( 265 ) 



of uhicli pass ihntiigii the [)üiii( L*. \\\ ;u'l)ilr;u-,v ray llintugii P is 

 intersected outside P 'u\ 84 ])oiiits. As P hears 27 C' the degree 

 of {P) is 84 + 27 -=111. 



According u» the notation of SciirMKirr we lui\'e thus 



H'- = 42 and ft r = 111. 



From the well known i-elatious ^) 



ov ^= 2ri + f^ 4- "^l^ '•'"f' 3r> = ^; + 2f/ -f 2fi, 



where in our ease ?; is e(iual to we deduce l>y symbolic luidli- 

 plication the following system of relations: 



3 fi r = d ft + 4 ft' , 3 f t (> = 2 d f t + 2 IÏ' , 



3 r' = ff V -f 4 ft r , 3 r (> = 2 rf r -}- 2 ft i? , 



3 r () = ff () -}- 4 ft o , 3 o' = 2 () ff 4- '- fK' • 



We have here six equations for nine characteristic numbers of 

 which we ha\"e already determined two. 



But the number rlr we can tind directly. For on the arbitrary 

 riglit line / rest 42 right lines of o" ; each of these right lines is 

 intersected bij 10 right lines of the surface S'^ to which it l)elongs ; 

 so it furnishes 10 pairs of lines restuig on /. Consequently 



év = 420. 

 We now tind successive!}" 



15^ = 288, ffft=:lG5, r(> = 354, 



[iQ = 138, (f<5 = 510, q'=z4:'ó2. 



3. Out of r"-^ r= 288 follows fJuft the surfda' A formed Ini the conies 

 cuttiiKj the r'ujht line I is of de(jree 288. 



Illvidently / is a 27 fold right line of A and a chord of42co]iics 

 lying on A. It is evident that on A lie 462 right lines, which are 

 situated three by three in 210 planes. 



If / is a trisecaut of 11", thus a right line of a surface S^\ then 

 ^•288 |)i.(.^.^i^t; ,,p jjjiQ ji)(. surface .S/' counted double and the loci of 

 the cojiics |)assing through each (»f the three jtoiiits of intersection 

 of the trisecaut. 



The conies haviiKj a point of the Itusecnrre It' in row man, fonn 

 thus (( siu'jdee of de(jree 94. 



Tiie surfaces 7'"^ l)elonging to the |)oints of intersection 1\ and 7\ 

 of the trisecaut lia\e evidently the 10 conies in eommou which are 

 determined by the 10 Irisecants lhr(uigh the thii-d point of inter- 

 sec tioJi 7 '3. 



1) ScHUBEKT, Kalki'U der abzahlemleu Geometric, p. 02. 



18* 



