248 
PEOFESSOE BOOLE ON THE THEOEY OF PEOBABILITIES. 
Now let a.jpt+aajpa • • • +««i>«+ 5 = 0 . 
This, as is positive, is to suppose that coincides with one of its own inferior limits. 
We must suppose this to be the highest of those limits, as otherwise some of the other 
limiting conditions would be violated. Now, since all the terms in H are affected with 
positive powers of ^i, while those in K do not contain the first equation of the system 
(6.) will be satisfied by Xi=0, provided that the remaining n — 1 equations give finite 
positive values for ^ut the vanishing of Xi reduces these equations to the form 
K3_ K„_ 
K K • K — 
( 7 .) 
It is therefore necessary to show that jp^, in this system are actually subject to 
the conditions to which the application of the method of Proposition IV. to the system 
itself would lead. 
The n quantities ^i, ^3 . . ^^re by hypothesis subject to the conditions furnished by 
the application of the method of Proposition IV. to the original system ( 1 .). In 
applying this method each of the original equations yields an equation of the form 
^1+^2 • • ■ ( 8 -) 
and to the equations thus formed are added the inequations 
^1 + ^2 • • • 
Xj > 0, ^2 > 0, . . . > 0 ; 
Xj, X2 . . . X^ having reference to the whole system of original equations. 
Now the satisfaction of the equation 
H 
H + K 
=0 
by the value a^, = 0, involves the vanishing of all those quantities of the system Xj, X2 . . . X^,, 
which are derived from terms in V that are also found in H. Hence the X quantities 
that do not vanish are those derived from terms in V which appear in K. 
Again, the condition 
«iPi+«2P2 • • • +«„Pn+^=0 
shows that the system of equations of which (8.) is the type are not independent. They 
must, under the particular circumstances of the case, be such that the above equation 
shall be derivable from them. Hence one of these equations may be rejected. If we 
reject the first, viz. the one which contains and then reduce the others by making the 
X quantities which are not derived from K to vanish, the system typified by (8.) evidently 
reduces to the system which we should have to employ if we applied the method ot 
Proposition IV. directly to the system of w— I equations ( 7 .). Hence the quantities 
P21 Pa? • • -Pa satisfy the final conditions to which that application would lead, and therefore 
by hypothesis the equations ( 7 .) admit of solution by a single system of finite positive 
values of^a, 
Now in general 
Oi 
Xf—X^ 
