Note on Unimoclular and other Persymmetric Determmants. 97 
(5) The result in §3 is identical with the last of the three above 
mentioned when the latter has been corrected as to sign. Strictly speaking, 
therefore, Mr. Datta's contributions are two in number, namely, one regarding 
P {an, aii-u ' ■ ' . , ai, ao, . . . . , au-\, a,i) 
printed on p. 201 of his paper with minus instead of plus signs in the last 
frame-line of the second pfaffian ; and the other regarding 
P (a,i, . . . . , ai, ai, ao, . . . . , an-i) 
as established above. 
(6) In the case of the second mode of resolution, namely, into deter- 
minants, the factors are accurately given, serving indeed as a check on the 
pfaffian forms. I would only note in this connection that a companion to 
Zehfuss' theorem regarding the resolution of centrosymmetric determinants 
applies directly to P a^^-i, . . . , ai, ai, a^, . . . , an-\), namely, the theorem 
that any n-line determinant having the array of its last n-1 rows centrosym- 
metric is expressible as the jjroduct of two determinants ; for example, when 
the order is even— 
' U V V) X 
lu X y z 
ai a.i a- ctg 
C'j^ Ok) C\}^ C'!^ 
^5 ^4 ^3 ^3 ^1 
dg (Xg a^ a^ a.y 
Co 
Co 
h + ^1 ^5 + ^3 + ^3 
ag + a^ a- -\- a^ a^ + a.^ 
and when the order is odd — 
u — z V 
% — ^5 — ^3 — % 
y w 
X 
U V 
IV X y 
rtj a.2 «3 a^ a- 
^1 h h h ^5 
h- 63 h 
a- a^ a.^ ao a 
— ^1 h, — ho 
a- — a^ 
a. 
ao 
V -\- X u -\- y 
3 aJ^ -f- ao a- + i 
h h + h h + ^1 
(7) Unconnected with the foregoing save in that the subject is still 
persymmetric determinants are the three following results : 
P(l„, 81, So, 73,....) = !, 
e 
V t' — j ^' 
P (lo, li, 2„ 3., 4o, 53, 63, . . . ) = 1, 
where n stands for r!/s! (r — s) ! The first two determinants are seen to 
P (h ^ 
U' 3' 
