1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
287 
Further, it will be observed that only one substitution is here 
involved, and that consequently in connection with the other 
substitution there must be analogous results, beginning with 
pio +p'w -\-p"w" + 2 qw'io" + 2<2 'w"w + 2 q'ww' -■ 
9 9 , 9 9 2 2 
GY + G' u + G" v . 
All of them, manifestly, may be described as transformations 
simultaneous with the main transformation, and, like one which 
appeared earlier in the paper, may be usefully enunciated in 
modern form as follows : — 
The linear orthogonal substitutions which change 
■r !]_ 
A B 
A' B 
A" B 
C 
C’ 
C" 
into Oast 4- G’s'u + G "s"v 
will at the same time change 
x 
y 
l n m 
n m V 
m l' n 
y 
A A A A tA 
into G s + G' s' + G' s" » 
mn — l' I'm — nn nl' — mm 
I'm — nn' nl — m mn - ll' 
nl' — mm mn - ll' Im - n'“ 
into G' G" s + G" 2 G s + G G' s"~ > 
w 
p Q ( 2 
q" p' q 
<1 2 P 
into G 
*t* 
G ' u + G" v > 
The second result, however, is seen to follow from the first, and 
a fourth from the third by the previously enunciated theorem of 
this kind. 
A,B,C a,/3,7 A , B , C . a , 0 , y ^ A ,B,C < d ' , 0' , y" 
a , j8 , 7 x ,y , z a, 1 8,7' x , y , z a" , 0 ' , 7" x , y , z 
_ a , b . a 
~ X ,y , z • 
