OF ARTS AND SCIENCES : NOVEMBER 8, 1865. 41 



at (x x y 1 Zj) and (x 2 y 2 z 2 ) respectively, either point is on the other's 

 polar as to each quadric ; or, 



2?1 %2 I i &1 3-2 I 1 



22 + k? -T - A2 _^_ k * f - 



whose difference, 



#1^2 I | Z l Z 2 __ A (A\ 



( n-2 _1_ L 2\ ( (12 I h 2\ '» ^ ' 



is the condition that the two tangent-planes are perpendicular. 



Or thus. The pairs of tangent-planes drawn from the line {x x y x z x ), 

 (x 2 y 2 z 2 ), to the different surfaces, are known to form an involution, 

 whose double planes, namely the tangents at (x 1 y 1 z x ) and (x 2 y 2 z 2 ), 

 must form a harmonic pencil with the tangent-planes to any one sur- 

 face, e. g. to the spherical circle at infinity, and hence are orthotomic. 



Particular cases of this orthotomism are that of the three quadrics 

 through one point, and the circularity of the cone mentioned in § IV. 



Now this orthotomism would preclude the existence of a common 

 developable ; but it fails for the envelope's rays ; for since every 

 tangent-plane from a finitely-near point to the circle at infinity has 

 some infinite direction-cosines, while 



j>„ « • d„, u t • co S «, • co S „, = ( ^, + ^, +V) . *«, 



are finite, \_U X , U 2 , Nx, N 2 , a 1} . ... . y 2 , being the quadrics, their 

 normals, and the direction-angles of these,] — it must be that 



D Ni U l -D Ni U 2 = 0; 



hence (4), which is the same as 



D Nl U x ' D Ni U 2 ' (cos ax cos a 2 -\- cos ft cos ft + cos y x cos y 2 ) = 0, 



becomes merely identical. And the other demonstration fails, because 

 the two tangent-planes to the circle at infinity coincide. 



III. According to Salmon, each ray to a spherical point at infinity 

 is " perpendicular to itself" (whatever that may mean) ; which would 

 extend the above orthotomism to the developable ; a result opposite to 

 ours in statement, but probably the same in its actual consequences. 

 Such lines must often thus simulate self-perpendicularity, from their 

 infinite direction-cosines having zero co-factors ; and this jnay make 



VOL. VII. 6 



