QA 6451 HC3 il!' .. ERIODIC CONJUGATE NETS A DISSERTATION presented to the Faculty of Princeton University IN Candidacy for the Degree OF Doctor of Philosophy BY EDWARD S. HAMMOND Reprinted from the Annals of Mathematics, Second Series, Vol. 22, No. 4, June, 1921 Accepted by the Department of Mathematics, May, 1920 PERIODIC CONJUGATE NETS A DISSERTATION presented to the Faculty of Princeton University IN Candidacy for the Degree OF Doctor of Philosophy BY EDWARD S. HAMMOND Reprinted from the Annals of Mathematics, Second Series, Vol. 22, No. 4, Juno, 1921 a 3- [Reprinted from Annals of Mathematics, Vol. XXII., No. 4, June, 1921.] PERIODIC CONJUGATE NETS.' ""*• '^^*o.... By Edward S. Hammond. Introduction. If n functions of u and v, x''^\ x''"\ • • •, z'-"\ which satisfy an equation of Laplace of the form . . dH _ d log a dx d log h dx dudv dv du du dv ' be interpreted as the homogeneous coordinates of a surface in (n — 1) space, the parametric curves on this surface are said to form a conjugate net. Where no ambiguity arises, this system of curves or the surface on which it lies will be called simply the net A^". Equation (1) will be called the point equation of A''. Now the functions* Xi and X-i, given by . . _ dx d log a _ dx d log b are also homogeneous coordinates of nets, Ni and A''-!, which are called the first and minus first Laplace transforms of N. Ni has as its first and minus first Laplace transforms nets A^2 and N itself; A^2 is called the second Laplace transform of N. Developing these transforms in both senses we get a series of nets • • • N's, • • •, N-i, N, Ni, • -, Nr, • • •, called a sequence of Laplace. f This sequence will be called the sequence Nr. In the first section of this paper general properties of this sequence will be developed. In section 2, we impose upon the sequence Nr the condition that it shall be periodic; that is, that a certain Laplace transform Np of A'^ shall coincide with A^ itself. After transformation of parameters it is shown that the identity of A^p and A^ involves the identity of A^p_i and A''-! and in general, of N p-k and A^-^, k = 0, 1, 2, • • •,p. Necessary conditions on the coefficients of the point equation of A^ are derived and it is shown by discussion of the completely integrable systems of partial differential equations involved that these conditions are also sufficient. It is also shown that if an equation of Laplace of form (1) is the point equation of one periodic net, it is the point equation of an infinity of others of the same period. The remainder of the paper is taken up with other sequences of Laplace * Here xi indicates any or all of a;i, . . . , a;/"\ A similar usage is followed throughout, t Darboux, Logons sur la thcoric generale des surfaces, 2d ed. (1915), vol. II, chap. 2. 238 4 4 4 o n r> 239 EDWARD S. HAMMOND. closely related to the sequence Nr. The sequences studied in section 3 involve certain properties of families of lines in higher ordered spaces which we pi:ocee<;i to develop. The lines joining corresponding points df a.'n^iyV aad its first Laplace transform A^i form a two-parameter family G, each line of which is a common tangent of these surfaces. Consider for the sake of definiteness the line joining the points on A^ and A^'i with parameters Uo and Vq. Through this line pass two developable surfaces all of whose generators are lines of G, namel}'', the tangent surfaces of the curve u = Uoon N, and of the curve v = Vo on A^i. WTien a two-parameter family of lines in higher ordered space possesses either of these equivalent properties, namely, that each line of the family is a common tangent to two surfaces, and that through each line there pass two developable surfaces all of whose rectihnear generators are lines of the family, it is called a congruence. In 3-space any two parameter family of lines possesses these properties, but in space of higher order this is not the case. The surfaces to which all the lines of G are tangent are called the focal surfaces and the nets A^ and A^i the focal nets of the congruence. Levy* has shown that the functions ^ and v, defined by dv du where 6 is any solution of (1), may be interpreted as the coordinates of nets which will be called Levy transforms of A^ by means of d. The points of these nets lie on the lines joining the corresponding points of A^ and A^i, A'_i and A^, respectively, and the developables of the congruences so generated cut the surfaces of the nets in the curves of the nets. In section 3, it is shown that these nets, there called A^o, i and A^_i. o, are Laplace transforms of one another. It is also shown that A^o, i is a Levy transform of A^i by means of 6i, a solution of the point equation of A^i formed from 6 by the same process by which the coordinates of A^i were formed from those of A'^. From these properties follows a very intimate connection between the two sequences of Laplace, Nr, the original sequence, and Nr,r+i of which A^_i, and A^o, i are two nets. The sequence A^,r+i is called the first Levy sequence. On it may be formed a first Levj^ sequence, A^, r+2 which is called a second Levy sequence of A^. The treatment given in section 4 of these sequences and of the Lev}- sequences of higher orders which are analogously formed, indicates their close dependence upon the Laplace transforms of A^. They are actually the sequences of derived nets of higher orders studied by Tzitzeicaf and others. * Levy, Journal dc I'Ecolc Polytechnique, Vol. LVI (1S8G), p. 67. t Tzitzeica, Coinptes Rendus, vol. 156 (1913), p. 375. PEEIODIC CONJUGATE NETS. 240 In section 5, the results of section 2 are applied to these Levy sequences and conditions for their periodicity are derived. Two interesting geom- etric configurations arising under special conditions are discussed. As a property of the Levy transforms of a net A^, it was mentioned that the developables of the congruences of tangents to the parametric curves of A^ cut the surfaces of the Levy transforms in the curves of the nets. Whenever this relation holds between a congruence and a net, they are said to be conjugate. Two nets conjugate to the same congruence are said to be in relation T and the transformation which carries one such net into the other is called a transformation T, to use the terminology of Eisenhart* who has developed a general theory of such transformations. The congruence to which both nets are conjugate is called the conjugate congruence of the transformation. In section 6, it is shown that similar Laplace transforms of two nets in relation T are also in relation T, and hence that two sequences of Laplace may be developed such that corre- sponding nets of these sequences are in relation T. The problem of finding a sequence Nr so related to the original sequence Nr is reduced to the problem of finding a solution of the adjoint equation of (1) and quadratures. Owing to arbitrary constants arising in the quadratures, their integration gives a multiple infinity of such sequences between which certain geometric relations exist. The results of section 2 are then applied to these sequences, and it is found, first, that if equation (1) has periodic solutions, so has its adjoint; second, if such a solution be used in the determination of Nr, the se- quences Nr are also periodic of period p. 1. Sequences of Laplace. In the study of these sequences, two func- tions of the coefficients of equation {!), H and K, defined by _ 5- log ad log a d log b dudv dv du (4) _ 3- log b d log a d log b dudv dv du ' are of constant occurrence. Their most important property is in connec- tion with the transformation to other coordinates x', such that (5) X = Xx', where X is a function of u and v. Since the coordinates x are homogeneous, evidently this transformation has no effect on the net. The coordinates x' do not satisfy equation (1), however, but are solutions of * Eisenhart, Trans. Amer. Math. Soc, vol. 18 (1917), p. 97. 241 EDWARD S. HAMMOND. d^d d , add , d , b dd = — los; log — ■ dudv dv '^X du du ^\ dv (6) \ dudv ^ dv '^Xdu ^\ dv du / ' as may be shown by differentiation. If the functions H and K be formed from the coefficients of (6) and the resulting expressions reduced, it is found that they are identical With (4), that is, H and K are invariant under the transformation (5). They are called the Laplace-Darboux invariants of the equation (1) or of the net A'^. If the independent variables are changed by a transformation (7) U = (/.M, V = yp{v'), the invariants H' and K' of the new equation are given by (8) H' = '{u'W{v')H, K' = 4>'{u'W{v')K, where 0' and ip' are the first derivatives of ^ and 4/ with respect to their arguments. Consider the coordinates dx d log a (9) ^' = a". - ~^ir ^' of the net A^i mentioned in the introduction. If we differentiate with respect to u, we get /im ^^1 a log 6 (10) -d^ - -du- ""' = ^""^ a relation confirming the statement of the introduction that the lines joining corresponding points of A'' and A'"! are tangent to the curves v = const, on A^i. Then if H vanishes equations (9) and (10) reduce the solution of equation (1) to quadratures; also in this case, the surface A^ degenerates into a curve. But if H does not vanish, we differentiate with respect to v and find that the coordinates Xi satisfy the equation of Laplace d^d _ d log aH d_e d_\ogh dd dudv ~ dv du du dv ,f_d^, h d log aH a log 6 a log a 3 log 6 \ '^{dudv^^^a dv du + dv du +^;^' which proves that A^i is also a net, as stated in the introduction. This equation has invariants Hi and Ki, analogous to H and K, defined by d^ , a-H , a log a a log 6 , „ d'~ , aH dudv ^ b dv du dudv ° b (12) Ki = — ^-^ log a H 5 — - ~z \- c = H. dudv ^ dv du PERIODIC CONJUGATE NETS. 242 Since frequent use is to be made of the point equation of nets associated with a net hr.ving the point equation (1), for the sake of brevity we denote such an equation by the expression (13) [Xi] Gi, hi, d], which means that the coordinates Xi of the net Ni satisfy the equation JiL - ^^^±i ^ 4_ ^ log ^ ^ dudv dv du du dv (14) / ^Mog Cj _ d log a^ d log b , ajog^ d log h \ V dudv dv du dv du } ' Also the net N i has invariants (15) dudv ^ Ci dv du ' d'~ . bid log a a log 6 , log - + —^ ^t;- + c. ' dudv ^ Ci dv du In this notation (11) becomes (16) [xi] aH, b, b/a], and the effect of the transformation (5) on the point equation is expressed by ^ ^ iJ' X' \' X J' The minus first Laplace transform, N-i, is the second focal surface of the congruence of tangents to the curves v = const, of A^. For, by the definition of its coordinates given in equation (2), the lines joining corresponding points are tangent to N, and the equation obtained by differentiating these coordinates with respect to v and using (1) shows them to be tangent to A_i. The point equation of this net is denoted by (18) [a:_i; a, bK, a/b]. Consider now the congruences of tangents to the parametric curves of Ni. We have seen that the congruence of tangents to the curves V = const, has N and Ni as its focal nets; that is, N is the minus first Laplace transform of iVi. This is also obvious as a consequence of equa- tion (10), whose left member is the expression for the coordinates of the minus first Laplace transform of Ni formed by analogy with (2). The second focal Surface of the congruence of tangents to the curves u = const, of Ni is the net No, the second Laplace transform of N. Its 243 EDWARD S. HAMMOND. coordinates , , dxi d log aH or, using (9), _ d-x d log a^H dx ( alogg d log aH _ d- log a \ ^^^~d?~ dv dv^\ dv dv dv- J^' satisfy the equation denoted by [.r2; ciHHi, h, 6-/a-i7]. Continuation of this process in both the positive and negative senses gives the nets of the sequence Nr. The coordinates of the rth Laplace transform A^ are (20) - = 1r- a log aHHi • dv • • Mr—'> Xr-1, or, by repeated substitution, (21) Xr = J^^r + ^^r, r-l ^ ^^, + --■ +Ar, O-T. h^ where the Ap, g are functions of a, H, Hi, ■ ■■, Hr-i and their derivatives. The point equation of Nr is (22) yxr; aHHi ■ ■ ■ Hr-„ b, ^Tjj^fj ^^-^ • • • Hr-2 J ' The equations analogous to (10) and (19) are /OON ^^'^ TJ , a log 6 dXr _ a log Or ^^^^ du ^ ^'-'"^--^ + -^du^'^-' dv ~ dv "-'- + '''+" and they will be used as formulas for the partial derivatives dXridu and dXr/dv. On the negative side of the sequence, the general Laplace transform N-s has coordinates _ a.r_,-+i a log hKK-i • ■ • K-s+2 ^-' ^ ~du du ^-'+'' or d^r a*~H' dr (24) X., = ^„. + S., ._, ^^ + . . . + B.. , ^ + B,. .., which satisfy the equation x-s', a, hKK-i ■ • • K.s+i, ^»j^s-i . . . j^_^^^ J • The formulas corresponding to (23) are dx-s . a log h-s dx-s a log a PERIODIC CONJUGATE NETS. 244 From (23) and (26) it follows that if Hr-i or K_s+i vanishes, the sequence terminates; for the surface Nr or N-s degenerates into a curve. This is a special case of great importance* but it is not before us in this paper. 2. Periodic Sequences of Laplace. In the introduction, a periodic se- quence was defined as a sequence such that a certain net Np coincided with the original net A^. When this is the case, the coordinates Xp and X must satisf}^ the relation (27) Xp = \(u, v)x, where X is a function of u and v at most, and is the same for all ?i coordi- nates. The coordinates Xp satisfy the equation denoted by (28) W' ''™' • ■ • "■"'' ''' «?ffi=r^^TH;:;k] this result being obtained when r in (22) is replaced by p. From (17), (27), and (28), the coordinates x must satisfy (29) r aHHi-^-JIp_, h h^ 1 L^' X 'X' aPR^-' ••• Hp.i\y as well as the fundamental equation (1). Since in every case which we shall consider there are at least three coordinates x, the coefficients of dx/du, dx/dv, and x in (29) and in (1) must be equal. We have therefore ,oAN ^1 aH ■ ■ ■ Hp-i a log a d b dlogb (30) ^log ^^ ^ -^ , -log-=--^^, From the equations (30), we get alogX ^ d log HHi • • • gp_i alogX ^ ^^ dv dv ' du ' and from these Using (32) and (33) in (31), we get (34) a^„ log a'H'-^'- ■ ff ,_, = ^' which can also be obtained immediately from the equality of H, the in- variant of (1), and Hp, the corresponding invariant of (28). * Darboux, I.e., p. 33. 245 EDWARD S. HAMMOND. Equation (33) may be integrated, giving HH, ■ ■ ■ H,., = UV, where U and V are functions of u and v alone respectively. From equation (8) we recall the effect of the transformation (7) on the invariants H and K. Likewise under this same transformation H/ = '{u'W{v')Hi. By giving to i values from to 79 — 1 and multiplying, we get H'H,' ■ • ■ ^,_/ == HH,--- H,.A4>'4^'Y = U{u')VW)['^p'V, where U and V are the transforms of U and T' under (7). Hence and \p may be determined so that H'Hi • • • H p-i =1; then from (32), X equals a constant,* w, since d log X _ ^ogj^ _ du dv In the remainder of this section, we shall assume that this transformation has been made, dropping primes for convenience. After this change of variable, there are two necessary conditions for a sequence of Laplace of period p, namely (36) HHiH. ■ • • //p-i = 1 and equation (34). To show that these conditions are sufficient, we pro- ceed as follows. Differentiate (37) Xp = mx with respect to u. Using (23), (2), and (37), we find (38) Hp-iXp-i = 771X-1, which states analytically the fact, evident from geometrj^, that if Np coincides with N, then A^p_i coincides with A^_i. Differentiating the last equation with respect to u, and using (23) and (26), we have d log hHp-i , d log bK (39) Hp-iHp-2Xp-2 + Hp-i ' -Q-— ^P-i = ^"'^-2 + m - ^^ — X-i. Now K = H-i and i/_i = i/p-i, since (38) is a transformation of the type (5). The equality of K and H p-i may also be derived from (34) and (36), using the values of these invariants given by (15). Then by (38), * Tzitzeica, Comptes Rendus, vo.l 157 (1913), p. 908. PERIODIC CONJUGATE NETS. 246 equation (39) reduces to (40) Hp-iHp-2Xp-2 = mx_2. If we continue this process we have in general, (41) Hp-iHp-2 • ■ ■ Hp-iXp-i = mx-i, or, by (36), (42) Xp-i = 7nHHi ■ • • Hp^i-iX^i and finally (43) X = nix-p showing that A^ is identical with its minus pth. Laplace transform as well as with the pth transform. We observe that this process is reversible, that is, by starting from (43), differentiating with respect to v, and reducing step by step, we may reproduce this same set of equations. If we refer to (21) and (24) it is evident that the p -\- 1 equations given by (41) when i takes integral values from to p inclusive, are a system of linear partial differential equations of various orders which, with (1), must be satisfied by the coordinates of the fundamental net A^ of a periodic sequence. From this point of view, let us examine in detail the trans- formation from equation (38) into (40), as this is entirely typical of the change from any one of (41) into the next. The substitutions for dXp-i/du and dx^i/du from (23) and (2) first engage our attention. The value of dXp-i/du used depends on the definition of Xp-^ and on the use of the point equation of Np-2. But this point equation is essentially the result of differentiating (1) p — 2 times with respect to v, a fact which becomes evident on consideration of the result of substituting the value of Xp-2 from (21) in the point equation denoted by (22). The value of dx-i/du used is merely the definition of the minus first Laplace trans- form. The rest of the reduction may be based as indicated on the two equations (34) and (36). From these considerations and from the re- versibility of the process we conclude that, by virtue of (1), its derivatives, and the conditions (34) and (36), any one of equations (41) or (42) is equivalent to any of the others. If the period be odd, let us set p = 2n -\- 1, and i = n in (42), so that it becomes Xn+l = mHHi • • • HnX-n- Also by setting p = 2n + 1 and i = n + 1 in (41), we get X—n—l = ~" ll2ntl2n—\ ' ' ' tlnXn- The differential equations to which these are equivalent give values of 247 EDWARD S. HAMMOND. d"+^:c/(9y"+^ and <9"+^x/5i6''+^ in terms of the 2n + 1 or p quantities d''x d"x d"-^x d"-'^x dx dx dv/" ' ay"' dvT-^ ' d v"-'^ ' ' ' ' ' du' dv ^, X. All other derivatives of order n + 1 may be obtained in terms of these same p quantities by differentiation of (1). Similarly, when the period is even, let p = 2n and i = n, n + 1, giving the two equations Xn = mHHi ■ ■ • Hn-lX-n, X—n—1 — ~ tl 2n-\tl -211-2 ' ' ' H n-lX n—\. By means of these equations and (1) all derivatives of the «th order but d^'x/du'', and all derivatives of higher orders may be expressed in terms of the 2n or p quantities d"x a"~^x d'^-'^x dx dx ai?" du"-' ' ai'"-^' '"' du' Jv' ^' In either case we have a completely integrable system of equations which possesses but p independent solutions. From this result follows the theorem stated by Tzitzeica: A sequence of Laplace of period p can exist in space of no higher order than p — 1. In particular we note: The only nets of period three are planar nets. It will be observed that the conditions (34) and (36) do not involve the constant m. Neither is it involved in the above discussion of the com- plete integrability of equations (1) and (37). Again, the equations them- selves show us that m is a significant constant, that is, one which cannot be reduced to unitj^ by any change of parameter. Then the solutions of the system, which are the coordinates x of our fundamental net N, may be written x' {u, v; m), i = 1,2, • • • , p. If we replace m by another constant, m' , so that we have the equation Xp = m'x', instead of (37), this equation forms with (1) another com- pletely integrable system with p independent solutions, which we may call x' {u, v; m'). A similar set may be obtained for every value of the constant. We may state this result as follows: // an equation of Laplace he the point equation of a net whose Laplace sequence is periodic of period p, it is the point equation of an infinity of nets having the same propcrti/. 3. Levy sequences. The first Levy sequence. In equation (3) of the introduction, functions ^ and rj are defined as the coordinates of the Levy PERIODIC CONJUGATE NETS. 248 transforms of N by means of a solution d of the point equation (1). For the study of these transforms in connection with the Laplace sequence, it is advantageous to denote the coordinates by Xq, i and a;_i, o, defined by ^0,1 1 X Xi X-1, — 1 as they indicate by their form that the points given by any parameter values lie on the line joining the points of N and its Laplace transforms with the same parameters. The functions 6i and d^i, defined by (44) dd dv d log a dv dd_ dli d log b du are called the first and minus first Laplace transforms of 6 and are solutions of the point equation of Ni and A^_i respectively. As we have the relations P'^O, 1 — ay"' 7_i.r_i, du V, the functions Xo, i and :r_i, o differ from ^ and rj only by factors of pro- portionality, and consequently are coordinates of the Levy transforms. We shall accordingly denote the Levy transforms by A^o, i and A^_i, ol their point equations are denoted by (45) and ^0, i] I adi ad b, Let the net Nr, r+i be defined by its coordinates Xr, r+i namely. (46) -T, r+1 r Xr r+1 ^r+l where r is any positive or negative integer or zero, and where di is formed from d by (21) and (24) as Xi is formed from x. The order of the sub- scripts in Nr, r+1 indicates that the points of these nets are to be considered as situated on the tangents to the curves u = constant of the net Nr', that is, on the line between any net Nr and its positive Laplace transform, Nr+i. The application of (23) and (26) to nets of this type leads to the theorem : Any Laplace transfor7n Nr of a net N has the nets Nr-i, r and Nr, r+i as its Levy transforms by means of Or, the rth Laplace transforrn of a solution 6 of the point equation of N; or, Nr, r+i is a Levy transform of Nr by means of dr, and of Nr+i by means of 6r+i. 249 EDWARD S. HAMMOND. Let us express the coordinates of the first Laplace transform of A^^-i, o, following (2) and simplify them. We find dx-i a d . ad _ ~dv 'd'v ^""^ ~eZ, ""-'• ' - ^'"- '' that is, the Levy transforms of a net by means of the same solution of its point equation are Laplace transforms of one another. From the last two theorems A^_i, r and Nr, r+i are Laplace transforms of one another for every value of r. Then A>, r+u (r = • • •, — 2, — 1, 0, 1,2, • • • ), is a sequence of Laplace; it will be called the first Levy sequence. In the expressions for the point equations and the formulas for the partial derivatives of the coordinates of the nets of this sequence, it is necessary to distinguish between positive and negative subscripts. If r and s be positive integers, we have r aHH, ■ ■ • Hr-idr+i i)^+' 1 r — ^ hk^j^ K ^ 1 dU - tlr-X,rXr-Ur+ ^^ Xr.r+l, dXr, r+1 d log ar, r+1 dv dv ^r, r+1 ~r •'^V+1, r+-2', dx-s-i.-s K^s-i.-s , (9 log 6_s_i. du ~ K-s ''-'-'-'-' ' du dx -s-i, -s _ d log g-s -i. _ dv dv ,-.-1+ ^^ X-s-l,~S, X—s—l, —s ~\~ ■ts. — s+lX—s, — s+1- 4. The second Levy sequence. Levy sequences of higher orders. The first Levy sequence is built up from the fundamental sequence of Laplace by the use of a solution 6 of (1) and its Laplace transforms. On this Levy sequence which is itself a sequence of Laplace, we may build a second Levy sequence and so on indefinite!}'. Let ^0. 1 be a solution of equation (45). The Levy transforms of A^o, i by means of this solution are the nets A^o. 2 and A^_i, 1, whose coordinates are defined in accordance with (46) as follows, 1 ^0, 2 '0, 1 ^0, 1 '1, 2 •'^1, 2 X-i. 1 = /-1, X-i, h, 1 ^0, 1 '0, 1 where 61,2 is the first Laplace transform of ^o, 1, and ^_i, is its minus PERIODIC CONJUGATE NETS. 250 first transform divided by /f_i, o, a quantity which occurs similarly in X-i, 0- The point equations of A^o, 2 and A^_i, 1 are denoted by ,.-, r adid,,2 . &' 1 r joedo.i , ab 1 ^^^^ L'^°' '' T^TT ' ^' ^lo7i J ' L^'-^- ^' 0:7:0 ' ^' OITo J • The same pair of theorems which established the first Levy sequence and the fact that it is a sequence of Laplace are valid here. We denote by Nr, r+2 and N-s-2, -s the general nets of the second Levy sequence and give their coordinates, namely 1 ■r, r+2 -s—2, -s — 'r, r+1 1 'r, r+1 ^r, r+l 'r+1, r+2 ^r+1, r+2 -s—2, — s— 1 -s-2, — s— 1 I t'-s— 1, X—s—2, — s-1 •^—s—l, — s Using the second Levy sequence and a solution of (47), a third Levy sequence may be formed. We shall not give the details of this sequence but pass at once to the A;th or general sequence. Here the net correspond- ing to A^o, 1 and A^o, 2 is A^o, U Its coordinates and point equation and the accompanying differentiation formulas can be written down by analogy with the corresponding expressions for A'o, 1 and A^o, 2, and their accuracy established by induction. Similar methods may be applied in the study of the other nets of the general sequence. The coordinates of the general net A^r, r+k are defined by •^r, r+ k 'r, r+k— I r, r-{- k — 1 3^r, r+A;— 1 r+1, r+k ^r+1, r+k where r is any positive or negative integer or zero, and k any positive integer. In forming the second Levy sequence, we made use of a solution ^o. i of equation (45); we now investigate the nature of this function. Sup- pose 6' to be a solution of (1) such that there is no linear relation con- necting 6, 6' and the coordinates x. If (52) 1 '0, 1 — then ^0, 1 is a solution of (45). It will now be proved that, conversely, to a solution ^o, i of (45), not linearly dependent on the coordinates Xo, i, there corresponds a solution 6' of (1) linearly independent of the x's and of 6. Consider the net ^o, i as the projection in {n — 1) space of a net A^o. 1 in ?z-space whose coordinates are 3:0.1^^', ^0, i^~\ • • •, ^0, i^'*\ ^0, i- Then the congruence G, composed of the lines joining corresponding points 251 EDWARD S. HAMMOND. of N and Ni, is the projection of a congruence G in n-space conjugate to the net A^o, i- One of the focal nets of this congruence, say N, projects into the net N'. Now the solutions x'^'^ of (1) are coordinates both of N and of N and, with 6, play the same role in both spaces in forming the coordinates Xo, /*^ of No, i and Xo, i. But in order to form the last co- ordinates of A^o, 1, namely ^o. i, there must be an (n + l)st coordinate of A", a solution of (1) which may be called 6'. Again the third Levy sequence depends on a solution ^o, 2 of (47) for its formation. The argument of the last paragraph then demands as a necessary and sufficient condition for the existence of this solution a second solution ^0, 1 of (45) not linearly dependent on those already obtained. In the same manner, ^0, 1 calls for a third solution, say 6", of (1) not linearly dependent on the solutions already used, such that 1 '0, 1 The final effect of this argument is to base the A'th or general sequence on k solutions, 6, 6', • • •, d'-^~'^^ of (1) such that there is no linear relation between them and the .t's. For further developments, we must prove, as a lemma, a property of determinants. Consider the general determinant of the ?ith order D = \ai,. I, m = 1, 2, Subtract from each element of the iih row the product of the corre- sponding element of the {i — l)st row bj' a,, i/ai_i, 1, (i = n, ?i — 1, • • •, 2,) and develop the result by the elements of the first column. We have D = ai,i 1 «i,i ^1, 1 ^1, 2 tto, 1 ^2, 2 J. fll, 1 ^2, 1 C(2,3 _J^_|«1,1 «1, n (2l, 1 Cto, 1 (^2, n 1 dn-l, 1 On— 1,2 Ctn—1. 1 ! Cln, 1 Cln, 2 1 Cln-1, 1 (ln—1, 1 ^n-1, n Ctn, 1 (^n, n = ai.iA where A is of order n — 1, so that (53) A = — D. rtii The coordinates of A^o, 2 are ^0, 2 — '0, 1 ^0. 1 ^*o, 1 "1, 2 ^1, 2 Using r = 0, 1 in (46) and the analogous expressions for ^0. 1 and ^1, 2 the coordinates Xo, 2 become determinants of the form of A. On applying the PERIODIC CONJUGATE NETS. property expressed in (53) to them, we find 252 1 •'^0, 2 '0.1 6' X 1 h 6.' x.\ ^^''' the latter expression being an abbreviated form in which only the elements of the main diagonal are shown. Consider Xo, k, the coordinates of the net N'o, k- By definition ^0, it — 1 Then by (53) '0, A--1 1 '0, A--1 •'^0, A:-l 'l, A; ^'l, k ^0, k — a a I "o, fc-2 "l, A--1 ^2, fc I , "O, A--2"0, fc-l and by its repeated use ^0, k = 'O, iCO, 2 • • • t^O, fc-1 h'd.:' ■ ■ ■ e^tl'xk As this method of exhibiting the coordinates of the nets of the Levy sequences is a purely algebraic matter, we have at once, (54) Xr, r+k 1 h"r, T+1 • 'r, r+k—1 'rUr+l Ur + k-lXr+k where r may be any positive or negative integer, or zero. We shall call the determinant in the above equation Xr, r+k', a determinant like it but for the last column, in which the Laplace transforms of x are replaced by those of d^^'\ a {k + l)st solution of (1), we shall call Or, r+k- Then (55) 1 Cr, r+k From (54) and (55), we have (56) drdr, r+1 ' and (57) drdr, r+l e r'Jr, r+l 'r, r+k-l r, r+k- r+k—l^r, r+k — Xr, r+k, %, ?■+ A; — "r, r+k- These equations are valid for any integral value of r, and for any positive integral value of k. If we replace k in (57) by /j — 1 and use the result in (56) and (57), we get (58) Qr, r+k-\Xr, r+k = Xr,r+k and e. , r+k—l^r, T+k = G r, r+k- Since the Xr,r+k are proportional to the Xr,r+k, the former may serve equally well as homogeneous coordinates of the nets Nr, r+k- 253 EDWARD S. HAMMOND. In the preceding paragraph we have used solutions 6 Unearly inde- pendent of the coordinates x. The following theorem states the situation under the opposite condition. // a solution 6 of the equation (1) used in the formation of any Levy sequence be linearly dependent on the coordinates x, all the nets of this Levy sequence lie in (i^r- 2) space; ifi such solutions he used, in (j^ — ^ — 1) space. For, supposed = Y.]='\g^^^x'-'^ where the g'-'^ are constants not all zero; then dk = Hlz.'lg'-'^Xk'''^ for every k. Now using these values of the Laplace transforms of 6, we have i:^(^)Z 0. 5. Periodic Levy sequences. If a sequence of Laplace is of period p, and in (p — 1) space, we shall now develop certain conditions under which its Levy sequences have this same period. If the first Levy sequence is PERIODIC CONJUGATE NETS. 254 to be periodic, we must have 'P. p+ 1 — X.To, 1, where X is an undetermined factor of proportionality. By the use of (37) and the value of Xp+i found by differentiating (37) with respect to v, we obtain m dp X X 9 X Op dp^i Xi ~ 6 Bx Xi that is, f7ndp+i \di\ xi{m -\) - xy —^ d' ) ^ Now there are at least three coordinates x, and accordingly the coefficients of x and .Ti must be zero. We have w = X, and dp+ijdp = 6i/d, which, because of the definition of dp+i and di, and equation (36), becomes (59) |l°g7 = 0- Let us now differentiate Xp, p+i = inxo, i with respect to u. By applying formulas already derived for such derivatives, we get Xp-i, p = m.r_i, 0- In this case, using (37) and (38) and expanding, we have Hp^idp-i/dp = d-i/d. Then (2) and (23) give (60) |,'o4' = 0- In consequence of (59) and (60), we see that dp = Cid, where Ci is a constant whose value is to be studied further. To show that under this condition the nets Np-i, p-i+i and N-i,-i+i are identical, consider the coordinates 1 I ^ • r • I ^p— 1, p—i+l = a \ a „ "p—i\Op—i+l Xp_i+i 1 Since (41) and (42) are true for 6 if 7n be replaced by Ci, we have Xp—i^ p—i-\-i = 77iHHi • • ' H p—iX—i, —i+i. For the second Levy sequence also to be periodic, it is necessary and sufficient that do, i the solution of (45) by which the second Levy sequence is formed from the first, shall be such that dp, p+l = Codo, 1. 200 EDWARD S. HAMMOND. By referring to equation (52), we see that this will be the case if Qp = C\B' . In general, we conclude that if the {k — l)st sequence is periodic, the necessary and sufficient condition for the Aih sequence to be periodic, is that ^p, p+^-l = C kdo, k—l and that this condition will be fulfilled if 6, 6', • • • ^^^""^^ are such that their 2;th Laplace transforms are constant multiples of them. Evidently under these last conditions not only is the A-th Levy sequence periodic, but also all the sequences of order less than k. The disposition of the constant multipliers in various ways leads to some interesting results. By the last theorem of section 2, there are p solutions 6 for every value of the constant occurring in (37). First, let us suppose that Ci is equal to m. Then 6 must be a linear combination of the a-'s and therefore the theorem of section 4 applies and the nets of this periodic Levy sequence lie space of order p — 2. This result was noted by Tzitzeica. For sequences of higher orders, it may be generalized into the following theorem: // a sequence of Laplace of period p lie in (p — 1) space, arid has co- ordinates such that Xp = ?fix, and if a Levy sequence of order k, periodic or not, based on this sequence of Laplace, he formed by the use of k solutions 6 of the original point equation, of which one is such that dp = mO, the nets of this Levy sequence lie in space of order p — 2; if i such solutions be used, the Levy sequence is in space of order p — i — 1. To prove this theorem, we need first to recall that there are but p solutions of the system of partial differential equations satisfied by the coordinates of a periodic sequence of Laplace; therefore, if dp = 7nd, d is a linear function of the coordinates x. The proof is then completed by the application of the last theorem of section 4. Consider now the Levy sequences which can be formed on a periodic sequence of Laplace by the use of the set of p solutions 6, 6', • • • , ^(p-i> such that dp'^'^ — 7n'6'''\ m' =t= in. There will be p periodic first Levy sequences, p{p — l)/2 periodic second Levy sequences, in general, as many of the A'th order as the number of combinations of p things taken A- at a time and finally, one periodic sequence of the pth order. It will now be shown that this pih. sequence coincides with the original sequence of Laplace. For, consider the coordinates of the net Nq, p, namely, A'o, p. We have X 0, p )(;'-!) ' p Up • • • [/ ^^ Xp 6^"-'^ X m u m m'e^P-^^ 771X PERIODIC CONJUGATE NETS. 256 Subtracting m' times the first row from the last, then -^0, p = ('^^ ~ ^n )9o, p-iXj so that the coordinates Xo, p are proportional to the x-'s. In general the coordinates Xr,r+p of the net A^r, r+p are determinants such that in each the elements of its last row may be made all zero but the last, which will be a constant multiple of Xr. Therefore Xr, r+p is proportional to xv, and the nets Nr, r+p coincide with the original Laplace sequence. 6. Nets in relation T and their Laplace transforms. In the introduction a geometric definition of the relation T was given; Eisenhart has shown that, if A^ be a net in relation T with N, their analytic relation is expressed in the statement that the homogeneous coordinates x of A^ may be ob- tained from quadratures of the form dx d fx\ dx d /x\ where ^ is a solution of (1) different from any x\ The factors r and a are not entirely arbitrary for the conditions of integrability of (61) show that they must be solutions of the equations dr , da da d b (62) -7- = (o- — r) — log ^ , ^~ = (t- — 0") ^- log -^ , or their equivalents, d . ar ad, a d , ha r d , h — log -^ = - -- log - , ^^ log — = - ^- log - . dv ^ 6 T dv ^ 6 du *^ B a du '^ 6 In connection with the derivation of the integrability conditions of (61) it is readily shown that the net A^ has the point equation d^x d , ar dx , d . ba dx = ^ log ^ x;: + ^ log dudv dv ^ d du ' du ^ 6 dv' For an equation of this special type in which the term involving x is missing we shall use a symbol similar to (13) except that the last of the quantities within the brackets is omitted_to indicate that the term in x is lacking. Thus, the point equation of A^ will be denoted by (63) [-^'T'TJ- Using the fact that ^ is a solution of (1), the invariants of A^ have the va'ues (64) H = H-'^^, K^K-'^f^. ^ ' dudv dudv 257 EDWARD S. HAMMOND. From these developments, it appears that the determination of a net in relation T with N depends on a solution of (1), a pair of functions r and 0- which satisfy (62), and the quadratures (61). Eisenhart has shown that the problem may be given another aspect by the introduction of a function 0, defined by the equation (65) T - d. By differentiation and the use of (62) and (1) it may be shown that is a solution of the equation denoted by But this is the adjoint of (1). Accordingly, the problem is reduced to the finding of a solution of (1) and a solution of its adjoint equation, and two sets of quadratures, namely dr d log b(h dr ^ d . a — = 00 — ^^ - , — = - 00 — log - , du • du ' dv ^ dv ^ 6' (67) da ^^ d . b da 5 log a0 -TT- = 00 ^r- log - , -T- = — 00 7. , du ^ du ^ 6' dv dv ' which follow from (65) and (62), and (61). A discussion of the effect on the net N of the arbitrary constants arising from these quadratures is in order at this time. If x be the coordinates of the net A' when the additive constant ^To r and a is set equal to zero, then for any other value of c, the coordinates of the T transform become X- + ex; 0. This point is on the line joining corresponding points of A" and N. Conse- quently, we may say that the variation of this constant leaves the conju- gate congruence of the transformation unchanged but moves the points of the net along the lines of this congruence. Again if x'^'^ and x'-'^ + c, are the coordinates of nets obtained by different values of the constant of integration in (61), the line of inter- section of the tangent planes to the nets is the same for all values of d. This is a result of equation (61) since the coordinates of the Lev}^ trans- forms of N by means of may be taken as d{x/6)idu and d{Xjd)jdv. The totality of such lines of intersection, or the joins of corresponding points of the Levy transforms form a congruence which has been termed by Guichard* the harmonic congruence of the transformation. We may say then, that the variation of the constant arising from (61) leaves the harmonic congruence of the transformation unchanged. Conversel}', all nets harmonic to this congruence are so determined, since it has been * Guichard, Annales de I'Ecole Normale Sup., 3^ Serie, t. 14 (1897), p. 4S3. PERIODIC CONJUGATE NETS. 258 shown by Eisenhart* that two nets harmonic to a congruence are in relation T. Now if Xi and di be the first Laplace transforms of x and 6, the co- ordinates of A^i, a T transform of A^i will be given by quadratures similar to (61), namely, The integrability conditions of this quadrature give equations for n and cTi analogous to (62), "^ = (o"! — •^i) -T" iog ^fl"> ii = (^1 ~ (^i) T~ log ;r dv dv ^ ^1 ^2* ^ ^ du ^ 01 and we also find that the point equation of A'^i is denoted by (69) L^-^'-^' 7rJ- In order to determine the relation between ri, o-i and r, o-, we proceed as follows. If A^i is the first Laplace transform of A^, the invariants Hi and H of these nets should be related as are the invariants H^ and H in (12). Forming the corresponding relation, we have - d- gtH - Hi = — ^—^ log -T h H. dudv ^ OCT On reducing this equation by the use of (69), (64), and (12), we find that Similar reckoning performed with Ki and K shows that d'^ log di d' log r (71) dudv dudv Now the equations denoted by (63) and (69) are of the form which must be satisfied by the non-homogeneous coordinates of a net, and this suggests that the same relation may hold between the coordinates of Ni and A^ that holds in the non-homogeneous case. This is shown to be true, for if we substitute - _ - _ 1 dx Xi X ^ 1 > d , ar dv ■ A result as yet unpublished. 250 EDWARD S. HAMMOND. and T 'H — particular solutions of (70) and (71) — and the values of .ri and ^i given by (2) and (44) in equations (OS), they are identicallj^ true. We have proved then that the T transforms of a net A'' and its first Laplace trans- form whose coiirdinates are obtained from the quadratures (01) and (OS), where ^i is the first Laplace transform of 6, and where n and ai have the above values, are Laplace transforms of one another. We find that the difTereuce n — ci, when reduced by the use of (04) and (07), is equal to - 0_i dilH, where d4> d log b 0-1 ~ ~^ — — '\ 4>f ^ da dii ' following (2) and (00). Now- ., J_ 1 1 ] IH ' aH' b' a'Hi' But this equation is the adjoint of (10); so that we have the net .Yi based on a solution of the adjoint of the point equation of .Vi which is proportional to the minus first Ltiplace transform of 0. In general, we have that, if A'r is a T transform of .Yr, the rth Laplace transform of A", whose coordinates are given by the quadratures then the nets vYr (/* = 0, 1, 2, • • •), form a sequence of Laplace. In this general case, w'e find, as in the particular cases we have considered, that —r d . no' — du IIUx ■ • • Hr-1 du r» = — jjTT jj 3-loga0_,.. dv HH X • • • Iir-1 dv ^ 7. Periodic sequences of T transforms. As a preliminary step in the question of the i)eriodieity of the T tninsforms, we investig;ate the adjoint equation of (1) when (34) and (36) are satisfied. These equations are necessary and sufhcient conditions for a periodic sequence of Laplace whose coordinates satisfy (1) and (37). Now if the invariants of (66) are foimed, it is found that they are the same as H and K but are inter- changed. Similarly the invariants of (72) are those of (16), that is, Hi and 7vi, but intei-clianged; and in general, the invariants of (75) are II r and Kr interclianged. We also notice that in working with (72), a and h are replaced by 1/a and 1/6, respectively. Then the conditions on the coefficients of (()6) which assure solutions such that are equivalent to (34) and (36), and we may state the following theorem: If an cqualion of Laplace has periodic solutions, so has its adjoint. Suppose that the fundamental sequence is periodic of period p, and that Xj, = 7nx, the conditions (34) and (36) being satisfied. Let 0, the solution of the adjoint equation of (1) which determines the quadratures 261 EDWARD S. HAMMOND. (67) and (61), be such that = n0_p; also let the solution ^ of (1) involved in these quadratures be such that dp = m'd. Then if we set r = p the quadratures (73) and (76) which determine a pth Laplace transform of A'^ become dtp m' b . ^ Bt-p w' ^ 5 , a — -P = — ^0 — log 60, —^= ^0 log-, d(Tp m' d , h dcTp m' d . "a ~ = — ^0 -^ log T , ^-" = ^0 — log a4>, du n du ^ 6 dv n ^ dv ^ ^' (79) and , . dxp _m d / x\ dxp _m d fx\ ^^ 'du ~ n'"du\d)' ^ ~n'^&u\d)' which differ from (67) and (61) only by constant factors. Then the coordinates Xp can differ from x only by a constant factor arising from the factors appearing in (79) and (80), or by an additive constant from the final quadratures. But these additive constants are entirely arbitrary, and since we are dealing with homogeneous coordinates the factor is immaterial. We have, now, the following theorem: Let (1) he the point equation of the fundamental net N of a sequence of Laplace of period p, whose coordinates X are such that Xp = mx] if 6 he a solution of (1) such that dp = 7n'd, and if ({)he a solution of the adjoint equation of (1) such that = n(t)^p, then each T transform of N determined hy quadratures from 6 and 4) is the fundamental net of a sequence of Laplace of period p; moreover, each of the nets of these sequences is a T transform of the corresponding net of the original sequence. The author wishes to express his gratitude to Professor Eisenhart for the suggestions which led to this paper and for continued helpful advice and criticism during its preparation. Gaylord Bros. Makers Syracuse, N. Y, PAT, JAN, 21, 1903 "_.: -1 fZJTS 425^ v UNIVERSITY OF CALIFORNIA LIBRARY