420 Proceedings of the Royal Society of Edinburgh. [Sess. 
Spottiswoode, W. (1853, August). 
[Elementary theorems relating to determinants. Second edition, . . . . 
Crelles Journ., li. pp. 209-271, 328-381.] 
Just as Spottiswoode viewed an axisymmetric determinant as the 
determinant of an n-&ry quadric, so he closely associated a persymmetric 
determinant with an even-ordered binary quantic. Taking, for example, 
the binary quartic which Cayley would a year later have denoted by 
(a 0 , a 1 , . . . , a 4 jj x , yY, namely, 
a 0 £ 4 + + 6a 2 x 2 y 2 + ia 3 xy z + « 4 ?/ 4 , 
Spottiswoode writes it in the form * 
(a 0 x 2 + 2a 1 xy + a 2 y 2 )x 2 
4- 2 (aye 2 + 2 a 2 xy + a 3 y 2 )xy 
-1- (a 2 x 2 + 2a z xy + a 4 y 2 )y 2 ; 
and calling it U points out that 
32 tt 
d ~2 = 12(a 0 x 2 + 2aixy + a 2 y 2 ) 
^ = 12 (a,* 2 + 2a 2 xy + a 3 y 2 ) ■ 
32TT 
= 1 2(a 2 a: 2 + 2a 3 xy + a 4 y 2 )J 
and thus like Sylvester concludes that the evanescence of 
CLy 
CLy &2 ^3 
a 2 a 3 a 4 
is the condition that the second differential-quotients of U shall simul- 
* A preferable form, because making tbe “ catalecticant 55 still more prominent, is 
X 2 
2 xy 
y 2 
a 0 
«i 
a 2 
a; 2 
<h 
a 2 
a 3 
2 xy 
a 2 
a 3 
a 4 
y 2 - 
Similarly an odd-degreed function may be represented so as to bring the “ canonizant” into 
prominence: for example (a, b, . . . , f^x x yf may be written ( ax + by , bx + cy, . . . . 
. . . , ex+fyfyx, y)\ or 
x l 2xy y 1 
ax + by bx+ cy cx + dy 
bx + cy cx + dy dx + ey 
cx + dy dx + ey cx + fy 
x 2 
2 xy 
