456 Transactions of the South African Philosophical Society. 



11. In order to find the actual quadrate expression in (XII.) we 

 note that £, £ being both axisymmetric determinants with a vanishing 

 principal minor are themselves expressible as squares, namely, 





d e 



e 3 



d e 



e j 



- n 



m 



/ o.| 

 d f 

 f o 



+ i 



+ h 



j k 



k o 



j k*\ 



k o 



We thus have the determinant with which we started, and which 

 has been shown equal to - (b^ - c£*) 2 , 



r 



bj~l 



J-l\q 



d e 



d e 



— n 



m 



[ 



b c 



q r 



d e 



b c 

 m n 



d /I* 

 / o\ 



d f> 



f o 

 d /'* 

 / o 



+ * 



+ h 



+ 



j k\*\ 



k o\ f 



j k\n 



k o\ j 



b c\ \j k I * 



h i\ ' \k o 



i 



I 



as required. 



It may be noted that the form of the result suggests the following 

 alternative proof : By performing the operations 



" c row 5 — b row 6 , 

 we have the given determinant 



c coL — b col 6 



c 



d e f ch - bi i 



e j k cm -bn n 



f k o cq-br r 



ch — bi cm - bn cq - br c 2 v - 2bciv + bx 2 civ - bx 

 i n r civ -bx x 



d e f ch — bi 



e j k cm — bn 



f k o cq — br 



ch — bi cm — bn cq — br c 2 v — l 2bav + bx 2 



-rC 2 , 



r 7 k i \d f i 



\(ch - bi) J k Q - {cm-bn)\, J Q + (cq-br) 



! J j ,,., l,.\ d e \*l 



e j | J (XIII.) 



by the theorem just used in regard to I and £. 



12. Eeturning now to the norm 



N(fl x/S + WV + c VX) 



