THOMAS MUIE ON DETACHED THEOREMS ON CIRCULANTS. 641 



and therefore by multiplication 



C(a',b',c', d', = 5 5 ABCDE, 

 as was to be proved. 



4. If the elements of the first row of a circulant of the nth order be multiplied 

 by « n , w n ~ 1 ) . . . . } w respectively, the elements of the second row by w n-1 , w' 1-2 , . . . , 

 », o n respectively, the elements of the third row by « n ~ 2 , w' 1-3 , . . . , w, w n , w"" 1 

 respectively, and so on, ivhere « is any nth root of unity, the circulant is unaltered 

 in value. 



If we take the circulant as thus changed outwardly, and multiply all the 

 elements of the second row by o>, all the elements of the third row by co 2 , and so 

 forth, the elements of the first column will have &>" for a common factor, the 

 elements of the second column will have m*" 1 , the elements of the third column 

 a) 0-2 , and so on. We thus can strike out of the columns the very factors we 

 introduced into the rows (with the immaterial addition of &>", i.e., 1), and leave 

 the circulant as it originally stood free of u> ■ — which proves the theorem. 



5. If we take two skew circulants, of the 5th order say, C'(a v . . . ,a 5 ), C'^,. . . ,6 5 ) 

 and write the first in the form 



— «r 



— a A — cu 



— «o — CI, 



do 



■ a* — a. — cu 



and the other with its rows in reversed order, then the determinant whose every 

 element is the sum of the corresponding elements in these two determinants, 

 that is to say, the determinant 





a i — h 



«2~ & 3 



a — b, 



u i -b b 



«„-&! 





-a 5 -h 



«j — b i 



a 2 -b 5 



a s + K 



a 4 +6 2 





-"-i-h 



-a 5 -& 5 



d x + \ 



a. 2 + b 2 



<?>3 + h 





-«3~ & 5 



-«4 + & l 



-a 3 + b. 2 



a x + b 3 



tt-2 + K 





— «2 + &l 



— «3 + &2 



-«A + h 



-«5 + &4 



p l + h 



for a factor 













(a 1 +a>~ 1 a 2 +(o~ 2 a s +to 3 « 4 + co 4 « 5 ) (« 1 + wa 2 + a) 2 « 3 + co 3 « 4 + co 4 rt 5 ) 

 -(& 1 + ft ,-i& 2 +a)- 2 &3 + ft)- 3 & 4 + &)-*& 5 ) (& 1 + aA 2 + <A + co 3 & 4 + a> 4 & 5 ) 



where <o is an imaginary fifth root of — 1, the linear factor which remains being 



a 1 — a 2 + a 3 -a i -\-a b + b 1 -b 2 + b 3 — h i -\-b b . 



