AND DOUBLE THETA-FUNCTIONS. 
983 
they are such as to give for the ratios of the 16 functions expressions depending upon 
two arbitrary parameters, x, y. Or taking the 16 functions as the coordinates of a 
point in 15-dimensional space, these coordinates are connected by a 13-fold relation 
(expressed by means of the foregoing system of 72 quadric equations), and the locus 
is thus a 13-fold, or two-dimensional, locus in 15-dimensional space. 
Hence, taking any four of the functions, these are connected by a single equation ; 
that is regarding the four functions as the coordinates of a point in ordinary space, 
the locus of the point is a surface. 
In particular the four functions may be any four functions belonging to a hexad : 
by what precedes there is then a linear relation between the squares of the four 
functions: or the locus is a quadric surface. Each hexad gives 15 such surfaces, or 
the number of quadric surfaces is (16x 15 = ) 240. 
The U>-nodal quartic surfaces. 
121. If the four functions are those contained in any two pairs out of a tetrad of 
pairs (see the foregoing “Table of the 120 pairs ”), then the locus is a quartic surface, 
which is, in fact, a Rummer’s 16-nodal quartic surface. For if for a moment x.y and 
z.w are two pairs out of a tetrad, and r.s be either of the remaining pairs of the 
tetrad ; then we have rs a linear function of xy and zw : squaring, rV is a linear 
function of x~y~, xyzw , z 2 w 2 ; but we then have r 3 and s' 2 , each of them a linear function 
of x 2 , y 2 , z 3 , wr; or substituting we have an equation of the fourth order, containing 
terms of the second order in ( x 2 , y 2 , z 2 , vr), and also a term in xyzw. It is clear that 
if instead of r.s we had taken the remaining pair of the tetrad we should have obtained 
the same quartic equation in (x, y, z, w). And moreover it appears by inspection 
that if xy and ziv are pairs in a tetrad, then xz and yw are pairs in a second tetrad, 
and xiv and yz are pairs in a third tetrad : we obtain in each case the same quartic 
equation. We have from each tetrad of pairs six sets of four functions (x, y, z, iv) : 
and the number of such sets is thus (^6.30 = )60 : these are shown in the foregoing 
“ Table of the 60 Gopel tetrads,” viz., taking as coordinates of a point the four 
functions in any tetrad of this table, the locus is a 16-nodal quartic surface. 
122. To exhibit the process I take a tetrad 4, 7, 8, 11 containing two odd functions ; 
and representing these for convenience by x, y, z, w, viz.: writing 
d 7 , d 8 , d u (u)=x, y, z, w 
we have then X, Y, Z, W linear functions of the four squares, viz., it is easy to 
obtain 
/ 3 
a(ar 
+» J )-Sfe 2 +to 2 )=2(« 2 - 
=S 3 )X, 
H 
„ )•“-&( ,, ) "“2 ( ,, 
)W, 
(i{x 2 
—z 3 ) 4 - y (r - w ~)=2 (/ 3 - - 
y)Y, 
y( 
>> ) J rfi( ,, ) —2 ( „ 
)Z. 
