1911-12.] The Theory of Circulants from 1861 to 1880. 
145 
Three results of computation made in the course of the paper are also 
worth recording, but in a shorter notation, namely, 
0 0 0 
C (a ,b,c,d,e) = ^ a 5 - 5 ^ a 3 (pe + cd) + 5 ^ a(b 2 e 2 + c 2 d 2 ) - 5 abcde, 
C ^l~^ e = ±a*-±aH b + c + d + e ) + ±aW + c>) 
and 
lb c d e 
1 c - x d e a 
1 d e - x a b 
1 e a b-x c 
1 a b c d — x 
where, as before, 
4- 2« 2 (26c + 2bd - 3 be - 3cd + 2ce + 2 de) - ^ abcd y 
= x* - x 
! ( 2 2“ 2 ~ 2“) 
J C (a ,b,c,d,e) 
(L-\rb-\-C-\-d-\-e 
2 a3 (be + cd) = a B (be + cd) + b B (ca + de) + c B (db + ea) + . . 
Glaisher, J. W. L. (1878). 
[On a special form of determinant, and on certain functions of n 
variables analogous to the sine and cosine. Quart. Journ. of 
Math., xvi. pp. 15-33.] 
Part of this paper (§§ 8, 9, 19, 20) is a continuation of the paper of 
1872, and the other part a continuation of that of 1877-78. 
The proof that the product of two circulants is expressible as a circulant 
is effected by substituting Spottiswoode’s equivalent for each. Similarly 
there is established, though somewhat imperfectly, the fresh theorem 
C(tq , $2 j . . . ,Ct>2 n) ~ G(Bi , -B2 j • • • 5 b»i)> 
where 
Pi = , <*2 , • • . j d^n \ > — 0"2n J ®2«-l 5 • • ' 5 — ^ 2 ) 
f>2 = (ttj ) ^2 ’ . * • ) ^2 n $ ^3 J — a 2 > ’ * • • ’ — ^4) 
P« = j d>2 ) • • . j d 2n $ <^2n— 1 » — ^2n—2 ) ^2n—3 5 • • • ) ~ ^2n)> 
the special point to be noted being that since x 2n = (x 2 ) n the (2n) iYi roots of 
1 are the square roots of the n th roots of 1 and therefore may be grouped 
in pairs whose sums vanish. For example, such a pair of sixth roots of 1 
being 0,-6, the product of the corresponding pair of factors of 
C (a ,b ,o ,d ,e ,/) is 
(a + Ob + 0 2 c + 0 s d + 6*e + O b f)(a -6b + 6 2 c - 6H + <9 4 e - Of), 
the development of which we might obtain by changing it into 
(a + 6 2 c + <9%) 2 - (b + m + 6f) 2 6 2 
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