202 Transactions of the Boyal Society of South Africa. 
Proceeding similarly with the next case we see that 
X y z 
X a^ + b. + b. -b^ -b^ 
y a. + c^-^b. -c^ 
^ -^4 -^4 «'4 + ^4 + C4 
— aji^x'^ — a^a^y'^ — a^a.z'^-^a^J^ . y z 
^y c, + b. -c. 
X 
X b^ + b^ 
z -b. 
b, + c. 
X y 
X + b^ — Z). 
y -^3 ^A + ^i 
X b. + b^ -b. -b^ 
y -b. c, + b. -c, 
^ -^4 -^4 ^^4 + ^4 
where each of the three-line determinants is an example of the previous 
case. As for the four-line determinant, we have only to perform the 
operations 
row^ + row3 -f roWs, col^ -f col. + C0I2 
to find that it is equal to 
-{x-^y + zy 
b, + b, -b. 
-b, c,f^3 
i.e., -{x-^y + zY i^-fi^ + ^3^4 + ^4^1). 
The conclusion thus again is that all the terms of the final development 
are negative. 
That this is generally true is due to the fact that, having proved it true 
when all the a's vanish, we can then make each case dependent on those 
preceding it. (V.) 
