333] VARIOUS EXERCISES. 361 



If the corresponding impulsive screws are reciprocal, then 



o = xv^v + /*y SfV/Si 2 + * v Spi v + (*&amp;gt;&quot; + vy ) SiVaA 



+ (XV&quot; + XV) SlvXTi + (pv&quot; + ^ V 

 and two similar equations. 



Take the two conies whose equations are 



= p a 



= a 



these conies will generally have a single common conjugate triangle. If the 

 co-ordinates of the vertices of this triangle be X , ///, v ; X&quot;, // , z/ ; X &quot;, //&quot;, v&quot; \ 

 then the equations just given in these quantities will be satisfied ; and as there 

 is only one such triangle, the required theorem has been proved. 



It can be easily proved that a similar theorem holds good for a pair 

 of impulsive and instantaneous cylindroids. 



333. Impulsive and Instantaneous Cylindroids. 



If a given cylindroid U be the locus of the screws about which 

 a free rigid body would commence to twist if it had received an impulsive 

 wrench about any screw on another given cylindroid V, it is required to 

 calculate so far as practicable the co-ordinates of the rigid body. 



Let us take our canonical screws of reference so that the two principal 

 screws of the cylindroid U have as co-ordinates 



1, 0, 0, 0, 0, 0, 

 0, 0, 1, 0, 0, 0. 



The co-ordinates of any other screw on U will be 

 a lt 0, 3 , 0, 0, 0. 



The cylindroid F will be determined by four linear equations in the 

 co-ordinates of rj. These equations may with perfect generality be written 

 thus ( 77), 



where A, B, A , B , A&quot;, B&quot;, A &quot;, B &quot; are equivalent to the eight co-ordinates 

 defining the cylindroid V. 



The screw on U with co-ordinates 



1, 0, 0, 0, 0, 



