640 



THOMAS MUIR ON DETACHED THEOREMS ON CIRCULANTS. 



we have 



C (a b c d e) ■■ 



e + coa + co 2 b + oo s c + co 4 d a 



d + u>e + wa + w 3 & + o) 4 c c 



c + wd + co 2 e + co 3 a + oo 4 b d 



b + wc + w 2 d -f w 3 e + co 4 a c 



= (« + wb + uPc + u?d + w 4 e) 



c 



d 



c 



b 



c 



d 



a 



b 



c 



e 



a 



b 



d 



e 



a 



1 



b 



c 



d 



e 



CO 



a 



b 



c 



d 



CO 2 



e 



a 



b 



c 



O) 3 



d 



e 



a 



b 



e a 



= (a + wb + w 2 c + ooU + a)V)(A + wE + co 2 D + w 3 C + eo 4 B) , 



as was to be shown. 



3. If the linear factors of a circulant, say of the fifth order, be a, ft, y, . . ., 

 and the complementary minors of the elements of the first row be A, — B, C, — . . ., 

 then 



C(a(3yS, a/3ye, a/3Se, aySe, fiySe) = 5 5 ABCDE . 



Let us denote by p one of the imaginary fifth roots of unity, the other roots 

 being p 2 , p 9 , p 4 , 1, and we have from § 2 



C(a,b,c,d,e) = (a + w b + co 2 c + w 3 d + w 4 e) (A + co E + W 2 D + a> 3 C + » 4 B) , 

 = (a+ w 2 b+w 4 c + co d+oo 3 e) ( A + w 2 E + « 4 D + w C + W 8 B), 

 = (a + w s b + w c + w 4 d + co 2 e) (A + w 3 E + w D + o) 4 C + w 2 B), 

 = (a + oo 4 b + u> 3 c + w 2 d + w e) ( A + a> 4 E + co 3 D + a) 2 C + w B), 

 = (a + b+ c+ d+ c) (A+ E+ D + C+ B). 



From these by division there result 



A + co E + w 2 D + a) s C + o) 4 B = /3y^\ 

 A + a> 2 E + a) 4 D + co C + a) 3 B = <yc5ea| 

 A + a) 3 E + ft) 1 D + a, 4 C + arB = <5fa^ 

 A+ft>*E+ft) s D+a) 2 C+« 1 B = ea J 8v 

 A + E + D+ C+ B = aPyS). 



Calling the right-hand members here a', I', &, d', e', we see readily that 



a'+ b'+ c'+ d'+e' = 5A\ 

 oo a + co 2 b' -f a>V + w 4 d' + e = 5B 

 u>a' -f co'i' + a>V + cohl' + c— 5C 

 uPa' + u? V + a> V + wW + e' = 5D 

 a> V + cfb' + W V + w rf' + e' = 5 E / , 



