DYNAMICS OF A^RIGID BODY. 99 



polar of a screw 17, with respect to this complex, must be 

 also completely independent of the screws of reference. 

 It is, therefore, obvious that the polar of r\ must be a 

 screw which lies in the same straight line as j, for sym- 

 metry will not permit any reason to be assigned in 

 favour of any other position. The co-ordinates, there- 

 fore, of a screw ?, which lies in the same straight line 

 as rj, but which has a different pitch p$ y must be equal 

 to: 



h dR \ ./ h dR \ 



~7~7T ' --> A ( 1 -rv ) 

 pi arii J \ A driQ J 



where A and h are constants to be determined. 



We sacrifice no generality by making the pitch of 

 n zero. We shall now write two identical equations. 

 Of these the first expresses that the pitch of is p^ and 

 the second expresses that the virtual co-efficient of % 

 and TJ is p$ : 



h 



* 



h dR\ I h dR\ 



~ 



Remembering that pw * + . . . +/ 6 r? 8 2 = o, and also that 

 R is an homogeneous function of the second order, and 

 that, therefore, by Euler's theorem* : 



dR dR 



j +....+ 7/ 6 = 2R = 2, 



dr\\ arje 



we have : 



* Williamson's Differential Calculus, 2nd Ed., p. 1 13. 

 H 2 



