26 
ON THE THEORY OF PERMUTANTS. 
[104 
POSTSCRIPT. 
I wish to explain as accurately as I am able, the extent of my obligations to Mr Sylvester in' 
respect of the subject of the present memoir. The term permutant is due to him—intermutant and 
commutant are merely terms framed between us in analogy with permutant, and the names date from 
the present year (1851). The theory of commutants is given in my memoir in the Cambridge Philo 
sophical Transactions, [12], and is presupposed in the memoir on Linear Transformations, [13, 14]. It 
will appear by the last-mentioned memoir that it was by representing the coefficients of a biquadratic 
function by a = 1111, b = 1112 = 1121 = &c., c=1122 = &c., c£= 1222 = &c., e = 2222, and forming the 
commutant ( 1111 ] that I was led to the function ae - 4bd+3c 2 . The function ace + 2bed - ad 2 - Ire — c 3 
I 2222 J 
or 
a, b, c 
b, c, d 
is mentioned in the memoir on Linear Transformations, as brought into notice by 
c, d, e 
Mr Boole. From the particular mode in which the coefficients a, b,... were represented by symbols 
such as 1111, &c., I did not perceive that the last-mentioned function could be expressed in the 
commutant notation. The notion of a permutant, in its most general sense, is explained by me in 
my memoir, “ Sur les determinants gauches,” Crelle, t. xxxvn. pp. 93—96, [69] ; see the paragraph 
(p. 94) commencing “ On obtient ces fonctions, &c.” and which should run as follows : “ On obtient 
ces fonctions (dont je reprends ici la théorie) par les propriétés générales d’un determinant défini 
comme suit. En exprimant &c. the sentence as printed being “ défini. Car en exprimant &c.,” 
which confuses the sense. [The paragraph is printed correctly 69, p. 411.] Some time in the present 
year (1851) Mr Sylvester, in conversation, made to me the very important remark, that as one of a 
class the above-mentioned function, 
ace + 2bcd - ad 2 - b 2 e — c 3 , 
could be expressed in the commutant notation f 0 0 ] , viz. by considering 00 = a, 01 = 10 = b, 
11 
12 2 J 
02 = ll = 20 = c, 12 = 21 — d, 22 = e; and the subject being thereby recalled to my notice, I found 
shortly afterwards the expression for the function 
a 2 d 2 + 4ac 3 + 4b 3 d - 3b 2 c 2 - Qabcd 
(which cannot be expressed as a commutant) in the form of an intermutant, and I was thence led 
to see the identity, so to say, of the theory of hyperdeterminants, as given in the memoir on 
Linear Transformations, with the present theory of intermutants. It is understood between Mr Sylvester 
and myself, that the publication of the present memoir is not to affect Mr Sylvester’s right to 
claim the origination, and to be considered as the first publisher of such part as may belong to him 
of the theory here sketched out.