242 
.PEOFESSOE BOOLE ON THE THEOEX OE PEOBABILITIES. 
or r 
dV„ 
v-v,-v, V/ 
Let the common value of these fractions be represented by — df, then we have a system 
of w+1 equations of which the first is 
Ydt-4-dV=0, 
while the n others are of the type 
The complete system, therefore, on efiecting the total difierentiations, becomes 
VJt+^'dx,..+^dx„=0. 
Now from the nature of the function V we have 
^_Vi ^_Yi ^_Vy- 
dxi Xi' dxi a?i’ dxj x/ 
so that the above equations become 
Ydt +v. ^+v, . . +v. *"=0, 
V,<?<+V. +v„^=o, 
v,*+v,.^+v, +v,.^=o, 
cvj u/g *^n 
v.<^<+v..^+v.,^^. +v„^*=o, 
and the elimination of dt, ^ from these equations gives the sought condition of 
Xy X, 
V, 
possibility of a maximum value of consistently with the satisfaction of the last w—1 
equations of the system (1.). 
This condition is therefore expressed by the equation 
: 0 . 
V 
V, 
V, .. 
v; 
V, 
V,.. 
V, 
V,, 
V, .. 
V„ 
V„, 
V„,.. 
V„- 
