Ball — On 'Properties of the Cylindroid. 521 



plete character. Since any transversal 6 across a, fB, and y may be a 

 reciprocal screw, if its pitch be equal and opposite to those of a and (3, 

 it will follow that each such transversal must be at right angles to y. 

 This will at once restrict the situation of y, for it is obvious that 

 unless it be specially placed with respect to a and /?, the transversal 6 

 will not always fulj8.1 this condition. Imagine a plane perpendicular 

 to y, then this plane contains a line I at infinity, and the ray 9 must 

 intersect / as the necessary condition that it cuts y at right angles. As 

 9 changes its position, it traces out a quadric surface, and as /is one 

 of the generators of that quadric, it must be an hyperbolic paraboloid. 

 The three rays a, (3, y, belonging to the other system on the paraboloid 

 must also be parallel to a plane, being that defined by the other gene- 

 rator /', in which the plane at infinity cuts the quadric. 



Let PQhe a common perpendicular to a and y, then since it inter- 

 sects y at right angles, it must also intersect /; and since PQ cuts the 

 three generators of the paraboloid a, y, and /, it must be itself a 

 generator, and therefore intersects ^. But a, ft, y are all parallel to 

 the same plane, and hence the common perpendicular to a and y must 

 also be a perpendicular to /3. We hence see the important result, that 

 all the screws on the surface S must intersect the common perpen- 

 dicular to a and (3, and be at right angles thereto. 



The geometrical construction of 8 is then as follows : — Draw two 

 rays a and /?, and also their common perpendicular A. Draw any third 

 ray 9, subject only to the condition that it shall intersect both a and /3. 

 Then the common perpendicular p to both 9 and A will be one of the 

 generators of the cylindroid, and as 9 varies this perpendicular will 

 trace out the surface. 



In the language of modern geometry, 9 is one of the rays of the 

 congruence defined by a and /5. A congruence is a doubly infinite 

 system of right lines, and it might at first sight appear that there 

 should be a doubly infinite series of common perpendiculars p to A 

 and 9. "Were this so, of course S would not be a surface. The diffi- 

 culty is removed by the consideration that evert/ transversal across 

 p, a, ft is perpendicular to p ; thus for each p there is a singly infinite 

 number of screws of 9. And thus all the rays p form only a singly 

 infinite series, i. e. a surface. 



A simple geometrical relation will now be very easily proved. Let 

 the perpendicular distance between p and a be di, and the angle be- 

 tween p and a be ^1 ; let dz and A^ be the similar quantities for p and 

 ft, then it will be obvious^ that 



di : (?o : : tan A^ : tan A^ ; 

 or dx-{- d^-.di- d^:: sin (^i + A^) : sin (Ai - A^), 



if 2 be the distance of p from the central point of the perpendicular h 



1 It is easy to make a rough model of the paraboloid with elastic threads, which 

 is an assistance in the study of the surface. 



