PEOFESSOR BOOLE ON THE THEORY OE PROBABILITIES: 
247 
which has the peculiarity that for every term. x, Xf. . . which appears in the numerator 
H the particular condition 
. . . -{-5> 0 
is satisfied. 
Let K be the aggregate of those terms] in V for which the remaining particular con- 
dition 
. • -\-h — 0 
is satisfied. Then V=H+K. If we now substitute (5.) in place of the first equation 
of the systepi (d.)„and then write H+K for V, Hi-f-Ki for Vj, &c., the system becomes 
converted into the following one, viz. 
H I I 7 Hq-j-Kq 
h -^t k =i^2, H + ~ K =-^3, • • 
H„ + K" 
• H + K • 
. ( 6 .) 
Now let us transform the above equations by assuming 
^2 y^l ^3 3^3 •• • 'y»* 
The general type of these equations is 
Xi=:X"%, 
and it includes the particular case of ^=:l, provided that we suppose, as we shall do,, 
3'.=1- 
Then representing, as before, any term of V by AXr x^Xf . ., we have 
AXrX.Xt... —Ax, y^y.yt.,. 
Let this substitution be made in the difierent terms both of the numerators and deno- 
minators of the fractions which form the first members of the above system, and then 
h_ 
let each numerator and denominator be multiplied by x^'. The result will be the same 
as if for each term Ax,, x, Xt. . .in. numerator or denominator we substituted the term 
a,-^ag+at. , . +b 
A^i y,y,yt... 
In considering the effect of this transformation we will first suppose a, positive, and 
afterwards suppose it negative. 
Case I ; the coefl&cient a, positive. Here, since for all the terms in H and in 
Hj, Hg . . . we have 
a, + a +at...+b „ 
^ ’ 
all such terms in the transformed equations will be affected with positive powers of x,. 
And since for all terms in K, Kj, . . . K„ we have 
(it ‘ & Q 
all such terms in the transformed equations will be free from x,. 
