428 Proceedings of the Royal Society of Edinburgh. [Sess. 
thus implying that A r s /( — l) r(s_1) as is independent of r , — a result suggested 
to Brioschi by an old proposition of Euler’s which is referred to under 
Continuants. 
Cayley, A. (1856, March). 
[A third memoir on quantics. Philos. Trans. R. Soc. {Loud.), cxlvi. 
pp. 627-647 : or Collected Math. Papers, ii. pp. 310-335.] 
Among Cayley’s tables of invariants there naturally appear the 
catalecticants of the binary quartic, sextic, and octavic ; so that we have 
from him the final expansions of the persymmetric determinants of the 
3rd, 4th, 5th orders. Of canonizants only that of the quintic is given. 
The four results are those which he numbers 10, 34, 35, 16. The first an 
last we need not reproduce. The second is 
; a 
b 
e 
d 
( 
i b 
i 
c 
d 
e 
c 
d 
e 
f 
= 1 
| d 
e 
f 
9 
l 
aceg - acf 2 - ad 2 g + 2 ddpf - ae 3 - b 2 eg + b 2 / 2 
+ 2 bcdg - Tbcef - 2bd 2 f + 2bde 2 - c 3 g + 2 c 2 df 
+ c 2 e 2 - 3 cd 2 e + d 4 , 
where the terms are arranged, as with Cayley, in alphabetical order, 
third, altered in form, is 
The 
b c d e f 
c d e f g 
d e f g h 
e f g h i 
e b - e 3 (ai + 3 eg + 4 df) 
+ e 2 {2(afh + bdi) + ( ag 2 + c 2 i) + 4 (bfg + cdh) + 3(c/ 2 + d 2 g)} 
- e{(acli 2 + b 2 gi) + 2 (adgh + befi) + 3 (af 2 g + cdH) 
+ 4 (bdg 2 + c 2 fK) + 2 (bf 5 + d B h) - aegi 
- 2adfi - b 2 h 2 - 2bcgh - 2e 2 <? 2 + 2cdfg + 3 d 2 f 2 } 
+ { - ( acfH + ad 2 gi) + 2 (acfgh + bedgi ) - (aeg 3 + c 3 gi) 
+ (, ad 2 h 2 + ft 2 / 2 /) - 2 {ad.ph + bdji) + 2 (adfg 2 + cHfi) 
+ ( a P + dH) - 2 (b 2 fgh + bedh 2 ) + (b 2 g 3 + c 3 h 2 ) 
+ 2 (bcf 2 h + bd 2 gh) - 2(bcfg 2 + c 2 dgh) + 2 (bdf 2 g + cd 2 fh) 
+ (c 2 f 2 g + cd 2 g 2 ) - 2 (cdf 3 + d 3 fg)}. 
It is the term, L, independent of X in the almost persymmetric deter- 
minant which Cayley * on Sylvester’s suggestion calls the lamdaic, namely, 
a b c d e-12A 
b c d e + 3A f 
c d e - 2A / g 
d e + 3 A / g h 
e-12A / g h i 
* Cayley, A. Memoire sur la forme canonique des fonctions hinaires. Crellds Journ ., 
liv. pp. 48-58, 292 : or Collected Math. Papers, iv. pp. 43-52. If “ lamdaic 55 be not used as a 
noun, “lamdaic canonizant ” would be better than “lamdaic determinant.” 
