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 Cooper. 
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 the Theory 
of simultaneous simple homogeneous Equations. By J. j. 
Sylvester, F.R.S. R.A.S., Professor of Natural Philo- 
sophy in University College ^ London^-. 
Art (l ) 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 {n), 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 subtracted from each that follows) ; thus 
PD {a h c) indicates {b — a) (c—a) (c — b). 
PD (o a b c) indicates a b c {b — a) {c — a) {c—b.) 
PD {o a b c ... 1) indicates a b c ,,,l x PD {a b c ... 1). 
Art. (3.) For want of a better symbol I use the Greek letter 
f 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 f will never be prefixed except to factors, each 
of which 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 power of its 
base; multiply according to the rules of ordinary algebra; 
after the multiplication has been do?2e fully out depress all the 
* Communicated by the Author. Part I, appeared in L. & E. Phil. Mag. 
Dec. 1839, vol. xv. p. 428. 
