178 
Transactions of the Boyal Society of South Africa. 
But the four-line determinant here is seen to be divisible by — 0.2) 
tts - Oi, as - ai ; and, these factors being removed in the ordinary way, 
we have 
! 3a^ Saj 1 
al + asOi + a] 3(ai + ag) 3 
03 + 02 + 01 3 
D. 
Further simplification is effected by performing the operations 
roWi - ai.row2+ OiOs-roWg, 
row2 - (oi + a2).row3, 
the penultimate and final results being 
Stti 3 
-I3,ft,l3, S/3,/32 
1, 
0,0203 + /3,/32/33 2/3,/32 2/3, 
2ox02 . - 3 
So, -3 . 1. 
Proceeding in exactly similar fashion we obtain 
1 0,020304 + /3,/32/33/34 S/3,/32/33 2/3,/32 2/3, | 
D4.4 1 20,0203 . . -4; 
2o,02 . -6.1 
So, -4 . . I 
and so on, generally. 
3. The form of quotient obtained is manifestly invariant (1) to the 
interchange of any two o's, (2) to the interchange of any two /3's, and 
(3) to the simultaneous interchange of every o with the corresponding /3 ; 
and this, as we know, is what ought to be. 
4. On account of the number of zero elements in the quotient, it is 
possible to put it in a simple non-determinant form : thus 
ZlZl = - 2(«xa2 + /3,/32) - Sa,.S/3„ 
1^3 ; 3 - KlZl = - 3.3(o,02a3 + /3,/32/33) - 3(So,02.S/3, + So,.S/3,/32), 
