40 PROCEEDINGS OF THE AMERICAN ACADEMY 



which is a homofocal system. Three of the plane curves in (1) nave 

 become the focal couics in (2). And the fourth curve in (1) becomes 

 in (2) - the spherical circle at infinity, i. e. the intersection of any finite 

 and finitely-distant sphere, with the plane at infinity, P. 

 For since the cone, 



x 2 + y 2 + z 2 = 0, (3) 



which envelops that circle, is sensibly asymptotic to the surfaces (2) 

 when k is indefinitely large, the common tangent-planes to those sur- 

 faces then differ not sensibly from the cone's tangent-planes, and hence 

 envelop the circle. 



Hence, as Salmon shows, a quadric's three focal curves, with the 

 spherical curve at infinity, are the intersections of non-consecutive rays 

 of that developable which envelops both it and the spherical curve. 

 And they are the only intersections ; for, being of the fourth degree, 

 the developable cannot cross one of its own rays more than four times. 



Any deformation that destroys a sphere, destroys with it the circle 

 at infinity and the homofocalism of system (2), unless it replace that 

 circle by one of (2)'s focal conies ; hence the system has but four pro- 

 jective forms. And since, as its tangential equation shows, any four of 

 its quadrics divide each ray of the developable in the anharmonic ratio 

 of their k 2 s, this ratio must remain after deformation ; hence in the re- 

 spective forms, k 2 is, at the respective plane curves, 



= — A 2 , — B 2 , — C 2 , a> 2 , 



— B 2 , — A 2 , oo 2 , — C 2 , 



— C 2 , oo 2 , — A 2 , — B 2 , 

 oo 2 , — C 2 , — B 2 , —A 2 ; 



hence the four plane conies exactly replace one another ; and so do 

 their four included groups of quadrics, since projection breaks no 

 cyclic order. 



II. The system (2) touches every point of space three times, every 

 line twice, every plane once; except that it meets each ray of the 

 developable throughout. A line's two planes of contact with the system 

 are known to be mutually perpendicular, so that the boundaries of two 

 homofocals are seen from any point to intersect at right angles if at 

 all. For if the line touch 



x i _l_ - 1 •'-" _|_ - i 



^ 2 + *i 2 ' A 2 + k 2 ^ 



