942 Proceedings of Royal Society of Edinburgh . [sess. 
He also uses effectively such umbrae as 
a n , a n ~ 1 b , a~ n b 2 ,...., 
— a usage which is most appropriately illustrated by taking a 
commutant having two lines of three umbrae each, that is to say, 
the determinant of the third order 
a 2 ab b 2 
a 2 ab b 2 . 
This is conveniently taken to represent 
a 4 • a 2 b 2 • 6 4 - a 4 • ab B • ab 3 - a B b ■ a B b • 5 4 + a z b • ab 3 • a 2 b 2 
+ a 2 b 2 -a z b-ab z - a 2 b 2 • a 2 b 2 . a 2 b 2 . 
and consequently if 
(ckc + fa/) 4 = A« 4 4- 4Bx 3 y + 6C x 2 y 2 + 4D xy B + Ey 4 , 
it is an expression for 
ACE + 2BCD - AD 2 - B 2 E - C 3 , 
that is to say, for the special determinant 
ABC 
BCD 
C D E . 
When he comes to speak of 1 partial ’ commutants, which are 
identical with intermutants, he devotes a page (pp. 88-89) to the 
subject of his relations with Cayley. As it would appear that he 
was not quite satisfied with the wording of the postscript above 
referred to, Cayley published a modified form of it as a note, 
headed “ Correction of the Postscript to the Paper on Permutants,” 
and there the matter between the two friends happily rested. 
Betti, E. (1852, Feb.). 
[Sulla risoluzione delle equazioni algebriche. Annali di sci. 
mat. ejis., iii. pp. 49-115.] 
In this important memoir dealing with the theory of substitu- 
tions, and with the application of the same towards finding the 
conditions of resolvability of algebraic equations, the author defines 
on page 80, in the manner afterwards so familiar, the expression 
