346 THE THEORY OF SCREWS. [318, 



If the former had been applied to M^ it would have generated about a 

 ( 280) a twist velocity represented by 



, sin (A/T - &amp;lt;f&amp;gt;) J^ cos(0a) 

 r sin (6 r -&amp;lt;/&amp;gt;) M t p a 



If the latter had been applied to the body M 2 , it would have generated a 

 twist velocity about the same screw, a, equal to 



, sin (6 ^r) J^ cos ((fta) 

 * sin (0-0) J/7 &quot;~ Pa 



Suppose that these two twist velocities are equal, it is plain that the original 

 wrench on ty would, if it had been applied to the composite rigid body, 

 produce a twisting motion about a. The condition is 



sin (^r &amp;lt;/&amp;gt;) _ sin (6 -fy) 

 MI cos ((pa) ~ M 3 cos (0a) 



We thus obtain tan i/r in terms of MI : M a . As the structure of the 

 composite body changes by alterations of the relative values of p l and p. z , 

 so will T/T move over the various screws of the cylindroid (6, &amp;lt;p). 



This result shows that, if three screws, a, ft, 7 be given, then the possible 

 impulsive screws, 77, , which shall respectively correspond to a, /3, 7 in a 

 given system of the third order P, are uniquely determined. 



For, suppose that a second group of screws, 77 , % &amp;gt; &amp;gt; could also be deter 

 mined which fulfilled the same property. We have shown how another 

 rigid body could be constructed so that another screw, ^, could be found on 

 the cylindroid (77, 77 ), such that an impulse thereon given would make the 

 body twist about a. It is plain that, for this body also, the impulsive 

 wrench, corresponding to ft, would be some screw on the cylindroid (f, ). 

 But all screws on this cylindroid belong to the system P. In like manner, 

 the instantaneous screw 7 would correspond for the composite body to some 

 screw on the cylindroid (, &quot;) Hence it follows that, for each different 

 value of the ratio p^ : p 2 , we would have a different set of impulsive screws 

 for the instantaneous screws a, ft, 7. We thus find that, if there were 

 more than one set of such impulsive screws to be found in the system P, 

 there would be an infinite number of such sets. But we have already 

 shown that the number of sets must be finite. Hence there can only be 

 one set of screws, 77, , , in the system P, which could be impulsive screws 

 corresponding to the instantaneous screws, a, ft, 7. We are thus led to the 

 following important theorem, which will be otherwise proved in the next 

 chapter. 



Given any two systems of screws of the third order. It is generally pos 

 sible, in one way, but only in one, to design, and place in a particular position a 



