116 



oo' raj6 /' passing tlirongii the poiiil of intersection of' / and F, so 

 that S./ contains also ft four dimensional sheaves of rays. If l)esides 

 aSj, we have anothei- system *SV wiili the characteristic numbers 

 ƒ!, and I'j, this has in r a I'liled surface ot the order i\ and it 

 contains /tj straight lines of each of the fotir-dimensional sheaves in 

 '^/ ■ ""^V ^'i*' '^»' liiive accordingly n fi^ -\- v v^ rays in common. 



Two sj/sft'ins S.^ (f, ^0 (^^i(f '**>/ (.'S. *')) ^"ive n fi^ -\- v i\ rays in 

 cotinnon. 



§ 5. Bij means of complete induction the following results ma}^ 

 be easily proved, through which the problem of the characteristics 

 is solved for the straight line in /**„. 



The characteristic nuiid)ers of a system S/, of go'' i-ays in R,i 



indicate how many straight lines of ^S*^, there are in an /i?„_„_j_i lying 



in R„ which cut an Rn-\-y. -/, •> i" 'be aforesaid /?„_^^_l_i for all values 



of fi satisfying the inequalities: n^ 0, n -\- ^i — p — 2<^n — f* -|- 1 



p -{-'S 

 or n <^ —- — and n -\- i.i — /; — 2 ^ — 1 or {^ ^ p — n -j- 1- 



From this follows that the />-fold number of characteristics for 



v-\-^ p 

 the straight line in A^,, if p <^ n, is equal to — - — or — -f~ ^ > 



according to whether p is odd or even, and for p'^n equal to 



2(n— 1)— /;+! 2(7i— 1)— /> , ^ ^. , , . ,^ 

 — or -|- 1 according to wliether p is odd 



or even. The ^;-fold number of characteristics is, therefore, equal to 

 the 2(n — 1) — /?-fold number. 



The expression indicating how many lines an S^, and an aS2(,j— i) -;l. 

 have in common, is a polynome of which all the terms are found 

 by multiplying each time those characteristic numbers of /S]y and 

 S-i(^n—\)-p that belong to conditions which together define a straight 

 line in Rn- 



^ 6. It is clear that the indicated method may also be applied to 

 the case when we have to do with figures composed of a definite 

 number of points, straight lines, planes etc. If the parts of these 

 figures are independent of each other, it will often be desirable 

 to transform them by different homographic representations. 



The system e.g. of the go" groups of n points [P^ P,, . . . , Pn) of 

 a straight line / may in the following way be represented homo- 

 graphically on itself. We assume on / 2?i arbitrary points (\, . . . , C», 

 Tj, . . . . /"„ and associate to a point Pi of a group of ?i points 

 (/2-group) the point P'i defined by. {Ci Fi PiP'i) =z ?.i . 



