30] SCREW CO-ORDINATES. 35 



When the co-ordinates of a screw are given, the screw itself may be thus 

 determined. Let e be any small quantity. Take a body in the position A, 

 and impart to it successively twists about each of the screws of reference of 

 amplitudes ea ly &amp;lt;z. 2 , ... e 6 . Let the position thus attained be B\ then the 

 twist which would bring the body directly from A to B is about the required 

 screw a. 



35. Identical Relation. 



The six co-ordinates of a screw are not independent quantities, but 

 fulfil one relation, the nature of which is suggested by the relation between 

 three direction cosines. 



When two twists are compounded by the cylindroid ( 14), it will be 

 observed that the amplitude of the resultant twist, as well as the direction 

 of its screw, depend solely on the amplitudes of the given twists, and the 

 directions of the given screws, and not at all upon either their pitches or their 

 absolute situations. So also when any number of twists are compounded, the 

 amplitude and direction of the resultant depend only on the amplitudes and 

 directions of the components. We may, therefore, state the following general 

 principle. If n twists neutralize (or n wrenches equilibrate) then a closed 

 polygon of n sides can be drawn, each of the sides of which is proportional 

 to the amplitude of one of the twists (or intensity of one of the wrenches), 

 and parallel to the corresponding screw. 



Let a n , b n , c n , be the direction cosines of a line parallel to any screw of 

 reference &&amp;gt;, and drawn through a point through which pass three rect 

 angular axes. 



Then since a unit wrench on a has components of intensities a,,... a 6 , 

 we must have 



(!! 4 . . . + tt (i a fi ) 2 + (&ii + . + &) 2 + (Cjfli 4- . . . + c (i a (i ) 2 = 1, 

 whence ^X 2 -f 2Sai&amp;lt;x,cos (12) = 1, 



if we denote by cos (12) the cosine of the angle between two straight lines 

 parallel to m 1 and &&amp;gt; 2 . 



36. Calculation of Co-ordinates. 



We may conceive the formation of a table of triple entry from which the 

 virtual coefficient of any pair of screws may be ascertained. The three argu 

 ments will be the angle between the two screws, the perpendicular distance, 

 and the sum of the pitches. These arguments having been ascertained by 

 ordinary measurement of lines and angles, the virtual coefficient can be 

 extracted from the tables. 



Let a be a screw, of which the co-ordinates are to be determined. The 



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