358 THE THEORY OF SCREWS. [329- 



As 77 belongs to the three-system V we must have the six co-ordinates of 

 i) connected by three linear equations ( 77) ; solving these equations we have 



772 = ^77! +.8773 +775, 

 774 = A i ll + B rj 3 + G rj s , 



The nine coefficients A, B, C, A , B , G , A&quot;, B&quot;, G&quot; are essentially the co 

 ordinates of the three-system V. We now seek the co-ordinates of the rigid 

 body in terms of these quantities. 



Take the particular screw of U which has co-ordinates 



1, 0, 0, 0, 0, 0. 



Then the co-ordinates of the corresponding impulsive screw are 77!, 772, ... 

 where 



+ ecu?! = p! 2 - a 2 ; + e&77 3 = az - bz - Z 3 2 ; + eC77 5 = - ay + cy - I? ; 

 - ea77 2 = pj 2 - a 2 ; - &ii 4 = az + bz - / 3 2 ; - ec77 B = a y - cy Q - 1 2 2 . 



Since by hypothesis this is to belong to V, the following equations must 

 be satisfied : 



2 ~ Pi 2 = A a *_P* + B az &amp;lt;&amp;gt; ~ bzp - 1 3 2 c cy - ay, - 1* 

 a a b c 



- az - bz + 1 3 2 _ , ttM-p 2 az - bz - 1 3 2 cy - ay, - I? 



T -a- - h -O - j- ----- HO - - 



b c 



R ,, az - z - 



cy - ay - 



cab c 



In like manner by taking successively for a the screws wiih co-ordinates 



0, 0, 1, 0, 0, 

 and 0, 0, 0, 0, 1, 0, 



we obtain six more equations of a similar kind. As these equations are 

 linear they give but a single system of co-ordinates x , y 0&amp;gt; z , If, I*, 1 3 2 , 

 P*&amp;gt; P*, Pa 2 for the rigid body. The theorem has thus been proved, for of course 

 if three screws of U correspond to three screws of V then every screw in U 

 must have its correspondent restricted to V. 



330. Construction of Homographic Correspondents. 



If the screws in a certain three -system U be the instantaneous screws 

 whose respective impulsive screws form the three-system V, then when three 

 pairs of correspondents are known the determination of every other pair of 

 correspondents may be conveniently effected as follows. 



