286 THE THEORY OF SCREWS. [270- 



Equating these two values of the pitch we ought to have for every screw S 



- n^ + M 3 2 &amp;lt; 2 2 + w 3 2 &amp;lt; 3 2 -f n 4 2 &amp;lt; 4 2 + m* 

 But it can easily be seen that this equation is impossible. 



Let H be the screw to which all the screws of the five-system are re 

 ciprocal, and let us choose for 8 the screw reciprocal to A 1} A 2 , A 3 , B 6 , H. 

 The fact that S is reciprocal to H is of course implied in the assumption that 

 8 belongs to the five-system, while the fact that S is reciprocal to each of the 

 screws A lt A 2 . A 3 , B 5 gives us 



1= =0, 2 = 0, 3 = 0, 8 = 0. 

 Hence we would have the equation 



which would require that all the co-ordinates were zero, which is im 

 possible. 



In like manner any other supposition inconsistent with the theorem of 

 this article would be shown to lead to an absurdity. The theorem is there 

 fore proved. 



We can hence easily deduce the important theorem that three of the 

 screws in a complete co-reciprocal system of six must have positive pitch 

 and three must have negative pitch*. 



For in the canonical system of co-reciprocals the pitches are + a, a, 

 4- b, b, + c, c, i.e. three are positive and three are negative, and as in this 

 case the w-system being the six -system includes every screw in space we see 

 that of any six co -reciprocals three of the pitches must be positive and three 

 must be negative. 



271. Identical Formulae in a Co-reciprocal System. 



Let any screw a be inclined at angles cxi, a2, ... aG to the respective six 

 screws of a co-reciprocal system. 



Then we have for the co-ordinate a n 



_ (p a +p n )cos al d al sin al 

 ^ ~~~ 



* This interesting theorem was communicated to me by Klein, who had proved it as a 

 property of the parameters of &quot;six fundamental complexes in involution&quot; (Math. Ann. Band. 

 i. p. 204). 



