16 THE THEORY OF SCREWS. [9, 



of p. Suppose that a and ft be varied while their ratio is preserved P 

 and Q will then be transferred to P&quot; and Q&quot; while by the property just 

 proved P, P , P&quot; will be collinear and so will Q, Q , Q&quot;. It therefore follows 

 that as P, P , Q, Q are collinear so will P, Q, P&quot;, Q&quot; be collinear. The line 

 PQ will therefore be displaced upon itself for every pair of values a and 

 $ which retain the same ratio. The position of the resultant screw is thus 

 not altered by any changes of a! and ft , which preserves their ratio. 



Let f be the angle between a and ft. We take the case of a point P 

 at an infinite distance on the common perpendicular to a and (3. This 

 point is displaced through a distance equal to 



h \fa!- + ft 2 + 2a /3 cos &&amp;gt;, 



where h stands for the infinite perpendicular distance from P to a or to ft. 

 This displacement of P is normal to p which itself intersects at right angles 

 the common perpendicular to a and ft. As the perpendicular distance from 

 P to p can only differ by a finite quantity from h 



hp = h Va 2 + /3 2 + 2a ft cos a&amp;gt;, 

 or 



p = Va 7 - + ft - 2 +2a ft cos^a. 



This determines the amplitude of the resulting twist which is, it may be 

 noted, independent of the pitches. 



Let &amp;lt;j) be the angle between the directions in which a point Q on p is 

 displaced by the twists about a and ft, then the square of the displacement 

 of Q will be 



(pa 2 + h a a ) a 2 + (pi + hi) ft 2 + 2 Vp a 2 + h* \?pi + hi aft cos cf&amp;gt; ; 

 but this may also be written 



whence we see that p p depends only on the ratio of a to ft . 



The pitch and the position of p thus depend on the single numerical 

 parameter expressing the ratio of a and ft . As this parameter varies so 

 will p vary, and it must in successive positions coincide with the several 

 generators of a certain ruled surface. Two of these generators will be the 

 situations of a and of ft corresponding to the extreme values of zero and 

 infinity respectively, which in the progress of its variation the parameter 

 will assume. 



We shall next ascertain the laws according to which twists (and wrenches) 

 must be compounded together, that is to say, we shall determine the single 

 screw, one twist (or wrench) about which will produce the same effect on the 



