( -202 ) 



and " intersecting a^. As we can lat 0' pass still throug-li two arl)ilrai'y 

 points, there is a possibility of bringing tlirongh the bisextuple an 

 0^ with sixteen lines. To this end we have bat to repeat the above 

 consideration for e.g. the lines a^, b.^, h^. 



7. An 0* through a hi/perOoloidical quadruple contains a secon»! 

 quadruple consisting of four quadrisecants of the former. For, througli 

 an arbitrary point of the intersection of 0^ with the hyperboloid 

 containing the given qnadruple we can draw a line of the second 

 system of the hyperboloid, wiiich then contains live points of 0^ 

 and lies therefore on 0' ; the intersection of the two surfaces con- 

 sists then of two hyperboloidical quadruples. 



Let us suppose an ()' to be laid through six lines ai; of which 

 a^,a^,a^,a^ and at the same time a^,a^,a.^,a^ lie hyperboloidically. 

 The hyperboloids bearing these quadruples have still two lines it and 

 /' in common which are evidently intersected by the six lines a 

 and are therefore situated on 0*. 



Besides these two transversals 0^ contains still two transversals of 

 the first (piadruplo and two of the second. In all 0^ contains there- 

 fore twelve lines ; they form a configuration in which the six trans- 

 versals appear 'm the same manner as the six lines a. For, the 

 six transversals form two hyperboloidical quadruples with a^, a^ as 

 transversals to six lines. 



It is evident that again cc^ surfaces 0* can be made to pass 

 thi-ough this configuration of twelve lines. So we can obtain an 0^ 

 with fourteen lines by drawing a transversal of «i, «^3, «j and a trans- 

 versal of a^,a^,ag, and by assuming on each of these lines two points 

 through which we make 0* pass. 



The six lines f/k can be chosea also in such a way that they form 

 three hyperboloidical quadruples. Let a^, a^, a^, a^ be such a qna- 

 druple, «5 an arbitrary line. The hyperboloids {a^ a.^ a^) and {a^ a^ a^) 

 have still two lines t and t' in common resting on the five lines a. 

 The hyperboloids ia^a,_a.J and {a^a^a^) have now the lines a^, t 

 and t, therefore one line a^ more, in common, resting on t, t' . 

 Consequently also the quadruples a^,n^,a^,a^ and a^, a^, a^,a^ lie 

 hyperboloidically. 



Each surface 0* containing this sextuple of lines passes at the 

 same time through the two transversals t, t' and through the three 

 pair of quadrisecants belonging respectively to the three quadruples; 

 the surface contains therefore at least fourteen lines. 



If we do not take t, t' into consideration we have a configuration 

 of twelve lines, showing the same structure as the configuration of § 2. 



