4 MATHEMATICS: H. S. WHITE 
in which n = v-i- kxiX2X3, while the quotients 
dxi^ ' dx2C>X3 ' ' dxibx2 
have integral coejSicients. The points of inflexion of ;^ = 0 are its 
intersections with C = 0. Although C is not a covariant, it forms with n 
a covariant pencil, since C is transformed into a linear function of n 
and C. 
iHurwitz, Arch. Math., Leipzig, ser. 3, 5, 25, (1903). 
2 Dickson, Madison Colloquium Lectures, American Mathematical Society, (1914). 
3 Dickson, Trans. Amer. Math. Soc, 15, 497, (1914). 
^An advance in the theory of seminvariant leaders of covariants of quadratic forms 
has been made recently by the writer. Bull. Amer. Math. Soc, January, 1915. 
^Dickson, Trans. Amer. Math. Soc, April, 1915. 
6MS. offered Aug. 4, 1914 to Amer. J. Math. To appear April, 1915. 
THE SYNTHESIS OF TRIAD SYSTEMS At in t ELEMENTS. 
IN PARTICULAR FOR t = 31 
By Henry S. White 
DEPARTMENT OF MATHEMATICS. VASSAR COLLEGE 
Presented to the Academy, December 3, 1914 
Purely theoretical interest first led to the study of triad systems 
in t elements; systems of threes or triads, that is, in which every possible 
pair of elements is found in some one triad, but only one. Their relation 
to other objects of research in algebra and geometry began to appear 
when Noether (1879) pointed out the peculiar nature of a resolvent 
equation of the seventh degree which had been found by Betti, Hermite, 
and Kronecker in discussing the transformation of the seventh order of 
elliptic functions. This modular equation has roots related in triads 
like the A/. Hesse had shown earlier, in plane curves of the third order, 
that the nine inflexion points lie by threes on twelve lines, thus exempli- 
fying a A 9. Noether succeeded also in connecting the A 7 with the im- 
portant sets of double tangents to the plane quartic curve which Aron- 
hold had introduced under the name of Siehenersysteme. With those 
important appHcations in hand, and ten or twelve known Ais's to serve 
as further data, mathematicians took up with renewed interest the 
question whether there are actual triad systems for every suitable 
member t of elements, i.e., for / = 13, 15, 19, 21, 25, 27, etc.; or precisely, 
^ = 6k -f- 1 or 6k H- 3. 
