144 Proceedings of the Koyal Society of Edinburgh. [Sess. 
so that, if x r , y r , %r. and £ r , rj r , g r be rectangular co-ordinates of points in 
space, the result reached gives an expression for thirty-six times the 
product of the volumes of two tetrahedrons in terms of the distances of 
the angular points of the one from the angular points of the other. By 
proceeding to the case where the two tetrahedrons are coincident, and 
thence to the case where the four remaining points are situated in the 
same plane, we reach Cayley’s relation connecting the mutual distances of 
four such points. 
It is thus seen that whereas Cayley’s vanishing axisymmetric deter- 
minant was originally got as a multiple of a peculiarly obtained square of 
the determinant 
X 1 
Vi 
0 
1 
25 
X 2 
y<i 
0 
1 
2*8* 
y s 
0 
1 
^4 
2/4 
0 
1 
1 
0 
0 
0 
0 
Sylvester arrives at it by squaring 
*1 
Vi 
0 
1 
x 2 
y 2 
0 
1 
y * 
0 
1 
2/4 
0 
1 
in a special fashion, and then performing certain transformations on the 
result. 
Sylvester (1852, Nov.). 
[Sur une propriety nouvelle de l’equation qui sert a determiner les 
inegalites seculaires des planetes. Nouv. Annales de Math., xi. 
pp. 434-440 : or Collected Math. Papers, i. pp. 364-366.] 
This paper of composite authorship probably originated in a letter from 
Sylvester giving his theorem and demonstration, with a remark or two 
additional. To these, which were made §§7, T, 8, the editor prefixed an 
introduction (§§ 1-6) on determinants and determinant-multiplication.* 
The theorem is an extension of one which is the basis of his paper 
in the Philosophical Magazine of the same year, and may be shortly 
enunciated as follows: If | (11) (22) . . . (nn) | be axisymmetric and have 
[ [11] [22] . . . [nn] \ for its p th power, then the roots of the equation 
* In the Coll. Math. Papers §§ 1-6 are omitted, and §§ 7, T, 8 are numbered §§ 6, 7, 8. 
The theorem of the original § 6 is incorrect. 
