292 THE THEORY OF SCREWS. [276, 



three directions of o, j3, 7. For real and finite rays this is impossible ; for 

 real and finite rays could not be perpendicular to each of three rays which 

 were themselves mutually rectangular. This is only possible if the rays 

 denoted by 1} ... 6 K are lines at infinity. 



It follows that the three equations, L = 0; M=Q; N=Q, obtained by 

 equating the three fundamental pitch invariants to zero, must in general 

 express the collection of screws that are situated in the plane at infinity. 



We can write the three equations in an equivalent form by the six 

 equations 



., ,,COSi COS&j 7 COSCi 



0i =/ + g- -+A- -, 



Pi Pi Pi 



n , cos a s cos 6 6 7 cos c 6 

 t/6 =/ - + g + fi - , 

 Pa p 6 p s 



where f, g, h are any quantities whatever; for it is obvious that, by substi 

 tuting these values for 1 , ... 6 in either L, or M, or N, these quantities are 

 made to vanish by the formula of the type 



X cos a 6 cos 6 



Pi 



We have, consequently, in the expressions just written for B lt ... 6 , the 

 values of the co-ordinates of a screw which lies entirely in the plane at 

 infinity. 



277. Expression for the Pitch. 



It is known that if a, 0, 7 be the direction-angles of a ray, and if P, Q, R 

 be its shortest perpendicular distances from three rectangular axes, then 



P sin a cos a + Q sin /3cosfi + R sin 7 cos 7 = 0. 



Let 77, , be three screws of zero pitch, which intersect at right angles, and 

 let 6 be another screw, then, if vr^ be the virtual coefficients of 77 and 0, 



2&quot;5T,,0 = p 6 cos a P sin a, 

 whence, by the theorem just mentioned, we have 



p e = Zvr^o cos a + 2tjy| e cos + 2^ cos 7. 



Let a 1} ... a 6 be the angles made by 77 with the six co-reciprocal screws of 

 reference, then 



cos a = L = 6 1 cos a x + . . . + 6 cos a a , 

 and, similarly, for the two other angles, 



cos yS = M= e i cos &J + ...+ 6 cos 6 6 , 

 cos 7 = N = l cos d + . . . + B 6 cos c 6 , 



