428 



Hydrogen and nitrogen undergo no change of volume when exposed 

 to the action of either form of discharge. Cyanogen is readily de- 

 composed by the spark, but presents so great a resistance to the 

 passage of electricity, that the action of the silent discharge can 

 scarcely be observed. Protoxide of nitrogen is readily attacked by 

 both forms of discharge, with increase of volume and formation of 

 nitrogen and hyponitric acid. Deutoxide of nitrogen exhibits the 

 remarkable example of a gas which, under the action either of the 

 silent discharge or of the spark, undergoes, like oxygen, a diminu- 

 tion of volume. It also is resolved into nitrogen and hyponitric 

 acid. Carbonic oxide has given results of great interest; but the 

 nature of the reaction has been only partially investigated. The 

 silent discharge decomposes this gas with production of a substance 

 of a bronze colour on the positive wire. The spark acts differently, 

 destroying, as in the case of oxygen, the greater part of the con- 

 traction produced by the silent discharge. The authors are engaged 

 in the further prosecution of this inquiry. 



II. " On the Equation of Differences for an Equation of any 

 Order, and in particular for the Equations of the Orders 

 Two, Three, Four, and Five." By ARTHUR CAYLEY, Esq., 

 F.R.S. Received March 2, 1860. 

 (Abstract.) 



The term equation of differences, denotes the equation for the 

 squared differences of the roots of a given equation ; the equation of 

 differences afforded a means of determining the number of real roots, 

 and also limits for the real roots of a given numerical equation, and 

 was upon this account long ago sought for by geometers. In the 

 Philosophical Transactions for 1 763, Waring gives, but without de- 

 monstration or indication of the mode of obtaining it, the equation 

 of differences for an equation of the fifth order wanting the second 

 term : the result was probably obtained by the method of symme- 

 tric functions. This method is employed in the ' Meditationes Alge- 

 braicee* (1782), where the equation of differences is given for the 

 equations of the third and fourth orders wanting the second terms ; 

 and in p. 85 the before-mentioned result for the equation of the fifth 

 order wanting the second term, is reproduced. The formulae for 



