276] EMANANTS AND PITCH INVARIANTS. 291 



discover the direction of the resultant screw and the magnitude of the 

 resultant twist velocity by proceeding as if the twist velocities were vectors. 

 Neither the pitches of the component screws nor their situations affect the 

 magnitude of the resultant twist velocity or the direction of the resultant 

 screw. This principle is, of course, an immediate consequence of the law 

 of composition of twist velocities by the cylindroid. 



Let any ray a make an angle A, with the ray 6, and angles a 1} ... a 6 with 

 the six screws of reference. The twist velocity $ on 6 if resolved on a has 

 a component 0cos\. This must be equal to the sum of the several com 

 ponents 1} # 2 , ... resolved on a; whence we have 



cos X = #! cos a-i + ... + G cos a 6 . 

 If we make = unity, we obtain 



cos \=0 1 cos a t + ... +0 6 cos a s . 



This gives a geometrical meaning to the pitch invariant. It is simply the 

 cosine of the angle between the screws 6 and a. As, of course, the pitch 

 is not involved in the notion of this angle, it is, indeed, obvious that the 

 expression for any function of the angle must be a pitch invariant. 



We now see the meaning of the equation obtained by equating the pitch 

 invariant to zero. If we make 



1 COS !+...+ # 6 COS C1 6 = 



it follows that a. and must be at right angles. The equation therefore 

 signifies the locus of all the screws that are at right angles to a. 



The two equations 



1 cos !+... + 6 cos a 6 = 0, 



0! cos &!+... -f # 6 cos b 6 = 0, 



denote the screws perpendicular to the two directions of a and /3. In other 

 words, these two equations define all the screws perpendicular to a given 

 plane. 



276. Screws at infinity. 



Let us now take the case where a, ft, 7 are three rectangular screws, and 

 examine the conditions satisfied by 1} ..., 6 when subjected to the three 

 following equations : 



#1 cos (&!+ ... + 6 cos a 6 = 0, 



0! COS h + ... + 6 COS b 6 = 0, 

 l COS G! + ... + 6 COS C 6 = 0. 



The screws which satisfy these conditions must all be perpendicular to the 



192 



