( '<;4 ) 



is an arbitrary- line through point ^ and 



is the condition expressing tliat this Hne (1) tlirough point ^contains 

 at the same time the point — X and is thus the ray of the pencil 

 looked for, corresponding to the parameter\aliie P.. If for shortness' 

 sake we write here 



Pi ^ Xqi + f.Ov + Xs;) I 



pi — ^qi I ~ ^' 



I'i — ^ *i 



direct calculation affords 



— ft 



— ft 



n + Xs, 



n — y. Si 



Pi - ). <Ii 



I'i 



Si 



Pi - ;. -;, 



= 0, 



= 0, 



= 0, 



_s^-\-xn^ 



so the equation of tlio I'ay under consideration is 



(Q: + ^ A) (/' + y- 7). + ('">'. + y- ^'':) ('• + ^- •^■). = 0' 



which is reducible to 



as the coeflicient of the first power of A identically vanishes. 



If now, for u = p, (^, r, .y, we understand by Uj:, u,/, 'I'z, »„ the forms 

 into which u^ x -\- u^ y -\- ?«, x ~\- ii^ passes by omitting successively the 

 term with x, the term with y, the term with : or the constant term 

 and if fc is substituted for P." tlie four prqjectively related pencils arc 

 represented in their ))iaiios liy liie equations 



