Fill and Sign the Informed Consent Psychology

1898.] NOTE ON THE TETRAHEDROID. 327 Ga is of the same form. Hence it remains only to consider the number of those which have no operator besides identity in common with Ga. All these subgroups may be divided into two classes, viz : (1) those which are transformed into themselves by Ga, and (2) those which are transformed into different groups by the operators of Ga> The number of the former class may evidently be written in the form ap + bq, a and b being positive integers. If a group of the latter class occurs, all its operators must be commutative to every operator of Ga * and hence r > p ( g — 1). I n this case the given theorem is evidently true. I t may be observed that the number of sel f-con jugate subgroups of G is not necessarily of the given form, e. #., the direct product of two non-commutative groups of order 21 contains only two self-conjugate subgroups of this order. CORNELL UNIVERSITY, February, 1898. NOTE ON T H E TETEAHEDEOID. BY DR. J . I. HUTCHINSON. (Read before the American Mathematical c ociety at the Meeting of February 26, 1898. ) I N a brief paper, " A special form of a quartic surface," Annals of Mathematics, vol. 11, p. 158, I have called attention to an interesting special form of the locus of the vertex of a cone passing through six points. I wish to point out in this note the connection between this special surface and the tetrahedroid. Given six arbitrary points in space 1, 2, 3, 4, 5,6. These determine a system of GO 3 quadric surfaces each of which pass through the six points. Denote this configuration by ]£• Choose any arbitrary point P and consider the polar planes of P with respect to the system of quadrics. There are determined in this way oo3 planes forming a configuration £ r To a quadric in V corresponds a plane in 2 r The vertices of the cones of 2 have for locus a surface iTof the fourth order. The planes of Xi corresponding to the cones of 2 envelope a Kummer surface. The point in each plane corresponding to the cone vertex is the point of tangency. * Dyck, Mathematische Annalen, vol. 22, p . 97. 328 NOTE ON THE TETRAHEDBOID. [April, To the twisted cubic h determined by the six basis points corresponds a single point O in Xi > which point is a node of the Kunimer surface. To the fifteen lines 12, 13, •••, correspond 15 points (12), (13), •••. These are the remaining nodes of the Kummer surface.* Suppose now that the six points 1, 2, •••, 6 form an involution on the cubic k, and that the lines 12, 34, 56, join the points that are paired in the involution. The quadric Q determined by these three lines will then contain the cubic h, and hence the corresponding plane in Xi will pass through the four nodes 0, (12), (34), (56) of the Kummer surface. The Kummer surface accordingly becomes in this case a tetrahedroid.f A quadric of 2 , since it contains six given points, may be required to pass through the three lines 13, 14, 23, being thereby completely determined. I n the case of involution this quadric also contains the line 24. The corresponding plane in Xi contains the four nodes (13), (14), (23), (24). Similarly, the lines 35, 45, 36, 46, lie on a quadric of X, to which corresponds a plane containing the nodes (35), (45), (36), (46). Finally, a quadric containing 15, 16, 25, 26, corresponds to a plane containing (15), (16), (25), (26). An interesting question is suggested in this connection. I t is well known that the Kummer surface is determined by six arbitrary points chosen for nodes. What relation exists among these nodes when the surface becomes a tetrahedroid ? The answer is, The six nodes form an involution on the twisted cubic determined by them. I doubt whether this result can be proved in a simple manner from the correspondence established between X and Xv I will limit myself here to showing analytically the existence of this geometric property for the Fresnel wave surface (the projective equivalent of the tetrahedroid). Taking the equation of the wave surface in the usual form and introducing a fourth variable w to make it homogeneous, consider, for example, the six nodes (i6/î, iaa, 0, ± ^ ) , (cy, 0, dz aoc, t/5), («, =fc /5, y, 0), * A detailed study of the correspondences between 2 and 2j is given by Reve ; " Ueber Strahlensysteme zweiter Classe u n d die Kummer'sche Flâchë," etc. (Crelle, vol. 86, pp. 84.) fCf. Cay ley : " S u r un cas particulier de la surface du quatrième ordre aTTec 16 points singuliers.'' (Crellc, vol. 65, p. 284.) Reye remarks (1. o , pg. 106) that the wave surface is apparently a special case (ein ziemlich speci elles Beispiel) of the 2, 2X configurations, bui> carries his remark no further. 1898.] NOTE ON INTEGRATING FACTORS. 329 where Transform to a new system of coordinates xv x2, #8, xv by means of the equations x i == ( c — &) [aöCA& — ( ^ 2 — cY2)y + iacayw"] , ic2 = (6 + c) [aa/2# — (6/52 + c^2)?/ + iacayw], #3 = (a + c) [aa# — 6/5^/ + cyz], #4 = (c — a) [aax — bfty — c/z]. With respect to this new system, the coordinates of the six chosen nodes are ( 1 , 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (ev e„ ez, e4), (e2, ev ev e8), where 61== p (62 - c2) (c - a), e3 = (a2 - c2) (c + 6), e2 = 13 (62 - c2) (c + a), e4 = (a2 - c2) (c - b). These six points form an involution of the kind described. (See Annals of Mathematics, vol. 11, p. 159.) NOTE ON INTEGRATING FACTORS. BY MR. PAUL SATJREL. (Read before the American Mathematical Society at the Meeting of February 26, 1898.) I F the differential equation Xxdxx + X2dx2 + - + Xndxn = 0, n ^ 3, (A) be integrable, and if u = constant be the integral of this equation, then, as is well known, there exists a function M such that du = MX1dx1 + MX2dx2 + ••• + MXndxn. And as du du du ,..« lir^r lirv