50 THE THEORY OF SCREWS. [54- 



If twist velocities about the three screws A, B, C animate a rigid body, 

 then these twist velocities will neutralize if they are respectively proportional 

 to the opposite sides of the triangle. 



55. Decomposition of Twists and Wrenches. 



The theorems we have just enunciated lead to simple rules for effecting 

 the composition, or the decomposition of twists or of wrenches. Let a twist 

 on a screw X be given, and let it be required to find the components of this 

 twist on any two given screws A, B, all three, of course, lying on the same 

 cylindroid. Let o be the amplitude of the twist on X. Then, by the last 

 article, the following triad of twists on the screws X, A, B, respectively, will 

 neutralize : 



BX AX 

 w &amp;gt; &quot;AB &quot;AB 



whence the components on A a,nd B of the twist about X are, so far as 

 magnitudes are concerned, 



AX 

 and &quot;AS 



A similar proposition holds of course for wrenches. 



56. Composition of Twists and Wrenches. 



Let two twists, of amplitudes a and /3, about the screws A and B, respec 

 tively, be applied to a rigid body. It is required to find the single resultant 

 screw X, and the amplitude CD of the resulting twist. Divide AB (Fig. 9) 

 in the point /, so that the segments AI and BI shall be in the inverse ratio 

 of a to /3. Bisect the arc AB at H, and draw HI, which will cut the circle 

 in the required point X. 



The value of &&amp;gt; is obtained from the equations 



