Note on the Porism of the in-and-circumscribed Polygon. 99 



1000 100-0 



Caproate of Magnesia crystallizes in small starry groups of fine 

 needles, and appears like the propionates to contain one equi- 

 valent of HO after drying over sulphuric acid. 0*1808 grm. of 

 the dried salt gave, on treating with sulphuric acid, 0*083 grm. 

 of sulphate of magnesia =15*3 per cent, of MgO. The formula 



MgO.C 12 H n 3 + HO 

 requires 15*1 magnesia. 



XII. Note on the Porism of the in-and-circumscribed Polygon, 

 By A. Cay ley, Esq.* 



THE equation of a conic passing through the points of inter- 

 section of the conies 



U=0, V=0 

 is of the form 



wU + V=0, 



where w is an arbitrary parameter. Suppose that the conic 

 touches a given line, we have for the determination of w a qua- 

 dratic equation, the roots of which may be considered as para- 

 meters for determining the line in question. Let one of the 

 values of w be considered as equal to a given constant k, the line 

 is always a tangent to the conic 



HJ+V=0; 



and taking w=p for the other value of w, p is a parameter 

 determining the particular tangent, or, what is the same thing, 

 the point of contact of this tangent. 



Suppose the tangent meets the conic U = (which is of course 

 the conic corresponding to w* = co ) in the points P, P', and let 

 6 } oo be the parameters of the point P, and &, go the parameters 

 of the point P'. It follows from my " Note on the Geometrical 

 representation of the Integral y*dx-*- V (x -f- a) (x + b) (x + c) f/' 

 and from the theory of invariants, that if □£ represent the 

 " Discriminant ,# of f U + V (I now use the term discriminant in 

 the same sense in which determinant is sometimes used, viz. the 



* Communicated by the Author. 



t I take the opportunity of correcting an obvious error in the note in 

 question, viz. a?+b 2 -\-c? — 2bc — 2ca — 2a1> is throughout written instead of 

 (what the expression should be) 6 2 c 2 -f cV+a 2 ^— 2a?bc— 2bHa— 2&ab. 



H2 



