= 0; 



174 Prof. Boole on certain Propositions in Algebra 



r — bpz, —ep, ap'z+.dp'^, 

 B = < bq'z, — eq, aq'z + dq' > , 

 I bt'z, —et, at'z —dt J 

 r—ep, ap'z + dp 1 , cj)'z^ 

 C=< —eq, aq'z + dq', —cqz >. 

 [.—et, at'z —dt, ct'zJ 

 From the consideration of these forms, we see that (18) may 

 be expressed in the form 



P^+Q^+B^+S^O, .... (19) 



P, Q, R, and S not involving z, but being functions of t and the 

 constants of the original system (15). 



The limiting forms of the above equation, as t approaches the 

 respective limits and infinity, are evidently 

 S = 0, P = 0; 



the former of these gives, on constructing the expression for S 

 by means of (18) and of the expressions for A, B, C, D, 

 — bp, —ep, dp' ■> r— ep, dp', cp'- 

 bq', -eq, dq' I X 1 -eq, dq', —cq 

 bt', —et, —dt J l-et, -dt, ct' . 

 whence, developing and dividing by the constant factors bed and 

 edc, we get 



(p + t-q){q + t-p)=0, 



.-. t=p — q or q—p (20) 



In precisely the same way the equation P = gives for the 

 values of t corresponding to z= oc, the relations 



ap', cp', -fyn f cp', —bp, —ep-\ 

 aq', —cq, bq V x <j — cq, bq', — eq j-=0j 

 .at', ct', bt' J [ ct', bt', -et) 

 which on development and division by the constant factors abc 

 and cbe, gives 



(l-t)(p + q-t)=0, 

 whence 



t=l oi- p + q (21) 



But the values (20) and (21) thus determined for the first 

 member of the final equation of the system (15), when z assumes 

 the respective values and infinity, are precisely the values 

 before determined for the limits of r. 



Further, it appears, as was to be shown, that if we assume 

 x, y, z to be positive quantities satisfying the two first equations 



