22 
Transactions of the Royal Society of South Africa. 
metric orthogonant involving superficially the full number of arbitrary 
constants. For example, when n is 3, and 
A ~ 
— v a 
fx — X 
we have the axi symmetric orthogonant 
A 2 — fX 1 — V 2 2XfX 
2ty _\2 _J_ ^2 
= a(a 2 + A 2 + ^ + > 2 ) 
2\r 
2\, 
_X2 _ ^ + v * 
with (/\ 2 + /x 2 + i' 2 ) 2 for its basic-constant. 
(4) The procedure fails when n is even, because then the expansion of 
A has its last term independent of a and cannot be divided by a prepara- 
tory to putting a = 0. Nevertheless if we introduce the condition that the 
said last term vanishes, and we so, in effect, diminish the number of arbi- 
trary constants by 1, a result worthy of note is arrived at when n is 4 ; 
namely, If af + bg + ch = 0, the axy symmetric determinant 
_ a 2 _ h 7 _ c 2 
+ P + 9* + h* 
2(bh - eg) 
2(bh 
eg) 
- a 2 + b 2 + c 2 
+ p-gl-h* 
2(cf-ah) 
2{cf- ah) 
2(ag-bf) 
2(fg ab) 
2(fh-ac) 
2{fg - ab) 
+ a 2 — b 2 f c 2 
2(gh-bc) 
2(ag 
-¥) 
2(fh 
— ac) 
%{gh 
- be) 
a° 4 
b 2 - c 2 
-}'- 
g 2 + h 2 
+ f- 
+ g 2 + 
*«)» 
What is equally interesting is the fact that this resolves into the product of 
two orthogonants 
f- 
v-f 
b — g c - 
c-h (J- 
f 
h — c 
b-g 
f- a 
a-f 
■ b-g 
■ c — h 
f + a 
c + h 
-g-b 
g + b h+ c 
■ h — c b + g 
— f — a 
a+f 
which are both skew and have the same basic-constant 
a 2 + h 2 + C 2 + p + g 2 + 7,2. 
October 20, 1920. 
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