42 



PROCEEDINGS OF 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 6 with itself, 

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

 should have 



cos^ a -\- cos^ /3 -\- cos^ y = cos ^, 



while cos^ a -)- cos^ jS -|- cos^ y = 1, 



. • . • cos^ a -|- • cos^ jS -|- • cos^ 7=1 — cos ^, 



. • . some of (cos a, cos /3, cos •)/) = <», 



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

 But the converse fails ; for 



cos a ^ CO 

 does not imply 



COS^ a -\- COS^/3 -j- COS^ y =jz 1. 



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 



( X cos a -\- V sin a = 1, 

 ^^ ^ ' S- found thus, -< x cos a 4- w sin a = 2, 



^2 _l_ ^2 __ 4 ) ' ) ' -^ ' 



^^ ^ ' V . • . cos a = GO , 



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

 each other's ordinary tangents, namely, 



60°, and (log2 + v/3) 'v/— 1 ? 



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

 the circle's angle t with its radius vector, we need only regard it as 

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

 may be a function of the independent polar angle ^. 



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

 a common developable envelope, so have f7-\-XY, F-j-/xY, W; 

 fi being a linear function of X ; for the equation 



lU-{-m F+ W=0, 

 implies 



l{U-^XY)-\-m (v—^--y)-\- W=0. 



