148 Proceedings of the Royal Society of Edinburgh. [Sess. 
144V 2 = (bc) 2 (da) 2 {(ca) 2 + (< db) 2 + (ab) 2 + (cd) 2 - (be) 2 - (da)*} 
+ (ca) 2 (db) 2 { (ab) 2 + ( cd ) 2 4- (be) 2 + (da) 2 - ( ca ) 2 - (db) 2 j- 
+ (ab) 2 (cd) 2 {(bc) 2 + (da) 2 + (ca) 2 + (db) 2 — (ab) 2 - (cd) 2 j 
- (bc) 2 (ca) 2 (ab) 2 - (bc) 2 (db) 2 (cd) 2 - (ca) 2 (cdf(da) 2 - (abf(da) 2 (db) 2 
— W say, 
and 
F = - (bef - (cdy - (db)* + 2 (cd) 2 (db) 2 + 2 (db) 2 (bcf + 2 (bcf(cd) 2 
Gr = - (ac) 4 - (cdy - (day + 2 (cd) 2 (da) 2 + 2 {da) 2 (ac) 2 + 2 (ac) 2 (cd) 2 
H = - (aby - (bdy - (day + 2 (ab) 2 (bd) 2 + 2 (bd) 2 (da) 2 + 2 (da) 2 (abf 
K = - (aby - (bey - (cay + 2 (bc) 2 (ca) 2 + 2 (ca) 2 (ab) 2 + 2 (ab) 2 (bc) 2 . 
With this notation Sylvester points out that the condition for the vanishing 
of the surface of the tetrahedron is 
JF + VG+n/H+ = 0, 
and that this when freed of root-signs is 
or say 
2 F4 - 42,F 2 G + 6 2,F 2 G 2 + 42f 3 GH - 4°FG H K . = 0, 
N = 0 , 
where N consequently is of the eighth degree in the squared edges. His 
reasoning then is that as the vanishing of the surface and the vanishing 
of the volume are necessarily coincident, it follows that W, having no rational 
factors, must itself be a factor of N ; and that, W being of the third degree 
in the squared edges, the quotient N /W must be of the fifth degree. Rely- 
ing on this he proceeds to determine the quotient by considering the cases 
where (1) ab = 0 = cd , (2) ab == 0 = ac , (3) ab = ac = ad — be = bd = cd = 1 , his 
result being 
^(abf(bc) 2 (ca) 2 
(d a y+(dby+(d c y 
{(da) 2 + (db) 2 + (dc) 2 }{(ab) 2 + (be) 2 + (ca) 2 } 
+ (ab) 2 (bc) 2 + (bc) 2 (ca) 2 + (aa) 2 (ab) 2 
+ 2 ^(ab) 2 (bc) 2 (cd) 2 (da) 2 (ac) 2 , 
where there are four expressions under the first 2 and six under the second ; 
or 
y' i (ab) 2 (bc) 2 (ca ) 2 
(day + (dby + (dcy 
- {(da) 2 + (db) 2 + (dc) 2 } {(ab) 2 + (be) 2 + (ca) 2 } 
+ (da) 2 (db) 2 + (db) 2 (dc) 2 + (dc) 2 (da) 2 
+ (ab) 2 (be) 2 + ( bc) 2 (ca ) 2 + (ca) 2 (ab) 2 
where there are four expressions under the 2 , one corresponding to each 
face. 
Although there is no explicit mention here of determinants, it being 
