Method of finding the impossible Roots of an Equation. 283 



alcohol was left in contact with aqueous ammonia ; this in the 

 course of a few hours was perfectly converted into a white, nearly 

 insoluble powder. This was well washed on a filter with water 

 and then with alcohol : a portion burnt with oxide of copper gave 

 the following numbers : — 



•2590 of substance gave -4055 of carbonic acid and "1295 of 

 water, 



equal to 42-7 per cent, of carbon and 5"5 of hydrogen. The 

 formula C^H^N^Q-^ requires 42-1 of carbon and 5-26 of hy- 

 drogen. This substance is therefore fumaramide. 



From this it evidently appears that, when pentachloride of 

 phosphorus and malic acid are heated together, oxychloride of 

 phosphorus and chloride of fumaryle are formed; thus, — 



H3 }OHPCP= H2 ^0^ + PCF02+2HCl, 



V -, ' I ^ > 



Malic acid. Fumaric acid, 



and then 



H2 ^0'» + 2PCP=C«H2 0'»", CP + 2PC13 0H2HC1. 



Chloride of fumaryle. 



Fumaric acid, chloride of fumar3de, and fumaramide bear a 

 very close relationship to succinic acid, chloride of succinyle, and 

 succiuamide, as will be seen from the following Table : — 



Fumai-ic acid C^ H* O^. Succinic acid C^ H« 0^. 



Chloride of fumanle. C^ H- 0\ Cl=. Chloride of succinyle. C« H^ O^, CP. 

 Fumaramide ..."... C^ H« N^ O^ Succinamide C^ H^ N- 0^ 



the only difference being that the derivatives of fumaiyle con- 

 tain 2 equivs. of hydrogen less than those of succinyle. In fact 

 we think that fumaric acid may be considered as the member of 

 a sei'ics of acids running parallel to the ordinary oxalic series of 

 which succinic acid is a member. 



We are now engaged with the investigation of the action of 

 pentachloride of phosphorus on tartaric acid. 



XLV. On a Method of finding the impossible Roots of an Equa- 

 tion, in reply to the Astronomer Royal, By Professor 

 Challis*. 



IN the course of the proof which I gave, in the Philosophical 

 Magazine for Februaiy, of the theorem that every equation 

 has us many roots as dimensions, 1 argued, from antecedent alge- 



* Commuuicatcd by the Autlior. 

 U2 



