226 
ME. A. CATLET ON THE POEISM OP 
sides in order twice over is a form of hexagon ; the condition for an octagon should in 
like manner include that for a quadrangle ; and so in other cases. Let the cubic fonc- 
tion, disct. (U + |V), be represented by l+i3|+7^+^S^ the coefficients A, B, C, D, E, &c. 
are functions of (3, y, Write 
c 
=(3), 
D 
=(4), 
c, 
D 
=(5), 
D, 
E 
E 
=(6), 
E, 
F 
c, 
E, 
E 
=(7), 
D, 
E, 
F 
E, 
F, 
G 
&:c. 
then (3), (4), (5) are respectively prime functions of j3, 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 w'ell 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-men- 
tioned equation has the geometrical signification that the relation for a dodecagon is 
the aggregate of the relations for a proper dodecagon, a proper hexagon, a quadrangle, 
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 forai which puts in evidence the property in question, that is, 
the series of equations 
(3) =[3], 
( 4 ) =[ 4 ], 
(5) =[5], 
(6) = [6] [3], 
(7) =m, 
(8) = [8] [4] 
(9) =[9] [3]. 
To do this, the discriminant is represented, not as above in terms of the constants 
y, but in a somewhat different form, by means of the constants d, c, d, the last two 
