1864] Mr. Russell on the Calculus of Symbols, 



227 



which numbers give the vapour-density 2*235, whilst the theoretical 

 vapour-density of chloride of ethyl is 2'233. 



The boiling-point of the residue left after the first distillation rose 

 quickly up to 60° C, whilst nearly the whole distilled over between this 

 temperature and 70° C. By a few more fractional distillations of this latter 

 portion, monochlorinated chloride of ethyl, Clg, boiling between 



62° and 65° C, was isolated. 



0*1270 grm. of this compound gave 0*3530 grm. of chloride of silver 

 and 0095 grm. of metallic silver, which corresponds to 71*43 per cent, 

 of chlorine, whilst the formula requires 71*71 per cent. 



Eight litres of methyl yielded about 8 grammes of the mixed chlorides, 

 or only about one-third of the theoretical quantity of chloride of ethyl 

 which should have been obtained. This is easily explained by the vola- 

 tilization of the liquid, and its solution in large quantities of water, as well 

 as by the formation of higher substitution-products, in consequence of 

 which a considerable quantity of methyl is left uncombined. 



From these results it appears that the lowest term of the series of 

 alcohol radicals behaves with chlorine exactly in the same manner as T 

 have shown in the paper above referred to is the case with its homologues 

 ethyl-amyl, H^g (which gives chloride of heptyl, H^g CI), and amyl, 

 (from which chloride of decatyl, C^q 112^ CI, is obtained). If an 

 excess of chlorine is avoided, the principal products consist of the chlo- 

 rides of monatomic radicals containing the same number of atoms of 

 carbon as the original hydrocarbon contained, whilst at the same time 

 chlorine substitution-products of these chlorides are formed in smaller 

 quantities. 



As there is no reason why those terms of the series which are placed 

 between Hg, C^ H^g, C^^ should show a different deportment, it 

 becomes obvious that, beginning with marsh-gas, C H^, the lowest term in 

 the series C^Ha^+g, the most simple of all hydrocarbons, and one which 

 can easily be obtained from its elements, we are now not only in a position 

 to prepare all the members of this series, but likewise to build up by 

 simple synthesis the series of mono-, di-, and polyatomic alcohols, acids, 

 compound ammonias, ethers, &c. &c. of which each of the marsh-gas hy- 

 drocarbons forms the starting-point. 



III. " On the Calculus of Symbols (Fifth Memoir), with Applica- 

 tions to Linear Partial Differential Equations, and the Calculus 

 of Functions.^' By W. H. L. Russell, Esq., A.B. Communi- 

 cated by Professor Stokes, Sec. R.S. Received April 7, 1864. 



In applying the calculus of symbols to partial differential equations, we 

 find an extensive class with coefficients involving the independent variables 

 which may in fact, like differential equations with constant coefficients, be 



