an interesting Case in Equations. 1 9 1 



tomizing the present argument, but in the hope of supply- 

 ing students with a pretty addition to their collections of ex- 



ec 



amples. Reducing (1.), (2.), and makings = ^, s m b -f- c, 



p = b c 9 the equations to be solved are 



x 3 — (s + 3y)x* + (p + 2sy) x — py = (A.) 



*<-{,- f) x+ ( P+ *M) =o (ft) 



Eliminating y by the method of the common measure, we find 



(3a— s)x*— %(as— p)x + ap = (C.) 



as determined by Mr. Moseley. But if we also eliminate x 

 by the method of combinations, we obtain a formula in y 

 which, when arranged, resolves itself into the factors 



(♦P-rfW ji. Q (D>) 



a 3 v 



and 

 3(3a — s) j/ 2 — 2(3a 2 — 2as+p)y + (a*— as-\-p) a = (E> 



Of these, equation (D.) shows that y = yields two solu- 

 tions, which are readily found to be x = b 9 x = c. But when 

 these values are placed in (1.), (2.), the conditions appear to be 

 rather eluded than satisfied, the equations being reduced to 

 a balance of infinity. 



Equations (C.) (E.) being compared, in the event of s = 3 a, 

 which leads to infinite values of x and y 9 which answer to 

 each other, and finite values which likewise correspond, give 

 in the former case x = 3 y. The same result is obtained di- 

 rectly from either (A.) or (B.), their mutual independence and 

 their dependence upon a, b 9 c 9 being simultaneously destroyed 

 by the same hypothesis, viz. that of the infinity of x 9 neces- 

 sarily involving that of y. 



jy 2 ft 3 a 



The corresponding finite values are y = * 2 __ . — , and 



x = , /__ . , which appear, therefore, to furnish the only 



legitimate solution of the pair of equations in the event which 

 has been insisted upon. 



It is for Mr. Moseley to determine how far his principle is 

 affected by these results. My concern has been to solve a 

 curious difficulty connected with equations in a case where 

 the application of some of the ordinary rules has proved illu- 

 sive. The general equation, of which (3.) is a partial case, 

 deserves attention. 



