A Treatise on the Theory of Screws

 292 [276, three directions of a, ß, y. 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 0lt ... 0e are lines at infinity. It follows that the three equations, Z = 0; M =0; N = 0, 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 THE THEORY OF SCREWS. cos a, cos 7 cos c, ------ + g-----+h------- Pi Pi Pi h „cosci6 cos 6« 7 cosc6 -j--------+ g-----+ h-------, P» Pi Pi where f g, h are any quantities whatever; for it is obvious that, by substi- tuting these values for ... 06 in either L, or M, or N, these quantities are made to vanish by the formulae of the type cos cos cos a6 cos L ----------1 + ... +----------- = 0. Pi Pi We have, consequently, in the expressions just written for 0l3 ... 06, 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, ß, y 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 ß cos ß + R sin y cos y = 0. Let £ be three screws of zero pitch, which intersect at right angles, and let 0 be another screw, then, if be the virtual coefficients of p and 0, = pe cos a — P sin a, whence, by the theorem j ust mentioned, we have pe = 2wve cos a + 2ct<!ö cos ß + cos y. Let alt ... a6 be the angles made by y with the six co-reciprocal screws of reference, then cos a = L = 01 cos «j + ... + 0O cos ae, and, similarly, for the two other angles, cos ß = M = 01 cos . + 06 cos be, cos 7 = N = 01 cos c, + ... -I- 06 cos c6,