1901-2.] Dr Muir on the Theory of Orthogonants. 279 
2 2 2 2 2 2 
p = G a 4- G 7 b 4 G" c , 
‘2 fi ,^fiy2 y/fi ' l 
p = G a 4* G b + G c , 
9 2 9 ,9 9 9 
p" = G a" + G r + G c , 
2 2 2 
q = G a a" 4- G' bb" + G" c'c", 
= G d'a 4* G b b + G c c, 
2 , fir tfi t 
q G ad + G bb + G cc ; 
the first three being got by use of the second part of theorem (0), 
but all of them readily verifiable by merely substituting the said 
values of A, A', A", ... In exactly the same way from another 
set of nine equations, viz., those beginning 
^ = (B'C"-B"C> + (B"C - BC")a' + (BC'-B'C)a", 
there is obtained 
pp" - q 2 
a 2 
4 - 
fP 
+ 
c 2 
9 > 
A 2 
G 
G' 2 
G" 
pp- q 2 
fi 
7 fi 
2 
a 
4- 
b 
+ 
C 
9 
T 
9 i 
A 
G 
G r 
G" 
pp - q 
r/2 
b" 
„2i 
a 
+ 
C 
2 
~ 2 
.9 
+ 
9 ? 
A 
G 
G 
G" 
qq -pq 
da" 
b'b" 
cc" 
o 
A 
G 
4 
¥ 2 
4 * 
qq - pq 
a"a 
4- 
b"b 
n 
C C 
A 2 
)<N 
I o 
II 
9 
G' 
+ 
G" 2 
qq -pq 
ad 
4 - 
bb’ 
+ 
cc 
9 
— n 
^ 
9 • 
A^ 
G 
G' 
G" 
Then, by mere addition, half of the first derived set gives-- 
G 4 - G' 4 - G" = p 4 - p 4 - p"\ 
and the corresponding half of the second set 
_L + 1 + 1 _ VP + V P + PP ~ q - q -q 
G G G A 2 
which on putting GG'G" for A becomes 
