1906-7.] Dr Muir on Axisymmetric Determinants. 
165 
Salmon (1859, July). 
[On the relation which connects the mutual distances of five points in 
space. Quart Journ. of Math., iii. pp. 282-288.] 
Salmon starts from the elimination of a , b , c from the set of trigono- 
metrical equations 
a = b cos C + c cos B 
b = c cos A + a cos C 
c = a cos B + b cos A 
and proceeding to similar eliminants of higher order reaches Cayley’s 
results of 1841 and others related to them. The interest of the paper is 
mainly geometrical. Use is made of the proposition that if the quadric 
ax 2 + by 2 + cz 2 + dw 2 
+ 2 lyz + 2 mzx + 2 nxy + 2 pxw -f 2 qyw + 2rzw 
be increased by (ax + /3y + yz + Swf, the discriminant 
is increased by * 
a n m p 
n b l q 
m l c r 
p q r d 
a /3 y $ 
a a n m p 
[3 n b l q 
y m l c r 
8 p q r d 
* The new discriminant being 
is easily seen to be equal to 
a+ a 2 
n + a# 
wl + ay 
p + a8 
n + ai 8 
b + 0 2 
l + P y 
q + l 88 
m + ay 
l + &y 
C + y l 
r + yS 
p + aS 
q + & 5 
r + 78 
d + 8 2 
- 
— 1 a 
£ 7 5 
a n m p 
q 
r 
d 
fi n b l 
7 m l c 
5 p q 
and therefore to be separable into the two expressions referred to. 
