224 



MR. R. HARIiEY ON IMPOSSIBLE 



existence of an impossible equation, I may licreafter address 

 some observations. They will, however, probably find, as 

 I have done, that their attempts are unsatisfactory, and their 

 results not philosophically admissible. But I shall here 

 Content myself with remarking, that by any system of 

 rules, however artificial, the difficulty is only thrown further 

 back. Thus, the equation 



V^+ Va;4- 1 =: 

 is utterly intractable." 



After what I have already written, it is scarcely neces- 

 sary to say, that in the opinion so elegantly expressed in the 

 passage cited I entirely concur, and that I cannot but 

 consider the existence of impossible equations an undoubted 

 fact. With regard, however, to the equation proposed by 

 Mr. Cockle, 1 may remark that, peculiar as it is, the 

 method of solution explained in this paper, is applicable 

 even to it. For, since 



»Jx -f V^+l ~ (1) 



.'. hjx 1Z n \^x -j- 1 ; 

 .'. X n: n^x -f- 7i^ ; 



•••" = 1-^:::^^ (2); 



substituting (2) in (1), we have 



I 1 Vl 4" ^ ^o 



0, 



Vl— «^ Vl — ?? 1 — n 2 



which verifies the solution (2). Still, it will be remarked, 

 that the condition 



*/x zn 



which ha^been admitted into the verification, is incompa- 

 tible with the restriction imposed on the symbol of radicality ; 

 and that, therefore, if the views which we have taken of 

 the office of the signs {-\- and — ) prefixed to -\/ be correct, 

 the numerical value of (2), viz., ^, cannot be accepted as a 



