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 



A0? + B0 2 * + CB? + 2F0A + 200,6, + 2H0A = 0, 



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 which 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 



But the condition that Q and a shall intersect will now submit to 

 modification. We sacrifice no generality by making a of zero pitch, so 

 that if 6 has a given pitch p e , the condition that a and shall intersect 

 is no longer of the third degree. It is the linear equation 



2t3- ae = p e cos (a0). 



If therefore the co-ordinates of 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 



