65] THE REPRESENTATION OF THE CVLINDRO1D BY A CIRCLE. 59 



AO.AB 



but we know ( 59) p l 



whence the virtual coefficient is 



280 



AO.BX 2m sin A . AO OG 



= in 



280 2SO OS 



as already determined. This is an instructive proof, besides being much 

 shorter than the other methods. 



Fig. 15. Fig. 16. 



64. Properties of the Virtual Coefficient. 



If the virtual coefficient be given then the chord envelopes a circle with 

 its centre at the pole of the axis of pitch. 



Two screws can generally be found which have a given virtual coefficient 

 with a given screw. 



Let A (Fig. lo) be a given screw, and X a variable screw ; then their 

 virtual coefficient is proportional to OG, and therefore to the sine of A, that 

 is, to the length BX. Thus, as X varies, its virtual coefficient with A 

 varies proportionally to the distance of X from the fixed point B. 



65. Another Construction for the Pitch. 



As the virtual coefficient of two coincident screws is equal to their pitch, 

 we shall obtain another geometrical construction for the pitch by supposing 

 two screws to coalesce. For (in Fig. 1C), let AG be the chord joining the two 

 coincident screws, that is the tangent, then, from 61, we have for the pitch, 



OG 



m OS 

 whence the following theorem : 



The pitch of any screw is proportional to the perpendicular on the tangent 

 at the point let fall from the pole of the axis of pitch. 



