168 Prof. Boole on certain Propositions in Algebra 

 whence bp 



x -W=p-y 



and this single value is positive if a, b, and p are positive, and 

 the last fractional. 



Now the general proof of the proposition consists in showing, 

 that if it is true for a particular value of n, it is true for the next 

 greater value, and so on ad infinitum. Wherefore, since it is 

 actually true for the case of ra=l, it is true universally. 



We shall therefore begin by assuming that the proposition 

 is true for n— 1 variables. And making this assumption, let us 

 suppose the first n—\ equation of the system (1) to be satisfied 

 while z varies from to oc, and seek under these circumstances 

 the nature of the variation which the first member of the final 

 equation of the system undergoes. 



Now if a particular positive value be given to z, the n — 1 first 

 equations of the system (1) will assume the form of a correspond- 

 ing system relative to the n — 1 variables x, y, ... which remain. 

 Thus if in the equations (2) and (3) we give to z a particular 

 positive value, and then make 



az + d—A, cz + e = B, bz+f=C, gz + h = D, 



those equations will become 



Axy + Bx 

 Axy + Bx + Cy + D ~~ P ' 



Axy + Cy 



Axy + Bx + Cy + D ~ H ' 



A, B, C, D being known positive constants ; and this is a binary 

 system derived from the function 



V = Axy + Bx + Cy -f D, 



just as the system (2), (3), (-1) was derived from the expression 

 for V by which it was preceded. 



Hence if the proposition be true for a system involving n— 1 

 unknown quantities, it will be true for that system which is 

 formed by giviug to z in the n— 1 first equations of (1) a parti- 

 cular positive value. And therefore to each value of z there will 

 correspond a single set of positive values of the n — 1 unknown 

 quantities x, y, ... determined by the solution of that system. 

 That set of values will of course vary as z varies. Moreover, 

 when z = 0, the first member of the final equation will be 0. 

 This is evident from the form of that member as developed in 

 (1), or in any other special instance. When ~= cc, the first. 

 member of the final equation is 1, as is in like manner made evi- 

 dent. Hence as z varies from to oc, while the n — 1 first 



