648 PROFESSOR BOOLE ON THE COMBINATION 
it must once, and only once, become equal to g. Wherefore the system (3) (4) 
admits of one, and only one, solution in positive values of # and y. 
The reasoning might also be presented in the following form. The condition 
of Y having a maximum or minimum value is expressed by the equation 
abexy +abda+acdy+bed=0 : : é , (10) 
It is obvious that this, as all the terms in the first member are positive, can 
never be satisfied by positive values of « and y Hence Y has no maximum 
or minimum, consistently with (3) being satisfied, and thus it never resumes a 
former value, and is only once, in the course of its variation, equal to q. 
In the case of n=3, we have 
V =aanyit bya+caxt+day+eatfytgzuth 
and the system to be considered is 
AXYs + C#2Z + day +ex i, 
V Pp ; H : ; : (11) 
anya _ day+fy _ 7 (12) 
anyet byat cant ge _ (13) 
Vv 
Let the first number of the last equation, considered as a variable function of 
2, y, « be represented by Z, and suppose 2, 7, and z to vary in subjection to the 
conditions (11) (12). Just as before, it may be shown that Z increases continu- 
ously with z. The condition of Z having a maximum or minimum value, will be 
expressed by the following equation: 
(D+H+E+F) (ABC+ACG+ ABG + BCG) 
+(A+B+C+G) (DHE+ DHF + DEF + HEF) 
+(AC+BGQ) (DF + DH+EF + EH) 
+ (AG+BC) (DF +EH+ DE+FH) 
+ (AB+CG) (DE+DH+FE+FH) 
+4AGFE+4 BCDH = 0 j : : 3 : (14) 
Wherein 
A=axyz B=byz C=cza D=day 
E=ee B=fy G=gz H=h 
And as this equation has positive values only in its first member, it cannot be 
satisfied by positive values of z, y, ¢; whence, by the same reasoning as before, 
the system (11), (12), (13) cannot have more than one solution in positive values 
of x, y, 2. 
To show that it will have one such solution, let z vary from 0 to «, then Z 
continuously increases from 0 to 1, and once becomes equal to 7. At every stage 
of its variation we may give to (11) and (12) the form 
