( i7 ) 



P + 2 A Q + A' 72 = 0. 



From (Iiis ensues lliat all Ihe surfaces of this svsloni have llic eiji,-ht 

 common jjoints (laiiueutial planes) of /^:=0, Q=(), y4*=() in common. 



The (le«»enerations of this system are fotii- iiuures consisting' each 

 of two planes as locus of points and of two points as locus of 

 taniicnlial planes. One of Ihose lium-es is formed l»y Ihc |)hines 

 (( and «lEEUl'i'') Jii>^l tlie [)oijits .1 ainl Ai~(aa'). 



The eight common points A^, JJ.^, C^, D.„ A^, B^, (^, D^ and llie 

 eight common tangejitial planes «2, ^2, y^, f^j, «,, /?.,,y.,, ^:, of the scrolls 

 are suKjiihir for the congruence (2,2). The remaining singulai- points 

 and planes are evidently A,B, C,D,Ay,B^, C\, D^ and a,i3,y,ö,a^,ii^,Y^,<f^. 

 These 16 points and 16 planes form the well known configuration 



of KUMMER. 



We can choose the notation in such a way, that .l^ bears (he 

 planes /?, 7, d, «j and A^ the i)lanes /?i, 7i, ^i, «, etc. Let us hear in 

 mind that three osculating planes of R^ ijitersect each other in a 

 point of the plane of their points of cimtact and let us further mark 

 the symmetry of the figure, we can then easily deduce from the 

 preceding, that 



in « the points A, A^, A^, B^, C\, B,, 



1/ ((^ It If A, A^, A^, B^, C3, A^;,, 



// «. // If A„ A„ A,, B, C, D, 



II f^z II II A, A^, A^, B^, C'l, Z)i, 



are situated, whilst 



^1 bears the planes «, u^, «;,, (?„, 7,,, <f^, 



A, II I, If ((, «J, a.,, /?3, 7.,, ff.,, 



A, If If !f «1, «„, «.,, ii, 7, Ö, 



A; If If If «, t(.,, «.,, /?!, 7,, ffj. 



It is clear that for each of these 1(5 |)oinls the comitlcxcone is 

 composed of a plane counteil double and a cone of degree four. 



Mathematics. — ''The shu/ulfiritiei? of the focul cnrrc i>f k currc 

 ill space." By Dr. W. A. Veksluvs. (Commuiiicaled Ity Prof. 



P. H. SCHOUTK.) 



In paper X". 5 of the "K. A. v. W." at Amslerdam, Vol. Xlli, 

 1 ha\e deduced some formulae e.\|)ressijig the singularities of the 

 focal developable and of the focal curve in function of the singulari- 

 ties of a plane curve. 



In like manner it is possible to deduce the following fornndae 

 Avhich express the singularities of the focal developable and of the 



2 



Proceedings Royal Acad. Amslcidam. Vul. VI. 



