, p, q), b(x, y , z, p, q) in th e new co-ordinates [AT, Y, Z , P , Q \, while F. Engel [15] o b tain ed th em w ith th e aid of th e bilin ear co­

v a ria n t b y a dev elo p m en t of a m etho d used by G. D arboux.

I t is read ily seen t h a t th e e q u a tio n h r ~ Y ^k s-\-lt-\-n i-\-n (rt—s 2) = 0 becom es H R + 2 K S + L T + M + N ( R T - S ) = 0 , w here

H = H P, + k ( P b + P q) + IP a + m P r + n P c,

2 K = h ( R p + Cp) + k ( C b + R b + Cq + R q) + l { R a + C.)

+ m ( R r + Cr) + n ( R c + Cc), L = hA p -f- k ( A b + A q) -j- IA a -f- tnA r T u A c,

M = hQp -f- k{Qb + Q,,) + IQ a + tnQr + nQc, A' = hBp + k ( B b + B q) -T l B a T- m B r T n B c, w here

2 8 8 H . B A T E M A N [V o l. I , N o . 4

A a = U UY2 - u2y u, A i — U u Y i - U \ Y U, A c = U2Yi - UxIT, A p = UyYv - U VY x, A q = U2Y v - u q y2, A r = U VY U- E T F „ Ba = X2Y U - x uy2, B b = X iT « - ^{X PY h} B c = X i Y2- AT2Fx, B P = X v Y i - AT ,F„ Bq = ^{X vY2} - x 2y v, B r = X UY V- ATIT, C a = X2U U - AT U 2, c b = X i U u - X u U u Cc = X1U2- x 2u h C p = X vU i - X i U v, C, = ^{x vu2} - X2U V, Cr = X UU V- x vu u, Pa = XuVa - X 2V u, P b = ^{x uv l - x 1v u,} Pc = X2Vx- X i V 2, P p = ATF„ - X vV u Pq = ^{X2V v} - X vV 2, Pr = X vVu - X uV v,

Qa = U2V U - U UV 2, <36 = ^UiV„ - _{U uV u} Qc = U2V2- u2v h Qp = U vv 1 - U i 7 „ _{Qq = U vV}2 - UiVv, Qr = U uV v - UvVu, Ra = Y uV2 - y2 i t , Rb = Y uv1 - Y iV u , Rc = Y2V1- F xF 2, R P = Y i V v - Y vV i, _{Rq = Y}2V v - YvV 2, Rr = Y vV u- Y uV v.

T hese eq u atio n s give th e relation

1944] T R A N S F O R M A T I O N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S 289

' K2- H L + M N = J ( k2- h l + m n ) , if ï(C& + i2|,-t-Cf! + i?ij)2— {Pb~\~ Pq){A b~\~A q) + (Qî> + <25) (Bb~\~Bq) —JI

\r(Rr+ C r ) ( R c+ C c) + (QrBc+QcBr) ~ (P rA c+ P cAr) = 7 , 1CRp+Cp) (Ra + C a) + ( Q p B a+ Q aBp) - (PpSl a+ P a A p) = - / , ï ( R p + C p )2+ Q PB p - P p A p = 0, l ( R a+ C ay + Q a B a- P „ A „ = 0,

\ ( R r + C ry + Q r B r - P rA r = 0 , KRc + Ccy + Q cB c - P ' A c = 0,

(58) i(-Z?p+C,p)(C j,+i?i,+C a+ i2 s)-f-Q p(5i,-|-3!I)+ 5 p (Q (,-|-0 a)

= P p ( A b+ A q) + A p ( P b+ P q),

■i{Ra-\-Ca)(Cb-\-Rb-\-Cq-\-Rq)-\-Qa(Bb-{-Bq)-\-Ba{Qb-\-Qq)

= P a ( A b+ A q) + A a ( P b + Pq), h(Pr + Cr)(C6 + i?i, + Ca + i?a) + ( ) r (BbA-Bq)-\-Br (06 + (?a)

= P r ( A b+ A i ) + A r ( P b + Pq), KR c + C c) (C b + R b + C q + R q) + Q C ( B b + B q ) + B c (Qb+Qq)

= P c ( A b+ A q) + A c ( P b+ P q), K R P+ C p ) ( R r + C r ) + Q p B r + Q r B p - P p A - , - P rA p = 0,

i( R p A - C p ) ( R cA-Cc)ArQpBcA-QcB p — P p / l c— P cA p = Q ,

%(R* + C a) (R r+ C r) + Q aB r+ Q r B a ~ P aA t - i V l „ = 0, f (i?a+ c „ ) (J?c+ C c) + ^ + < 2 A - P o A - P cA . = 0.

T hese relations m ay be established by using a p a ra m e tric re p resen ta tio n of th e q u a n titie s satisfy in g L ie’s conditions for a c o n ta c t tran sfo rm atio n , we th erefo re w rite

X \ — d\e -f- die', X2 = b\e -\~ b2e', X u — c\e T c2e', X v = die -T d2e', U i = a i f + a2f , U2 = b i f + b2f , Up = c i f + a f ,

— Y u = a2p -f- dip', — Y v = bzp -f- bip', Y i = Cap -f- Cip',

— V u = az q + diq', — V v = b3q + b ^ ' , V i = c3q + Ciq', w here th e q u a n titie s

U v = d i f + d2f , Y2 = dzp + dip', V i = dzq + d iq ',

(59)

d ^{1 0 2} & ³ &\

b1 #2 ^4

Cl C2 Cl C\

d\ d% dz di

form an orthogonal m atrix an d th e q u a n titie s e, p, p ', q, q ' are such t h a t (ciß2 —xC2ßi T dpbi — d2b i ) ( e f T e'f) = {d2di — CI4C3 T bzdi — bidz)(pq' — p'q) = cr. (60) I t is th en found t h a t

- H = eq{ 1, 3) + e'q(2, 3) + eq'{\, 4) + e'q'{2, 4), - L = f p ( 1, 3) + f p (2, 3) + / / ( ! , 4) + / y ( 2 , 4),

2IT = («/' - < /) ( 1, 2) + G Y - p 'q ) (3, 4), (61) M = / ç ( l , 3) + A ( 1 , 3) 4) + / Y ( 2 , 4),

= ep{ 1, 3) + e'p{2, 3) + 4) + e'ÿ '(2 , 4),

where

(1, 3) = li(dic3 + dibz) — l(bifl3 -|- Cydz) -f- k(d \d3 -f- 6163 — d\d3 — C1C3) + n (a id3 — bi.Cz) + m ( d ia3 — Cib3),

(2, 3) = h{dzCz "I- <22^3) — l(bzd3 4* c3dz) -|- k (d3d3 4“ ¿263 — <2203 — C2C3) 4~ ti(d3d3 — bzCz) 4- m(dzdz — c3bz),

(1, 4) = h(diCi 4- 01^4) — l(bidi 4* Cidi) 4- k ( d id t — bibi — 0104 — C1C4) 4- n (d \d \ — J1C4) 4- m (d \d \ — C\bP),

(62) (2, 4) = h(d3d.\ 4- 02^4) — lifizdi 4- c->di) 4- k(dzdi 4~ b3b\ — <220-4 — C2C4)

4- n (d3d* — ¿2C4) 4" m^dzdi — C2J4),

(1,2) = h(did3 — dzd\) 4- l(b\Cz — ¿2^1) 4- k(diCz — O2C1 4- d\bz — dzbi) 4~ u(d\bz + <22^1) 4- m{c\dz — diCo),

(3, 4) = h(bzCi — b.iCz) 4” l(d.\d3 — <23^4) 4" k(c3d3 — C4O3 4" b3di — bid3) 4- n (d 3Ci — dAc3) 4- m{b3di — b4O3).

I t is readily seen th a t

M N - I I L = [(1, 3)(2, 4) - (2, 3)(1, 4)](e’f - e f ) ( p ' q - pq') (63) an d th a t, on ac co u n t of the p ro p erties of an o rthog on al m atrix

( e 'f - e /') 2 = {p'q - Pq'Y: (64)

T h e expression for K also sim plifies considerably an d th e proof m ay be re ad ily com ­ pleted. T h e q u a n tity J as in K u rs c h a k ’s analysis, is equal to a2 a n d so is n o t zero.

C o n ta c t tran sfo rm atio n s are n o t th e only ones in w hich th e con ditio n gp^O is in v a ria n t. In th e th eo ry of th e ste a d y tw o-dim ensional m otion of an inviscid elastic fluid th e eq u atio n s satisfied b y th e velo city p o te n tia l z an d strea m -fu n ctio n z are respectively

puur 4- 2p uvs 4- p w t = 0, pxivX 4" 2p uvs -f- p vvi = 0. (65) In this case d z = udx-{-vdy, d z = ii dx-\-v d y = p td x — p ud y an d so

u = p v, 5 = — p u , g = g = p l v — p Uupw - c2(u2 - f i>2 — c2). (66) T h u s g = 0 e ith e r w hen c = 0 or w hen q = c. T h e supersonic region is ch a rac te rized by th e condition g > 0 an d th e subsonic region b y the condition g < 0 . T h e curve for w hich c = 0 is a b o u n d a ry for th e flow ju s t as in th e case of th e P ra n d tl-M e y e r flow ro u n d a corner. T h e tran sfo rm atio n u n d er consideratio n is a special B acklund tra n sfo rm a tio n an d is included in th e g roup of B ack lu n d tran sfo rm atio n s

x = X ( x , y, u, v), y = Y ( x , y, u, v), u = U (x, y, u, v), v = V ( x , y, u, v), (67) for which th e Jac o b ia n d ( X , Y, U, V ) / d ( x , y , u, v) is n o t zero. T h ese tran sfo rm atio n s have been stu d ied carefully by G o u rsa t [16]. T h e re q u irem en t th a t

u d x -f- v d y

should be ex a ct leads to an eq u a tio n of th e M onge-A m pere ty p e in w hich z does n o t

2 9 0 H . B A T E M A N • [V o l. I , N o . 4

occur explicitly. I t is show n, how ever, t h a t th e general M onge-A m pere e q u a tio n of th is ty p e c a n n o t be o b tain ed in this w ay an d a sim ilar re su lt has been found b y J.

C lairin [ l 7 ] in his stu d ies of m ore general B iicklund tran sfo rm atio n s. C lairin has stu d ied in p a rtic u la r tran sfo rm atio n s of ty p e

1944] T R A N S F O R M A T I O N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S 291

x ' = f i ( x , y, z, it, v, z'), y ' = / 2(x, y, z, u, v, z'),

u ' = f3(x, y, z, u, v, z'), v' = / ¿ x , y, z, u, v, s'). (68) Som e of C lairin ’s w ork is su m m arized in th e book of F o rsy th [18] an d illu stra te d b y m eans of exam ples.

A n o th er tran sfo rm atio n of ty p e (67) w hich preserves th e condition 0 is o b ­ tain ed b y w ritin g

d U = R d x + S d y , d V = S d x + T d y ,

w here R , S , T are th e functions of u an d v used in section 2. M ak in g use of th e e q u a ­ tions .

(■u p u + v p v) d x = p udz + vdz, (iipu + v p v) d y = p vdz — udz, (69) we find t h a t

( T n - S v ) d U + (Rv - S t f i d V = ( R T - S2)dz, ( S p u + T p v) d U - ( R p u + S p v) d V = ( R T - S2)dz.

H ence, if

= u , y ' = V , u ' = d z / d U , v' = d z / d V , ii' = d z / d U , v' = d z / d V , (71) we have th e re la tio n s

u ' = ( T u - S v ) / ( R T - S2), ' / = (Rv - S u ) / ( R T - 5 2), u' = (Spu + T p l) / ( R T - 5 2), v' = - ( R p u + S p t) / ( R T - S 2)

(70)

(72)

w hich, in co nju nction w ith the preceding relations define tw o B acklun d tra n s fo rm a ­ tions. T h e tran sfo rm atio n s considered in m y p ap e r on th e lift a n d d ra g fu n c tio n s are of this ty p e [19] an d are n o t generally c o n ta c t tran sfo rm atio n s as is a p p a re n tly im ­ plied b y a s ta te m e n t re la tin g to the correspondence of the supersonic regions in th e tw o associated ty p es of flow.

In th e case in w hich p is a fun ction of q only th e relatio ns betw een u ' , v' , u, v are u ' = - u / p , v' = — v /p , q' = q / p

an d , if p ' - — 1 I p , p ' + p ' q ' 2= — 1 / ( P + p q 2) we have

, pP

p = --- >

P + p q 2

d p ’/ d q ' = — qp/('p + pq2) = — p'q',

1 — q'2/ c' 2 = (1 — q2/ c 2) [ p / ( p + pq2) ] 2, where c '2 = d p '/d p '.

T h is tran sfo rm atio n m ay be com pared w ith th a t o b tain ed b y m eans of H a a r ’s a d jo in t v a ria tio n problem s [20]. In this case

u* = p j ( p - Upu ~ v p v) , v* = p j ( p - upu ~ v p v), p* = 1 / ( p - Upu - v p v),

and, w hen p depends only on q,

u* = - p u / ( p + pq2), v* = - p v /(p + pq2), P* = + I / O + q2), q* = ' q / ( p + ?2)- D efining p* b y the eq u a tio n d p * = —q*dq*, we have

p* = (p + pq2) / p p , p* + q* 2 = 1 / p ,

c * 2 = p tp2q*(q2 — c2) / ( p + pq2)2[c2(p + pq2) 2 + p2(q2 - c2)], c*2 _ q*2 = _ p Y c2/ [ c 2(p + pq2) 2 + p2(q2 - c2)].

H ence c*2 = q* 2 w hen c2 = 0 and c*2 = 0 w hen q2 = c2. I t should be n oticed, how ever, th a t

c*2{c* 2 - q2) = c2(c2 - q2) p2Py / ( p + pq2)2[c2(p + pq2) 2 + p2(q2 ~ C2) ] 2.

T h is m ay be com pared w ith H a a r’s general relation*

(p*.v.p * .v. - p t l v')(puupvv - p lv) = p ^ p * - * .

T ra n sfo rm atio n s m ore general th a n those of B ack lund have been considered by G au [21] b u t so fa r no hydro d y n am ical app licatio n s h ave been found for these so fa r as I know . M en tio n should be m ade, how ever, of th e equ iareal tra n sfo rm a tio n s from th e E ulerian to L ag ran g ian co-o rdinates in th e tw o-dim ensional flow of an inco m pres­

sible fluid. T hese tran sfo rm atio n s have been m uch used in m ap p in g b u t th e hy dro- dynam ical app licatio n s are b eset w ith form idable difficulties.

N o m ention has been m ade of th e use of tran sfo rm atio n s in th e th eo ry of surfaces, congruences, etc. T h is is a su b je c t w hich is well tre a te d in th e books of D a rb o u x [22], F o rsy th [ l 8], G o u rsa t [ l 6], B ianchi [23] an d E ise n h a rt [24].

4. T ra n sfo rm atio n of th e lin e a r equation. In th e special case in which h, k a n d I are fu nctio ns of x an d y only, n — 0 an d m is a lin ear hom ogeneous fu nction of u, v a n d z w ith coefficients d epending o nly on x an d y , th e M onge-A m pere e q u a tio n reduces to a linear eq u atio n . T h e behav io r of th is e q u a tio n in tran sfo rm atio n s of ty p e

X = X ( x , y ) , Y = Y ( x , y ) , Z = zF (x, y) (73) has been stu d ied b y D arb o u x [22], C o tto n [25], R iv ereau [26], J. E. C am pbell a fte r th e case F = 1 h ad been discussed b y L aplace [27], C hini [28] an d o th ers [ l 6].

C am pbell uses th e eq u a tio n in a form in w hich g = 1, a form to w hich th e general e q u a ­ tion can be reduced b y m u ltip ly in g it b y a su itab le factor. H e th en shows t h a t th ere are tw o in v a ria n ts I , / an d an ab so lu te in v a ria n t J / I w here if suffixes d en o te p artia l d eriv a tiv e s

I = h a x T k ( a y T b x) T lbv T (lix T k y) a T (k x -j- l v)b + Ita2.-f- 2kab + lb2 — m x, J — a y — b x, 2a = l ( h x + k y — m u) — k ( k x + l y — w„)>

2b = h ( k x + /„ — m v) — k ( h x + k y — m u).

* This is a consequence of the relations

= P P ' ^ P - - P-), ^ = P ' P K P i ^ - P i -0.

In the correspondence between the two hodograph planes complications arise on account of the relation of PuuPui Puvto these Jacobians.

292 H . B A T E M A N [V ol. I , N o . 4

(74)

1944] T R A N S F O R M A T I O N O F P A R T I A L D I F F E R E N T I A L E Q U A T I O N S 293

L ap lace's in v a ria n ts are 1(1 — 7 ), ^ ( I + T ) . T hese are used w ith a d ifferent n o ta tio n , in D a rb o u x ’s T heorie des Surfaces, t.2. C am p b ellsh o w s th a t in th e case of th e e q u a ­ tion of E uler an d Poisson th e in v a ria n ts I an d 7 are c o n sta n t. T h is m ay be co m pared w ith C o tto n ’s result. T h e h arm o nic eq u atio n s belong to th e g ro up ch a rac te rized b y th e relation 7 = 0 . T h e eq u a tio n s considered by B u rg a tti [29] are such t h a t 7 = 0.

In m a th em atica l physics th e sim ple solutions of lin ear eq u atio n s p lay an im po r­

ta n t p a r t an d th e p rim ary problem is t h a t of sep a rab ility . E ven in th e case of th e eq u a tio n w ith tw o in d e p e n d e n t v ariables th ere are som e unsolved problem s. A good idea of the progress w hich has been m ade m ay be derived from D a rb o u x ’s boo k [22].

T h e m eth o d of L aplace provides an im p o rta n t w ay of reducing eq u a tio n s by a c a s­

cade process w hich is p a rtic u la rly useful in th e tre a tm e n t of eq u a tio n s arising in th e th eo ry of plane w aves of finite am p litu d e . R eference m ay be m ade to a p a p e r of Love and P id d u c k [30], an article b y P la trie r [31 ], som e p apers b y B ech e rt [32] an d to tw o p apers b y Oseen [9] in w hich th e tran sfo rm atio n a n d redu ction is given for e q u atio n s occurring in th e th e o ry of e a rth pressure an d in th e th eo ry of p lasticity .

In th e th eo ry of th e ste a d y m otion of an inviscid com pressible fluid th e eq u a tio n s in th e hodograph plane are linear. T hese equ atio n s are

PvvlVuU 2p uvWuc ~\~ puu^vv — 0, pvvWHU 2puv^uv “1“ Puu'&VV b) 2 = 1115, + VWv — W = qivq — W, Z = UWu + V~d>v — w = q w q — w.

W hen p is a function of q only th e eq u atio n s u = p v, v — —p u ta k e th e form

u = — pv = — q sin r, i' = pu = q cos r, q = pq, (76) a n d th e eq u atio n s becom e

Wr, + q(qwQ)q = 0, w TT + q(qWq)q = 0 (q a function of q). (77) T hese are consequences of sim ple relations betw een w an d w

w T + qw^ = 0, w r — qw^ = 0. (78)

T h e corresponding relations betw een z an d s are found to be

q % = qzr, q \ = - qzT (79)

an d so th e equatio ns for z an d z are

(? 7 S )](2 2/?)Z5L + Srr = 0, 1 (q.*/q){(q2/ q ) z q h + ZrT = o. j

T hese are e q u iv a le n t to th e eq u a tio n s o b tain ed b y M olenbroek [33] an d T schap lyg in [34] for th e case in w hich th e relation betw een p an d p is of th e p o ly top ic o r a d ia b a tic ty p e. T h e sym m etrical form s of th e eq u atio n s are easy to rem em ber.

I t is som etim es useful to in tro d u ce o th e r q u a n titie s w hich sa tisfy lin ear relation s.

T h u s we m ay o b tain th e desired relations betw een z, z, w, w b y w riting w = eT = qeq, w = eT = — qeq,

z = — eT — (q /q )eTT, z = {q/q)e TT — eT, w here

cTt “h ^tt ~h q

294 H . B A T E M A N [V ol. I , N o . 4

T h e lite ra tu re dealing w ith th e tran sfo rm atio n of linear eq u a tio n s in several v a ri­

ables is v ery extensive an d only a brief su m m ary can be a tte m p te d here. B e ltra m i’s w ork on differential p aram eters [35] was ex ten ded by Ricci a n d L e v i-C iv ita [36], C o tto n [25], L ev i-C iv ita [37] an d m an y o th e r w riters. T h e d ev elo p m en t of general re la tiv ity , electrodynam ics a n d the th eo ry of elasticity has m ade this w ork m ore or less know n. T h e w ork of L am é on sim ple solutions of the p o te n tia l eq u a tio n [38] was m uch developed b y la te r w riters an d a good su m m a ry of resu lts up to 1893 is given in th e book of B ôcher [39]. T h e use of a v aria tio n a l principle for o b ta in in g th e tra n s ­ form atio n of th e eq u a tio n w as recom m ended b y L arm o r [40], V o lterra an d o th ers [41]. Since th e a d v e n t of th e new q u a n tu m th eo ry the in te re st in sep arab le equ atio n s an d separable system s has m uch increased. M en tio n m ay be m ad e of th e w o rk of S taeckel [42], E ise n h a rt [43] an d R ob ertso n [44].

In ad d itio n to th e sim ple solutio ns of p a rtia l differential eq u atio n s th ere are solu ­ tions having th e form of pro d u cts in w hich one or m ore o r th e facto rs satisfies a p a r­

tial differential eq u atio n in stead of an o rd in ary differential eq u a tio n . C o m p arativ ely little w ork has been done on th is problem . In th e case of L ap lace’s eq u a tio n V xx+ V vu

+ F*2 = 0, th e aim is to find a solution of form [45]

V = Z F ( X , Y ) , (generalized binary potential) where F satisfies a p artia l differential eq u atio n of the second order in th e variab les X and Y. T h e problem seems to depend on the form ation of a relation of type

(p2 + q2 + r2) ( d x - + d y2 + dz2) - (p d x + qdy + rdz) 2 = a d X2 + I h d X d Y + b d Y2 in which a , b and h are functions of X an d Y only. T h ere is a sim ilar relation for the corresponding problem in a n y n u m b e r of variables.

5. T h e tran sfo rm atio n of L e g e n d re ’s equation. L eg en d re’s eq u atio n 3C(R, S , T ) = 0

is u n altered in form by a L egendre c o n ta c t tran sfo rm atio n

X ' = U , Y ' = V , U ' = X , V = Y \ Z ' = U X + V Y - Z , w hich m akes

R ' — T / ( R T — S 2), S ' = - S / ( R T - S 2), T ' = R / { R T - S 2).

In p a rtic u la r, th e eq uation

R ' T ' - S '2 = F { R ' + T ' ) becom es

f R + T \

an eq u a tio n of th e sam e general type. A gain, if a , b and h are c o n sta n ts th e c o n ta c t tran sfo rm atio n

Z ' = U a X 2 + 2 h X Y + 5 F 2) + Z , X ' = X , Y ' = Y , U ' = a X + h Y + U , V = h X + b Y + V

m akes R ' = a + R , S ' = h + S , T ' = b + T and so tran sfo rm s an eq u atio n of L egendre’s ty p e in to a n o th e r e q u a tio n of th e sam e ty p e. E q u a tio n s of th e preceding ty p e usually go in to L egendre eq u a tio n s of a slightly d ifferen t typ e.

O th e r tran sfo rm atio n s m ay be found b y first tran sfo rm in g th e eq u a tio n to the M onge-A m père form by L egen d re’s device. If, for instan ce, th e eq u a tio n is

T = F ( R , S )

an d we d ifferen tiate w ith respect to x using th en th e new n o ta tio n z = U , R = U x = u, S = U y = v, T x = S y = t , R x = r, S x = s, the new eq u a tio n is

i = F u(u, v)r + F v(u, v)s.

C om parin g this w ith th e eq u atio n p u u r + 2 p uvs - \ - p vvt = 0 we find th a t F u Puu/Pwt F v ~ 2p u t ' / p w

E lim in a tin g F we find th a t the eq u a tio n for z is n o t a general eq u atio n of th e ty p e considered in §2 because th e fu nctio n p ( u , v) satisfies th e condition

d D /d v = 0 where D — p uup vv — p uv.2

A n eq u a tio n for w hich D is c o n s ta n t satisfies th is condition an d th e e q u a tio n of ty p e

X ( R , S , T ) = 0

associated w ith it b y th e analysis of section 2 m ay be regarded as a tran sfo rm of th e original eq u a tio n 3C(2?, S, T ) = 0 . In th e case w hen D = 1, we m ay w rite

puu = ea sec b, p vv = e~a sec b, p uv = tan b, w here a an d b are functions of u an d v w hich m u st be chosen so th a t

sec2 bbu = e2 sec b(a t) + e° sec b tan b(bv), sec2 bbv = — e~a sec b(au) + e~a sec b tan b(b„).

Also, since F u = —e2°, F v— — 2ea sin v, we m u st have th e ad d itio n al eq u ation 2e“ cos b(bu) — 2eu { a l) + 2e“ sin v(au) = 0

which is seen, how ever, to be a consequence of th e o th e r two. E lim in atio n of the d eriv a tiv e s of a gives th e eq u atio n

6 bw I e buu 2 sin bbuv — H or puubw *E pwbuu 2puvbuv — d

an d it is readily seen th a t a satisfies the sam e eq uation . T h e eq u atio n 19 = 1 is given as an exam ple in F o rs y th ’s book, p. 220, Ex. 11.

Re fe r e n c e s

[1] J. E. Ca m p b e l l, Proc. London Math. Soc. (2) 10, 406-422 (1910).

[2] C. Ca r a t h é o d o r y, Bull. Greek Math. Soc. 3, 16-24 (1922); Math. Ann. 86, 272-275 (1922);

85, 78-88 (1922).

[3] G. Mo n g e, Application de l’analyse d la géométrie, Bernard, Paris, 1809.

[4] A. Ha a r, J. f. Math. 149, 1-18 (1919); see also T. Rad6, Hungarian Univ. Acta, Szeged, 2, 147-156 (1925).

[5] A. M. Le g e n d r e, Histoire de l’Académie des Sciences 1787, 309.

[6] H. H e n c k y , Zeit. f. angew. Math. u. Mech. 3, 241-251 (1923).

[7] L. Pr a n d t l, idem., 1, 15-20 (1921); 3, 401-408 (1923).

[8] C. Ca r a t h é o d o r y a n dE. Sc h m id t, idem. 3, 468-475 (1923).

[9] C. W. Os e e n, Arkiv f. Math., Ast., o Fysik 20A, Nos. 24, 25 (1928).

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