179] 



FREEDOM OF THE THIRD ORDER. 



181 



system. Since P R^ is to be reciprocal to PT 2 , it is essential that ^ be 

 a right angle; similarly ,R 2 is a right angle. The reciprocal cylindroid, whose 

 axis passes through P , will be identical with the cylindroid belonging 

 to the system whose axis passes through P ; but the two will be differently 

 posited. If the angle at P be a right angle, the points T l and T 2 are at 

 infinity ; therefore, the plane touches the quadrics at infinity ; it must, 

 therefore, touch the asymptotic cone, and must, therefore, pass through the 

 centre of the pitch quadric ; but P is the centre of the cylindroid in this 

 case, and, therefore, the centre of the cylindroid must lie in the plane A. 



The position of the centre of the cylindroid in the plane A is to be 

 found by the following construction : Draw through 

 the centre a diameter of the pitch quadric 

 conjugate to the plane A. Let this line intersect 

 the pitch quadric in the points P 1} P 2 , and let S, 

 S (Fig. 38) be the feet of the perpendiculars let 

 fall from P 1} P 2 upon the plane A. Draw the 

 asymptotes OL, OM to the section of the pitch 

 quadric, made by the plane A. Through S and S 

 draw lines in the plane A, ST, ST , S T, S T , 

 parallel to the asymptotes, then T and T are the 

 centres of the two required cylindroids which belong 

 to the two reciprocal screw systems. 



This construction is thus demonstrated : 



Fig. 38. 



The tangent planes at P^ P 2 each intersect the surface in lines parallel 

 to OL, OM. Let us call these lines PI-//I, P\Mi through the point P 1} and 

 P.,L,, P Z M., through the point P. 2 . Then P^, PM&amp;lt; are screws belonging 

 to the system, and P l M l , P. 2 L 2 are reciprocal screws. 



Since OL is a tangent to the pitch quadric, it must pass through the 

 intersection of two rectilinear generators, which both lie in a plane which 

 contains OL ; but since OL touches the pitch quadric at infinity, the 

 two generators in question must be parallel to OL, and therefore their 

 projections on the plane of A must be S T, ST . Similarly for ST, 

 S T ; hence ST and S T are the projections of two screws belonging to 

 the system, and therefore the centre of the cylindroid is at T . In a similar 

 way it is proved that the centre of the reciprocal cylindroid is at T. 



Having thus determined the centre of the cylindroid, the remainder of 

 the construction is easy. The pitches of two screws on the surface must be 

 proportional to the inverse square of the parallel diameters of the section 

 of the pitch quadric made by A. Therefore, the greatest and least pitches 

 will be on screws parallel to the principal axes of the section. Hence, lines 



