380 
Transactions of the Boyal Society of South Africa. 
and first with columns 9 and 7 interchanged. Then, having thus got the 
first element of every column, substitution 83 suffices to give all the other 
elements ; for example, the first element of the fourth column being 
the next two are 
12348+12857, 
45672 + 15681 and 78915+78924 : 
and the fourth being the sum of the first and second with columns 3 and 6> 
interchanged, namely, 
12648 + 12657 + 45372 + 45381, 
it follows that the remaining two are 
45972 + 45981 + 78615 + 78624, 
78315 + 78324 + 12948 + 12957. 
10. Further, since each of the given equations may be written in 
bipartite notation. 
a^ a 
a 
5 ^6 
8 a. 
0, 
= 0, 
and as each row of the eliminant consists of the coefficients of an equation- 
in rf, 4/7, 77<^, 4;^, and each column consists of the coefficients of an» 
equation in u', v^, w^, uv, vw, wu, it would not be difficult to devise a rule- 
for writing down any element of the eliminant quite independently of the 
others. Thus, the element in the (3,2)th place being the coefficient of 
in the one set of equations and of in the other set must have its column- 
numbers taken from the mutilated bipartite 
u 
V 
IV, 
and, as a matter of fact, it is 25789. 
