248 
MATHEMATICS: L. E. DICKSON 
functions in the solution of the problem of 0^,2 as has been done in the 
case of the sextic. 
The developments sketched above suggest a number of inquiries 
which promise results of interest in various directions. Also some of 
the problems considered such as that of the invariants of Pn are worthy 
of closer study than has been given them in the account here reported. 
^ This investigation has been pursued under the auspices of the Carnegie Institution of 
Washington, D. C. 
2 For references see Study, Math. Ann., Leipzig, 60, 348 (footnote). 
2 E. H. Moore, Amer. /. Math., 22, 279 (1900). 
* Trans. Amer. Math. Soc, 9, 396 (1908), and 12, 311 (1911). The relation of these 
papers to the earlier work of Klein and others is there set forth at length. 
^ S. Kantor has used this device for an in his crowned memoir: Premiers Fondements 
pour une Theorie des Transformations Periodiques Univoques, Naples (De Rubertis), 1891. 
^ This will appear as a joint paper by Mr. C. P. Sousley and A. B. Coble. 
^ Letter to Hermite, /. Math., Paris, Ser. 4, 4, 169. See also Math. Ann., Leipzig, Witt- 
ing, 29, 167; Maschke, 33, 317, 36, 190; and Burkhardt, 38, 161. 
THE STRAIGHT LINES ON MODULAR CUBIC SURFACES 
By L. E. Dickson 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF CHICAGO 
Presented to the Academy, January 21, 1915 
L In ordinary space a cubic surface without singular points contains 
exactly 27 straight lines, of which 27, 15, 7, or 3 are real; there are 45 
sets of three coplanar lines, the three of no set being concurrent. In 
modular space, in which the coordinates of points and the coefficients 
of the equations of lines or surfaces are integers or Galois imaginaries 
taken modulo 2, it is interesting to notice that three coplanar lines on a 
cubic surface may he concurrent (§2). A point with integral coordinates 
is called real. A Kne or surface is called real if the coefficients of its 
equations are integers. In space with modulus 2, the number of real 
straight Hues on a cubic surface without singular points is 15, 9, 5, 3, 
2, 1, or 0. 
We shall give here an elementary, seK-contained, investigation of 
some of the most interesting cubic surfaces modulo 2. A complete 
classification of all such surfaces under real linear transformation will 
appear in the Annals of Mathematics, but without the present investiga- 
tion of the configuration of their lines. 
2. Every real point of space modulo 2 is on the surface'^ 
ocyipc -\- y) = zw(z -f w). (1) 
since each member is an even integer when x, y, z, w are integers. 
