214] FREEDOM OF THE FOURTH ORDER. 221 



only movement which the body can receive, so as to fulfil the prescribed 

 conditions, is a twist about the screw X. For X is then reciprocal to 

 AU...AS, and therefore a body twisted about X will do no work against 

 forces directed along A 1} ... A s . 



From the theory of reciprocal screws it follows that a body rotated 

 around any of the lines A ly ... A 5 will not do work against nor receive energy 

 from a wrench on X. 



In the particular case, where A l} ... A 5 have a common transversal, then 

 X is that transversal, and its pitch is zero. In this case it is sufficiently 

 obvious that forces on A^...A 5 cannot disturb the equilibrium of a body 

 only free to rotate about X. 



214. Screws of Stationary Pitch. 



We begin by investigating the screws in an ?i- system of which the pitch 

 is stationary in the sense employed in the Theory of Maximum and Minimum. 

 We take the case of n = 4. 



The co-ordinates O l ,... S of the screws of a four-system have to satisfy 

 the two linear equations denning the system. We may write these equations 

 in the form 



The screws of reference being co-reciprocal, we have for the pitch p e the 

 equation 



SM -.Rp^O, 



where R is the homogeneous function of the second degree in the co 

 ordinates which is replaced by unity ( 35) in the formulae after differ 

 entiation. 



If the pitch be stationary, then by the ordinary rules of the differential 

 calculus ( 38), 



As however belongs to the four-system, the variations of its co-ordinates 

 must satisfy the two conditions 



Following the usual process we multiply the first of these equations by 

 some indeterminate multiplier \, the second by another quantity p, and then 



