381] THE THEORY OF PERMANENT SCREWS. 417 



screw. We ought, therefore, to find that the expressions for the co-ordinates 

 of ?) remained unaltered if we substituted for 1} ... 6 , the co-ordinates of 

 any other screw on the same straight line as 6. These are ( 47) 



04), 04 - y (0 3 



where H is arbitrary. 



Introducing these into the values for rj^, it becomes 



- aca) 2 (ft + -- a) 3 } + abw. A U% + -?- j + (& 2 - c 2 ) &&amp;gt; 2 &&amp;gt; 3 , 

 y C .. / \ / 



from which .H&quot; disappears, and the required result is proved. 



The restraining screw is always reciprocal to the instantaneous screw, 

 and, consequently, if e be the angle between the two screws, and d their 

 distance apart, 



(Pi + Pe) cos e - d sin e = 0. 



We have seen that this must be true for every value of p g , whence 



cos e = ; d = ; 



i.e. the two screws must intersect at right angles as we have otherwise shown 

 in 376. 



This also appears from the formulae 



Vi + W = 26p 2 &&amp;gt; 3 2c/9 3 &&amp;gt; 2 , 

 1)3&quot; + -n&quot; = 2c/o 8 a&amp;gt;i - 2a/&amp;gt; 1 o&amp;gt; 3 , 



multiplying respectively by &&amp;gt;,, &&amp;gt; 2 , eo s , and adding, we get 



til + 17s) (01 + #2) + (% + ^4&amp;gt; (03 + 4 ) + (^75 + 17) (05 + 6 ) = 0, 



which proves that rj and ^ are rectangular ; but we already know that they 

 are reciprocal, and therefore they intersect at right angles. 



381. A Particular Case. 



The expressions for the restraining wrenches can be illustrated by taking 

 as a particular case an instantaneous screw which passes through the centre 

 of gravity. 



B. 27 



