42 



So these form a net. If we put z = it, this is changed into a net 

 of spheres. So all the H's, just as all the spheres of a net, pass 

 through two points i2, and £i^. Hence the curves which are their 

 intersections also pass all through these points. 



Through the joining line F of Si^ and Si^ pass oc^ planes. As 

 there are a^ curves, oc^ curves lie in each of them. 



All of these pass through £2^ and S2^, and through the points of 

 intersection of their plane with A'ao , and so they form a sheaf. 

 Now the general shape of the sj^stem is determined. 



Tlie following cases may now be distinguished. 



I. r lies at infinity. 



a. i2j and i2.^ do not lie both on /vx . Then the plane at 

 intinity has a conic and another point in common with each of the 

 H's, and so it constitutes part of it. So the H's degenerate into 

 planes, the curves into straight lines. 



h. i2j and 12^ both lie on Ky: . The cyrves in each plane have 

 two pairs of coinciding points at intinity in common, and so tliey 

 are concentric, similar, and similarly placed. The centres in tiie 

 successixe planes are the centi-es of parallel sections of oue of the 

 H's and lie therefore on one straight line. 



II. r does not lie at infinity. Now the following cases are 

 possible : 



1. The angle of F w^ith the z-axis is ^ 45° 



2. ,, ,, ,, F ,, ,, j-axis ,, = 45 

 o. ,, ,, ,, F ,, ,, j-axis ,, <^ 45 



a. iij and i^^ are real. b. They coincide. 



c. They are conjugate imaginary. a. One of them lies at intinity. 



e. Both lie at intinity. 



In the cases J (7, le, 3(/, and 3e the plane at intiuity has besides 

 Kao another point in common \vith each of the H's, so that the H's 

 degenerate into planes. 



In the cases 2a, 2h, and 2c F intersects /v x, and has therefore 

 3 points, viz. this point of intersection, ii^ and i^.^ i" common with 

 each of the H's; so F is a common generatrix of it. As the H's 

 have moreover K go in common, their further sections, i.e. the con- 

 sidered curves, are straight lines. 



At Id the curves in every plane have a twopoint contact, at 2c 

 a three-point contact at infinity. 



In the cases J b and J c the tangents of all the curves form angles 

 ^ 45° with the 2:-axis. If they are considered as world lines, the 



