and called the quadrantal solution. 4. For the linear element (II) the expressions for the Christoffel symbols which are different from zero are: where B22 =L, B23 = M − x2 L, B33 = L x22 − 2 M x2 + N, We consider the system of equations (3) as in § 3, taking up separately the five cases where V is independent of x2 and x3; of x2; of x1; of x3; involves all three variables. Only the first two cases lead to solutions. For the first case we have the linear element of S and V = √a a -1/2 — 1 ' where a is a constant. This is the longitudinal solution obtained by Levi-Civita." For V independent of x2 two cases arise, according as ẞ can be made equal to zero by a transformation of coördinates, or not. When ẞ = 0, the functions a and y must satisfy the three equations When k γ dy2+ a dx2- (ax2 + r) dx3 (14) 1, we solve the second of (12) for a'/a and substitute in the third of (12). The first integral of the resulting equation is ± where a is an arbitrary constant and b = (1 − k) √1 + k + k2. B'a - Ba' = kô1/4, ¿12 + 28 (x'y' – B′2) + 24 a 8 = 0, where k denotes an arbitrary constant. (18) If we introduce dependent and independent variables t and @ by To the evident solutions of this equation, t = 5/2, t = - 5/3, correspond the solutions: excluded since we assume that > 0 in accordance with the theory. If we put t dt/de = y, and take y for dependent variable, the above equation may be replaced by dy y+y (7+13)+(2+5) (3+5)(1 + 3) = 0. dt When a solution of this equation is known, the corresponding functions a, ẞ, y can be found by quadratures. 5. Making use of the formulas of Bianchi we find the following expressions for the principal curvatures of the spaces (7), (8), (11), (14) 2a Κι a (7*) (8*) (11*) K2 = 2 = 2 a3/2; a 2 a3/2; K1 = − (1 + √3), K2 = 73, K3 = −3 (1−√3). (14*) a 23 The principal curvatures for k± 1 in (12) can be obtained explicitly, but their forms are quite involved. At each point of space the principal curvatures correspond to three directions, mutually perpendicular to one another. When the curves tangent to these directions are the curves of intersection of a triply-orthogonal system of surfaces, the space is called normal by Bianchi. All the spaces referred to above are normal. For the cases (7) and (8) the tangents to the curves of intersection x; = const., x; = const. are the principal directions. 1 Mem. Soc. Ital., 1896, p. 347. 2 Levi-Civita, Rend Lincei (ser. 5), 26, 1917, sem. 1 (460). 'Bianchi, Lezioni, 1, 377; Cotton, Ann. Fac. Toul. (ser. 2), 1, 1899 (410). 4 Science, 54, 1921 (305). 5 Rend. Lincei (ser. 5), 27, 1918, sem. 2 (350). • Lezioni, 1, 354. GEOMETRIC ASPECTS OF THE ABELIAN MODULAR FUNC TIONS OF GENUS FOUR (II) BY ARTHUR B. COBLE DEPARTMENT of MathematiCS, UNIVERSITY OF ILLINOIS Communicated by E. H. Moore, June 21, 1921 8. The form().-This form, written symbolically as (pz) (rx) (sy), where z is a point in S3, x a point in S2, and y a point in S2', has 36 coefficients and therefore 35-15-8-8 = 4 absolute projective constants. Points x, y determine a plane which becomes indeterminate for six pairs x, y = Pi qi; (i = 1,...., 6) which form associated six points. They are the double singular points of a Cremona transformation T of the fifth order between the planes S., Sy. A given plane u is determined by ∞1 pairs x, y which lie, respectively, on the cubic curves, (pp'p''u) (rx) (r'x) (r''x) (ss's'') = 0, (pp'p''u) (rr'r'') (sy) (s'y) (s''y) = 0 These curves pass, respectively, through the six points p; and the six points q;. Thus the given form is associated with a general cubic surface, (pz) (p'z) (p''z) (rr'r'') (ss's'') = 0, with an isolated double-six of lines and separated line-sixes. The mapping of the surface from the planes S, and S, given by the above systems of cubics. is 9. The form (1).-In the space figure just described we insert a quadric Q with generator, t, T. The point coördinates z can be replaced by bilinear expressions in t, and the form (1) becomes a form (pr) (πt) (rx) (sy) general of its type, with thirteen absolute constants corresponding to the four absolute constants of the above cubic surface and the nine additional constants for the inserted quadric. The quadric meets the cubic surface in a normal curve of genus 4 so that we have in space the figure of the normal sextic and a particular one of the ∞ cubic surfaces through it. In the planes St, Sy we have projectively general (with thirteen absolute constants) sextics of genus 4 with nodes at p; and q;, respectively, transforms of each other under T. The canonical adjoints of these sextics, with isolated gi"'s, are obtained by substituting for u in 8 the proper bilinear expressions in t, 7; while a similar substitution for z in the equation of the cubic surface gives the equation on the quadric of the normal curve. We may therefore regard the general form (1) as a definition of the projectively general plane sextic of genus 4. 10. The forms F, F on the conic K(7). Counter sextics.-We mark the conic K(7) on the plane S, and plot with reference to it the two counter sextics S2(t) and S2(t). For each there is a cusp locus of perspective cubics ĞC(7) and GC(7). Similarly we mark on a plane S, the conic K(t) and plot with reference to it the two counter sectics Si(7), S1(7) which are paired with the above counter sextics, and which have for cusp loci of perspective cubics the sextic curves GC(t), GC(t), respectively. We now consider the form F = (at)3 (ar) = 0 where r is a tangent of K(T). For variable t we have ∞ 1 triangles circumscribed about K(7) whose vertices run over a sextic curve. If t determines 71, 72, 73 then the point t, 71, 72, of this sextic curve is birationally related to the solution t, T3 of F = 0. Hence the sextic curve is birationally equivalent to GC(7) and as a result of the algebraic discussion of the next two sections we prove that this sextic curve is actually GC(7). This amounts to the effective elimination of t from (ar1)3 (at)3 = 0, (αT2)3 (at)3 = 0 and the separation of the factor (7172)3. Hence we have on the sextic GC(7) two gi3's such that the ∞ 1 t-triads are cusp triangles of perspective cubics of S2(t) and the ∞ T-triads are triangles circumscribed to K(7) for which the intersection of the lines joining each vertex to the contact of the opposite side is the point t of S2(t). Of course a similar statement is true of any one of the four cusp loci. If we polarize the form F into (AT) (AT1) (AT2) (at) (atı) (at2) and replace the pairs 71, 72; t1, t2 by points x; y referred in Darboux coördinates to the conics K(7); K(t) in Sx; Sy, respectively, we have a form (πx) (dт) (ôt) (py) of the type discussed in 9. Hence the sextics GC(7) and GC(t) are trans |