25 



§ 5. The transformation used here, gives also the representation 

 of another congruence [(>']. Let us consider the image of the sheaf 

 that has J/' for centre. A raj' r' through il7' cuts each of tlie scrolls 

 (t*) and {t/kY twice and is therefore the image of a curve (>' through 

 the fixed point J f that cuts 0' and the lines ak twice. This [^*] is 

 a special case of a congruence described by Vknekoni ^). 



Through a point there passes one o' of this congruence. A curve 

 fx', the image of a straight line m, sends 07ie chord through M' ; 

 hence in is a chord of one curve (j*. Also this [9'] is therefore 

 bilinear. 



If r' intersects the curve o*, (/ consists of a straight line i and a 

 q'' through J/, which intersects 0' twice and which rests on a^, a„ 

 rtj and t. The cone k^ projecting 0^ out of J/', has two points of 

 0' in common with a ft'; there are accordingly seven o* resting 

 on m. The conies of the degenerate figures in question form there- 

 fore a surface if''; on this a^, a^, a^ are double lines (each straight 

 line t defines one point S, hence one ray M' S, and cuts tp^ for this 

 reason besides in ajc in one more point) and 0' is a threefold curve 

 [t meets three generatrices of k^). 



The surface ip' is represented on a plane by the chords of 0' ; 

 it is therefore a rational surface and belongs to the group of lioma- 

 Idids to which I have drawn attention in a communication of 

 Vol. XX, p. 419 of these Proceedings. 



If r' rests on a^, (>* degenerates into a straight line t^^ (the image 

 of the point «j r') and a 9* of the plane « corresponding to the 

 plane a'^M'a^. The conies p* form a pencil with base points M, 

 the points A^ and A^ of a, and a^, and the intersection of « with 

 0', which point does not lie on «,. Each (>* is connected with a 

 straight line t^^ and this rests on a,, a^ and 6*. 



There are accordingly in all four systems of compound figures (>'. 



The chord of 0' passing through M' , is the image of a (>' com- 

 posed of two straight lines t and the straight line through 71/ which 

 cuts them and which is at the same time a chord of <j'. 



The transversal of a, and a, through M' is the image of a (>* 

 formed by a straight line /,,, a straight line t^^ and their transversal 

 through M which rests at the same time on a^ and a^. 



The transversal through M' of a^ and 0' is the image of a o' 

 formed by a straight line t, a straight line i^^ and their transversal 

 through M which rests at the same time on a^ and on o^. 



There are therefore in all seven figures (>' consisting of three 

 straight lines. 



1) Rend. Palermo XVI, 209. 



