147 
1906-7.] Dr Muir on Axisymmetric Determinants. 
he first uses the multipliers 
1, xyz, x 2 y 2 z 2 , x 2 z, y 2 x, z 2 y , x 2 y , y 2 z , z 2 x, 
the eliminands then being 
x , y , z, y 2 z 2 , x 2 yz , y 2 zx , z 2 xy , z 2 x 2 , x 2 y 2 ; 
next he uses the said eliminands as multipliers, the new eliminands being 
x 2 , y 2 , z 2 , yz , za? , xy , xy 2 z 2 , 2 /z 2 ^ 2 , za? 2 ?/ 2 ; 
and finally using the new eliminands as multipliers, he eliminates 
1 , xyz , x 2 y 2 z 2 , a; 2 £ , y 2 x , 2 2 ?/ , x 2 y , y 2 z , z 2 x , 
that is to say, the first set of multipliers. Only in the case of the second 
elimination is the determinant axisymmetric, namely, 
. c b ... 1 . . 
c . a .... I . 
b a 1 
. . . a ... 1 1 
. . . . b . 1.1 
c 1 1 . 
1 . ..11... 
.1.1.1... 
..111.... 
which must thus be equal to (a + b-\-cf — 27abc. In the two other cases 
the determinants are not essentially distinct,* the rows of the one being 
columns of the other ; and this is said to be true of two of the three forms 
whatever be the number of variables. 
Sylvester (1853, March). 
[On the relation between the volume of a tetrahedron and the product 
of the sixteen algebraical values of its superficies. Cambridge 
and Bub. Math. Journ., viii. pp. 171-178: or Nouv. Annates de 
Math., xiii. pp. 203-209 : or Collected Math. Papers, i. pp. 
404-410.] 
Denoting the vertices of a tetrahedron by a , b , c , d , its volume in terms 
of the edges by V , and the areas of its faces by \ J F > i JG , i JH , i \/K > 
we know that 
* Cayley fails to notice, however, that each of those is readily transformable into the 
second. Thus, taking his first form, we have only to multiply the 4th , 8th , 9th columns of 
it by a , and divide the 2nd , 4th , 7th rows by a , when we obtain a determinant which by mere 
/1 2345 6 7 89\ j ^ i /1 23 4567 8 9\ 
permutation of the rows ( 74 ^ 935352 ) an< ^ su hsequently 01 the columns ( 105493732 ) 
becomes identical with the axisymmetric form. 
