122 THE THEORY OF SCREWS. [134 



We thus see that p^ and p., are linear functions of /?/ and p. 2 } the several 

 coefficients A B , A B, &c., in these two equations being constant. The 

 equation for u is thus to be transformed by a linear substitution for p 1 and 

 p.,. Of course ti e , being dependent only upon the position of X, is quite 

 unaffected by the change of the screws of reference. We can therefore 

 apply the well-known principle that the invariant of this binary quantic 

 can only differ by a constant factor from the transformed value. The 



invariant is 



(\ - u e -) (v - u e 2 ) - (yu, + / cos e) 2 . 



This must be true for every point X, and therefore for all values of u g -. 

 It is necessary that the coefficients of the terms in the expression 



itg 4 sin 2 e UQ&quot; (X + v -f 2/t cos e) + \v p? 

 shall be severally proportional to those in the transformed expression 



u e 4 sin 2 e u g - (X + v -f 2// cos e ) + XV // 2 . 

 We thus obtain the two equations of condition, 



sin 2 e _ X + v + 2// cos e _ X i/- // 2 

 sin 2 e X + v + 2/it cos e Xp /A 2 



The four quantities, X , /A , v , e , may now be chosen arbitrarily, subject to 

 these two equations, which are the necessary as well as the sufficient 

 conditions. Indeed it is obvious that there must be but two independent 

 quantities corresponding to the two positions of A and B . 



We may impose two conditions on the four quantities, and for our present 



purpose we shall make 



X = v ; /u/ = 0. 



The equations of X and e are then 



sin 2 e _ 2X _V* 



sin 2 e X -4- 2/A cos e + v \v fi 2 

 and we obtain 



x/= 2 (^-A^ 

 X + 2/A cos e + v 



4 (\v - /u 2 ) 



sin - e = sin 1 e 



- s r, 



(X + 2/LA COS 6 + V) 2 



X is thus uniquely determined, and the expression for sin 2 e gives for e four 

 values of the type e , + (TT e ). The negative values are meaningless, and 

 the two others are coincident, because the arc which subtends e on one side 

 subtends IT e on the other. 



There is thus a single pair of screws of reference which permit the 

 expression for u e - to be exhibited in the canonical form 



