16] 



THE CYLINDROID. 



23 



parallel to OR. It is desired to find the cylindroid determined by these two 

 screws. 



In the plane FOR draw OS perpendicular to OP and denote Z ROS 

 by X. 



The translation of length p along OR may be resolved into the components 

 p sin X along OP and p cos X along OS. 



Erect a normal OT to the plane of POR with a length determined by 

 the condition 



coOT = p cos X. 



The joint result of the two motions is therefore a twist of amplitude to 

 about a screw 6 through T and parallel to OP. 



The pitch p e of the screw is given by the equation 



(op g = wp + p sin X, 

 whence p e p=QT tan X. 



Fig. 3. 



In Fig. 3 we show the plane through OP perpendicular to the plane POR 

 in Fig. 2. The ordinate is the pitch of the screw through any point T. 



If p 6 = then OT= OH. Thus H is the point through which the one 

 screw of zero pitch on the cylindroid passes, and we have the following 

 theorem : 



If one screw on a cylindroid have infinite pitch, then the cylindroid 

 reduces to a plane. The screws on the cylindroid become a system of parallel 

 lines, and the pitch of each screw is proportional to the perpendicular distance 

 from the screw of zero pitch. 



