130 
ON THE SURFACES THE LOCI OF 
[503 
where I recall that for (X,...), (X(X", ...), (L'",...) the suffixes are (a, /3), (7, 8), 
(a, 7), and (8, /3) respectively. The values of the first two determinants thus are 
p 3 aa/3.78 and p 3 aa7.8/3 respectively: that of the third is p 3 aa/3. ay • viz. this is 
=p 2 a(3y .paa\ similarly, that of the fourth is p 3 ay8.8/3, which is = — p 3 a8y. 8/S = + p 3 ci8/3.8y ; 
or finally this is = p 2 8/3y.pa8. And we have thus the before-mentioned equation of the 
surface. 
47. Singularities. The equation of the surface shows that 
(0) The point a is a 2-conical point: in fact, we have for this point p 3 aa/3.78 = 0, 
p 3 aa7.8/3 = 0, paa = 0, pa8 = 0. 
(2) The line a is a 4-tuple line: in fact, for any point on this line p 2 a/3e = 0, 
p 3 aa/3.78 = 0, p 2 aye = 0, p 3 aa7.8/3 = 0, p 2 a/3y = 0, p 2 aa = 0. 
(4) The line (a, /3, 7, e) is a 2-tuple line: in fact, for any point on the line 
we have p 2 af3e = 0, p 2 aye = 0. 
(10) The excuboquartic a(3e.y8.a is a simple curve: in fact, for any point of 
this curve we have p 2 a/3e = 0, p 3 aaf3.78 = 0, these two surfaces inter 
secting in the lines a, (3 and the curve. It is, moreover, obvious that 
the surface is touched along the curve by the hyperboloid p 2 a/3e. 
I notice that the surface meets the quadric p 2 a/3y in 
lines (a, /3, 7) each 4 times, 12 
„ (a, /3, 7, 8) „ twice, 4 
>> (®> y» *0 » }> ^ 
curve aa/3y. 8e „ „ 8 
14 x 2 = 28 
Surface a/3y8e£. 
48. The equation of the surface may be written 
p 2 a/3e .p 2 y8e .p 2 ay£. p 2 8/3%—p 2 a/3%.p 2 y8%. p 2 aye.p 2 8/3e = 0, 
where p 2 a/3e = 0 is the equation of the quadric through the lines a, /3, e; viz. p 2 a/3e 
has the value already mentioned. 
The form is one of 45 like forms depending on the partitionment 
of the six letters. 
{Surface afiyoef.}