Mathematics. — "A.res of Rotation of Qaadnitlc Surfaces through 

 4 Given Points''. By H. J. van Vkkn. (Coinmuiiicated by 

 Prof. Jan de Vries). 



(Communicated at the meeting of March 25, 1922). 



§ 1. If we assume (hree points in space, any straight line / may 

 1)6 oonsidered as the axis of rolalion of a quadratic surface of revo- 

 lution through these j)oinls. For the ciirles which Ihe three j)oiuts 

 desci'ibe during the revolution round /, cut a plane through / in 

 six points. Tliese lie apparently on a conic /f" which has / as axis 

 of symmetry. Revolution of k^ round / gives a quadratic surface of 

 revolution (in what follows to be indicated by 0'), which has / as 

 axis of rotation (briefly axis) and which passes through Ihe 3 given 

 points. 



As a rule an W^ is defined by its axis and three points; if, 

 however, during the revolution round / two (or three) of the given 

 points describe the same circle, there exists a pencil {net) of O^'s 

 (hat have / for axis and pass through the 3 points. 



An W' is always defined by 3 cii'cles with the same axis, provided 

 these circles do not all lie in the same plane. 



§ 2. The axes of Ihe O^'s through 4 given points ^, (/ = !... 4) 

 form a complex of rayes r, which will be investigated in what 

 follows. By 0^ we shall understand a quadratic surface touching 

 the sphere-circle y' twice; the line p joining the points of contact, 

 will be called chord of contact; the conjugated polar line of 

 p — defined as the locus of the points the polar planes of wliich 

 pass through p — is ihe axis of ()^ Asa rule this locus is a straight 

 line p' passing through the pole P of p relative to y' ; if /;' is 

 indefinite only the straight line (or lines) conjugated to p and passing 

 through P will be consideied as axis. 



As special quadratic surfaces which according to the aforesaid 

 must be considered as 0% I mention : a parabolical cylinder with 

 a plane pencil of axes in Ihe plane Vod at infinity and a pair of 

 parallel planes with a sheaf of // axes. 



