Prof. Sylvester on Derivation of Coexistence. 37 



the purity of the nitric acid is a matter of indifference, strong 

 commercial acid answering every purpose. 



I remain, dear Sir, very truly yours, 



John Thomas Coopek. 

 82, Blackfriars Road, London, Dec. 10, 1839. 



P.S. I have found within these few days that some kinds of 

 common coke, such as is very brilliant, close-grained, and 

 has a columnar fracture, is equally good with the other varieties 

 of carbon, and admits of being selected of almost any variety 

 of form and size. J. T. C. 



VI, On Derivation of Coexistence : Parti. Being tlie Theory 

 of simultaneous simple homogeneous Equations. By J. J. 

 Sylvester, F.R.S. Sj- R.A.S., Professor of Natural Philo- 

 sophy zn University College, London*. 



A , ,-. V "^TSTE shall have constant occasion in this paper to 

 •\ ') T T denote different quantities by the same letter 

 affected with different subscribed numerical indices. 



Such a letter is to be termed a " Base." 



Every character consisting of a base and an inferior index I 

 call an argument of the base, viz. the first, second, or wth ar- 

 gument, according as 1, 2, or in general (?z), be the number 

 subscribed. 



Art. (2.) I use the symbol PD to denote the product of 

 the differences of the quantities to which it is prefixed (each 

 being to be subtractedy?'om each that follows) ; thus 



PD (a 6 c) indicates [h — a) (c — a) {c—h). 



PD (o a b c) indicates a h c (b — a) (c — a) (c—b.) 



PD (o a b c ... I)' indicates ab c ...I x PD [aba ... /). 



Art. (3.) For want of a better symbol I use the Greek letter 

 ^ to denote that the product of factors to which it is prefixed 

 is to be effected after a certain symbolical mannner. This I 

 shall distinguish as the zeta-ic product. 



The symbol ^ will never be prefixed except to factors, each 

 of wMch is made up of one or more terms, consisting solely of 

 linear arguments of different bases, i.e. characters bearing in- 

 dices below but none above. 



I am thereby enabled to give this short rule for zeta-ic mul- 

 tiplication : " Imagine all the inferior indices to become supe- 

 rior, so that each argument is transformed into a jjower of its 

 base; multiply according to the rules of ordinary algebra; 

 after the multiplication has been done fully out depress all the 



* Communicated by the Author. Part I, appeared in L. &-E. Phil. Mag. 

 Dec. 1839, vol. xv. p. 428. 



