63] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 57 



The negative sign has no significance for our present purpose, and hence we 

 have the following theorem : 



The virtual coefficient of two screws is equal to the cosine of the angle 

 subtended by their chord, multiplied into the perpendicular from the pole of the 

 chord on the axis of pitch. 



This is, perhaps, the most concise geometrical expression for the virtual 

 coefficient. It vanishes if the perpendicular becomes zero, for then the 

 chord must pass through the pole of the pitch axis, and the two screws be 

 reciprocal. The cosine enters the expression in order that its evanescence, 

 when 0= 90, may provide for the circumstance that the perpendicular is then 

 infinite. 



This result is easily shown to be equivalent to that of the last article by 

 the well-know ti theorem : 



If any two chords be drawn in a circle, then the cosine of the angle sub 

 tended by the first chord, multiplied into the perpendicular distance from its 

 pole to the second chord, is equal to the cosine of the angle subtended 

 by the second chord, multiplied into the perpendicular from its pole to 

 the first chord. 



It follows that the virtual coefficient must be equal to the perpendicular 

 from the pole of the axis of pitch upon the chord joining the two screws, 

 multiplied into the cosine of the augle in the arc cut off by the axis of pitch. 

 This is the expression of 61, namely, 



nr AS 

 OG 08 



63. Application of Screw Co-ordinates. 



It will be useful to show how the geometrical form for the virtual coefficient 

 is derived from the theory of screw co-ordinates. Let a }&amp;gt; 2 &amp;gt; and ft l} #&amp;gt; be the 

 co-ordinates of two screws on the cylindroid ; then, if the screws of reference 

 be reciprocal, the virtual coefficient is ( 37) 



Let A, B (Fig. 14) be the screws of reference, and let C and C be the two 

 screws of which the virtual coefficient is required. Let PQ be the axis of 

 pitch of which is the pole, then lies on AB, as the two screws of reference 

 are reciprocal ( 58). 



As AB is divided harmonically at and H, we have 



AO : OB :: HA : HE :: AP : BQ :: p, : p 2 ; 



whence is the centre of gravity of masses , at A and B, respectively. 



