140 
MAJOR P. A. MACMAHON ON A CERTAIN CLASS OF 
1 . 
‘■xy 
Py^ 
1 1, 
^xz^lxz 
^y~ly~ 
P= 
= V:cy. . . . 
In the determinant the quantities a occur in an inversely symmetrical manner, and 
the determinant becomes inversely symmetrical on putting the quantities p and q 
equal to unity. 
Art, 43. The determinant is transformable in the same manner as the corresponding 
inversely symmetrical form, and the foregoing “ Digression ” establishes the fact that 
the quantities a will then occur in only some or all of ^ combinations y, where 
_ ^xy 
y nr Oi Ci 
X, .•c+l“^:+l, «+2 • • • y-1. 2 / Pxy 
Hence we are presented with 
C)ll 
I — n 
n 
equations 
involving ^ ) quantities y. 
Art. 44. Eliminating these ( ) quantities, we find 
n - 1 
2 " “ 1 — n 
n - 1 
- O/i 
IV -p n — 2 
relations or syzygies between the coaxial minors 
p.ryx 
of the determinant 
a 
In 
Art. 45, This shows that the coefficients of V„ must satisfy 
independent conditions. 
2 " — IV n — 2 
