133 



Then (3), (4), (5) are respectively prime functions of /3, y, S; that 

 is they cannot be decomposed into factors, rational functions of these 

 quantities ; and it is convenient to denote this by writing (3) = [3], 

 (4) = [4], (5)={5]. But by what precedes, (6) contains the factor 

 (3), that is [3] ; and if the other factor, which is prime, is denoted by 

 [6], then we have (6) = [6] [3]. The next term (7) is prime, that 

 is we have (7) = [7] ; but the term (8) gives (8)= [8] [4] ; the 

 term (9) gives (9)= [9] [3], and so on. Thus we have (12) = [12] 

 [6] [4] [3], the numbers in [ ] being all the factors, the number itself 

 included, and as well composite as prime, of the number in ( ), the 

 factors 2 and 1 being however excluded. To make this clearer, it 

 may be remarked that the last-mentioned equation has the geome- 

 trical signification that the relation for a dodecagon is the aggregate 

 of the relations for a proper dodecagon, a proper hexagon, a qua- 

 drangle, and a triangle ; that is, the relation for a dodecagon implies 

 one or other of the last-mentioned relations. The relations for the 

 several polygons up to the enneagon are in the memoir obtained in a 

 form which puts in evidence the property in question, that is, the 

 series of equations 



(3)= [3], 



(4)= [4], 



(5)= [5], 



(6)= [6] [3], 



(7)= [7], 



(8) = [8] [4], 



(9) = [9] [3]. 



To do this, the discriminant is represented, not as above in terms of 

 the constants /3, y, 3, but in a somewhat different form, by means 

 of the constants b, c, d, the last two whereof are such that c= is the 

 the relation for the triangle, d0 the relation for the quadrangle ; 

 thus [3] = c, [4]=e?, and for the particular cases considered, the ana- 

 lytical theorem consists herein, that c is a factor of (6), and of (9), and 

 that c? is a factor of (8) . I have, for the sake of homogeneity, intro- 

 duced into the formulae the quantity (=!), but this is a matter of 

 form only. 



The functions [3], [4], &c. have been spoken of as prime ; they 

 are so, in fact, as far they are calculated ; and that they are so in ge- 

 neral rests on the assumption that for a polygon of a given number of 



VOL. XI. L 



