449 
1910-11.] Dr Muir on Boole’s Unisignant. 
one and the same series of variables, the multipliers in the diagonal 
elements being all positive, and if the multipliers of any one of the 
variables in any row be in order proportional to the multipliers of the 
same variable in any other row, then the final development of the 
determinant contains nothing but positive terms. 
The variables being denoted by a,b, c, ... and the multipliers by Greek 
letters, the first column of the determinant must be of the form 
< xyct + cufb + a 3 c + ... 
(3-pt + f3 2 b + /3 3 c + ... 
7i a +72 b + 7s c + 
where we have only to bear in mind that a ± , a 2 , a 3 , ... are positive. 
This being the case, the axisymmetry and proportionality posited in the 
enunciation entail that the whole determinant must be 
a 1 a+a 2 b+a 3 c + ... A a + /3 2 b + /3 s c + ... yya + y 2 b + 73 ^ + ••• 
P 1 a + /3 2 b + P z c+ ... &? a +££b +^c + ... bh a + ^hb + ^ y M + ... 
a, a 2 a 3 a 2 a 3 
y 1 a+y 2 b+y 3 c + ... Siha + Pith + + ... + Z& + l s -c + ... 
a l a 2 a 3 a i a 2 H 
where the order is restricted to the third merely for shortness’ sake in 
writing. This, however, is seen to be the product, 
a Y a 
af) 
a g c 
a yd 
1 
1 
1 
1 
@2 b /3 3 c 
Pi d . 
• 
A 
a l 
A 
a 2 
A 
a 3 
A 
a 4 
yi« 
if 
73 c 
7i d ■ 
7i 
a i 
72 
a 2 
73 
a 3 
74 
a 4 
and therefore by Binet’s theorem 
a -pi 
a 2 b 
a 3 c 
1 
1 1 
a,a 
a 2 b 
a 4 d 
1 
1 
1 
P 2 b 
• 
A 
a l 
@2 A 
a 2 a 3 
+ 
Pl a 
fi 2 b 
M 
A 
a i 
A 
a 2 
A 
a 4 
7i a 
7 2 b 
IS 
— 
a l 
7_2_ 7s ! 
a 2 a 3 
7l a 
y 2 b 
7i d 
7i_ 
a i 
72 
a 2 
74 
a 4 
= abc l-ifesl! + gbd I “iAsYi 1 2 + .... (I.) 
a l a 2 a 3 a l a 2 a 4 
3. The advantage of this proof does not lie merely in its brevity ; for 
we have to note that whereas Boole’s three pages of the Philosophical 
Transactions establish only the fact that the terms of the final develop- 
ment must be all positive, what is here obtained is the actual positive 
terms themselves. 
VOL. XXXI. 
29 
