296 



line a/ci as corresponding singular line, llie /, lias as appears from 

 the above, moreover four otlier singular poirits A,,,, which are in 

 pairs collinear to tiie points Aki, and that in such a waj that A,n 

 and A„ are connected with A^i by the singular straight line ajci- 

 In other words, there are ten .singuhtr points and teii shu/ular straight 

 lines, which form a fourside and a complete quadrangle, in which 

 the former is inscribed in such a way to the latter that a configu- 

 ration 10, of Desargues has arisen. ') 



The curvt^ of coincidences is now a conic, as the fonr sti-aight 

 lines (Ik form part of the Jacohian. This may moreover also be 

 confirmed by paying attention to the common tangents of the curves 

 {p)^ and {ifj^\ they have besides the two straight lines x indicated 

 by the point pq and the 10 singular straight lines, moreover 4 

 straight lines ./■ in common, which each connect a point A' of p 

 with a |)oint A' of q. To a line p as locus of A corresponds 

 therefore a curve p'' as locus of the pairs A', A" and the latter 

 intersects p in two coincidences. It is eas}^ to find now that the 

 sti-aight line A';::^A''A^" describes a plane pencil. 



The /, here described has been known longest; it may properly 

 be called the involution of Reyk. 



7. With this Jive of the involutions /, found in the above 

 mentioned paper have been deduced from nets of cnbics. The sixth 

 /, is obtained if each c' passing through the points E, F^, F,, F^ is 

 intersected by each c' passing through the points F, G^, 6r,, G^. 

 This /j was amply discussed in my paper "A quadruple involution 

 in the plane". (These Proceedings XIII, 82—91). 



When the base-points B^, B„ B, of a [c'J lie on a straight line 

 (6,33 and the base-points B^, B^, B^ on a straight line b^^^, this net 

 contains a pencil, each figure of which is composed of the two 

 straight lines mentioned and a ray s of a plane pencil whose centrum 

 be indicated by A. 



On each ray s [c'J determines an /j ; here we have therefore a 

 cubic involution in the plane, which contains collinear triplets only, 

 and consequently was excluded from the investigation mentioned 

 above. Neither is it of the first class, for on an arbitrary straight 

 line does not lie a single pair. 



The Jacobian of this net consists of the lines 6i,,, b^^^ and a 



^) In a more symmetrical way the points and lines of the IO3 are indicated by 

 the symbols kl and Mm; the points kl,km,lm, lie on the straight line /c^m {k,l,m 

 to be replaced by 1,2,3,4,5), 



