170 Prof. Boole on certain Propositions in Algebra 



manner from the ternary, quaternary, and higher systems derived 

 from (1) by the assigning of the successive values of 3, 4, &c. 

 to n, is possessed of remarkable properties. Representing by 

 H„ the function thus formed from the system of n equations, I 

 have obtained by a laborious analysis, the relation 



H„=V"- 1 K, (8) 



where K is a rational and integral function of the n unknown 

 quantities x, y, ... z t consisting wholly of positive terms. Thus, 

 in the above case of n — 2, where H 2 is expressed in full by the 

 first member of (7), it would follow that 



H 2 =V(abcxy + abdx + acdy + abd), ... (9) 

 wherein 



V = axy + bx + cy + d, (10) 



as may easily be verified. In this case the functions V and K 

 are reciprocally related. If to V we assigned the actual form of 

 K, we should find for K the form of V, with the addition of a 

 constant factor. 



Whentt = 3, we have, in like manner, 



H 3 =V*K, (11) 



wherein 



V = axyz + byz + cxz + dxy + ex +fy -\-gz + h; 



and here K is a wholly positive function of the sixth degree, 

 which, if we exhibit only its initial and final terms, may be thus 

 expressed, 



K=abcdx z y' 2 z' 2 + {adfc + abdc)x 2 y' 2 z + (abce + acdg)x i yz i 



+ {abcf+abdg)xy^... +efgh (12) 



The proof which I have obtained of the simply positive cha- 

 racter of the function K is rigorous up to the case of n = 4, and 

 it is further of such a nature as to leave upon my mind no doubt 

 of its general truth. Assuming, as I venture to do, that it is 

 generally true, the argument stands thus : 



Since H„ is wholly positive, the equation 



H =0 (13) 



admits of no solution in positive values of x, y, ... z. Hence the 



V 



function ~ in the final equation of the system (1) admits of no 



maximum or minimum value, while x, y, z are positive, and the 

 n — 1 first equations of the system are satisfied. Hence in the 

 variation of z from to oc, there is one point, and only one, at 

 which the entire system (1) is satisfied by positive values of 

 x, y, . ■ z. Hence, finally, that system admits of one, and only 

 one, solution in positive values of those quantities. 



