Prof. Sylvester on Derivation qfCoexistenee, 43 



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 a, b, c ... /, and let the arguments 

 of each be "recurrents of the «th order*," that is to say let 



.,..,. I = w ( cos . 1. 



Let R^ 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 



is equal to the product of the number 



Ti f ( 2 7r , / — - . 2 7r\ / 4 tt ^ /— - . 4 7r\ 



K I (cos — + '^^ — l.sin — I (cos — i-'v— l.sm — ) 



'■\\ n n ) \ n n J 



( cos — + V _ 1 . sui — I 



\ n n ) 



/{2n--l)^ ,—^ . 2(;2-l)7r\\ 



cos ( ^ — + ^ — 1 . sm — ^ ^— \ I 



\ n n } J 



multiplied by the zeta-ic phase 



^_^^PD {pabc..,l)\\ 



* I am indebted for tins 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 would require special but very simple printing types to be 

 founded for the purpose. 



In the next part of this paper an easy and symmetrical moAe will be given 

 of representing any polynomial either in its developable or expanded form. 



