( 360 ) 



^ 4, The drawing of the surfaces to be found offers no difficulties. 

 For R"^ 1 (§ 2) we must tal<;e C\ positive and then we have to 



distinguish the cases C = 0. 



So for C ^ the surface consists of two separated parts connected 

 by points forming parts of a double conic in the XOF-plane. The 

 planes .v = ±[^C touch both parts according to equal ellipses and 

 no points lie between with z'^O. 



The section with the XOZ-plane consists of two hyperbolae 



with centres lzz=± \/ L j on the Z-axis. At infinity they are 



connected twice, and intersect each other in the points of intersection 

 of the double conic with the X-axis. The hyperbolae coincide in 

 the planes y = ± V C-^, where the common vertex of the double 

 conic is lying. 



C becoming smaller, the two parts of the surface approach 

 each other and for 6'=0 the conies meet in the planes a' = ± l/ 6^. 

 The surface becomes a ruled surface, so it Dreaks up into two 

 cylinders with axes in the AY>Z-plane. 



The axes have for equation z ^=l ± x \X ■ . (Comp. ^ 3). The 



section perpendicular to these axes is a circle which is in accordance 

 with the signilication of the axes as found in § 3. 



^ 5. It is known that the normals of a sj^stem of similar, con- 

 centric ellipsoids form a tetrahedral complex ^). So this sj^stem must 

 be a particular integral of the above-mentioned differential equation. 



Let us put C=^ gCi -{- h {g and h being constants) and let us 

 operate in the ordinary way; we find C and Cj as functions of the 



variables out of: 



qy^4-h — A'* 



-C^-g 



gy''-\-h - x' 



9-\-R 



Substitution in the complete integral furnishes: 

 -" — gy -\- A"* = n. 



9+R 

 equa 



along the Z-axis ; we shall then find if we take a' positively 



a 

 Let us put in this equation (/ = — — , and let c be the axis 



1) Dr. J. DE Vries : On a special tetrahedal complex. Proceedings of Febr. 25 

 1905, Vol. XIII, pages 572-577. 



