The ElUptic Functions Defined. 49 



same." Among the historical references which it contains to re- 

 searches bearing upon this problem, there are several to theorems 

 of Poncelet, which, as Jacobi shows, may easily be demonstrated 

 by the use of elliptic functions. In reversing the procedure of 

 Jacobi, and attempting an application of elementary geometry 

 to the elliptic transcendents, I make one of these theorems of 

 Poncelet my first objective point; but I propose to reach it, not 

 by the projective method of Poncelet himself, but by the less 

 elegant but more generally familiar processes of analytical 

 geometry. For this purpose I require a short series of intro- 

 ductory propositions, as follows: 



Pkoblem. — When two sides of a triangle inscribed in a circle 

 are represented by a single quadratic equation, to derive thence 

 the equation for the third side. 



The general equation of a pair of lines intersecting at the 

 origin is 



Ax' + 2Hxy + By' = 0; 



and to transfer the intersection to any point whose coordinates 

 are c, s, we must write 



A {x — c)- -{-2H (x — c) (2/ — s) + B' {y — s)' = 0; 



which becomes, on expansion, 



Ax' + 2Hxy + By- — 2 (Ac + Hs) x — 2{Bs + Hc)y+ ^ 

 (Ac- + 2Hcs + Bs-) = 0; 



This equation will represent two sides of a triangle inscribed 

 in the circle 



x' + f = R\ (2 



provided we take for R' the value & + si 



The loci of equations (1) and (2) have two coincident inter- 

 sections at the point c, s; our problem is to write the equation 

 of the line joining the two remaining intersections. 



It is a familiar principle that, if P = and ^ = are the 

 equations of two conies, then an equation of the form P + kQ 

 = 0, where k is an arbitrary constant, represents a conic passing 

 through the four points in which the first two conies meet; and 



