442 Mr. Sylvester on Absolute Criteria 



of the decomposition of nitric acid and persalts of iron, and 

 finally the direct experiments of the reduction of gold, silver, 

 platinum, palladium, nickel, copper, tin, and the decomposi- 

 tion of the persalts of iron and nitric acid inclosed within a 

 tube, appear to set the question to rest, and to point out in the 

 clearest manner that hydrogen is the cause of the reduction of 

 the metals. All the experiments which I have detailed only 

 add confirmation to the valuable researches of Faraday on 

 Electro-Chemical Decomposition, published in the Philoso- 

 phical Transactions. To the inquiring mind a question na- 

 turally arises as to whether the hydrogen reduces the metal 

 directly from the metallic solution, or whether it reduces its 

 oxide. The former opinion, from the above experiments, ap- 

 pears to be most worthy of credit, though should other facts 

 be discovered to elucidate that action, they will form sub- 

 jects hereafter of a separate communication to this Society. It 

 follows from these interesting experiments, that when a solu- 

 tion of metallic salt is subjected to the voltaic influence, the 

 water is decomposed, oxygen passing one way and hydrogen 

 the other; and that this hydrogen at the moment of decom- 

 position on the negative plate performs the same part to sul- 

 phate of copper and other metallic salts that a piece of iron 

 or zinc would to the same solutions. 



LXXIV. On the Existence of Absolute Criteria for determi- 

 ning the Roots of Numerical Equations. By J. J. Sylvester, 

 Esq., M.A.^FJl.S.'^ 

 T WISH to indicate in this brief notice a fact which I be- 

 lieve has escaped observation hitherto, that there exist, 

 certainly in some cases, and probably in all, infallible criteria 

 for determining whether a given equation has all its roots ra- 

 tional or not. 



In the equation of the second degree it is enough, in order 

 that this may be the case_, that the expression for the square 

 of the difference of the roots shall be a perfect square ; in other 

 words, \^ x^—px-\- q = have its roots rational, p'^ — 'lq must 

 be not only a positive number (the condition of the roots being 

 real), but that number must also be a complete square. In 

 this case it is further evident that (p) must be either prime to 

 q, or if not, the greatest common measure oi' p'^ and q must 

 be a perfect square; but this condition is contained in the 

 former, which is a sufficient criterion in itself. 



If we now consider the equation of the third degree, 

 x^—p a:'^ + qx—r = 0, 

 * Communicated by the Author. 



