2 
ON A FORM OF QUARTIC SURFACE WITH TWELVE NODES. 
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An easy way of satisfying the identity AF + BG + GH = 0 is to assume 
A, B, G, F, G, H = ayz, bzx, cxy, fxw, gyw, hzw, 
where the constants a, b, c, f, g, h satisfy the condition af+bg + ch = 0; this being so, 
the functions A, B, G, F, G, H, and consequently the functions A + F, B + G, C + H 
and A — F, B — G, C — H each of them vanish for the four points {y = 0, z — 0, w = 0), 
(z = 0, x = 0, w = 0), (x = 0, y = 0, w = 0), (x = 0, y= 0, z = 0), or say the points (1, 0, 0, 0), 
(0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1). It hence appears that the quartic surface 
il = a?y‘ i z’ 1 + b 2 z 2 x 2 + c 2 x 2 y- + f^x^w 1 + g 2 y 2 w 2 + h 2 z 2 w 2 = 0 
is a quartic surface with twelve nodes; viz. it has as nodes the last-mentioned four 
points, the remaining four points of intersection of the surfaces 
ayz + fxw = 0, bzx + gyw = 0, cxy + hzw = 0, 
and the remaining four points of intersection of the surfaces 
ayz — fxw = 0, bzx — gyiv = 0, cxy — hzw = 0. 
The above is the analytical theory of one of the two forms of quartic surface 
with twelve nodes recently established by Dr K. Rohn in a paper in the Berichte 
ü. d. Verhandlungen der K. Sächsische Gesellschaft zu Leipzig, (1884), pp. 52—60.