132 



the coordinates (x, y, z) (such discriminant being of course a cubic 

 function in regard to I, and also in regard to the coefficients of the 

 two conies U, V, jointly), then if we write 



the relations for the cases of the triangle, pentagon, heptagon, &c. 

 are 



C=, 



C, D 

 D,E 



= 0, 



= 0, &C. 



C,D, E 

 D,E, F 

 E, F, G 



respectively, while those in the cases of the quadrangle, hexagon, 

 octagon, &c. are 



D=0, 



D,E 



E, F 



=0, 



= 0, &c. 



D,E, F 



E, F, G 



F, G, H 



respectively. The demonstration of this fundamental theorem is 

 for greater completeness here reproduced ; but the chief object of the 

 memoir is to direct attention to a curious analytical theorem, which is 

 an easy & priori consequence of the Porism, and to obtain the relations 

 for the several polygons up to the enneagon, in a new and simple form, 

 which puts in evidence a posteriori for these cases, the analytical 

 theorem just referred to. The analytical theorem rests upon the 

 following considerations : the relation for a hexagon ought to in- 

 clude that for a triangle ; in fact a triangle with its 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 function disc* (U + V) be represented by 

 1 4- /3 + y 2 + 3 3 , the coefficients A, B, C, D, E, &c. are functions of 



/3, y, S. Write 



C =(3), 



D =(4), 



C, D 



D, E 



D, E 



E, F 



C, D, E 



D, E, F 



E, F, G 



&c. 



