320 Proceedings of the Royal Society of Edinburgh. [Sess. 
There is thus obtained 
* = (« - j8)(« - *) { (j8 - y)(r - y) - (8 - y)(p - /) } 
- (r - jS)(y - «•) | 08 - «)(«' - «') - (P - «')(* - a) ]. > 
= (a - P)(o! - i‘)( y - S)(/3' - y) + (y - f3)(y' ~ ~ « 0 (* ~ «) , 
= - (a - P)(y - S)(rt - $)(y ~ P) + (a! - p)(y - 8')(a - S)(y - /3) .* 
(y 2 ) SYLVESTER’S UNISIGNANT. 
Sylvester, J. J. (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.] 
In the course of Sylvester’s investigations a peculiar three-line deter- 
minant turns up, which he considers deserving of attention on its own 
account, namely, the determinant 
05 1 + $2 ^3 ~ 0>2 — a g 
-b x bj + b 2 + b s - b z 
-Ci -c 2 Cl + c 2 + c 8 , 
the final expansion of which consists of 16 terms, all positive. To obtain 
this expansion a “ simple rule ” is laid down, namely, to substitute 
a a b a c \ 
b a b b c > for 
c a c h c J 
and then multiply together the elements of the diagonal, rejecting every 
term such as a b b a , a b b c c a , .... in which the letters form a cycle. Two 
examples are given, but no justification of the “ rule ” is vouchsafed. The 
examples are — 
a + a b + a c - a b - a c 
-b a b + b c + b a - b c 
-C a -c b c + c a + c b 
+ c(a c b c + a c b a + b c a h ) , 
= abc + (c a + c b )ab + (a b + a c )bc + ( b c + b a )ca 
+ a (b a c a + b a c b + c a b c ) + b(c b a b + c b a c + a b c a ) 
* It will be seen that merely by accident the three ways in which ¥ can be expressed 
as the difference of two products have turned up in succession, and that they may be written 
|PQ'|,|QR'|,|PR/i 
if we put 
(a-) B)(7-5) = P, 
(a -7)(0 -8) = Q. 
(a-S)(B-y) — R, 
