A Treatise on the Theory of Screws

214] FREEDOM OF THE FOURTH ORDER. 223 is of course (Pe + P<t,) cos W) - sin W) de-l> = 2Pe cos (0<f>), or (P4> ~ Pe) cos W) ~ sin W) de4> = °- If then p<i, = p(i we must have sin(Ö</>) dfj,,,= 0, which requires that 0 and <f> must intersect at either a finite or an infinite distance. In the case where <f> is at right angles to 0 it follows from the formula = cos (#</>) p6 that ■sr^ = 0, or that 0 and <f> are reciprocal. But two screws which are at right angles and also reciprocal must intersect, and hence we have the following theorem. If 0 be a screw of stationary pitch in an n-system, then any other screw belonging to the n-system and at right angles to 0 must intersect 0. If <j> belongs to an n-system its co-ordinates must, on that account, satisfy 6 - n linear equations. If it be further assumed that </> has to be perpen- dicular to 0, then the co-ordinates of have to satisfy yet one more equa- tion, i.e. . dll , dll .. In this case </> is subjected to 7 - n linear equations. It follows (§ 76) that (p will have as its locus a certain (n — l)-system, whence we have the following general theorem. If 0 is a screw of stationary pitch in an n-system P then among the (n _ systems included in P there is one Q such that every screw of Q intersects 0 at right angles. These theorems can also be proved by geometrical considerations. If a screw 0 have stationary pitch in an n-system it follows a fortiori that 0 must have stationary pitch on any cylindroid through 0 and belonging wholly to the n-system. This means that 0 must be one of the two principal screws on such a cylindroid. Choose any other screw </> of the system and draw the cylindroid (0, <f>) then 0 is a principal screw, and if 0 and the other principal screw on the cylindroid be two of the co-reciprocal screws of reference, then the co-ordinate of </> with respect to 0 is cos (00) (§ 40). But that co-ordinate must also have the general form plt whence at once we obtain ww = cos (Ø^Pe- Let 0 be a screw of stationary pitch in a throe-system, and let </> and be any two other screws in that system. Then 0 is one of the principal screws on the cylindroid (0<f); let a be the other principal screw on that