1906-7.] Dr Muir on Axisymmetric Determinants. 
157 
Brioschi (1854, March). 
[La teorica dei determinanti, e le sue principali applicazioni ; 
del Dr Francesco Brioschi ; viii + 116 pp. ; Pavia. Translation into 
French, by Combescure; ix + 216 pp. ; Paris, 1856. Translation 
into German, by Schellbach ; vii + 102 pp. ; Berlin, 1856.] 
Unlike Spottiswoode, Brioschi in methodical manner defines “ nn deter - 
minante simmetrico,” and gives four known properties expressed in clear 
language, all within the space of one page (p. 70). 
Brioschi (1854, Dec.). 
[Sur quelques questions de la geometrie de position. Grelles Journal, 
1. pp. 233-238 : or Opere Mat.] 
The title here recalls that of Cayley’s maiden effort ; and, as a matter of 
fact, the paper of 1854 had its origin in the paper of 1841. Cayley, it will 
be remembered, obtained the relation connecting the mutual distances of 
five points in space by multiplying the two determinants 
2hq 2 - 2x 1 - 2 y 1 - 2 z 1 - 2 w 1 1 
1 aq y 1 z 1 w l 2aq 2 . 
2x 2 2 - 2x 2 - 2 y 2 - 2z 2 - 2 w 2 1 
1 *2 Vz Z 2 W 2 2 X 2 
2* 5 2 - 2* 5 - 2 y b - 2 z 5 - 2w 5 1 
1 *5 2/5 H W b 2 *5 2 
1 .... 
5 
1 
the first of which is — 16 times the second, and then putting the u>’s equal 
to zero. Brioschi now follows on the same lines, but with an interesting 
difference. Having shown that the determinant 
x i + V\ + z i 
( X 6 
x iY + (vq - 
-2/i) 2 + (%-%) 2 
1 
- 2x 1 
-2 y 1 
- 2 z 1 
X 2 4" Vz 4" Z 2 
( X 6~ 
x 2 ) 2 ■ t (y& ' 
••y 2 ) 2 + ( Z 6-%) 2 
1 
-2*2 
- 22/2 
CM 
1 
*5 +V *■+**. 
(*6~ 
x 5 ) 2 + (y& - 
~ S/s) 2 + ( 2 6 - %) 2 
1 
-2* 6 
-2% 
10 
• ?Si 
cq 
• 1 
1 1 
vanishes identically, the simple fact being that the second column is a sum 
of multiples of the other columns, he multiplies it by ^ of itself, namely by 
1 
S (* 6 -* i ) 2 
2 « q 2 
*1 
y \ 
z i 
1 
S (* 6 -* 2 ) 2 
3.X* 
X 2 
2/2 
Z 2 
L 
^(*6 “ X b ) 2 
w 
X 5 
Vb 
Z 5 
1 
1 
