143 
1906-7.] Dr Muir on Axisymmetric Determinants. 
X 6 4- x 4 
a 2 + h 2 + g 2 
— X 2 
a h 
2 1 
+ 1 
a g j 
2 
+ 
h g 
+ 
a 
h 
9 
. 
+ W + b 2 +f 2 
h b 
h f 
■ bf 
o 
h 
b 
f 
+ 9 2 +/ 2 + c 2 
+ 
a g 
^ + 
a g 
2 + 
h 9 
L 
► 
9 
f 
c 
N. J 
h f 
9 c 
\f e 
+ 
h g 
2 
+ 
h 9 
2 
+ 
b f 
2 
1 b f 
\f c 
f c 
1 J 
The fact that the coefficients here are negative and positive alternately 
is what Sylvester utilises for his main purpose, application being made of 
Descartes’ rule of signs. 
Sylvester (1852, Oct.). 
[On Staudt’s theorems concerning the contents of polygons and poly- 
hedrons, with a note on a new and resembling class of theorems. 
Philos. Magazine , (4) iv. pp. 335-345 : or Collected Math. Papers, 
i. pp. 382-391.] 
As an illustration of his mode of expressing the product of two deter- 
minants of the n tYi order as a determinant of the ('R + l) th order, Sylvester 
gives the identity 
x i 
Vi *1 
1 
x 2 
Hz % 
1 
X 3 
y% 
1 
*4 
1 
2x 2 ^i 
= 
^A 
1 
1 
(l 
Vi 
Ci 
1 
h 
Vi 
? 2 
1 
^3 
Vs 
is 
1 
Vi 
i* 
1 
?x 
A 
1 
2 x 2 ^3 
'Zx 
2C4 
1 
2^ 3 ^3 
3C4 
1 
2r £ 
c 4-3 
2 . 7 J 
4C4 
1 
1 1 
where 'Ex r £ s is put for x r £ s + y r rj s -f- z r £ s . Then performing on the last deter- 
minant the operations which we may denote by 
row x - J-2^ 2 . row 5 , ro\v 2 - J2& 2 2 . row 5 , . . . . 
coli - J2^ 2 . col 5 , col 2 - J2£ 2 2 . col 5 , . . . . 
2^-ft) 2 Sfo-f*) 2 2^-f,)* S^-O 2 1 
2(.t 2 -^) 2 2(^-£ 2 j 2 2(* 2 -f 8 ) 2 2(^-0* 1 
-ft) 2 -4) 2 2(z 3 -£ 4 ) 2 1 
S^-fO 2 2(* 4 -^)* 2(* 4 -f3)» 2(z 4 -£ 4 ) 2 1 
he obtains 
