148 
ilAJOR P. A. MACMAHON OX A CERTAIX CLASS OF 
equations involving tlie ^ quantities and the co-axial minors of the first s 
orders of the determinant [ [ . Hence, by elimination, we find 
relations between such coaxial minors. 
Special to the coaxial minors of order s, we thus find f 1 relations if n be greater 
n\ 
than 3. The one relation, special (from this standpoint) to the determinant of even 
order (greater than two), is obtained by eliminating the determinant itself from the 
two special identical relations above referred to. 
Art. 53. I take this opportunity of verifying the statements made in Art. 49 in 
regard to the systems of values of the quantities 
which satisfy the 
y-ry 
2 " — 1 — n 
equations. 
It is, in reality, a question concerning the properties of determinants. 
To ensure that the coefficients of the condensed form satisfy the requisite con¬ 
ditions, assume them to be derived from the determinant 
a 
n 
We have ( ) equations of the type 
1 
Jxz 
’y.ryjijx 
a 
a. 
This equation, being a quadratic for jx-. j yxyjyz, has only two roots, and it is easy to 
verify that the equation is satisfied by the values 
^zx 
In Art, 49, these values have been obtained by solving the quadratic, and it Mms 
found that the values corresponded to the positive and negative sign respectively. 
