448 Proceedings of the Royal Society of Edinburgh. [Sess. 
XXIX. — Boole’s Unisignant. By Thomas Muir, LL.D. 
(MS. received January 17, 1911. Bead February 20, 1911.) 
1. In two memoirs on the “ Theory of Probabilities,” * Boole was led 
to the consideration of a peculiarly interesting form of determinant which 
has only positive terms in its final form of development, and which in the 
case of the fourth order may be written 
v 
0V 
'V* 
0Y 
0Y 
Y 
dx 
V dy 
2 dz 
0Y 
0Y 
0 2 Y 
, 0 2 Y 
X dx 
X dx 
ih LJ 
y dxdy 
XZ - — — 
dxdz 
ev 
0 2 Y 
0Y 
0 2 V 
V by 
XV 
y dxdy 
y lT 
dy 
yz 
y dydz 
0Y 
0 2 Y 
0 2 Y 
0Y 
xz dxdz 
2 02! 
V standing therein for 
axyz + byz + czx + dxy + ex +fy + gz + h . 
In the first of the two memoirs the determinant is not explicitly 
referred to, but in the second it receives considerable attention, Boole, 
indeed, there saying that the memoir “ involves discussions relating to the 
properties of a certain functional determinant, and to the possible solutions 
of a system of algebraical equations of peculiar form, discussions which 
will, I trust, be thought to possess a value as contributions to mathematical 
analysis, independent of their present application.” The results, so far 
as determinants are concerned, occupy pages 235-240, and are summed 
up in a lemma and two propositions, the first proposition being of a general 
character, and the second concerning the special form just referred to. 
The object of the present note is to present these two propositions in 
an entirely fresh light, and to co-ordinate therewith some recent investiga- 
tions on similar functions.! 
2. Boole’s general proposition may be formulated as follows : If each 
element of an axisymmetric determinant be an aggregate of multiples of 
* Boole, G., “ On the Application of the Theory of Probabilities to the Question of the 
Combination of Testimonies or Judgments,” Trans. Roy. Soc. Edin., xxi. (1857), pp. 597- 
652. 
Boole, G., “ On the Theory of Probabilities,” Pliilos. Trans. Roy. Soc. Lond., clii. (1862), 
pp. 225-252. 
t Muir, T., “ A New Unisignant,” Messenger of Math . , xl. pp. 177-192. 
