DYNAMICS OF A RIGID BODY. 



I2 7 



which may be regarded as any plane parallel to A ; 

 then P is the intersection of the pair of screws be- 

 longing to the complex PT ly PT 2 , which lie in that 

 plane, and P is the intersection of the pair of reciprocal 

 screws P'R^ P'R* belonging to the reciprocal complex. 

 Since P'R^ is to be reciprocal to PT^ it is essential that 

 Ri be a right angle, similarly R 2 is a right angle. The 

 reciprocal cylindroid, whose axis passes through P' y 

 will be identically the same as the cylindroid belonging 

 to the complex whose axis passes through P; but the 

 two will be differently posited. If the angle at P be 

 a right angle, the points 7i and T 2 are at infinity; 

 therefore, the plane touches the quadric at infinity; it 

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

 therefore, pass through the centre of the pitch quadric O; 

 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 O a diameter 

 conjugate in the pitch quadric to the 

 plane A. Let this line intersect the 

 pitch quadric in the points P ly P 2 , and 

 let S, S' (Fig. 4) be the feet of the per- T 

 pendiculars let fall from P iy P 2 upon 

 the plane A . Draw the asymptotes OL, 

 OM to the section of the pitch quad- 

 ric, made by the plane A . Through 

 *$* and S f draw lines in the plane A, 

 ST, ST' y S'T y S'F, parallel to the 

 asymptotes, then T' and T are the 

 two required cylindroids which belong to the two reci- 

 procal screw complexes. 



This construction is thus demonstrated : 



Fig. 4. 



centres of 



the 



