Ijgap is symmetric it follows from a theorem in Algebra that this matrix

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1 VOL. 20, 1934 MA THEMA TICS: T. Y. THOMAS 215 fo which the quotient in the ight-hand side of (10) is nea to H(so) ae diffeent fo the diffeent functions. Then, applying the method that I d used in the case p() = P + lo we see that: VI. Given a subtigonometic function of ode p'h(0), which is always positive o zeo, and a poximate ode L p() tending to p, thee exists an entie function f(z) fo which we have the equality (10). 1 "On Cetain Points in the Theoy of Diichiet Seies," Ame. Jou. Math., 50, 73 (1928). 2"On Entie Functions Defined by a Diichiet Seies," Ibid., 53, 1 (1931). 3"Untesuchungen ube Liicken und Singulaitaten von Potenzeihen," Math. Zeitsch., 29, 549 (1929). 4 "M6thodes de sommation et diections de Boel," Annai R. Scuola no. di Pisa, 2, 2, (1933). a "Recheches su le Theoame de M. Boel," Acta math., 52, 67 (1928). 6 "Su la ep6sentation analytique des fonctions d6finies pa des s&ies de Diichlet," Ibid., 35, 253 (1911). I See Valion, "Su les diections de Boel des fonctions m6omophes d'ode fini," Jou. math., 9, 10, 457 (1931), and a note by Rauch, Bull. Soc. math, 61 (1933). 8 "Siu les fonctions entiwes d'ode fini,...," Annales fac. sc. Toulouse, 3, 5, 117 (1913). 9 "Fonctions entiaes et fonctions convexes," Bull. soc. math., 60, 117 (1933). Fo the definition of subtigonometic functions, see my cited pape of Jounal math. THE REDUCTION OF DEGENERATE QUADRATIC DIFFERENTIAL FORMS BY TRAcY YERKES THOMAS DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY. Communicated Febuay 2, Conside a quadatic diffeential fom E Ei gg,0dxadxb (1) a-1 D.1 in a domain T such that xl _ all < - bli..., len ail < bns whee the a's ae constants and the b's ae positive constants. Assume that the coefficients gao ae analytic functions of the vaiables xa in T. Let the ank of the matixijgaop Ibe such that 1. < n - 1. Since Ijgap is symmetic it follows fom a theoem in Algeba that this matix

2 216 MATHEMATICS: T. Y. THOMAS PROC. N. A. S. contains a diagonal deteminant D of ode which does not vanish identically in T. We can theefoe assume that D is not equal to zeo at any point of T by a suitable estiction if necessay of this domain. Without loss of geneality D can be taken to occupy the position in the uppe lefthand cone of the matix gpl since a mee eletteing of the vaiables will in any case accomplish this esult. Such a eletteing of the vaiables is equivalent to a coodinate tansfomation; in fact if the non-vanishing diagonal deteminant D of ode coesponds to the ows and columns a,1,..., a we can by the coodinate tansfomation xaj =- i Xa = x whee O, *.., I = 1..., bing this deteminant into the uppe left-hand cone of the matix. In this pape we investigate the existence of analytic tansfomations of the coodinates xa such that (1) is educed to the fom ZE habdyadyb, (2) a=1 b=1 whee the coefficients hab ae analytic functions of the vaiables y', y' in a domain ly - all < b*,..,ly" - an < bn* (3) contained in T. Such tansfomations will educe effectively the degeneate quadatic diffeential fom (1) to a non-degeneate fom in vaiables y1,.., y and n - paametes y+1,..., In 2. It is equied to find a solution of the pope type of the system pm g.p(x) 0y =*+ 'n n )(4) Now since the matix j g is of ank and futhemoe since the deteminant of ode in the uppe left-hand cone of this matix does not vanish in the domain T by hypothesis, thee must exist identical elations of the fom gjub E gabais (s + i, * *5n a=1 ~~a,b= 1 fomgaai=b gpaa,(.v=+1.. Eliminate the quantities gj,b and g;, in the left membes of (5) and (6) fom the system (4). We obtain Egab a ; + E E gab A ay ( 1 ) a,b= 1 a,b-l p=+l Egab 7+ E E gab A,Aa = o v = + 1,., ). (6)

3 VOL. 20, 1934 MATHEMATICS: T. Y. THOMAS 217 Since the deteminant Ig.bI is not equal to zeo in T, the equations (7) can be solved in this domain so as to yield a<= _ E Al ayv z- + ;9.,n When the left membes of (9) ae substituted into (8) these latte equations ae satisfied identically. Hence the poblem of obtaining a solution of the system (4) is educed to that of obtaining a solution of (9). The quantities Aa in (9) ae analytic functions of the vaiables xa in T; this follows fom the coesponding hypothesis on the coefficients gal, and the fact that the Aa ae uniquely detemined by the equations (5). 3. Take fist the case = n - 1. Putting x' = y' in (9) these equations become -=-A*n (xlw x'-, yn), (k = 1,...., n - 1) (10) and can in fact be egaded as a system of odinay diffeential equations. In accodance with the existence theoem fo such systems the equations (10) admit a solution xk = 4k(yl, y"-l y), (k = 1,..., n - 1), (11) such that xk = yk fo y' = a' whee the sp's ae analytic in a domain (3) and can be solved uniquely fo the n - 1 vaiables y1,..., y"'1 in this domain. Hence the equations (11) taken in conjunction with the equation x = y' define an analytic coodinate tansfomation of the domain (3) of T such that (1) is educible to the fom (2). THEOREM. If the ank of the matix lgabi is n - 1 thee always exists an analytic co6dinate tansfomation of the n vaiables xa which educes the degeneate fom (1) to the non-degeneate fom (2). As a matte of fact thee exists a one paamete family of such tansfomations since the functions (k can be consideed to involve analytically a paamete yo such that O- a"i < b*, the quantity y' being associated with the initial value of the vaiable y'. 4. Assume = n - 2. Then by a slight fomal modification of the equations of 3 we can show that (1) can be tansfomed into (2) with = n - 1. Now, howeve, the deteminant Ihabl vanishes identically in the domain (3). But the deteminant of ode n - 2 in the uppe left-hand cone of the matix Ilhabit does not vanish in (3); this follows diectly fom the type of coodinate tansfomation at which we aive as a esult of the pocess of 3 and the fact that the deteminant of ode n - 2 in the uppe left-hand cone of the oiginal matix Ig41p is assumed to be diffeent fom zeo in T. Now epeat the pocess of 3 with efeence to the above degeneate

4 218 MA THEMA TICS: T. Y. THOMAS PROC. N. A. S. fom (2) fo = n - 1 when this fom is egaded as a quadatic diffeential fom in n - 1 vaiables y1,..., y"'1 depending on a paamete yfl We thus aive at the fom (2) fo = n -2 with coefficients hab which ae analytic in a domain of the type (3) and with a deteminant Ih,b not equal to zeo in this domain. As this esult can be extended to cases < n - 2 we have the following: THEOREM. If < n - 1 it is possible to educe (1) to the non-degeneate fom (2) by a succession of coo-dinate tansfomations: Fist a co6dinate tansfomation in the n vaiables xg esulting in a degeneate fom in n - 1 vaiables, then a tansfomation of the n - 1 vaiables of this latte fom by which it is educed to afom in n - 2 vaiables,..., andfinally a tansfomation in + 1 vaiables by which the non-degeneate fom (2) is obtained. 5. Now assume < n - 1 and conside the question of eplacing the succession of tansfomations in the theoem of the last section by a single coodinate tansfomation in the n vaiables xg which will educe (1) to the non-degeneate fom (2). This is not always possible since integability conditions aise! Diffeentiating (9) with espect to y, intechanging the indices,u,v and subtacting in the usual manne, we obtain these conditions in the fom <,p-+l { bl [ axba Xb I - ax a (12) whee a, b = it...., and,u, v = + it..., n. Suppose now that at a point P of the domain T the deteminant Ibx'/by0I of ode n - is equal to zeo. Intoduce the notation B, = -A when i =1,...,;= + 1,., n and B = when i,,u = + 1,...,n. Then we have ~JB CYp = 1 by jys + it n fom (9), o in matix fom ylavl A=bY'il ial +l... in Since by the above assumption the ank of the n - owed squae matix IxI/ebypII is less than n - at the point P and since the ank R of the matix Wxi/AyP in the left membe of (13) cannot exceed the ank of the matix bx'/by'i it follows that R < n - at P. Hence the deteminant I bx'/lyji whee i,j = 1,..., n must be equal to zeo at the point P with the esult that the solution of (9) cannot, in the neighbohood of P, epesent an analytic tansfomation with unique invese, i.e., a coodinate tansfomation. Hence we must assume that the above deteminant Ix/lby'j is not equal to zeo at some point P of the domain T and we can (13)

5 VOL. 20, 1934 MATHEMATICS: T. Y. THOMAS 219 without eal loss of geneality assume that ox/lby'i does not equal zeo at the point with coodinates xa aa used in the definition of T. Hence the integability conditions (12) become b b=l [JaXb - xaa- Now put a + A 0= (14) x+1 = Y+l. xi' y (15) in (9) so that these equations become =-A (xl, X, Y+1 n) (k = 1,., (16) Assuming that (14) is satisfied in T, the system (16) is completely integable and hence admits a solution x = pk(y, * *,y, y +,.., y ), (k = 1,..., n), (17) such that xk=yk fo y7+l da+l,..., y" = an. Also the sp in (17) ae analytic functions in a domain (3) and can be solved uniquely fo the vaiables y1,..., y in this domain. Hence (15) and (17) define an analytic coodinate tansfomation of the domain (3) in consequence of which the fom (1) is educed to the non-degeneate fom (2). THEOREM. When the ank of the matix Ijgapi is less than n - 1 thee exists an analytic co6dinate tansfomation of the n vaiables x' which educes the degeneate fom (1) to the non-degeneate fom (2) if, and only if, the conditions (14) ae satisfied. Coesponding to the equations (11) the above equations (17) belong to an n - paamete family, the paametes being y+l,..., yo such that lyo+ _ a+ li< b*+.., Yo -a# I < b*. Since the A,, ae uniquely detemined by (5) when the matix IIg- is of ank, the conditions (14) ae immediately expessible as conditions on the coefficients of the fom (1). ERRATUM In the contibution by W. E. Castle, these PROCEEDINGS, 20, Febuay, 1934, on p. 102, line 15 fom the bottom, fo "any eggs" ead "many eggs."