164 Proceedings of the Royal Society of Edinburgh. [Sess. 
\/[22] + \/[33] + \/[44] + n/[55] = 0, 
— a result hitherto only obtained independently from geometry.* 
Cayley (1859, June). 
[Note on the value of certain determinants, the terms of which are the 
squared distances of points in a plane or in space. Quart. Journ. 
of Math., iii. pp. 275-277 : or Collected Math. Papers, iv. pp. 460- 
462.] 
The determinants referred to are those occurring in his first paper of 
the year 1841 ; but the expansions of them which are given do not assume 
that 12 = 21, etc. Sylvesters related paper of March 1853 is also referred 
to, the 2W of which is put in the formed 
. c b f 1 
c . a g 1 
b a . h 1 | 
f g h . 1 | 
1111.1, 
with the result that Q takes the form 
a 2 b 2 c 2 {f l + g i + h 4: + g 2 h 2 + h 2 f 2 +f 2 g 2 + b 2 c 2 + c 2 a 2 + a 2 b 2 - (/ 2 + g 2 + h 2 )(a 2 + b 2 + c 2 ) } 
+ a 2 g 2 h 2 {f i + b* + c 4 + b 2 c 2 + c 2 f 2 + f 2 b 2 + g 2 h 2 + h 2 a 2 + a 2 g 2 - (f 2 + b 2 + c 2 )(a 2 + g 2 + h 2 ) } 
+ b 2 h 2 f 2 {g^ + c 4 + a 4 + c 2 a 2 + a 2 g 2 + g 2 c 2 + h 2 / 2 +f 2 b 2 + b 2 h 2 - (g 2 + c 2 + a 2 )(b 2 + h 2 +Z 2 )} 
+ c 2 fg 2 {h^ + a 4 + & 4 + a 2 b 2 + b 2 h 2 + Ji?a 2 + f 2 g 2 + g 2 c 2 + c 2 / 2 - (h 2 4- a 2 + b 2 ')(c 2 +Z 2 + g 2 ) } . 
* Baltzer gives (p. 20) Cayley’s determinant form for 
- {Ja+ Jb + Jc) (- Ja + Jb + Jc) (Ja- Jb + Jc) (Ja + JF- Jc), 
placing in front of it what looks like a generalisation, namely 
a i W c i 
<h • c 2 ^2 
b\ c 2 . (%2 
C 1 \ a 2 ' 3 
but is not really such. We can easily show that if a x , b x , c x be multiplied and a 2 , b 2 , c 2 
be divided by x , y , % respectively, the determinant is unaltered ; consequently it 
&1& 2 CjC 2 
1 
1 
1 
*Jb]b 2 \Jc]C 2 
Ol&2 
1 1 
1 
C l C 2 
&1& 2 
s]a x a 2 
rj C X C 2 sjbf) 2 
b l b 2 
1 
1 
1 
C \ C 2 
®1^2 
\/&l&2 C 1 C 2 
• sj 
C 1 C 2 
1 
1 
1 
&1& 2 
Jc lC 2 \Jb]b 2 
N/V* 2 
