40 Prof. Sylvester on Derivation of Coexistence » 
This proposition admits of a great generalization*, but we have 
now all that is requisite for enabling us to arrive at a proposi- 
tion exhibiting under one coup deceit every combination and 
every effect of every combination that can possibly be made 
with any number of coexisting equations of the first degree, 
containing any number of repeated^ or to use the ordinary 
language of analysts, (variable or) unknown quantities. 
Art. (9.) For the sake of symmetry I make every equation 
homogeneous ; so that to eliminate n repeated terms, no more 
than n equations will be required. 
In like manner the problem of determining 7i quantities 
from n equations will be here represented by the case in which 
we have to determine the ratios of {pi + 1) quantities from n 
equations. 
Art. (9.) Statement of the Equations of Coexistence, 
Let there be any number of bases {ah c ... Z), and as many 
repeated terms {x y z ... Z), and let the number of equations 
be any whatever, say (w). The system may be represented 
by the type equation 
X ^h^,y ^ c^,z^- , . . + . Z = 0. 
In which r can take up all integer values from — oo to -f oo . 
The specific number of equations given will be represented by 
making the arguments of each base pei'iodic^ so that 
a -=■ a , h •=■ h , c '=■ c , . . , I I , 
r fjt.n-\-r r fjt.n-\-r r /an-f-r r 
fji being any integer whatever. 
Art. (10.) Combination of the given Equations , — 
Leading Theorem, 
Takey; g, ... k as the arbitrary bases of new and absolutely 
independent but periodic arguments, having the same index 
of periodicity {ii) as a b c ... Z, and being in number {n — 1), 
i. e. one fewer than there are units in that index. 
The number of differing arbitrary constants thus rnanufac- 
hired is w . (;z — 1). 
Let A jc + B + C 2 : + . . . + LZ = 0be the general 
prime derivative from the given equations, then we may make 
A = ? PD {p afg ... k) 
B= lfVD{obfg.„k) . 
C = rPD {ocfg,„k) 
L = ^FB(olfg,„k). 
Art. (11.) Cor. (1.) Inferences from the Leading Theorem, 
Let the number of equations, or, which is the same thing, 
' Sec the Postscript to this paper for one specimen. 
