PEOFESSOE BOOLE ON THE THEOET OE PEOBABILITIES, 
239 
Peopositiojst II. 
If Y he any rational entire function of the n variables x^, ... x^, hut involving no 
powers of those variables above the firsts and if further^ all the different terms of V have 
positive signs, then the determinant 
V 
V, 
V, . 
.. V„ 
V, 
V, 
V,, ., 
.. V.„ 
V. 
V,, 
V3 . 
.. 
.. V,, 
in ivhich V,- denotes the sum of the terms in V which contain x^, and the sum of the 
terms in V which contain Xi, Xj, will, when developed as a rational and entire function of 
x^, X2, . .. x„, consist wholly of terms with positive coefficients. 
From the definition it is plain that in general 
y _y y .=y. 
whence the above determinant is symmetrical. 
Again, all its elements are homogeneous linear functions of the terms in V. 
Again, if a, a^, a.^, . . . a„ represent the successive coefficients of any one of the terms 
of V in any row or column of the determinant, and /3, j3i, . . . j3„ the successive corre- 
sponding coefficients of the same term in any other row or column of the determinant, 
the one series of coefficients shall be proportional to the other. 
Let us compare the first column and the ^-column headed with the element V^. 
Selecting any term in V, suppose it to contain Xi, then in whatever element of the first 
column that term is found, it will be found in a corresponding element of the ^-column, 
and in each case with unity for its coefficient, since all the elements are mere collections 
of terms from V. But when it is not found in a particular element of the first column, 
it will not be found in the corresponding element of the ^-column. The entire series of 
coefficients in the one being then the same as that in the other, the common ratio of 
the corresponding terms is unity. 
Suppose, secondly, that the proposed term is found in V and not in ; then in all 
the elements of the '^-column its coefficient is 0, so that the series of coefficients in the 
4-column might be formed from those in the first column by multiplying the latter 
successively by 0. This again represents a common ratio. 
The same reasoning may be applied to the comparison of any two columns of the 
determinant. Thus in comparing the 4-column and the ^-column : — terms of V which 
contain both Xi and Xj will be found in corresponding elements of both columns — terms 
which contain Xi but not Xj will be wholly absent from the ji'-column. Thus in all cases if 
a, «„ a^, . . . «„ represent the coefficients of a term of V in one column, its coefficients in 
any other column, taken in the same order, will be of the form Xa, Xa ^ . . . Xct„, the 
coefficient X being either 1 or 0. 
■ Lastly, the principal elements consist, as do all the elements, of positive terms. 
