648 PROFESSOR BOOLE ON THE COMBINATION 



it must once, and only once, become equal to q. 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 

 abcxy + abdx + acdy + bcd = . . . . . (10) 



It is obvious that this, as all the terms in the first member are positive, can 

 never be satisfied by positive values of x 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 —aoayz + by% + gkz + dxy + ex +fy + gz + h 



and the system to be considered is 



axyz + cxz + day + ex 



-: P ... . (11) 



. r//r>)i 4- fit 



(12) 



axyz + byz + cxz + ,'/;. _ . . 



axyz + bi/z + dxi/ +fy _ 



v • 



Let the first number of the last equation, considered as a variable function of 

 x, y, z be represented by Z, and suppose ar, v, 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 + BG) (DF + DH + EF + EH) 

 + (AG + BC) (DF + EH -t DE + FH) 

 + (AB + CG) (DE + DH + FE + FH) 



+ 4AGFE + 4BCDH = (14) 



Wherein 



A = axyz ~B = byz C~czx D = dxy 



E=«c F=fy G=gz H = /i 



And as this equation has positive values only in its first member, it cannot be 

 satisfied by positive values of x, y, z ; whence, by the same reasoning as before, 

 the system (11), (12), (13) cannot have more than one solution in positive values 

 of x, y, z. 



To show that it will have one such solution, let z vary from to a , then Z 

 continuously increases from to 1, and once becomes equal to r. At every stage 

 of its variation we may give to (11) and (12) the form 



