1915-16.] The Theory of Circulants from 1880 to 1900. 153 
Weihrauch, K. (1880). 
[Ueber doppelt-orthosymmetrische Determinanten. Zeitschrift f. Math, 
u. Phys., xxvi, pp. 64-70.] 
Weihrauch establishes Spottiswoode’s property by using Fiirstenau’s 
theorem of 1879; that is to say, he multiplies each element in the (r, s) th 
place by co r ~ s , where go is any one of the n th roots of unity, and then takes 
the sum of the rows. 
More interesting is the fresh proposition that the circulant of a ± ,a 2 , . . . , 
a n , is equal to the sum of the coefficients of the equation whose roots are the 
n th powers of the roots of the equation 
a-pc n ~ l + a 2 x w ~ 2 + . . . + a n = 0 . 
By way of proof it is noted that in the case where n is 4 we have 
C (a , b , c ,d) 
= (cl + ba)^ + c co 2 + + bw 2 + cu>2 "l - d(o\ j)(cj + &oo 3 "I" doi\^(cL + 6o) 4 + coi\ + dc^j , 
so that, if be the roots of the equation 
ax 3 + bx 2 + cx + d = 0 , 
there results 
C (a ,b , c jd) 
= a( 1 - «i£i)(l - <*q£ 2 )(l - «i£ 3 ) 
. a(l - <o 2 Q(l - <o 2 C 2 )(1 - w 2 4) 
. a(l — co 3 £i)( 1 — a> 8 f 2 )(l — w 3 £ 3 ) 
. a(l - w 4 Ci)(l - w 4 £ 2 )(l - w 4 £ 3 ) 
= clP( 1 - «i£i)(l - a) 2 f x )(l - co 3 ^)(l - w 4 ^) 
. (1 - 0,^(1 - co 2 C 2 ) (1 - a> 3 £ 2 ) (1 - a> 4 £ 2 ) 
• — — w 2^3)(^ — w 3^3)(^ — 
. 1 Ia 4 (l-£)(1-©(1-S), 
as desired. 
Instead of this, however, Weihrauch in effect proposes to substitute a 
wider proposition, namely, that the equation whose roots are the 4 th powers 
of the roots of the equation ax 3 + bx 2 + cx + d = 0 is 
abed 
b c d ay 
c cl ay by 
d ay by cy 
= 0 , 
the previous result being thence obtained on putting y = 1 . To establish 
this y * is substituted for x in the given equation and rationalisation is 
