A Treatise on the Theory of Screws

388] THE THEORY OF PERMANENT SCREWS. 425 We now examine the points on the cylindroid intersected by the axis of the screw F =pX+yZ — zY, G =pY+zX— xZ, H = pZ + xY — yX. We write the equations of the cylindroid in the form « = jRcos</>; 7/= 72 sin </>; .? = m sin2</>; then, eliminating p and R, and making U=X* + P +Z\ V = FX+ GY+HZ, we find, after a few reductions, tan3 <f>(YV-GU) + tan2 </> (XV-FU + 2mXU) + ta,n <f>(YV—GU— 2mYU) +XV — FU=0. This cubic corresponds, of course, to the three generators of the cylindroid which the ray intersects. If we put FY - GX = 2mXY, then the cubic becomes, by eliminating m, (Ftan </> + X) {X (YV - GU) tan*</> + (XV-FU) Fj = 0. The factor Y tan <j) + X simply means that the restraining screw cuts the instantaneous screw at right angles. The two other screws in which y intersects the cylindroid are given by the equation (XYV - XGU) tan2 </> + (XYV- FUY) = 0. These two screws are of equal pitch, and the value of the pitch is P1 (X YV - XG U) +p2 (FUY-X YV) U(FY-GX) where p1 and p2 are the pitches of the two principal screws on the cylindroid. After a few reductions the expression becomes 7 (Ip - XopJ sin 0 + (!■? + y0/>2) cos 0 U + x0 sin 0 — y0 cos 0 This is the pitch of the two equal pitch screws on the cylindroid which y intersects. If y is to be reciprocal to the cylindroid, then, of course, the pitch of y itself should be equal and opposite in value to this expression. Hence the permanent screw on the cylindroid is given by (Ip - xopi)sin 0 + (('2 + y0p2) cos 0—0.