378 Transactions of the Boyal Society of South Africa. 
difference in the result, the rows of the one eliminant being the columns 
of the other. 
8. The mode of effecting the preliminary set of eliminations is well 
w^orth a little attention. To make it clear, let us take the case where 
m, n = 3, 3, namely, the case of the elimination of ^, ?/, ^, u, v, w from the 
five equations 
ajlti + a^rj^i + a^l^ti + a^^v + . . . + agi^iu = O^j 
-0, 
e^^u + e^rjic + e^i^u + e^E,v + . . . + egC,io = Oj 
Arranging the equations in the form 
a^iu + a^i^u + a.pi + {a^l + . . .)v + {a^^ + . . .)id = 0 
we eliminate lu, r}U, i^u, v, w, and obtain 
a^ a. a^l + a.-q + aeC a^i ^- a^i-] + a^^ 
e, C2 63 + + e^t, + e^r) + e^c, 
whence there results 
12347£^ + 1^3mr) + 12349^^ 
+ 12357/?^ + 12358,7^ + 12359,7<: 
+ 12367^s^ + 12368^^/ + 12369^- ^ 0. 
By the elimination of u, ^v, rjv, '(v, w, we obtain a similar equation, and by 
the elimination of u, v, t,io, rjW, i^w a third. We have thus three equations 
in the six compound variables q^^^ have therefore to 
seek for other three. It is the obtaining of the latter three that is most 
interesting. 
Putting the given equations in the form 
{a^l + a^riyu + a^^'Cu + a^iv + {a^-q + ae'C)v + [a^i + ag'/ + ^qCjw = 0, 
eliminating u, i^u, h\ v, iv, and proceeding as before, we reach a cubic in 
^, T], 'C, every term of it involving r] except two, namely, 
13467r^, 13469tr. 
By ehmination of t,n, u, v, 'Cv, w there is reached in the same way a cubic 
of the same kind, alike even in the particular that all its terms involve rj 
except 
13467^^, 13469^^^ 
