246 
.PEOJFESSOE BOOLE ON THE THEOET OE PEOBABILITIEB. 
W 2 , CC 3 . . . Xn corresponding to the assumed positive finite value of x^. And these values 
V " . . V 
together make ^ ^ positive proper fraction. We may notice that, representing 
under the form ^ ( 
+ B’ 
it cannot be thg^t either A or B is wanting so as to reduce y to the value 0 or 1. For 
if A w'ere wanting, V would not contain x^^ at all, as by hypothesis it does; and if B 
were wanting, V w'ould contain x^ in every term. Thus Xy would divide out from the 
system (1.), which would thus become a system of w equations between n—1 variables, 
and would cease to be independent, as by hypothesis it is. 
But when a;i=0, or 0 ^ 1 = infinity, the form of V, considered as a function of x^, X 3 . . . x„y 
will not generally be the same as in the case last considered ; and the conditions con- 
necting will no longer be such that we can affirm the possibility of deducing 
from the last n—\ equations of the system (1.), as transformed, positive 'finite values of 
X<yi X^^ . . . Xjy. _ 
The theory of this case depends upon a remarkable transformation. 
The most general form of the inequations of condition connecting ^ 2 ? • • • Pn? 3s 
determined by Proposition IV., is 
^lPl'“]~<J52p2 • • • . (3.) 
Hence, from the nature of the system (1.), it follows that the function 
^ ........ (4.) 
must consist wholly of positive terms. Therefore V must consist of terms which would 
either appear in the development of the above function with positive signs, or not appear 
in it at all. Let Ax^x^Xt be any term of V. Then, as the coefficient of this term in 
(4.) would be > 
(X/,.A-^(i^A-\-(ZfA . . . -j-JA, 
and as A is positive, we have 
+ 0 , 
a general condition which determines not what terms have actually entered, but what 
could alone possibly have entered into the constitution of V. 
From the system (1.) we have 
+ +<^nVn+'^V J 
V =«iPi+«2P2 • • • 
Hence if we write 
«iVi-l-«2V2 . . . d-<3^reV„-l-^V =H, 
We have • 
H 
Y =®lPl+®2P2 • • • 4*<^nPn‘i'^5 (5.) 
an equation by which W'e may replace any one of the equations of the system (1.), and 
