96 
where 
ON THE SIXTEEN-NODAL QUARTIC SURFACE. 
[817 
a + /3 +7=0, 
a + /3' +y = 0, 
a." + (3" + 7" = 0, 
r, X y 
P = - + | + 
a /3 
7->/ x V 
a' + /3' + 
X = a{y'y"y -/3'/3"z), 
Y — ¡3 (a a" z — y y" x), 
Z = 7 (/3'/3"tf - a"2/)> 
X^a (7" 7 y-&"(3z), 
Y' = /3' (a" a z — y'yx), 
P" = + jp + T - ' ’ ^ = ? “ a " a 2/)> 
X"=a" (77'?/ -/3/3's), 
F" = /3" (aa^ — 77' ¿c ), 
Z" = y" {¡3[3'x — aa y ), 
and where the equations of the sixteen singular tangent planes are 
x — 0, y — 0, z = 0, w = 0, 
X —w — 0, Y —w — 0, Z — w = 0, P — 0, 
X' - w = 0, Y - w = 0, 0-10=0, P' = 0, 
X" — w = 0, Y" — io = 0, ZT-w = 0, P" = 0 ; 
see Crelles Journal, vol. lxxiii. (1871), pp. 292, 293, [442], and also Proc. Lond. 
Math. Soc., vol. ill. (1871), p. 251*. [454]. 
To identify the two forms, using x', y', z', £', y, for the new form, I assume 
*5 y', z\ %, y, f' = lx, my, nz, p(X -w), q(Y- w), r(Z - w), 
where Ip = mq = nr = 1; and so convert the equation 
Vx(X — w) + Yy(Y—w) + V'z (Z — w) = 0 
into 
\Zx'f + Yyy + fzf = 0. 
The constants (l, m, n, p, q, r) and (a, h, c, f g, li), where af=bg = 6h = \, are then 
to be determined so that we may have identically 
x'+ y' + /+ £' + y' + £' = 0, 
ax' + by' + cz' + + gy + h£' = 0, 
and we thus obtain 8 new equations to be satisfied by the 12 constants, viz. these are 
l + r. 7/3 / /3" — q . f3y'y" — 0, 
m +p.cLy'y" — r.ya'a" =0, 
n +q. (3a'a" — p. afi'fi" = 0, 
p + q + r =0, 
al + hr. 7/3'(3" — gq. /3y'y" = 0, 
bm + fp. ay'y" — hr. ya'a” = 0, 
cn +gq . (3a a” —fp. a(3'(3" = 0, 
fp +gq + hr = 0. 
[* This Collection, vol. vii., p. 282.]