440
ON THE TETRAHEDROID.
[700
meeting in points
0,
~ß,
7>
a" ßy'. a ,
a',
0,
-y>
«"ßy'.ß'
a",
ß",
0,
«ßy.y",
, addi',
a"./3/373",
7" • «'W.
0,
the four points being in fact the vertices of the tetrahedron formed by the four planes
of the tetrahedroid. Observe that, if the singular planes of the 16-nodal surface in
their original order are
1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13, 14, 15, 16,
then the planes forming the last-mentioned four systems of planes are
(1, 8, 11, 14),
(2, 7, 12, 13),
(3, 6, 9, 16),
(4, 5, 10, 15),
viz. they correspond each of them to a term which in the determinant formed with
the 16 symbols would have the sign +.
The equation a!¡3"y = a"fiy is evidently not unique. The triads (a, /3, 7), (a, /3', 7'),
(a", /3", 7") enter symmetrically into the equation of the 16-nodal surface; by taking
the singular planes of one of the surfaces in a different order, the equation would
present itself under one or other of the different forms
a'/3"y = a!'/3y', a"/3y = a/3'7", a/3'y" = a'/3"y,
a'/3y" = a"/3'y, a"/3'y = a(3"y', a/3"y' = a{3y".
Cambridge, 9 December, 1878.