50 
Proceedings of the Royal Society of Edinburgh. [Sess. 
It is readily seen that the left-hand member 
1 1 1 
2 b 
-2c 
and that therefore the right-hand member ought to be 
( - l) n 2 n ~ x abc. ..(- + £+-+ . . . Y 
\a b c J 
where n is the number of variables. 
Boole, G. (1862, June). 
[On the theory of probabilities. Philos. Transac. R. Soc. London , clii. 
pp. 225-252.] 
An important part of Boole’s investigation turns upon the solution of 
a peculiar set of linear equations, and he consequently devotes considerable 
space (pp. 235-240) to an examination of the determinant of the set. 
As the determinant is axisymmetric he begins by establishing the pro- 
position which results, in regard to such determinants, from subtracting 
X times the j th row from the i th row, and thereafter X times the j th column 
from the i th column. This operation does not do away with the axi- 
symmetry, and the diagonal element a u is thereby changed into 
a u — 2 Xa { j + h 2 a,jj. 
Following this lemma and made in part dependent on it comes the 
rather notable proposition that if an axisymmetric determinant have all 
its elements of the form 
Xcj + fLb + vc + . . . . , 
and the coefficients X, /ul, v, . . . in all the diagonal elements be positive, and 
generally be such that all those joined to any one of the variables in any 
row be in order proportional to the coefficients of the same variable in any 
other row , then the final development of the determinant will contain only 
positive terms. The proof is disappointingly lengthy, occupying very 
nearly three pages (pp. 226-238). 
The other proposition, which is a deduction from this, is to the effect 
that if ¥ be a rational integral function of n variables x 1? x 2 , . . . x n , having 
no powers of them above the first and having all its terms positive, then 
the final development of 
