292 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



stu + cstu 



Prob.aj 



stu+ stu + stu + stu + stu 



(8) 



To simplify this system of equations, change into s, into t, 



s t 



&c., and after the change let X stand for stu + s + t+l. We then 



have 



-T) . S + CStU 



Prob. x = j, (9) 



with the relations 



stu -t- s stu + t stu + u 



= = = stu + s + t + u + I = X. (10) 



p q r 



From these equations we get 



stu + s = \p, (11) 



stu + s = \ t-u- 1, 



Similarly, u + s = X (1 - q) - 1, 



and s + t = \ (1-r) - 1. 



From which equations we find 



(12) 



2 



Now, by (10), 



stu = \p - s. 



Substitute in this equation the values of s, t, and u above deter- 

 mined, and we have 



= 4 {( P + q + r-l)X + l}, (13) 



an equation which determines X. The values of s, t, and u, are 

 then given by (12), and their substitution in (9) completes the 

 solution of the problem. 



