DYNAMICS OF A RIGID BODY. 93 



ai cos / + /3i sin /,..., a 6 cos / + j3 6 sin /, 



reference being made as usual to any set of six co- 

 reciprocals. 



In addition to the examples of the use of these co- 

 ordinates already given ( 42), we may apply them to 

 the determination of that single screw 9 upon the cylin- 

 droid (a, /3), which is reciprocal to a given screw TJ. 



From the condition of reciprocity we must have : 



/i?/(ai cos / f )3i sin /) + + /6i?6(a 6 cos / + j3 6 sin /) = o, 



or, 



nr a7? cos / + w^ sin / = o. 



From this tan / is deduced, and therefore the screw 

 becomes known ( 28). 



In general if 2^0 be the virtual co-efficient between 

 any screw ij and a screw on the cylindroid, we have : 



cty = -<, cos / + TTfa sin /; 



whence if on each screw 9 a distance be set off from the 

 nodal line equal to half the virtual co-efficient between 

 TJ and 0, the points thus found will lie on a right circular 

 cylinder, of which the equation is ; 



x* + y 2 = w^x + v ftll y. 



Thus the screw which has the greatest virtual co- 

 efficient with ij is at right angles to the screw reciprocal 

 to TJ, and in general two screws can be found upon the 

 cylindroid which have a given virtual co-efficient with 

 an external screw. 



88. Relation between Two Cylindroids. We may here 

 notice a curious reciprocal relation between two cylin- 

 droids, which is manifested when one condition is satis- 

 fied. If a screw can be found on one cylindroid, which 



