158 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and, putting d Q1 , . . . for 2(a? 6 — xff, . . . , obtains the relation 
<V 
^61^62 "t* d\2 
• * ^61^65 + ^15 
•i 6 i + 1 
^61^62 **“ ^12 
d 2 
a 62 * ' 
' * ^62^65 4 ^25 
<i 62 + 1 
^61^65 
^62^65 + ^25 * ' 
d 2 
■ • a 65 
d 6 5 + 1 
d$ i + 1 
d 62 +1 
d 6 5+1 
1 
which degenerates into Cayley’s result when we put x Q , y Q , z 6 = x x , y 1 , z x , 
and make certain easy transformations. 
In a similar manner the relation between the distances of five points on 
an ellipsoid is found, and the relation “entre les plus courtes distances 
respectives et les inclinaisons mutuelle de sept lignes quelconques.” 
Sylvester (1855, April). 
[On the change of systems of independent variables. Quarterly 
Journ. of Math., i. pp. 42-56: or Collected Math. Papers, ii. pp. 
65-85.] 
Having reached in the course of his investigation (p. 55) a determinant 
of the form 
-f- Qq "t" $3 — &3 — C 2 
- « 2 \ + ^2 + ^3 “ C S 
— a 3 ~ ^2 C 1 C 2 "t c 3 ’ 
the final expansion of which, he says, contains only positive terms with the 
coefficient unity, Sylvester naturally notes that the number of such terms 
must be 
3-1-1 
-1 3-1 
-1-1 3 . 
He is thus led to the consideration of the ^-line axisymmetric determinant 
a 
- 1 
-1 . . 
. -1 
-1 
a 
-1 . . 
. -1 
- 1 
-1 
a . . 
. -1 
- 1 
-1 
-1 . . 
a ' 
to which he assigns the value 
(a - n + l )( a + 1) M_1 • 
