376 



Mr. A. Cayley on the Porism of 



U=cK 1 



V = (-)<cK 2 



W=(-)' +l cK 8 ; 



and so, in general, the ratios to one another of any number of 

 functions of one variable, of which the linear conjunctives for a 

 sufficient number of given values of the variable and of the 

 coefficients of conjunction are known to vanish, may be expressed 

 in terms of those values. 

 August 1853. • 



LVIII. Correction of two Theorems relating to the Porism of the 

 in-and-circumscribed Polygon. By A. Cayley, Esq.* 



THE two theorems in my " Note on the Porism of the in- 

 and-circumscribed Polygon" (see August Number) are 

 erroneous, the mistake arising from my having inadvertently 

 assumed a wrong formula for the addition of elliptic integrals. 

 The first of the two theorems (which, in fact, includes the other 

 as a particular case) should be as follows : — 



Theorem. The condition that there may be inscribed in the 

 conic U = an infinity of zi-gons circumscribed about the conic 

 V=0, depends upon the development in ascending powers of f 

 of the square root of the discriminant of fU + V; viz. if this 

 square root be 



then for w=3, 5, 7, &c. respectively, the conditions are 



| C | =0, 



c, 



= 0, 



c, 



E, 



D, 



E, 



= 0,&c.; 



and for n=4, 6, 8, &c. respectively, the conditions are 



D |=0, 



I), 

 B. 



=0, 



=0, &c. 



The examples require no correction ; since for the triangle and 

 the quadrilateral respectively, the conditions are (as in the erro- 

 neous theorem) C = 0, D = 0. 



Communicated by the Author. 



