1888-89.] Dr T. Muir on the Theory of Determinants. 
409 
K 
cos . 
67 r 
. 2 t r\ 
( ^ 7 — ; 
. 47T' 
sin — ) , 
, ( cos . — 4- J - \ 
. sin — 
n J 
\ n v 
n 
. 67 A 
sm — J . 
n J 
COS 
-( 
(2n— l)7r — . 2(n-l)ir 
v J-l.sm 1 
1 v 0 
)) 
multiplied by the zeta-ic phase 
£ e _£PD(0aZ>c. . . Z)!!” 
Unfortunately the meaning of the proposition is seriously 
obscured by misprints and inaccurate use of symbols. Instead of 
u r th » d e g ree we should have Z th degree ; £ preceding R* ( abc ... Z) is 
meaningless, and should be deleted ; £ preceding ^PD (0 abc . . Z) in 
the first member of the identity is unnecessary when a £ has already 
been printed at the commencement ; and the subscript e, although 
giving an appearance of greater generality, serves no purpose what- 
ever. Making the corrections thus suggested, and denoting 
2 tt — - . 2ir 
cos J-l sin 
n n 
47 T . . 47T 
cos- — + J - 1 sin — , 
n n 
which are the roots of the equation 
x n ~ l + x n ~ 2 + x n ~ s + ... +^+1 = 0, 
by a, /3, y, .... A., we are enabled to put the theorem in the more 
elegant form 
£ |R t (a,b,c . . . , Z) . S . PD(0,a,6,c, . . ., Z)]- 
=Z-,{M<hP>y. • • • . x).a.PD(0,a,6,c, • • 0} 
It is readily seen to be a generalisation of the first theorem of 
the paper, into which it degenerates when instead of being any 
function of a,b,c, ... Z, is a constant, and R*, instead of being 
any symmetric function, is one of the series 5a, 5a5, 5a&c, .... 
As, however, the constant R f (a,/?,y, ... A.) on the right-hand side 
will then be one of the series 5a, 5a /3, 5 a/3y , .... and will not 
therefore be + 1 unless when t is even, there must be an inattention 
to sign in one or other theorem. The matter can be more appro- 
priately inquired into when we come to the subject of alternants, 
because, as has been pointed out in a recent footnote, it is to this 
