A Treatise on the Theory of Screws

44 THE THEORY of SCREWS. [49, 50 In this form it is obvious that to increase each of the three quantities P«., pß, p by m does not affect the equation. This is of course the well-known property of the cylindroid. (§ 20.) If the quadratic have two equal roots, then all the screws of the cylindroid form a plane pencil, and all have equal pitches. The discriminant of the quadratic is cos- (aß)+(pa + Pßy - (pa +pß) cos (aß) - 4>papß sin2 (aß)+4^ sin2(aß) = l'^aß - (pa + pß) cos (a/3)j2 + (pa + Pßy sin-' (aß) _ 4pB/JjS sin2 - (daß2 + (pa ~Pß)2) sin2 (aß). Hence discarding the case when (aß) = 0 we have daß=0, Pa~Pß = 0, as was to be expected.