334 
MATHEMATICS: A. B. COBLE Proc. N. A. S. 
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-or- 
thogonal 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 %i — const., Xj = const, are the 
principal directions. 
For the cases of § 4 we put x 2 = e X2 3 . Then for (11), the curves of 
intersection of surfaces X\ = const., xi = const., x% = const, have the 
principal directions. For (14) and the case k + 1, the principal di- 
rections are given by x\ = const., x% = const., and the orthogonal systems 
of curves on X2 = const, defined by 
(■' - T ) dX > + 2 [" + 4 1 a ' 7 ' - " (T - Ta )] *• *> 
1 Mem. Soc. Ital., 1896, p. 347. 
2 Levi-Civita, Rend Lincei (ser. 5), 26, 1917, sem. 1 (460). 
3 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). 
6 Lezioni, 1, 354. 
GEOMETRIC ASPECTS OF THE ABELIAN MODULAR FUNC- 
TIONS OF GENUS FOUR (II) 
By Arthur B. Cobi^ 
Department oe Mathematics, University of Illinois 
Communicated by E. H. Moore, June 21, 1921 
8. The form (f \ ?). — This form, written symbolically as (pz) (rx)(sy), 
where z is a point in 5 3 , % a point in S 2 , and y a point in 5 2 r , has 36 co- 
efficients and therefore 35— 15— 8— 8 = 4 absolute projective constants. 
Points x, y determine a plane which becomes indeterminate for six pairs 
x > J = Pi °i\ (i = X,. . . 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 x , S y . A given plane u is determined by oo 1 
pairs x, y which lie, respectively, on the cubic curves, (pp r p"u) (rx) (r f 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 pi and the six points g t -. Thus 
the given form is associated with a general cubic surface, (pz) (p'z) (p"z) 
(rr f r") (ss's") = 0, with an isolated double-six of lines and separated 
