146 
MAJOR P. A. MACMAHON ON A CERTAIN CLASS OF 
— Xy 
ay2 
^h3 
CO 
Oi 
agy 
^'32 
ttgg Xq 
and herein writing for x^, &c., and multiplying by XiX.^x^ and by -- 1 Avhen the 
order is uneven, we get 
1 Ct-y-yX^ Ct'^t^X'y Ct^-yX-^ 
““ ClnyX.^ 1 Ct.-):)X.-y Clo:}X.-} 
®31^3 <^^33^3 
1 ®33^’3 
Thus the latent function is a particular case of the function V, 
In the discussion of the roots of the latent function we are concerned with the order 
of vacuity of the matrix which may be any integer of the series 0, 1,2,. . . In 
the case of the function V, which may be called the homographic function of the 
matrix, it is evident that a more refined nature of vacuity is pertinent to the discus¬ 
sion. We have to consider not merely the vanishing of the sum of all the co-axial 
minors whose order exceeds a given integer, but rather the vanishing of each separate 
co-axial minor. 
It may be remarked that the homographic function V vanishes for the system of 
values of Xy, x^, x^, wliich satisfies the equations 
= X, -r Xg = 1. 
§ 6. Digression on the General Theory of Determinants. 
Art. 51. The foregoing investigation has established the fact that the co-axial 
minors, of a general determinant of Order n, are connected by 2" — — 2 
relations, or in other words, that but + 1 of them can assume given values. 
Of these relations a certain number are connected in a special manner with the 
determinant of Order n, in that they are not relations merely between the coaxial 
minors of one of the principal coaxial minors of the determinant. 
Let this number be 
and put 
Then 
i// (a), 
2 " — ir n — 2 ■= (f) («). 
(/> (n) = xfj (n) -f Q xfj {n - 1} + xf; (n - 2) -j- ...-j- ^ xjj (4); 
