208 MR. R. HAELEY ON IMPOSSIBLE 



Then, multiplying by v/ — v/j we get 



/-/= 0, (2) 



12 



a rational equation. Now, for whatever value of x (1) ob- 

 tains, (2) must likewise obtain ; otherwise we should have 



v7- v7= (/-/) -^ (v7+ v7) = (/-/) -T- n «, ; 



18 12 is 12 



and V7+ V7= 0; 



1 1 



which is absurd. Hence the proposition in relation to (I) 

 is estabhshed. 



Similar reasoning will evidently apply to any surd equa- 

 tion whatever. Thus, if we take the equation 



V7+ V/+ V7=: 0, (3), 



where // f are rational functions of ar, we shall have 



113 



(v7+ v7+ v7) (v7+ v7~ v7) (v7- v7+ v7) 

 (- v^+ v7+ v7) = 0,..' ! (4), 



or,/» +/^ 4-/^ - 2 (//+// -f //) = 0, (5), 



133 121323 



a rational equation. Now, if there be a value of x which 

 will satisfy (3), and which will not also satisfy (5), or, which 

 is the same thing (4), one at least of the factors -v7+ ^7 — 

 V7 V7— V7 + V/ — ^7+ ^4- v7must be infiLite. 



3 1 3 3 I t 3 



Suppose 



V7+ ^7— v7= °o ; 



12 3 



then, subtracting this equation from (3), we get 

 2\7= — <» » ■*•/= 00 ; .-. a; = 00 ; 



3 3 



which is absurd. 



Cor. 1. Every value of ^ which will satisfy (2), will 

 also satisfy either (1) or its congener. For, 



v/-/=0; 



1 2 



•••(^7+v7)(^7~v7) = 0; 



