43 
Prof. Sylvester on Derivation of Coexistenee. 
author to play no secondary part in the development of some 
of the most curious and interesting points of analysis. 
Let there be (w — 1) bases c ... Z, and let the arguments 
of each be “recurrents of the 7Uh order that is to say let 
a 
I 
/ 2tt l\ 
( 2 7t A 
I cos . 1 
3 = 
( cos . I 
V n ) 
1 ^ 
V n ) 
II 
8 
/ 2 7T L\ 
{ COS . ) 
i 
V n J 
Let denote that any symmetrical function of the rth de- 
gree is to be taken of the quantities in a parenthesis which 
come after it, and let ^ indicate any function whatever. Then 
the zeta-ic product 
... Z) X ^ PD {p ah c ... Z)^ 
is equal to the product of the number 
T> ^ a/ T • 2 7T\ / 4 7T / r . 4 7 t\ 
R,| ( cos — + ^ — 1 . sin — ) ( cos — -t- ^ , sin — ) 
^\\ n n) \ n n) 
( 6 9T , / — r . 6 Tt\ 
I cos — + ^ — I . sin — 1 
\ n n ) 
cos , sin 
multiplied by the zeta-ic phase 
PD [o a b c .,,1) \\ 
* I am indebted for this term to Professor De Morgan, whose pupil I 
may boast to have been. I have the sanction also of his authority, and that 
of another profound analyst, my colleague Mr. Graves, for the use of the 
arbitrary terms zeta-ic, zeta-ically. 1 take this opportunity of retracting 
the symbol S P D used in my last paper, the letter S having no meaning 
except for English readers. I substitute for it Q D P, where Q represents 
the Latin word Quadratus. On some future occasion I shall enlarge upon 
a new method of notation, whereby the language of analysis may be ren- 
dered much more expressive, depending essentially upon the use of similar 
figures inserted within one another, and containing numbers or letters, ac- 
cording as quantities or operations are to be denoted. This system to be 
carried out v/ould require special but very simple printing types to be 
founded for the purpose. 
In the next part of this paper an easy and symmetrical mode will be given 
of representing any polynomial either in its developable or expanded form. 
