1912-13.] Dr Muir on Axisymmetric Determinants. 
53 
Walker, J. J. (1865, 1868). 
[Properties of the discriminant of the quaternary quadric form. 0. C. 
and B. Messenger of Math., iv. pp. 25-31, 189-190.] 
[Question 2730. Educ. Times, xxi. p. 139 : or Math, from Educ. 
Times , xi. pp. 107-108.] 
The subject here is mainly the vanishing of the primary minors of any 
axisymmetric determinant of the fourth order, the final result being that 
all the 'primary minors will vanish ivhen three of them vanish, provided 
that two, and not more than two, of the three be taken from the same rows 
of the adjugate. The number three is Sylvester’s ^(4 — 3 + l)(4 — 3 + 2) of 
the year 1850. 
In proving a simple case of the well-known theorem regarding a minor 
of the adjugate he incidentally finds 
a e f g 
e b h i 
f h c j 
g i j d 
e 2 - ab ef - ah eg — ai 
ef -all f 2 - ac fg-aj 
eg — ai fg — aj g 2 — ad 
and thence later shows that when | af)<p z d± | is axisymmetric and vanishes 
we have 
I | • | aqc 3 d 4 | - | af 3 | • | aqc 2 d 4 | + | a A 1 1 +c 2 d 3 | = 0 . 
Here, however, we must note that for this last identity axisymmetry is not 
necessary, and that the other condition may also be dispensed with if in 
place of 0 we put a 1 1 a 1 5 2 c 3 (7 4 1 . 
Caldarera, F. (1871). 
[Nota su talune propriety dei determinant^ in ispecie di quelli a 
matrici composte con la serie dei numeri figurati. Giornale di 
Mat., ix. pp. 223-232.] 
The main determinant dealt with is 
a 
a + 8 
a + 2 8 
a + 38 .... 
a 
2a + 8 
3a + 38 
4a + 68 .... 
a 
3a + 8 
6a + 48 
10a +108 .... 
or. 
a 
4a + 8 
10a + 58 
20a+ 158 .... 
say, D(a,8), 
it being first shown to be independent of S, and consequently to be 
equal to 
