159] THE GEOMETRY OF THE CYLINDROID. 149 



belonging to the other system on the paraboloid must also be parallel to a 

 plane, being that denned by the other generator / , in which the plane at 

 infinity cuts the quadric. 



Let PQ be a common perpendicular to a and 7, then since it intersects 

 7 at right angles, it must also intersect / ; and since PQ cuts the three 

 generators of the paraboloid a, 7 and /, it must be itself a generator, and 

 therefore intersects ft. But a, ft, 7 are all parallel to the same plane, and 

 hence the common perpendicular to a and 7 must be also perpendicular to ft. 

 We hence deduce the important result, that all the screws on the surface S 

 must intersect the common perpendicular to a and ft, and be at right angles 

 thereto. 



The geometrical construction of S is then as follows: Draw two rays a and 

 ft, and also their common perpendicular X. Draw any third ray 6, subject 

 only to the condition that it shall intersect both a and ft. Then the common 

 perpendicular p to both and X will be one of the required generators, 

 and as 6 varies this perpendicular will trace out the surface. It might 

 at first appear that there should be a doubly infinite series of common 

 perpendiculars p to X and to 6. Were this so, of course S would not be 

 a surface. The difficulty is removed by the consideration that every trans 

 versal across p, a, ft is perpendicular to p. Each p thus corresponds to a 

 singly infinite number of screws 9, and all the rays p form only a singly 

 infinite series, i.e. a surface. 



A simple geometrical relation can now be proved. Let the perpendicular 

 distance between p and a be d lt and the angle between p and a be A l ; let d z 

 and A 2 be the similar quantities for p and ft, then it will be obvious that 



rfj : d z : : tan A l : tan A 2 ; 

 or rfj + d 2 : d 1 d. 2 : : sin (A^ + A 2 ) : sin (A l A 2 ), 



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

 between a and ft ; and if e be the angle between a and ft, and 6 be the 

 angle made by p with a parallel to the bisector of the angle e, then we have 

 from the above 



z : h : : sin 2&amp;lt; : sin 2e. 



The equation of the surface 8 is now deduced for 



oc 

 tan 6 = - ; 



y 



whence we obtain the equation of the cylindroid in the well-known form 



z (# 2 + */*) = xy. 



sm 2e * 



The law of the distribution of pitch upon the cylindroid can also be deduced 



