180 
[664 
664. 
ON THE 16-NODAL QUARTIC SURFACE. 
[From the Journal fur die reine und angewandte Mathematih (Crelle), t. lxxxiv. (1877), 
pp. 238—241.] 
Prof. Borchardt in the Memoir “ Ueber die Darstellung u. s. w.” Crelle, 
t. lxxxiii. (1877), pp. 234—243, shows that the coordinates x, y, z, w may be taken as 
proportional to four of the double ^-functions, and that the equation of the surface 
is then Gbpel’s relation of the fourth order between these four functions: and he 
remarks at the end of the memoir that it thus appears that the coordinates x, y, z, w 
of a point on the surface can be expressed as proportional to algebraic functions, 
involving square roots only, of two arbitrary parameters £, £■'. 
It is interesting to develope the theory from this point of view. Writing, as in 
my paper, “ Further investigations on the double ^-functions,” pp. 220—233, [663], 
[a] = aa', 
[b] = bb\ 
[c] = cc', 
[d] = dd', 
[e\ = ee', 
[/]=//', 
[a b] = (gzjy abfcd'e' — V a'bf'cde) 2 , 
etc., 
where on the right-hand sides a, &,..., a', ... denote a — £, b — f, ..., a — £', ... (f, % 
being here written in place of the x, x' of my paper), then the sixteen functions