468 THE THEORY OF SCREWS. [422- 



and that joining 



p 



Fig. 47. 



Let a 1} 2 , 3 , 4 and /3 1} /9 2 , /3 3) /3 4 be the co-ordinates of the corners at p 

 and p&quot;. If we substitute a : + X,^, a 2 4- X/3 2 . . . &c., for x 1} x 2 &c. in fl = 0, 



2\ (a, J3, + a 2 /3 2 + a 3 /3 3 + 4 /3 4 ) = 0. 

 Let us make the same substitution in U, we have, in general, 



= (11) (a, + X&) + (12) (a, + \/3 2 ) + (13) (a, + X&) + (14) (a 4 + X/3 4 ) 

 = p otj + Xp&quot;/3i ; 

 whence, remembering that 



+ \p&quot;/3 2 ) + &c., 

 and as a and /S are both on O, we have, 



U=\(p+ p&quot;) ( ai /3 x + 2 /9 2 + a 3 yS 3 + a 4/ S 4 ) ; 



but since the line joining p and p&quot; is a generator of O, the last factor must 

 vanish, and the line is therefore also a generator of U. 



It is thus proved that U has four generators in common with fl. 

 423. Verification of the Invariance of Intervene. 



As an exercise in the use of the orthogonal system of co-ordinates, we 

 may note the following proposition : 



Let #!, x. 2 , x 3 , # 4 , and #/, a? 2 , x 3 , #/, be two objects which are conveyed by 

 the transformation to y lt y 2 , y a , 7/ 4 , and y/, y a , y 3 , y 4 , respectively, it is 

 desired to show that the intervene between the two original points is equal 

 to that between the transformed. The expressions for the cosine of the 

 intervene are 



(x* + x? + x? + xff (x^ + x? + x^ + 



