94 DYNAMICS OF A RIGID BODY. 



is reciprocal to a second cylindroid, then conversely a 

 screw can be found on the latter, which is reciprocal 

 to the former. Let the cylindroids be (a, j3), and (Xyu). 

 If a screw can be found on the former, which is recipro- 

 cal to the latter, then we have : 



cos / + J3i sin /) + ...+ pr^n(a n cos / + j3 sin /) = o. 



/ifti(ai COS /+ /3i Sin /) + ...+ p n ^n(cLn COS / + ]3n SHI /) = O. 



Whence eliminating /, we find : 



= O. 



As this relation is symmetrical with regard to the 

 two cylindroids, the theorem has been proved. 



89. Co-ordinates of Three Screws on a Cylindroid. The 

 co-ordinates of three screws upon a cylindroid are con- 

 nected by four independent relations. In fact, two screws 

 define the cylindroid, and the third screw must then 

 satisfy four equations of the form ( 22). These rela- 

 tions can be expressed most symmetrically in the form 

 of six equations, which also involve three other quan- 

 tities. 



Let X, ^y v be three screws upon a cylindroid, and let 

 Ay ByC denote the angles between ju, v y between v X, 

 and between X ^u, respectively. If wrenches of inten- 

 sities X", ju", v"y on X, ju, Vy respectively, are in equili- 

 brium, we must have ( 1 7) : 



Xrr rr rr 



UL V 



sin A sin B sin C 



But we have also as a necessary condition that if 

 each wrench be resolved into six component wrenches 

 -on six screws of reference, the sum of the intensities of 

 the three components on each screw of reference is zero; 

 whence : 



