PEOFESSOE BOOLE ON THE THEOET OF PEOBABILITIES. 
235 
Analytical Theorems relating to Functional Determinants and Systems 
of Algebraic Equations. 
A symmetrical determinant may be conveniently expressed in the form 
the conditions of symmetry being 
A. 
Ai2 
• • • A]„ 
A21 
A2 
• • • 
......... (1.) 
A„i 
A „2 
... A„ 
A, 
— A^-;, 
Aii=Ai. 
It is desirable to employ fixed language in referring to this. We shall therefore call 
the quantities Ai, Aa- ..A„ the ‘principal elements,’ and the diagonal series of terms 
which they form the ‘ principal diagonal.’ The elements A,y, when ^ andy differ, we 
shall call ‘ subordinate elements.’ The element A^, together with all the subordinate 
elements which occur upon the same horizontal or vertical line of the determinant, we 
shall designate the ‘^-system of elements ’ Lastly, in comparing two rows or two 
columns of elements together, those elements will be said to correspond which occupy 
the same numerical place in their respective rows or columns. 
The following Lemma will next be established. 
Lemma . — A symmetrical determinant expressed in the form (I.) will be unaltered in 
value, if from each subordinate element of its ^-system we subtract the corresponding 
element of its ^-system multiplied by a quantity X, which is invariable for the same 
system, — and for the principal element A^ substitute A^ — 2A.A,y4-A^A^-. 
It is known that a determinant vanishes if two of its lines or columns are iden- 
tical, and it is known as a consequence of this that if from a particular line or column 
of a determinant the corresponding elements of another line or column, multiplied each 
by the same constant, are subtracted, the determinant is unaltered in value. From the ^th 
line of the above symmetrical determinant subtract, term by term, X times the yth line, 
and then from the rih column of the resulting determinant subtract X times the yth 
column. As respects any subordinate element, the result will obviously accord with the 
statement in the Lemma. But the element A^ will be successively converted into 
Ai — >A.ji 
{Ai — XAji) — X{Aij — XAj). 
The last expression, since A,-i=Aij, is reducible to 
Ai — TkAij-\-^Aj. 
Upon this property the demonstration of the following general proposition will be 
founded. 
Proposition I. 
Let the symmetrical determinant {!.) possess the following properties., viz .’. — 
2 I 2 
