1904 - 5 .] 
Dr Muir on General Determinants. 
923 
Sylvester, J. J. (1850, Sep.). 
[Additions to the articles in the September number of this 
journal “ On a new class of theorems . ... ” and “ On 
Pascal’s theorem.” Philos. Magazine (3), xxxvii. pp. 
363-370 : Collected Math. Papers , i. pp. 145-151.] 
Of the three additions referred to in the title it is the last 
which concerns us, viz., that in which Sylvester introduces and 
explains his use of the term minor as applied to a determinant. 
Starting with the ‘square array’ of a given determinant, and 
leaving out one ‘line ’ and one ‘ column,’ he calls the determinant 
of the minor array which remains a ‘ First Minor Determinant ’ ; 
similarly ‘ Second Minor Determinant ’ is explained ; and then 
he adds, 
“and so in general we can form a system of r th minor deter- 
minants by the exclusion of r lines and r columns, and such 
system in general will contain 
j n(n- 1) ... (n — r+ 1) ( 2 
t 1*2 . . rr j 
distinct determinants.” 
It is thus seen that ‘ minor determinant * is used as ‘ partial 
determinant’ had already been used by Lebesgue (1837), and as 
‘ determinant of a derived system ’ had been used by Cauchy 
(1812), hut that, whereas Cauchy added a distinguishing epithet 
to specify the order of the determinant, Sylvester did so to indicate 
how many lines or columns fewer it had than the ‘ principal ’ or 
‘ complete ’ determinant originally started with. 
The following proposition or ‘ law ’ is next given, viz. : The 
whole of a system of v th minors being zero implies only (r+1) 2 
equations , that is , by making (r + l) 2 of these minors zero , all will 
become zero : and this is true , no matter what may be the dimensions 
or form of the complete determinant. Then, after some geometr cal 
applications concerned with first minors of a symmetrical deter- 
minant, there follows the explanation — 
“The law which I have stated for assigning the number of 
independent or, to speak more accurately, non-coe vanescent 
