642 THOMAS MUIR ON DETACHED THEOREMS ON CIRCULANTS. 



This is equivalent to saying that the determinant is equal to 



[2'V + 2 cos 36° (fl/f 2 + a 2% + a s a A + a i a $ ~ a b a \) + ^ cos 72° (ctjCi^ + a 2 a 4 + a s a 5 — « 4 a x — a s a 2 

 - 2V - 2 cos 36° (6^ + b.J) 3 + b l b i + bj) 5 - b 5 \) - 2 cos 72° (6^ + b 2 b t + b s b 5 - bfo - b 5 b 2 )] x 



[2«j 2 — 2 cos 72° (a x a 2 + « 2 a 3 + a 3 a 4 + a 4 « 5 — a & a x ) — 2 cos 36° (a x a 3 + a 2 a 4 + a 3 a 5 — a 4 a x — « 3 a 2 



-SV 2 + 2C0S 72° (V> 2 +¥3+¥4+¥ 5 -¥i)+ 2cos36 ° (¥ 3 +¥4+¥ 5 -¥i-¥i)] x 



[« 2 — « 2 + « 3 - « 4 + « 5 + \ — & 2 + &3 - &4 + & 5 ] . 



where the law of signs in 



a x a 2 + a 2 a 3 + a 3 a 4 + « 4 « 5 — a 5 a x and a x a 3 + a. 2 a t + « 3 « 5 — a i a x — a 3 a 2 

 is made clear by noting that each expression is got from two rows of 

 C (a v a 2 , a 3 , a±, a 5 ), the former being (a x , a 2 , a 3 , a p a 5 ,$—a 5 , a v a 2 , a B , a 4 ) and the 

 latter (-« 2 , — « 3 , -« 4 , ~a 5 , c^— « 3 , -a 4 ,« 5 , — 04, Og). 



This theorem is established exactly as its analogue in ordinary circulants. 

 (See Messenger of Math., xi. pp. 105-108.) 



6. One of the hardest problems connected with circulants is the finding of 

 the final expansion of the circulant of the nth order. Anything that has been 

 done towards a solution will be found in the following papers : Glaisher, Quart. 

 Journ. of Math., xv. p. 354, xvi. p. 33 ; Mum, Quart. Jouru. of Math., xviii. 

 pp. 176, 177; Forsyth, Mess, of Math., xiv. pp. 43-46.; Muir, Mess, of Math., 

 xiv. pp. 169-175. 



One plan which occurred to me of determining the law of the coefficients 

 was to try to hit upon a determinant of some more general form than the 

 circulant and having an easier law of formation for its final expansion, and then 

 to specialise. The determinant which seemed to offer fairest promise is 

 exemplified by 



aa bf3 cy dS ce 



e/3 ay bo ce du 



dy cS <>e bu c/3 



cS de ea a/3 by 



be ca d/3 cy aS 



It evidently degenerates into the circulant C(«, b, c, d, e) when a = fi = y=:S=:e. = l, 

 and, what is of importance, the letters a, /3, 7, S, e are themselves introduced 

 in cyclical fashion, the determinant, in fact, degenerating into the circulant 

 C(a, /3, 7, 8, e) when a = h = c = d = e — 1. This determinant I find equal to 



I a 1 + &6 _|_ c 5 +d 6 + e : ')a/3ySe 



- (a 3 bc + bha + chlb + d?ec + (Pad) (a s yS + /3 3 <5e + y'eu + (S ;i a/3 + e :, /3y) 



- (pficd + bhh + cha + d*ab + e%c) (a 3 /3e + /3 3 ya + y 3 <$/3 + S'ey + f?a8) 



+ \aWd + b\*c + c*d*a + Ah% + e*a?c) (a 2 /3y 2 + /3-yS 2 + y 2 Se 2 + S W + e 2 a/3') 

 + (r< 2 k 2 + Wed 2 + c 2 de 2 + d 2 ea 2 + e 2 ab 2 ) (a 2 /3 2 S + /3 2 y 2 e + y 2 c5 2 a + S'e 2 /3 + eVy) 



- 1 Oabcdeaftyde, 



