470 THE THEORY OF SCREWS. 



425. The Vector in Orthogonal Co-ordinates. 



Since, in general, 



cos 6 = pr , 



12&amp;gt; 



we have for the vector ( 419) the following conditions : 



(11) = (22) = (33) = (44), 

 and also, 



[425 



and the similar equations. In fact, U can only differ from fl by a constant 

 factor. 



The orthogonal equations require the following conditions 

 + (11) . (12) - (12) . (11) + (13) . (23) + (14) . (24) = 0, 

 + (11) . (13) - (12) . (23) - (13) . (11) + (14) . (34) = 0, 

 + (11) . (14) - (12) . (24) - (13) . (34) - (14) . (11) = 0, 

 + (12) . (13) + (11) . (23) - (23) . (11) + (24) . (34) = 0, 

 + (12) . (14) + (11) . (24) - (23) . (34) - (24) . (11) = 0, 

 + (13) . (14) + (23) . (24) + (11) . (34) - (11) . (34) = 0, 



h(14) 2 =l, 

 h(24) 2 =l, 

 + (13) 2 + (23) 2 + (11)- + (34) 2 = 1, 

 + (14) 2 + (24) 2 + (34) 2 + (II) 2 = 1. 



We now introduce the notation : 



(!!) = ; (12) = 0; (13) = 7; (14) = 8, 

 and the equations give us 



+ 7 (23)+ B (24) = (i), 



-(23) + a (34) = 



-(24)- 7 (34) = 



+ 7 +(24)(34) = 





+ a 2 + p 2 

 + /3 2 + a 2 

 + 7 2 + (23) 



-(23)(34) = 

 +(23)(24) = 

 + 7 2 + 8- = 1 

 + (23) 2 + (24) 2 =l 

 + a 2 + 



(34) 2 =l 

 a 2 -l) 



(ii), 

 , (iii), 



. (iv), 



(vi), 

 (vii). 



