1906-7.] 
161 
Dr Muir on Axisymmetric Determinants. 
CTj 2 
+ K 2 
- 2 M 
= (T 
cf 2 
+ a£ 2 
- 
2 gtf 
= 
= 0 
b £2+ 
arj 2 
- 
2 htn - 
= 0 
v 
- ye 
- 2 arj£ + 
■2 Kt+ 
2 g£v = 
= 0 
- 
ye 
+ 2 h v t- 
26ff + 
m - 
= 0 
— 2 ht, 2 
+ 2 gr,t + 
m- 
2c£r] = 
- 0, 
and therefore 
c 
b 
-2/ 
c 
a 
- 2 g 
b 
a 
-2 h 
-2/ 
- 2 a 
2 h 
2 g 
= u 
. 
- 2 g 
2 h 
-2 b 
2/ 
-27? 
y 
2 f 
-2c 
or, 
say, 
8A 
= 0 . 
But on account of the breaking up of the quadric into linear factors the 
discriminant must likewise vanish. It is thus suggested that the dis- 
criminant is a factor of A ; and by actual trial it is found that 
A = -2 
a h g 
h b f 
g f c 
2 
It has only in addition to be noted that A originally made its appearance 
in Sylvester’s second paper on dialytic elimination. 
Faa di Bruno (1857, April). 
[Sopra il volume della piramide triangolare. Annali di sci. mat. efis., 
viii. pp. 77-78.] 
With an eye on Sylvester’s paper of October 1852, Faa di Bruno first 
expresses the volume (V) of a tetrahedron in the form 
X 1~ X 2 V1-V2 Z 1~ Z 2 
X 1 ~ X 3 Vl - Vz Z \ — Z 3 
%i — V\~ z i ~ z 4 f 
and then by squaring obtains the result 
288 V 2 = 
2 dj d^+d^-d^ 
d\2 + ^ia 2 ~ ^23 2 2 g7 13 2 d u 2 + d u 2 — d 34 2 
^ 12 2 + ^ 14 2 — ^ 24 2 ^ 13 2 ^ 14 ** “ ^ 34 " 2dj 2 
VOL. XXVII. 
11 
