172 Proceedings of Royal Society of Edinburgh. [sess. 
lead to the result 
x, x. 
y_x_ , v£_ . y£_ 
Gf Gf G/' 
(4) Combining this with the result of § 2, we have the general 
theorem : — 
The orthogonal substitution which changes 
( x i > x 2 » x 3 )( m )( x i » X 2 ? x 3 ) into ^lYi 2 + G 2Y2 2 + G 3 y 3 2 
will change 
( x i 5 x 2 1 x 3 )( mp )( x i > x 2 5 x s) into G i p yi 2 + G 2 p y2 2 + G 3 p y 3 2 
where p is any integer , positive or negative. 
(5) Since G 1? G 2 , G 3 , are the roots of the equation 
«12 
®21 
a 22 
a s \ 
a 32 
Cl 23 
a 33 — x 
= 0, 
it is at once suggested from § 4 that the equation whose roots are 
the p th powers of the roots of this equation is got by substituting 
for a n , a l2 , . . . , the corresponding elements of the matrix 
which is the p th power of 
( a i2 a u ) 
I ®21 ^22 ® o 3 
! 
I CL 3\ a S 2 a 33 ! 5 
a theorem first formulated by Sylvester in 1852 (v. Nouv. 
Annates de Math., xi. p. 438). 
(. Issued separately March 17 , 1904 .) 
