699 
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700. 
ON THE TETRAHEDROID AS A PARTICULAR CASE OF THE 
16-NODAL QUARTIC SURFACE. 
[From the Journal für die reine und angewandte Mathematik (Crelle), t. lxxxvii. (1878), 
pp. 161—164.] 
In the paper “ Sur un cas particulier de la surface du quatrième ordre avec seize 
points singuliers,” Crelle, t. lxv. (1866), pp. 284—290, [356], I showed how the surface 
called the Tetrahedroid could be identified as a special form of Kummer’s 16-nodal 
quartic surface ; but I was not then in possession of the simplified form of the 
equation of the 16-nodal surface given in my paper ‘‘Note sur la surface du quatrième 
ordre douée de seize points singuliers et de seize plans singuliers,” Crelle, t. lxxiii. 
(1871), pp. 292, 293, [442]; see also my paper, “A third memoir on Quartic surfaces,” 
Proc. Lond. Math. Soc. t. ill. (1871), p. 250, [454, this Collection, t. VII., p. 281]. Using 
the equation last referred to, I resume therefore the consideration of the question. 
Taking the constants a, /3, 7, a.', /3', 7', a", /3", 7", such that 
a + /3 + 7 = 0, a. 4- ft + f — 0, a.” 4- /3" -1- <y" = 0, 
and writing also 
M=aa"(/3 —7 ) + /3' /3" (7 -a) + 7 V(« - /3 ) 
= a a (/3' - 7') + /3"/3 (7' - ) + 77 («' - /3') 
= a « 03" - 7") + /3 /3' (7" ~ «") + 77 (*" - /3") 
= - * {(£■-7)(/3' - 7)08"- 7") + (7“«)(7'- «')(7"~a") + (a -/8)(«'- /3') (a” -¡3'% 
(the equivalence of which different expressions for M is verified without difficulty): 
writing also X, Y, Z, W as current coordinates, the equation of the 16-nodal surface is 
( W*(X*+Y 2 + Z*-2YZ-2ZX-2XY) 
0 = J + 2 W [eux'a” (Y 2 Z- YZ 2 ) + ßß'ß" (Z 2 X -ZX 2 ) + 777" (X 2 Y-XY 2 ) + MXYZ\ 
(+ (aa'a "YZ+ ßß'ß" ZX + 77 yXY) 2 ,