271 
1909-10.] Dr Muir on the Theory of Orthogonants. 
respectively, and where by reason of the range of the two 2’s each element 
is the sum of n 2 binary products. Any said element may thus be represented 
as the product of two rows of n 2 elements each, and a little examination 
shows that only n rows of the latter kind are necessary for the representa- 
tion of all. In other words, the array of s’s can be represented by the 
product obtained by multiplying the array 
off 
n (0) 
w 12 . . . 
a (0) 
U 1 n 
n (l) 
a 2l 
„(°) 
u 22 
n (0) 
a 2n 
a m 
a 3 1 • • 
• • u nn 
<> 
ag/ 
a (1 > 
u ln 
a 2l 
n m 
u 2 2 * • 
a {1) 
• ^271 
n W 
a 3l . . 
o (1) 
. . u nn 
air 1 ' 
ag- 1) . . . 
a l n 
m 21 
n (n- 1) 
«22 • • 
n (n- 1) 
• a 2n 
„(n- 1) 
a 31 • • 
a (n-l) 
• • u nn 
by itself, and therefore is, by Binet’s theorem, expressible as a sum of 
squares. 
By way of illustration, Borchardt takes the case where n — 3. The 
product of the squared differences of the roots is then, in later notation, 
1 
1 
. 
1 
a 
h 
9 
h 
b 
/ 
9 
f 
c 
r i r i 
r l r 2 
r l r Z 
r 2 r l 
r 2 r Z 
r 3 r i 
r 3 r 2 
r z r z 
where r a means the product of the a th and /3 th rows of 
a li g 
h b f 
9 f c • 
By performing on the 3-by-9 array the operation 
row 8 - (a + b + c) row 2 + ( ab + bc + ca -f 2 - g 2 - h 2 ) ro w x 
there is obtained 
1 
. 
1 
. 
1 
a 
h g h 
b 
/ 
9 
/ 
c 
A 
H G II 
B 
F 
G 
F 
C 
which is readily shown to be equal to Summer’s sum of squares. 
It is a little curious that Borchardt nowhere draws attention to the fact 
that the determinant of the coefficients in the right-hand members of his 
m th set of equations is the m th power of the determinant of the corre- 
sponding coefficients of the original set. 
