A Treatise on the Theory of Screws

244 THE THEORY OF SCREWS. [228 a three-system which satisfy an equation of the nth degree must have as their locus a surface of degree not exceeding 3n. The most important application of this is when n = 2, in which case the screws form a quadratic two-system. The degree of this surface cannot exceed six, on the other hand, if the quadratic condition which we may write aø?+ve?+co*+2F0&+zgoj,+2se1e, = o, should break up into two linear factors each of these linear factors will correspond to a cylindroid, i.e. a surface of the third degree. Hence the degree of the surface must in general be neither less than six nor greater than six, and hence we learn that the surface whieh is the locus of the screws of a quadratic two-system is of the sixth degree. A particular case of special importance arises when the pitches of all the screws on the surface are to be the same. The statement of this condition is of course one equation of the second degree in the co-ordinates of the screw. In the case of canonical co-reciprocals, this equation would be Pi+ ■ • • +pA3=Pe {(0> + 02)2 + (0S + W + (05 + 06)2}. But the condition that 0 and a shall intersect will now submit to modification. We sacrifice no generality by making a of zero pitch, so that if 0 has a given pitch pg, the condition that a and 0 shall intersect is no longer of the third degree. It is the linear equation = pe cos (ad). If therefore the co-ordinates of 0 satisfy three homogeneous equations of degrees I, m, n respectively, in addition to the equation of the second degree expressing that the pitch is a given quantity, then the locus is a surface of degree not exceeding 2lmn. As the simplest illustration of this result we observe that if I, m, n be each unity, the locus in question is the locus of the screws of given pitch in a three-system. This locus cannot therefore be above the second degree, and we know, of course (Chapter XIV.) that the locus is a quadric. If I and m were each unity and if n = 2 we should then have the locus of screws of given pitch belonging to a four-system and whose co-ordinates satisfied a certain equation of the second degree. This locus is a surface of the fourth degree. In the special case where the given pitch is zero, the surface so defined is known in the theory of the linear complex. It is there presented as the locus of lines belonging to the complex and whose co- ordinates further satisfy both a linear equation and a quadratic equation. Mr A. Panton has kindly pointed out to me that in this particular case the surface has two double lines which are the screws of zero pitch on the