234 
ME. A. CATLET ON THE POEISM OF 
we have 
1234: 
= X 
f 
^ 
— 3 
(fed" 
— 12 
— 4 
-16 
— 14 
a^be’d^ 
— 3 
a^d'dP' 
— 8 
— 12 
- 6 
afbe^d 
— 1 
aV 
-79 
The conditions for the triangle and the quadrangle are c=0, d~^ ; those for the 
pentagon and hexagon are 12 = 0, 23 = 0; for the heptagon and octagon, 123 = 0, 
234 = 0; and that for the enneagon is 1234=0. The foregoing values show that 23 
and 1234 (which belong to the hexagon and the enneagon) divide by c (which belongs 
to the triangle), and that 234 (which belongs to the octagon) divides by d (which 
belongs to the quadrangle). But I was not prepared for the destruction which will be 
observed in the several determinants, of the terms involving the lower powers of a 
(that is, the terms of the highest orders in h, c, d), and which renders these expressions 
so much more simple than they would otherwise have been. 
Kepresenting the reduced equations for the several polygons, as before, by 
[3] =0, 
[4] =0, 
[ 5 ] = 0 , 
[ 6 ] = 0 , 
[7] =0, 
[ 8 ] = 0 , 
[9]-0, 
, &c. 
Then retaining the quantity «( = 1) for homogeneity, but rejecting the powers of a 
which dhide out, and reversing in some cases the signs, the values of the functions [3], 
[4], &c. are 
[.3]= [4] = 
II 
1 1 < 
1 1 
[6] = 
[ 7 ] = 
A 
[8] = 
[ 9 ] = 
A 
+1 c i + 1 d 
\ 
\ 
1 
1 
1 
1 
+ 1 ad?' 
+ 2 bed 
+ 1 c® 
' ^ 
+ 2 aeP 
+ 2 bed 
+ 1 c® 
( „ 1 
+ 1 a^d'^ 
+ 2 abed^ 
— 1 ac^d"^ 
— 2 bc^d 
— 1 c® 
^ A. ^ 
+ 1 (Pd^ 
+ 4 abed^ 
+ 4 
+ 4 PPd^ 
+ 6 bc*d 
+ 2 c® 
' + 3 a^d’^ 
+ 12 aHcd^ 
+ 4 aVd* 
+ 16 abVd* 
— 14 abc^d? 
+ 3 ac^d'^ 
+ 8 b^c^d^ 
+ 12 We-d? 
+ 6 b(?d 
+ 1 c® 
+ 1 1 +1 
+ 4 
+ 5 
— 1 
+ 21 
+ 79 
