226 Investigation of Determinants. (April, 
Now "C, being the number of combinations of things 
taken y together (as in Problem III.), it is clear that “C;, is 
the number of ways in which p rows can be selected out of 
nm rows, also that “Cy, is the number of ways in which 
fp columns can be selected out of ~ columns. But “My, is 
evidently the number of ways in which any p rows and any 
p columns can be selected simultaneously (for erasion) from 
the m rows and 2 columns in the determinant. 
nN nN n n | a 
nt es IG ec, Ss 2. 1(46). 
Eq. (46) shows that, as is also evident from the reasoning 
itself— 
"M, LS "Nig 4; a pa be uote Cie ve ke "pum (47). 
From the preceding reasoning, it may also be inferred 
that result (47) is a property common to all determinants 
whose minors are all finite (except in certain cases when 
these numbers ”M, ,”M,-» are unequally reduced, in conse- 
quence of the constituents being so related as to produce 
equality among some of the minors). 
As a particular case of Eq. (47), "M:= *Mu-z, 1.¢., “ The 
first minors are the same in number as the constituents 
(these being actually the »—1th minors).” 
XIV. Number of Minors in a Symmetric Determinant. 
In symmetric determinants it is clear that there is only one 
way in which a particular leading minor (which is itself 
also a symmetric determinant) can be selected, but that any 
other minor can always be selected in two ways, e.g., the 
same (non-leading) fth minor may be formed either by 
omitting a certain set of # rows and / columns from the 
original determinant, or by omitting the conjugate set of 
p columns and f rows. This amounts to saying that the 
non-leading minors occur in pairs of equal magnitude. 
Now the number of leading pth minors being those minors 
which contain (n—f) leading constituents of the original 
determinants in their own leading diagonals, is clearly equal 
to the number of ways in which (7—/) constituents can be 
selected from the whole 7 leading constituents, 7.e., is equal 
to CO or is 65 ° 
Also, the number of non-leading pth minors would in an ordi- 
nary determinant be (Problem XIII.) {("C,)? —"C, }, which 
reduces in a symmetric determinant to } {("C, )* —"C, } for 
