464 THE THEORY OF SCREWS. [419, 



419. On the Intervene through which each Object is Conveyed. 



Given an object, X 1} X 2 , X s , X 4 , find the intervene through which it is 

 conveyed by the transformation, when 



is the infinite quadric. 



In this equation substitute for X l} X l + 6Y 1&amp;gt; &c.; and, remembering that 

 Y l = p^X-L, &c., we have, 



(X, + pjx,) (X, 



or 6- (p 1 p 2 X l X 2 + \p 3 p 4 X 3 X 4 ) 



+ 6 (p^X^X^ + p 2 X l X 2 + Xp 3 X 3 X 4 + \p 4 X s X 4 ) 

 + X,X 2 + \X 3 X 4 = 0. 



We simplify this by introducing 



Pi 02 = PS Pl, 



and writing \X 3 X t -=- XX Z = (f&amp;gt;, whence the equation becomes 



ffifrpi (1 + 0) + 6 [ Pl + p. 2 + &amp;lt;/&amp;gt; ( Pa + p,)] + (1 + &amp;lt;/&amp;gt;) = ; 

 hence if 8 be the intervene, we have, 

 cos 8 = 



2 V/hp. 1 + &amp;lt; 



or, if we restore its value to &amp;lt;f&amp;gt;, 



cos g _ 1 ^1^2 (pi + p 2 H_(p 3 + p 4 



V^L/o \j . . . ^^ ^^ 



H Pt + P2 = Pi + P*, 



then cosS = ^il: 



i.e. all objects are translated through equal intervenes. This is the case 

 which we shall subsequently consider under the title of the vector, as this 

 remarkable conception of Clifford s is called. In this case, as 



and also, p l p 2 = p 3 p^ } 



we must have PI = p s , and p 2 = p 4 , 



or Pi = p t , and p a = p a . 



In either case the equation for p will become a perfect square. 



