236 
PEOFESSOE BOOLE ON THE THEOEY OF PEOB ABILITIES." 
1st. That all its elements are linear homogeneous rational functions of certain quanti- 
ties a, h, c, Sc., imlimited in number. 
2ndly. That if the coefficients of any one of these quantities a in the elements of any 
jgarticular line or column taken in order are a.^, and in any other line or column 
/la, ...j3„, then these tico series of quantities are resjgectively propoidional. 
3rdly. That the prineipal terms Aj, A^, ...A^ are 'positive, i. e. that the coefficients of all 
the quantities a, h, c, &c. which appear in these terms are positive. 
Then the developed determmant will be itself positive, and will consist of products of 
the quantities a, b, c, &c. without powers, each product affected by a 'positive sign. 
First, it may be observed that any letter a of the set a, b, c... which appears in the 
subordinate term Aij will appear in both the principal terms A^, Aj. 
For let m be the coefficient of a in A,y, and therefore also in Kji ; let I be the coeffi- 
cient of a in Ai, and n its coefficient in Aj. Thus to the elements A^, Aj^ in the z-column 
correspond Aji, Aj in theji-column. Hence, by the definition of the determinant, 
l:m::m:n, 
w?=ln, 
which implies that neither I nor n vanishes, so that a appears in A^ and Aj. 
Secondly, we shall show that the determinant can, without alteration of its final deve- 
loped value, be reduced to a form in which any letter a of the system a, b, c ... shall 
appear in only one system of elements, and therefore only in the principal term of that 
system, since every subordinate term is common to two systems. 
Let us suppose a to be contained in two at least of the systems of elements, and for 
convenience of expression, let these be the 1-system and the 7^-system. Let, then, 
«!, a ^,... be the successive coefficients of a in Aj, A 21 , • . . A„i, and therefore, by definition 
of the determinant, Xoji, ... Xa„, its coefficients in A„i, A„ 2 , ... A„. Any of the quanti- 
ties a,, a^, ... a„may be 0. But by the Lemma above demonstrated the determinant may, 
without alteration of value, be reduced to the following form, viz. : — 
? .^125 ... Ajn XAj 
.^ 21 ? .^2 ? • • . 
A„i — xAj, A„2 — ^Ai 2 ... A„ — 2?A.„i+X®Ai 
(B.) 
Now in the determinant thus transformed the quantity a will no longer occur in the 
w-system. 
This is obvious with respect to the subordinate elements of that system. With respect 
to the principal element, we observe that the coefficient of a is 
in Ai, equal to 
in A„i, equal to \a^. 
in A„, equal to X or 
whence the coefficient of a in A„ — 2x„i-l-A*Ai is equal to 0. 
