51 
1912-13.] Dr Muir on Axisymmetric Determinants. 
F 
x 1 F 1 
^F 2 ■ • 
x n F n 
^iFj 
xJFi 
X}X 2 F 12 . . 
• • XiX n F ln 
x 2 ¥ 2 
x+c-pE 2i 
•^F 2 . . 
• • '^‘P^nF 2n 
XnFn 
Vibi 
x n x 2 F n2 . . 
. . % n F n 
where F r , F rs stand for 0F/0x r , 0 2 F/0x r 0x s respectively , will contain nothing 
but positive terms. For the case of two variables the result is 
axy + bx + cy + d axy + bx axy + cy 
axy + bx axy + bx axy 
axy 4- cy axy axy + cy 
abcxhy 2 + abdx?y + acdxy 2 + bcdxy : 
and for the development in the case of three variables we are referred to a 
similar memoir * of the year 1857, where we find the expression 
(d 4- h 4- e 4- f)(abc + acg + dbg 4- beg) 
+ (a 4- b + c + g)(dhe + dhf+ def+ hef) 
+ (ac + bg)(df + dh 4 ef 4- eh) 
4- (ag + bc)(df+eh + de+fh) 
4- ( ab + cg){dc + dh+fe+fh ) 
+ 4 agfe + 4 bedh , 
and infer that it is the equivalent of 
£? + 0 + c + ... + 7i a + c + d + 6 
a+c+^+e a+c+d+e 
a b d f a + d 
a + b + c + g a + c 
a + b + d +f 
a + d 
a + b + d -\-f 
a + b 
a+b +c+g 
a + c 
a + b 
a + b + c + g 
Siebeck, F. H. (1862). 
[Ueber die Determinante deren Elemente die Quadrate der sechzehn 
Yerbindungslinien der Eckpunkte zweier beliebigen Tetraeder 
sind. Crelles Journ., lxii. pp. 151-159.] 
The determinant in question is brought forward as a companion to 
Sylvester’s of the year 1852, being, in fact, the complementary minor of the 
zero element in the latter. It is shown that the ratio of the one to the 
other is 
2rp cos (f> 
where r, p are the radii of the spheres circumscribing the tetraedra and 
(p the angle of intersection of the said spheres. The interest of the paper, 
like Sylvester’s, is mainly geometrical. 
* Boole, G., “ On the Application of the Theory of Probabilities to the Question of the 
Combination of Testimonies or Judgments,” Trans. Roy. Soc. Edin ., xxi. pp. 597-652 (see 
p. 648). 
