54 THE THEORY OF SCREWS. [60- 



whence is the centre of gravity of particles of masses and placed at 

 A and B, respectively. 



From the known property of the centre of gravity, 

 .1 1 /I 



Pa Pft \Pa 



but each of the terms on the left-hand side is unity, whence, as required, 



I 1- _?_ 



The second mode of representing the pitch also verifies this theorem. 

 For since ( 59) 



AO.AB 



_BO.BA 



Pft ~ ~20S~ ; 

 we have 



_AB&amp;gt; AB 2 .AO.BO 



p+pft- 



from which 



but OA . OB is constant for every chord through ; and, as OS is constant, 

 it follows that the sum of the reciprocals of the pitches of two reciprocal 

 screws on any cylindroid must be constant. 



61. The Virtual Coefficient. 



Let A and B (Fig. 12) be the two screws. Let, as usual, be the pole of 

 the axis of pitch PT. Let be the point in which the chord AB intersects 

 OT the perpendicular drawn from to the axis of pitch, and let FT be the 

 polar of , which is easily shown to be perpendicular to SO. From T let fall 

 the perpendicular TF upon AT , and from let fall the perpendicular OG 

 upon AB. 



As before ( 58), we have AT F = tT TF=e\ also, since 



^SAO =^AT , and ^SAO = ^ATO, 



we must have ^SAO -^SAO = /.AT O 1 -^ATO, or Z OAG = z TAF- 

 whence the triangles OA G and TAF are similar, and, consequently, 



