888] 
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888. 
ON A FOEM OF QUAETIC SUEFACE WITH TWELVE NODES. 
[From the British Association Report, (1886), pp. 540, 541.] 
Using throughout capital letters to denote homogeneous quadric functions of the 
coordinates (x, y, z, w), we have as a form of quartic surface with eight nodes 
' O = (*]£ JJ, V, W) 2 = 0; viz. the nodes are here the octad of points, or eight points 
of intersection of the quadric surfaces U= 0, V = 0, Ti r =0; the equation can, by 
a linear transformation on the functions U, V, W (that is, by substituting for the 
original functions U, V, W linear functions of these variables), be reduced to the form 
n = U 2 + V 2 + W 2 = 0. 
Suppose now that the function il can in a second manner be expressed in the 
like form O = P 2 + Q 2 + R 2 (where P, Q, R are not linear functions of U, V, W); 
that is, suppose that we have identically U 2 + V 2 + W 2 = P 2 + Q 2 + R 2 , this gives 
U 2 — P 2 + V 2 — Q 2 + W 2 — R 2 = 0; or, writing U+P, V + Q, W+R = A, B, C, and U—P, 
V — Q, W — R = F, G, H, the identity becomes AF + BG + CH = 0 ; and this identity 
being satisfied, the equation Q = 0 of the quartic surface may be written in the two 
forms 
il = (A + Ff + (B + G) 2 + (C + H) 2 = 0, 
and 
n = (A - Ff + (B - G) 2 + (C - Hf = 0; 
viz. the quartic surface has the nodes which are the intersections of the three quadric 
surfaces A+F= 0, B + G= 0, C+H—0, and also the nodes which are the intersections 
of the three quadric surfaces A—F= 0, B—G — 0, C—H= 0. We may of course also 
write the equation of the surface in the form 
Vi 
C. XIII. 
n = A 2 + B 2 + C 2 + F 2 + G 2 + H 2 = 0. 
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