Ibe tzmaixersito of Cbica0o A TYPE OF SINGULAR POINTS FOR A TRANSFORMATION OF THREE V ARIABLES A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY WILLIAM VERNON LOVITT PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. I915 tbe u'nivertitt of Chicago A TYPE OF SINGULAR POINTS FOR A TRANSFORMATION OF THREE VARIABLES A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY WILLIAM VERNON LOVITT PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. I915 A TYPE OF SINGULAR POINTS FOR A TRANSFORMATION OF THREE VARIABLES* BY W. V. LOVITT It is the purpose of this paper to discuss from as geometrical a standpoint as possible the character of a transformation (1) x = 40(u, v, w), y = (U, /u, v,), z =x(, v, Vw) near points of a special type at which the jacobian of the transformation vanishes. Let a particular one of the singular points in question be denoted by P, and let S denote the surface through P in the tvw-space defined by setting the jacobian of the transformation equal to zero. The point P and the surface S are transformed by (1) into a point P1 and surface Si in the xyz-space, and the initial assumptions in Section 1 are such that P and Pi are non-singular points of S and Si, respectively. A neighborhood of the point P is divided by S into two parts. In Section 2 it is shown that under the hypotheses of Section 1 each of these parts is transformed in a one-to-one way into a connected region with interior points in the xyz-space, and the two xyz-regions so obtained adjoin Si and lie on the same side of that surface. Moreover the single valued inverse functions of x, y, z defined by the transformation in either of the regions adjoining Si are continuous, and they have continuous first derivatives except possibly at points of the surface S1. In the first two sections the transformation is studied in a neighborhood of a particular singular point P. But most of the results there found can be extended to apply to a neighborhood of any surface consisting of singular points of the type of P. This is done in Section 3. The generalization is analogous to an extension of the well-known theorems on implicit functions due to Bolza.t The methods of Sections 1-3 are applicable to n-dimensional spaces with but little more than the necessary changes in notation. * Presented to the Society, April 2, 1915. t See his Vorlesungen iber Variationrechnung, p. 160; also Mason and Bliss, Fields of extremals in space, these T r an s a c t i o ns, vol. 11 (1910), p. 325; or Bliss, Princeton Colloquium Lectures, p. 19. Trans. Am. Math. Soc. 25 371 372 W. V. LOVITT: [October In Section 4 we show that the two-parameter family of directions through a point P on the surface S, with the exception of a certain critical direction, transforms into a single plane 7r1 of directions. This plane 7r1 is tangent to the surface S1 at the point Pi. Urner* showed that every plane of directions through P in the uvw-space transforms into a single direction in the plane 7rT. The direction coincident with instead of distinct from the critical direction I transforms into the two-parameter family of directions through P1, with the exception of those in the plane 7r1. All curves D having the critical direction and which have the same principal normal are transformed, as Urner has shown, into curves with directions in a common plane fi. We add that as the radius of first curvature of D varies from infinity through P on the principal normal and out to infinity on the other side again the direction of the transformed curve rotates in f1 about P1 through an angle of 180 degrees. The character of a transformation in the neighborhood of a point at which the functional determinant vanishes has been the subject of study in a number of papers of comparatively recent date. For a review of the papers in widely different fields bearing on this subject the reader is referred to Chapter II of the Colloquium Lectures referred to above. Bliss showed in these lectures that for a transformation in two variables, regions on opposite sides of the curve defined by the vanishing of the functional determinant of the transformation are transformed into two overlapping regions. The theorems deduced in Sections 1, 2, 4 below afford a generalization of this result to spaces of three or more dimensions. The paper of Urner mentioned in the preceding paragraph contains a discussion of the transformation effected by equations (1) upon the directions of curves through the point P, but he does not otherwise discuss the transformation of the region of space adjoining P, as is done in the present paper. 1. THE INITIAL ASSUMPTIONS Let us consider a real point transformation of three space (1) x = c (u,,w), y = (u,v,w), z = X( u, v,w) with determinant hu 04v <w J(u,v,w)= u {v w. Xu Xv Xw The functions 0, A/, x are not necessarily analytic but it will be presupposed * S. E. Urner, Certain singularities of point transformations in space of three dimensions,. these Tr an s a c t i o n s, vol. 13 (1912), pp. 232-264. 1915] SINGULAR POINTS OF TRANSFORMATIONS 373 that (a) the functions A, 4', x are of class C"'* in a neighborhood of the origin (u, v, w) = (0, O, 0); (b) the following initial conditions are satisfied: 5(0, 0,0) = (0,0,0) = x(0, 0,0) = 0; (c) J(0, 0, 0) = 0; (d) at the origin (u, v, w) = (0, 0, 0) at least one of the determinants of the matrix Ju Jv Jw (2) qu qVv qw Xu Xv Xw is different from zero. There is no loss of generality in assuming, as indicated in these conditions, that the singular point P to be considered for the transformation is at the origin in the uvw-space, and that the transform of P by (1) is the origin P1 in the xyz-space. Neither will generality be lost if we assume for convenience that the determinant Ju Jv Jw (3) H -i - v 1 w Xu Xv Xw is that one of the matrix (2) which does not vanish at the origin. Only a change of notation in x, y, z, 4, 4, X is required to bring this about if it is not already true. At least one of the derivatives Ju, JJ, Jw does not vanish at the origin on account of our assumption (d). Hence the equation J(u, v, w) = 0 can be solved for one of the variables in terms of the other two. There exists therefore a unique surface (S) u = U(a, ), v = V(a, ), w = W(a, f) satisfying J = 0, in which a and 3 are a suitably selected two of the parameters u, v, w. Since Ju, JV, Jw do not all vanish at the origin this point is a nonsingular point of the surface S. * We shall say that a single-valued function f of (u, v, w) is of class C"' if f (u, v, w) and its partial derivatives of orders one, two, and three are continuous in a region in which f is defined. 374 W. V. LOVITT: [October It follows from the derivation of equations (S) above that J(U, V, W) -0, whence J, U,+ JV U,-+ Jw Wa = 0, Ju U~ + Jv V, + JW W, 0. From these equations we obtain V TWa U, W, Ua V,,(4) kJu, =, kJ = V, W, Ua W" U, V, where k is a factor of proportionality. Moreover k + 0 since the surface S is non-singular and consequently the determinants in the second members -of equations (4) can not vanish simultaneously. The transform of the surface S by (1) is a surface x = f(u, v, VW) = X(a, ), ) y = ( U,V, W) = Y(a, 3). Z = x(u, T,) = Z(~, p). The origin P1 is a non-singular point for Si if the determinants of the matrix Xa Yr Za (5) X Ys Z do not vanish simultaneously. But iX', X,B | U( + )v 1Va + qfw Wa u UO- + v T-V + Cp. WTV Yr Ya = \ 7U Ua+ r u, + V I A V W W U v, + V J-+ /, T3 \VITa I' - VI W, 1T1, TFa - Ua TV, 3 a, - I U V7( =! )u <v W;u (6) 4u 4/v Jau Jv Jw = k (fU OV (P W I *1U Av +1 IV 1915] SINGULAR POINTS OF TRANSFORMATIONS 375 and by a similar argument Ju Jv Ju Jv J w (7) Z = k ou ov w in Z =k l u qv 1w. u Xv X Xw Xu Xv Xw Thus assumption (d) assures us that P1 is an ordinary point on the surface S1. We may now state the following theorem. THEOREM I. Under the assumptions (a), (b), (c), (d) the origin P in the uvw-space is a non-singular point of the surface S defined by the equation J(u, V, w) =0. Furthermore in the xyz-space the image P1 of the point P by means of equations (1) is a non-singular point of the surface S1 into which S is transformed by (1). By our assumptions (b) and (c) the equations (8) J(uv,,w) =0, y ' = (, ( v, ), = x(u,v,w) have the initial solution (u, v, w, y, z) = (0, 0, 0, 0, 0). The hypothesis (d) justifies the assumption that the determinant (3) is different from zero, as we have seen. Hence by the usual theorems of implicit functions there exists a neighborhood (0,0,0,0, 0)* in which no two solutions (u, v, w, y, z) of equations (8) have the same projection (y, z), and a neighborhood (0, 0)8 of the point (y, z) = (0, 0) in which equations (8) determine u, v, w as functions of class C" of y and z, (9) u= u(y,z), v = v, z), w = w(y, Z), wz) defining values (u, v, w, y, z) in the neighborhood (0, 0, 0, 0, 0)e. By substituting these results in the third of equations (1), a surface (10) = X(y,z) is found, which is the transform by (1) of the surface S. This is identical with the surface given by equations (Si) above. Let Al, A2, A3; B1, B2, B3; C1, C2, C3 denote respectively the cofactors of the elements of the first, second, and third rows in the jacobian J. Since at the origin H1 -- J A1 + JA A2 + Jw A3 + 0 * For these theorems see Bliss, Princeton Colloquium Lectures, pp. 8-9. By the notation (0, 0, 0, 0, 0 ), is meant a neighborhood ul <, vIv < E, IWI< e, lyl< e, IZJ <e of the point (0, 0, 0, 0, 0). 376 W. V. LOVITT: [October it follows that Ju, Jv, Jw are not all zero. Moreover some one of the terms Ju A1, Jv A2, Jw A3 is different from zero at P, let us say Ju A1 + 0. Only a change of notation in u, v, w is required to bring this about if it is not already true. Then at the point P A1= o(v ) 0, J(0, 0, 0) 0. Thus the tangent plane to the surface S at P is not parallel to the u-axis, and the matrix of the determinant J is of rank two at the point P. We now interpret equations (1) as a two-parameter family of curves with the parameters v, w. Under the assumption (d), the surface SI is the envelope of the curves (1).* For from equations (8) and (10) the expression U - Xy gu - Xz xu has the factor 0 JO JV J, ^.u ^u ^v ^I Xu Xu Xv Xw which vanishes on the surface Si on account of the first of equations (8). If we keep v and w fixed and let u vary we get a straight line in the uvw-space, and by varying v and w we obtain a two-parameter family of lines parallel to the u-axis. These lines are, by the above, transformed by equations (1) into a two-parameter family of curves for which S1 is the enveloping surface. Under the assumptions made, none of the curves of the two-parameter family mentioned cross the surface S1, at their points of tangency. For if we put W(x, y, z) x - X(y, z) and in place of x, y, z substitute their values in terms of u, v, w from equations (1), we find dW 1 du = I- - X A {u -Z Z Xu JJu, d2W 1 d 1 du2 - tH1( JJU + J) + JJdu H But dW/du vanishes on the surface S1 since J = 0, while d2 W/du2 does not vanish on S1 since J + 0 by hypothesis. We have thus proved th( following * Mason-Bliss, The properties of curves in space which minimize a definite integral, thes, T r a n s a c tions, vol. 9 (1908), pp. 440-466. 1915] SINGULAR POINTS OF TRANSFORMATIONS 377 THEOREM II. The two-parameter family of straight lines v = constant, w = constant parallel to the u-axis in the uvw-space is transformed by means of equations (1) into a two-parameter family of curves x =(uW), y, =w(u, ), y = (uv,), u, v,w) in the xyz-space, with the parameters v and w. Under the assumptions (a), (b), (c), (d) this family of curves has an enveloping surface Si which is the transform of the surface S. by means of the equations (1). The curves of the family do not cross the surface S1 at their points of tangency. 2. TRANSFORMATION OF A NEIGHBORHOOD OF A POINT ON THE SINGULAR SURFACE In this Section we show that under our assumptions (a), (b), (c), (d) there is a one-to-one correspondence between either of two regions in the uvw-space on opposite sides of the jacobian surface and its transform in the xyz-space, the two transformed regions being on the same side of the surface S1 which is the transform of the surface S by means of equations (1). Moreover each of the transformed regions is a connected region possessing interior points. The equations of transformation define inverse functions which are continuous at points of the xyz-space which are on the surface Si, and of class C' at points not on Si. The equations (11) y = 4(u,v, w), z = x(u, v, w) represent a system of curves in the uvw-space. Every solution (u, v, w) of these equations, when y and z are fixed, determines a value ~ of the determinant J such that equations (12) J(u, v, w) = i, (u, v, w) = y, (u, v, w) = z are satisfied. By our assumptions (b) and (c) the equations (12) have the initial solution (u, v, w, y, y, z) = (0, 0, 0, 0, 0, 0). Since the determinant (3) is different from zero the usual theorems of implicit functions justify the statement that there exists a neighborhood (0, 0, 0, 0, 0, 0), in which no two solutions (u, v, w,, y, z) of equations (12) have the same projection (i, y, z), and a neighborhood (0, 0, 0)5 of the point (S, y, z) = (0, 0, O) with a = e, in which equations (12) determine u, v, Iw as functions of class C' of i, y, z, (13) u = u({, y, z), v = v(, y, ), = w(, y, y). 378 W. V. LOV1TT: [October If in these equations ~ is placed equal to zero we obtain equations (9) of Section 1. As a result of the choice of notation mentioned above which assures us that J, (0, 0, 0) + 0, the equations of the surface S of Section 1 may now be written in the form = U(v,w), v= a, w S. Consider then in the uvw-space a neighborhood (K) 0 u- U(vw) a, Ivl -b, iw\ -c of the origin, where a, b, c are so small that the values (u, v, w, I, y, z) defined over the region K by equations (12) all lie in (0, 0, 0, 0, 0, 0). Further the region K is to be restricted so that to any (u, v, w) in K there corresponds by (12) a (S, y, z) which is in the 5-neighborhood (0, 0, 0)6 of the point (, y, z) = (0, 0, 0). Then there passes through each (u, v, w) in K one and but one of the curves (11). That there is one is seen from equations (12), for to every (il, v, w) there is defined by (12) a (S, y, z) in the aneighborhood (0,0, 0)s of (5, y, z) = (0, 0, 0). That there is only one follows from the fact that the only solutions (u, v, w, i, y, z) of equations (12) in (0, 0, 0, 0, 0, 0), corresponding to a point (~, y, z) in (0, 0, 0), must satisfy the equations (13). On any one of the curves there is but one point where the determinant J vanishes, namely the point defined by - = 0. We now substitute the values of u, v, w as given by (13), in equations (12), and by differentiation find that Ju e + Jv v + JwW = 1, (14) l/u u +, v + 4/ o = 0, Xu + XvV + Xw = 0, from which it follows that 1 tw it o e11 u t i (15) ut =. Hi Xv XW I i i Xw XU 'i Xu X From the first of equations (14) we conclude that the functional determinant J is univariant along the curve (11), and the curve (11) is not tangent to but actually cuts through the surface S. Thus we see that on the curve (11) the functional determinant J has opposite signs on the opposite sides of the surface S. In order to fix ideas let us consider J > 0 on that side of the surface S which contains the positive u-axis. Only a change in notation from u to - u is necessary to bring this about if it is not 1915] SINGULAR POINTS OF TRANSFORMATIONS 379 already true. The only restriction on the notation u, v, w preceding this was for the purpose of calling J, that one of Ju, J,, J, which was not zero. We propose to show that under the assumptions made, the equations (1) define a one-to-one correspondence between either of the regions (K) 0 u - U(v, w) a, lv b, Iw c, (M) - a u - Uv, ) 0, I b, |wl |c, on opposite sides of the surface S, and its image KI or M1, in the xyz-space. We note that according to our assumptions J = ~ 0 in K, and J c 0 in M. Suppose that the correspondence between K and K1 is not one-to-one. It will be shown that this leads to a contradiction with the statement that J varies univariantly on the curve (11). If two points (u', v', w') and (u", v", w") in K define the same (x, y, z) they must lie on that one of the curves (11) defined by the values y, z, since any (u, v, w, x, y, z) satisfying the original equations (1) determines also a solution (, v, w,,y, z) of equations (12), and the only solution of equations (12) for a (u, v, w) in K is given by equations (13). They must define distinct values i' and 4" satisfying i'. 0, v" > 0 if they lie on the same side K of the surface S, since distinct points of a curve (11) can only correspond to distinct values i', i". The function f is continuous with continuous first partial derivatives and takes by hypothesis the same value x on the curve (11) for 5 = i' and ~ = /". Hence its derivative for 4 must vanish for at least one value - = T between i' and a". But this derivative is dcq d4 = ut ui + 4fv V + vw q.With the use of equations (15) we can show that d~ J dt H1 But r is positive and 0 i' < r <, provided we assume i' the smaller of i' and k", and so the point of the curve (11) defined by r is not on the surface S. Hence we have indeed reached a contradiction with the statement that J is univariant and vanishes only for 0 = 0 on the curve (11). Therefore the correspondence between the regions K and K1 is one-to-one, and in a similar manner it can be shown that the correspondence between the regions M and Mi is one-to-one. We are now in a position to make some statements with respect to the inverse functions (16) u = f(X,,, ), v = g(x, y, z), w = h(x, y, z) defined by equations (1), and the character of the regions K1 and M1. 380 W. V. LOVITT: [October Let us omit from the regions K and KI the points on S and S1. Denote the six dimensional set of points so formed by K. Then every point p (u, v, w, x, y, z) of K is an ordinary point for the equations (1) and satisfies both those equations and the equations (16). By the usual theorems of implicit functions it follows that there exists a neighborhood p, of p in which no two solutions of equations (1) have the same projection (x, y, z), and a 5-neighborhood of the point (x, y, z) with _- e, in which equations (1) determine u, v, w as functions of class C' of x, y, z. Thus to every point (u, v, w) interior to K there corresponds a point (x, y, z) interior to K1 at which f, g, h are of class C'. In like manner we show that the region M contains interior points. It is evident that K is a connected region. We shall now show that K. is a connected region. To do this it is sufficient to show that any two points A1, B1 interior to K1 can be joined by a continuous curve also interior to K1. By equations (16) the points Al and B1 are transformed into two points A and B interior to K. But K is a connected region, and hence A and B can be joined by a continuous curve interior to K, the transform of which by equations (1) is a continuous curve interior to K1, joining A1 and B1, and thus our statement is proved. We may then state the following theorem: THEOREM III. Under the assumptions (a), (b), (c), (d) either of the two regions K and M in the uvw-space on opposite sides of the surface S is transformed in a one-to-one way by means of equations (1) into a region, K1 or M1, in the xyz-space. Each of the regions K1, M1i possesses interior points and is a connected region. From the proof just preceding, the functions f, g, h are of class C' in K1 or M1 except possibly at points of the surface S. We desire to show next that they are continuous even at points which are on the surface Si. To do this it is sufficient to show that any infinite sequence of points qi in the region K1 with a single limit point k on the surface S1, has corresponding to it in the region K an infinite sequence of points pi with a single limit point on the surface S, and that this limit point is the point wr of which k is the image by means of equations (1). The infinite sequence of points pi certainly has at least one condensation point. Let 7r' be such a point. Since the functions q$, X, 4 are continuous it follows that the image of -' must be k, and since the correspondence between K and K1 is one-to-one,r' coincides with -r. The point -r necessarily lies upon S since such points are the only ones which can go into the points of S1. We have thus proved that the functions f, g, and h are continuous at points of Ki or Ml on the surface S1, and may state the following theorem: THEOREM IV. Under the assumptions (a), (b), (c), (d) the inverse functions u =f(x, y, z), v = g(x, y, ), w = h(x, y, z) 1915] SINGULAR POINTS OF TRANSFORMATIONS 381 transforming K1 into K (or M1 into M) as described in Theorem III, and defined by means of equations (1), are continuous at all points of the xyz-space situated in the region K1 or on its boundary, and have continuous first derivatives except possibly on the part of the boundary of K1 which is coincident with S1. It remains for us to show that the regions on opposite sides of the surface S transform into regions on the same side of the surface Si. We take two points A and B within the regions K and M, respectively (see Fig. 1), on opposite sides of the surface S and on the same straight line parallel to the u-axis through a point P on S. The line segment APB transforms by means S1 B A: i A PP~~~~~P1 FIG. 1. of (1) into an arc Al P1 B1 which by Section 1 is tangent to the surface S1 at the point Pi and does not cross Si at P1. The determinant J does not vanish on the line segments AB except at the point P, and hence P1 is the only point of Al P1 B1 which is on the surface S,. Thus we see that the points Al and B1 are on the same side of the surface Si. The point A is in the region K which has been shown to transform in a one-to-one way into the region K1 containing the point Al, while the point B is in the region M which transforms in a one-to-one way into the region M1 containing the point B1. But we have just shown that A1 and B1 are on the same side of the surface S1. We may therefore state the following theorem: THEOREM V. Let (1) x = (u, v, w), y = (u,, v ), z =X(u, v, w) be a transformation which at a point P has properties (a), (b), (c), (d). The surface S on which the jacobian of the transformation vanishes contains P and is FIG. 2. 382 W. V. LOVITT: [October transformed into a surface S1, in the xyz-space containing the image P1 of P. The portions K, M of the uvw-space on opposite sides of S are transformed in a one-to-one way into portions K1, MA of the xyz-space on the same side of Si (see Fig. 2). '\ 3 kl I xx a l /! I /bi FIG. 2. 3. TRANSFORMATION OF A NEIGHBORHOOD OF A SINGULAR SURFACE OF ANY EXTENT In this section by the use of the assumptions (a), (b), (c), (d), (e), (f) below we extend most of the results of the preceding section to a transformation of a neighborhood of even an extensive part of a singular surface. Let us now consider the transformation (1) in a neighborhood Se of the set of points (u, v, w) on the surface (S) iu = U(a, d), v = 7(a, ), w = W(., a ) defined over a finite closed region I of the ao3-plane. By a region is understood a piece of the plane which is bounded by a finite number of regular curves. The region is said to be closed if it contains all of its limit points. By a neighborhood Se is understood the totality of points (u, v, w) for which u - Ul _ v, II - V _, l - fl E when (a, c) range over the region. It will be presupposed that (a) the surface S does not intersect itself and has no singular points in l, in other words, the determinants of the matrix U. V. W. do not vanish simultaneously at any point of:; (b) the image x = 4(U,. VW) = X(a a), (Si) i y )z = ( U, V, W,) = Z(a, 3), Z - X (U,V,W) =Z(a,) 1915] SINGULAR POINTS OF TRANSFORMATIONS 383 of the surface S by means of equations (1) does not intersect itself; (c) the functions U, V, W are of class C' in:; (d) the functions X, A, x are of class C" in a neighborhood of the surface S; (e) the functional determinant J = 0 (X, 4, x)/ (u, v, w) vanishes on the surface S; (f) at every point of the surface S at least one determinant of the matrix (2) is different from zero. The assumptions (a) and (b) are sufficient to assure us that there is a one-toone correspondence between the points of the surface S and its image Si. For from equations (S), on account of the assumption (a), there is for any point (u, v, w) on S one and but one pair of values of a, 0; and from equations (Si) by means of the assumption (b) there is for any point (x, y, z) on Si one and but one pair of values of ac, f. Hence for any point (u, v, w) on the surface S there is one and only one point (x, y, z) on the surface S1. Suppose that however small e is chosen two points in Se may be found on the same side of S whose images in the xyz-space coincide. For points on one side of the surface S the determinant in (e) is positive, while for points on the other side it is negative, since by the assumption (f) one at least of the derivatives J, J, Jv, Jis different from zero at each point of S. Consider now a sequence {en} (n = 1, 2, **, oo ) with limit zero, and the corresponding sequence of regions {S, }. In every region S, there is by hypothesis a pair of distinct points whose images in the xyz-space coincide. These two points may be denoted by P,(u,,, vn, n,,), P,(u, v, Vn, n) and their image by (xn, yn, zn). Consider the set of points pn = (Un, vn, n; ua:n v',, w; xn, n y n ) thinking of pn as a point in a nine-dimensional space. The set p, (n = 1, 2, ~*.) certainly has a condensation point which we designate by (ugo vo, w0,; U, v, w0; Xo, yo, z0), and for which the points Po (uo, vo, wo) and Po ( u, Vo, wo ) necessarily lie upon the surface S. Thus we arrive either at two distinct points (uo, v0, wo), (uo, v0, W4) on the surface S whose images in the xzy-space coincide; or if (uo, vo, wo) and (ua, v, Wo) are the same we arrive at a single limit point on the surface S in every neighborhood of which there are two points whose images in the xyzspace coincide. The first case is impossible since we have shown the one-toone correspondence between the points of the surface S and its image S1. The second case is the one considered in Section 2 and was there shown to be impossible. Thus we have shown that there must exist a neighborhood S, in which no two distinct points (u, v, w) (u', v', w') on the same side of S can define the 384 W. V. LOVITT: [October same point (x, y, z) by means of equations (1). It follows that there is a one-to-one correspondence between either one of two suitably chosen neighborhoods on opposite sides of the surface S in the uvw-space and its image in the xyz-space. In the same manner as in Section 2 we can now show that a region K in the uvw-space consisting of all points of Se for which J = 0, goes by means of equations (1) over into a region K1 in the xyz-space which contains interior points, and further that the inverse functions f, g, h defined by equations (16) are continuous at every point of K1 and of class C' except at points on the surface J1. Similar statements hold for the region M consisting of all points of Se for which J c 0 and its image M1 in the xyz-space. We may therefore state the following theorem: THEOREM VI. Let S be a surface of any extent satisfying the hypotheses prescribed at the beginning of this section. Every sufficiently small neighborhood Se of this surface will be divided into two portions K and M having points of S in common, but in which elsewhere the jacobian J of the transformation (1) has opposite signs. Each of the regions K and M will be transformed in a one-to-one way by the transformation (1) into a region of the xyz-space including the image S1 of S. In a neighborhood of any point of S1 the two xyz-regions K1 and M1 so defined are both on the same side of S1. 4. THE RELATION OF THE PRECEDING RESULTS TO THOSE OF URNER In a paper referred to above Urner has considered the transformation by (1) of all the curves (18) u = a(t), v = b(t), w = c(t) through a point P at which the functional determinant of the transformation (1) vanishes and is of rank two. His initial assumptions are consequences of our conditions (c) and (d) of Section 2. It is proposed in this section to study the relationship between the results of the preceding pages and the theorems which Urner has developed. It is shown in Urner's paper that every curve (18) through P, with the exception of those which have a certain critical direction, designated by I, is transformed into a curve tangent to a fixed plane 7r. The critical direction through P just mentioned is defined by the ratio I1: I2: I3 of the three rows of cofactors of J. The direction a: b': c through P1 in the xyz-space and corresponding to the direction a': b': c' at P of any curve whatsoever ot the form (18), always lies in the plane 71r of directions defined by the equation (7rx) Ai a' + Bi bi + Ci c' = 0. The plane 7rl is the tangent plane to the surface S1 at the point P1. For 1915] SINGULAR POINTS OF TRANSFORMATIONS 385 from equations (5), (6), (7) of Section 1 we find Ai: Bi: Ci = H1: H2: H3, where H1, H2, H3 define the direction-ratios of the normal to the surface S1 at the point P1 and are given by the equations H1 = Ju A1 +- J A2 + Jw A3, H2 = J, B1 + Jv B2 + Jw B3, H3 - Ju C1 + Jv C2 + Jw C3. This result permits of ready extension to n variables. The condition J (0, 0, 0) = 0 means that at P the three surfaces (u,V, w) =0, 4'(u, v, w)=0, (u, v, w) = 0 have their normals co-planar. The condition (d), Section 2, implies that J is of rank 2; in particular, if we assume H1 + O, then the second two surfaces have distinct normals and, by implicit function theory, a well defined curve of intersection. The direction of this curve of intersection is exactly I1:I2:I3, and the surface b = 0 is tangent to this curve of intersection at P. The assumption (d) means that the direction I: I2: 13 is not tangent to the surface defined by the equation J = 0. According to Urner, all directions not coincident with the critical direction, go into directions in a plane I7. This is the tangent plane to Si at P1 (see Fig. 3), as we have shown. According to him also all directions which are not./s I'pr:1! qb ct ~~~FS S3 FIG. 3. coincident with the critical direction I, but which are in a common plane a through I, go into the same direction tangent to S1. Let a be determined by I and the tangent to a curve C on S. The curve C goes into a curve Ci 386 W. V. LOVITT on S1, the direction of C at P being transformed into that of C1 at P1, and it follows that all directions in a go into the tangent to C1 on S1. Take now a curve D in the uvw-space with initial direction coincident with instead of distinct from the critical direction I, with equations of the form (18), and with the length of arc as the parameter t, and furthermore let the numbers I1, 12, I3 be modified by a common factor if necessary so that they are direction cosines. Then at P, a' = I,, b' = 2, c' = 13. Further, since t is the length of arc, the values a", b", c" define the principal normal perpendicular to the direction I, and this direction goes into a direction pi = Ou a" + fv b" + cOw ", ql = 41' a" + -v b" + — w c", r = Xu a" + Xv b" + x c", in the xyz-space. In Urner's equation (9) the constant k is unity under these circumstances, and these equations take the form a' = pi + U (~, I), b' = C1 + U(4, I), c'[ = ri + U (X, I), where (F, I) -= PF~u + I2 F1v + 2 Fww + 212 13 Fvw + 213 I1 Fw + 21, I2 Fuv. All of the curves D of the type just described which have the same principal normal are transformed into curves with coplanar directions, as Urner has shown. For if two curves D and D have the same principal normal, then a" = Xa", b" = b", c" = Xc". Hence for the transform D1 of D by equations (1) al'; = Xp + U( I, I), b'' = Xq + U(,, Z), C' = X- I t+ U (X, I). We get the directions of the transforms of all curves D with the same principal normal (i. e., the same osculating plane) as D by letting X vary, and we see with Urner that the transformed directions '': b:': C'' all lie in the plane fi determined by pi ql: 1r and U(4, I): U(4', I): U(x, I) = U1: U2 Us, thought of as directions through Pi. To get this plane geometrically take the plane through I and a": b" c" in the uvw-space. It cuts a curve T out of the surface S (see Fig. 3) which in turn determines a curve T1 on S1 with direction p: qi: r1. The plane /3 required is that through pi: q: ri and U1: U2: U3. As X varies from - co to + o, i. e., as the radius of first curvature of D varies from infinity through P on the line whose direction cosines are a": b": c", and out to infinity on the other side again, the direction 'al': 1': cl starts at - p -: - r and rotates in /3 about P1 to pi: qi: r1, passing through U': U2: U3 for X = 0. VITA I, WILLIAM VERNON LOVITT, was born near Whiting, Kansas, February 7, I88i. My high school work was completed at Shenandoah, Iowa, in I898. In September, I899, I entered the University of Nebraska and received therefrom in 1903 the A.B. degree. I was Principal of the school at Arcadia, Nebraska, I903-I904; teaching fellow at the University of Nebraska I904-I906; graduate student at the University of Chicago for three quarters of the year I906-1907, receiving the Ph.M. degree in June, I907; instructor in mathematics at the University of Washington, Seattle, Washington, I907-I912; graduate student at Harvard I912-1913; studied at the University of Chicago during the summers I9IO-II-I2-I4. Here I received instruction under Professors Blickfeldt, Bliss, Bolza, Dickson, Laves, Maschke, Moore, Moulton, Young, and Wilczynski, to all of whom I desire to express my thanks for their encouragement and help. I desire especially to express my gratitute to Professor Bliss for his encouragement and uniform kindness while guiding me in the preparation of this paper.