42 PROCEEDINGS OP THE AMERICAN ACADEMY 



Salmon's interpretation often convenient and suggestive, though I 

 think it is arbitrary. 



If a finitely-near plane [or line] should make an angle 8 with itself, 

 it would doubtless touch [or meet] a sphere's circle at infinity ; for we 

 should have 



cos 2 a -j- cos 2 /3 -J- cos 2 y = cos 6, 



while cos 2 a -\- cos 2 j3 -J- cos 2 y = 1, 



, , * . • COS 2 a -|- • COS 2 )3 — |— • COS 2 y = 1 — COS 5, 



. • . some of (cos a, cos /3, cos y) = co , 



which is the condition of such contact ; [nor need 6 nor co be real.] 

 But the converse fails ; for 



cos a = co 

 does not imply 



COS 2 a -j- COS 2 /3 -J- COS 2 y^:l. 



Nor, if 6 can be > 0, need it be always 90°. Various considera- 

 tions often suggest 90°, as in Salmon's beautiful instance of a circle's 

 tangent-radius. But should not the common tangent of circles 



f x cos a -\- y sin a = 1, 



~ y ~ ' l found thus, -< x cos a -j- y sin a = 2, 

 x 2 -I- y 2 = 4, ) V 



1 * V . ' . COS a = CO , 



make in like manner such angles with itself as the circles make with 

 each other's ordinary tangents, namely, 



60°, and (log2-f y/3) • v/— 1 ? 



Of course in either instance, to throw the self-contradiction, instead, upon 

 the circle's angle e with its radius vector, we need only regard it as 

 the limiting case of another curve or of an eccentric circle, so that e 

 may be a function of the independent polar angle cp. 



IV. If quadrics U, V, W, expressed in tangential coordinates, have 

 a common developable envelope, so have U -\- X Y, V -\- p Y, W; 

 H being a linear function of X ; for the equation 



lU+m V-j- W=0, 

 implies 



I ( U+ X Y) + m ( V— — y) + W=0. 



