290 THE THEORY OF SCREWS. [274- 



For, take a ray e, which makes direction-angles X, /A, v with a, b, c, then 

 we have 



COS j = COS X COS ! + COS /A COS &! + COS V COS Cj , 



cos e 6 = cos X cos a 6 + cos /u, cos 6 6 + cos v cos c 6 . 

 Hence 



then, for every ray, we shall have 



cos 



ft 



It thus follows that the coefficients of a linear function which possesses 

 the property of a pitch invariant must be subjected to three conditions. 

 There are accordingly only three coefficients left disposable in the most 

 general type of linear pitch invariant. Now, 



fL+gM + hN 



is a pitch invariant which contains three disposable quantities, f, g, h; it 

 therefore represents the most general form of linear function which possesses 

 the required property. 



We have thus solved the problem of finding a perfectly general expression 

 for the linear pitch invariant function of the co-ordinates of a screw. 



It is convenient to take the three fundamental rays as mutually rect 

 angular; but it is, of course, easy to show that any linear pitch invariant 

 can be expressed in terms of three pitch invariants unless their determining 

 rays are coplanar. We may express the result thus : Let L, M, N, be 

 four linear pitch invariants, no three of which have coplanar determining 

 rays. Then it is always possible to find four parameters, X, /*, v, p, such that 

 the following equation shall be satisfied identically : 



\L + fiM + vN + pO = 0. 



275. Geometrical meaning. 



The nature of the pitch invariant function can be otherwise seen. It is 

 well known that in the composition of two or more twist velocities we 



