248 THE THEORY OF SCREWS. [230 



relation must subsist between the six screws. This relation may be ex 

 pressed by equating the determinant of 39 to zero. The determinant 

 (which may perhaps be called the sexiant) may be otherwise expressed as 

 follows : 



The equations of the screw A k are 



oc XT, y i/ 1, z ZT. , . , , , 

 - - = 3 ho J2 = - - (P^ch Pk ). 

 * ft 7/fc 



We shall presently show that we are justified in assuming for T the 

 equations 



The condition that A k and T be reciprocal is 



77*) + ** (7& - P 



Writing the six equations of this type, found by giving k the values 

 1 to 6, and eliminating the six quantities 



P&amp;lt;*, p&, py, &amp;lt;*, ft 7) 

 we obtain the result : 



#i ~ a i2A&amp;gt; a i ft. 7i 



3/J3 + 732/3 - *3, 



4/&amp;gt;4 + 742/4 - ft^4, ft/&amp;gt;4 + a 4 ^4 ~ 744 , 74/&amp;gt;4 + ft4 ~ ^4 , 4, ft, 74 



a 5P5 + 75 3/5 - ftj^B, ftps + 4^5 - 75^5, 75/5 + ft^5 ~ ^S^/S, 5 , ft, 7 



6 2/6, 6 , ft, 7e 



= 0- 



By transformation to cmy parallel axes the value of this determinant is 

 unaltered. The evanescence of the determinant is therefore a necessary 

 condition whenever the six screws are reciprocal to a single screw. Hence 

 we sacrificed no generality in the assumption that T passed through the 

 origin. 



Since the sexiant is linear in x l} y 1} z i} it appears that all parallel screws 

 of given pitch reciprocal to one screw lie in a plane. Since the sexiant is 

 linear in a 1} ft, 7^ we have another proof of Mobius theorem ( 110). 



The property possessed by six screws when their sexiant vanishes may be 

 enunciated in different ways, which are precisely equivalent. 



(a) The six screws are all reciprocal to one screw. 



