56 



THE THEORY OF SCREWS. 



[62, 



at A, H, C, their centre of gravity must lie on the three lines AX, BY, GZ. 

 These lines must therefore be concurrent at /, which is the centre of gravity 

 of the three masses. 



Fig. 13. 



Let B Y intersect the circle again at H. Then, since AC is the polar of 

 Y, the arc AC is divided harmonically at H and B: consequently the four 

 points A, C, B, H subtend a harmonic pencil at any point on the circle. Let 

 that point be B, then BC, BI, BA, BZ form a harmonic pencil ; hence CZ is 

 cut harmonically, and consequently Z must be the centre of gravity of 

 particles, + a? at A, + b 2 at B, and - c 2 at C. 



Suppose the axis of pitch to be drawn (it is not shown in the figure), and 

 let h be the perpendicular let fall from Z on this axis, also let p lt p 2 , p 3 be the 

 pitches of the screws A, B, C. 



Then, by a familiar property of the centre of gravity, we must have 



Pitf + p^ - p.? = (a? + b 3 - c 2 ) k = 2abh cos C. 



We shall take A, B as the two screws of reference, and if p, and p,, be the co 

 ordinates of C with respect to A and B ; then, from the principles of screw 

 co-ordinates ( 30), we have 



where W|3 is the virtual coefficient of A and B. In the present case we have 



_ a _b 

 pl ~c p *~c 



whence 

 and, finally, 



Pi 2 



12 = h cos C. 



