an interesting Case in Equations. 189 



between two roots, neither of which has been proved to be re- 

 levant to the argument. What is still more strange, that root 

 which has been most strenuously claimed as auxiliary to one 

 side of the cause, proves to be quite adverse to it. Under all 

 the circumstances, it is not without a sentiment approaching 

 to diffidence that I venture to make these assertions; and 

 especially because the objections I have to offer to the method 

 employed in reducing the proposed equations are so palpable, 

 that it is difficult to dismiss the notion that they have been 

 considered and overruled; but on what sound principle of 

 reasoning I cannot conjecture. 



1. If the equations contain but one unknown quantity, one 

 of them suffices for the solution, and the other is either super- 

 fluous or contradictory ; and so, a fortiori, is the third equa- 

 tion, which has been derived from them. 



2. If two unknowns are involved in each equation, either 

 two new equations must be formed by elimination, or if only 

 one subsidiary equation is employed, the result must at all 

 events be introduced into the original statements. 



3. A third exception applies to the mode of obtaining the 

 subsidiary equation, namely, by taking the quotients of the 

 separate scales. If it were stated that X s = «, and x = b> 



would it follow that .\ x* = -^-, and x = + »/ -r-? As- 

 suredly not. 



4. This objection is still more valid when the quantities 

 exterminated by division are zero or infinite. That this im- 

 pediment exists in the present instance will appear on adopt- 

 ing a mode of reduction exempt from the faults above re- 

 cited ; e. g. 



Dividing by a, the equations become 



± + i + _L_ . M (i.) 



X x—b x—c u 



x—b x—c 

 Deducting (2.) from a times (1.), 



Ma 

 a 



(2.) 



«+«=»+«=?. (3.) 



x x—b x—c 



To prove this to be a complete cubic equation, it would be 

 abundantly sufficient to make x = ,f ' u and reduce. The 



