218 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



- _ 



- 



or 



as the equation of the ruled surface of the third order in which the 

 resultant screw must lie, whatever the values of R x , R y . This surface 

 is called the Cylindroid. 

 Since 



- sin 2 a 



Fig. 67. 



23) 



the surface is the locus of a line 

 which, always intersecting a fixed line 

 at right angles ; revolves about it, and 

 makes a harmonic oscillation along it, 

 making two complete oscillations for 

 each rotation. In this manner the model 

 shown in Fig. 67 was constructed. 



For every screw lying on the 

 cylindroid there is a definite pitch, 

 given by the equation 22). If we lay 

 off the square roots of the reciprocals 

 of the pitches on lines making angles a 

 with the X-axis in the plane of XY, 

 and call the coordinates of their ends xy } we have 



1 1 . 



x = -7=1 cos a, y = ~ sin a, 



yp VP 



and our equation is 



25) p x x 



representing a conic section, such that the pitch belonging to the 

 direction of any radius vector is inversely proportional to the square 

 of the length of the radius vector. This is called the pitch -conic. 

 If p x and p y are of the same sign, the pitch -conic is an ellipse, if 

 of opposite signs it is an hyperbola. In the latter case, there are 

 two lines of zero pitch, given by the asymptotes. In other words, 

 if one screw is right-handed, while the other is left-handed, there 

 are two screws on the cylindroid representing merely rotation. 



Any two screws determine a cylindroid. Let their pitches be 

 PtPz, let them make an angle 7 and let the length of their common 

 perpendicular be h. Then if they lie on a cylindroid we must have, 

 by 23), 22), 



