PEOrESSOE BOOLE ON THE THEOET OE PEOBABILITIES. 
241 
nature with respect to the n~l variables Xs.-.x^, as (1.) is with respect to the 
n variables Xi, X 2 , ... x„. This will be at once seen by taking any particular example. 
Hence by hypothesis X 2 , x^. , .x^ will be determinable as positive quantities, and their 
values substituted in the first member of the first equation of (1.) will reduce it to the 
form 
Aa^i + B’ 
V . 
A and B being finite and positive. Hence the function y will become 0. 
Secondly, let any finite positive value be assigned to x^. The last n — 1 equations of 
the system (1.) will again form a system of the same nature as before, and will by hypo- 
thesis determine a set of finite positive values for X 2 , X 3 ,... x„. These values again substi- 
V 
tuted in y*, will give to it again the form 
Aa^j 
Axj^ + B’ 
. . V . 
A and B being finite and positive. Hence as x^ is finite and positive, y will be a 
positive fraction. 
Lastly, let.r, be infinite. Still the last n—1 equations of the system (1.) will assume 
V 
the same form as before. Determining thence X 2 , x^.. . Xn, and substituting in -y , we have 
Vt_ A^, 
V A^Tj + B’ 
in which A and B are finite and positive and x^ is infinite. 
Hence 
It is seen 
then that as x^ varies from 0 to infinity, x^^ . . . x„ being at the same time always by 
hypothesis determined to satisfy the last n — 1 equations of the system (1.), the function 
y will vary from 0 through positive fractional values to unity. It is manifest, too, that 
it varies continuously. If then it vary by continuous increase^ it will once, and only 
once in its change, become equal toy?i, and the whole system of equations thus be satis- 
fied together. I shall show that it does vary by continuous increase. 
If it vary continuously from 0 to 1 and not by continuous increase, it must in the 
course of its variation assume at least once a maximum or minimum value. Let us then 
seek the condition of possibility of 
L 
V 
= a maximum or minimum. 
the variables being subject to the relations 
V,_ V3_ v„_ 
y — y — i^3 • • • y — ‘^n- 
Here, proceeding in the usual way by differentiation, we have 
YdY^-Y^dY ^ YdY„-YJY 
y2 t), yo U, . . . y2 
MDCCCLXII. 2 K 
