( 1084 ) 
Mathematics. — “Quadratic complexes of revolution and congruences 
of revolution”. By Dr. J. Worrr at Middelburg. (Communicated 
by Prof. JAN DE Vries). 
§ 1. The following treatise joins the investigation of Prof. Jan pr Vries 
(Proceedings Royal Acad. of Amsterdam Vol. LX 1906/7, p. 216—221),. 
If we choose the common centre O of the twe quadratic surfaces 
of revolution O,? and 0,°, forming together the singular surtace of 
a quadratic complex of revolution 2, as origin of a rectangular system 
of coordinates, then the roots of 42 —2 Fz 4 A—O must differ 
only in signs, the bisingular points B, and B, where O,? and O,* 
touch each other corresponding to those roots. Then we have F=f 
so that the equation of 2 becomes 
A tobe cs Ps’) ai Bp,” ne 2 CPsPs a Dp,” sE Elp: - Pis) RN (1) 
This equation can be written in each of the forms 
ANS hs AN? 
B(n tr] / 5) +2(e trl 5) + 
+ Bp, + 2(C + VAE)p,p, + Dp, =9, - - (2) 
ree Ay? 
E Pax Wi E ae BP | Tt 
+ Bp,? + 2(C — AE) p,p, + Dp” =0 . -. (2) 
As the first members of (2) and (2*) can be reduced to four 
squares, thus also to 2 products, out of each of the equations (2) and (2%) 
two systems of congruences (1,1) can be deduced, out of which 2 
is built up; we call them I and I’, resp. F* and I’*. Each T' bas 
a regulus in common with each I”; likewise each P* with each 
I’*, The directrices of all those congruences form 0,’ and Q,?, 
§ 2. We apply to the zOy-plane a screwing around Jz of which 
the angular velocity counted positively from Ov to Oy is in a ratio 
to the translation velocity as 1:4. We choose the right lines p, and p, 
along which coincide the velocities of two arbitrary points P, and P, 
lying in zOy as directrices of a congruence (1,1) in order to find 
the equation of the complex which originates when TI revolves 
about Oz. 
Let OP, be equal toa,, OP, =a,, 720P,=9, ~ cOP,=% Ha. 
The coordinates of p, are in order of succession of the indices: 
a, sind, — a, cos 9, —k, ka, sin P, — ka, cos 9, eee 
those of p, are deduced from these by substituting a, for a, and 
