( 6 ) 



c. The surface contains four separate double points, the four vertices 

 of «. By transporting the axes of coordinates parallel to theiijselves 

 to one of these points as origin and equalling the terms of the 

 second order to zero the equation of the corresponding osculating 

 cone is found. 



d. The curve limiting the projection of 1) on the plane of the 

 circles is obtained by elimination of z between 1) and its differential 

 quotient according to z. It consists of the intersection of 1) with the 

 plane of the circles and of the projection of a curve in space. The 

 first is the locus of the octuples of points common to the corres- 

 ponding pairs of circles of the two circle involutions 



(x ± a sec A)- -j- y^ = a^ tcf^ X ) 

 •■'■^ + (y ^ ^ fosec /)2 = 6^ coi^ A ] 



and as sucli a quadricircular octavic; the isolated double points 

 in the vertices of f excepted, all its points are imaginary. The 

 second is found by the elimination of u" between 1) and its diffe- 

 rential quotient according to ?r, which, v being substituted for u^j 

 comes to the elimination of v between 



4-— ^ — - — 1 



4a^ x^ A b^ y^ 



-f— -4-=2r + a2 + 62. 



By solution we find 



{v -j- a^)^ = 4: a^ c- or , (r + ^''^j^ = — Mfi c^ y" 

 and by elimination of v 



{2 a ,f 4- (2 b yf = r-^ , 



. 4 3"'"^ 4 1/^ 



i.e. the evolute of the ellipse — ;r H s"^^^- This result was to 



«■^ b" 



be foreseen. For the normal at the plane of the circles in a point 

 of this evolute meets two immediately succeeding rectangular hy- 

 perbolae and is therefore tangent to the surface on either side of 

 the plane X Y. The curve of contact itself, of which this evolute 

 is the projection, is of the twelfth order. The cylinder of the sixth 



