PEOFESSOE BOOLE ON THE THEOET OF PEOBABILITIES. 
249 
Hence since .Tj = 0 and is finite and positive for all values of ^ from 2 to n, we see that 
will be 0 for all values of ^ for which is positive, finite and positive for all values of 
i for which is 0, and infinite for all values of ^ for which is negative. 
Case 2; the coefficient negative. Here the inequation of condition (3.) must be 
supposed to determine the lowest of the superior limits of pj, and therefore when 
coincides with that limit we have 
The transformations remaining formally the same as before, the following results will 
present themselves. 
The terms in H and in Ha, Hg . . . H„ will be affected with negative instead of positive 
powers of Xy Hence the same determination of y^, y^. ,.y^ from the last 1 equa- 
tions of (6.), which before followed from the assumption a^i = 0, will now follow from 
the assumption x^-= oo, which at the same time satisfies the first equation of (6.). 
The equation 
shows, since «, is here negative and x^ infinite, that Xi Avill be infinite for those values of 
i for which is negative, finite for those values of i for which «,■ is 0, nothing for those 
values of ^ for which is positive. 
In all these cases the values 0 and oo appear as limits of finite positive values. This 
results from the connexion of the second member of the first equation of the system (6.) 
with the condition (3.). 
Lastly, as the incompleteness of form of V only causes certain terms of the developed 
determinant of Proposition II. to vanish, but leaves the signs of the terms which remain 
positive, it follows that as x^ varies from 0 to infinity, x^, X3,... x„ being always determined 
V 
by the last w— 1 equations of (1.), the function ^ will vary by continuous increase be- 
tween the limits above investigated, viz. from the highest inferior to the lowest superior 
limit of j?i. Once, therefore, in its progress it becomes equal to^i, and all the equations 
are satisfied together. 
The above reasoning establishes rigorously that if the proposition is true for n — 1 
variables, it is true for n variables. It remains then to consider the limiting case of 
n=l. 
Here, however, only the complete form of V, viz. Y=ax-{-b, leads to a definite value 
of X, and this, as has been seen, is finite and positive. If we give to V the particular form 
ax, the equation becomes 
ax _ 
~=p, orp=l, 
which determines p, but leaves x indefinite. If we employ the other particular form 
Y=b, we obtain no equation whatever, and here again x is indefinite. But as the 
MDCCCLXIl. 2 L 
