A Treatise on the Theory of Screws

134] PT, A NF. REPRESENTATION OF DYNAMICAL PROBLEMS. 123 We are now led to a simple geometrical representation for w82. Let A, B (fig. 19) be the two canonical screws of reference. Bisect AB in O', then P]? + p.i=BX- + AX2, = 2AO'*+2XOf = 2Z0'. YO’ + 2X0'2, = 2XY.X0'. Fig. 19. It is obvious that the point O’ must have a critical importance in the kinetic theory, and its fundamental property, which has just been proved, is expressed in the following theorem:— If a rigid body be twisting with the unit of twist velocity about any screw X on the cylindroid, then its kinetic energy is proportional to the rectangle XO' .X Y, where O’ is a fixed point. We are at once reminded of the theorem of § 59, in which a similar law is found for the distribution of pitch, only in this case another point, 0, is used instead of the point O'. Both points, 0 and O', are of much significance in the representative circle. We can easily prove the following theorem, in which we call the polar of O' the axis of inertia:— If a rigid body be twisting with the unit of twist velocity about X, then its kinetic energy is proportional to the perpendicular distance from Å to the axis of inertia. The geometrical construction for the pitch given in § 51 can also be applied to determine uf This quantity is therefore proportional to the perpendicular from O’ on the tangent at X. It thus appears that the representative circle gives a graphic illustration of the law of distribu- tion of m92 around the screws on a cylindroid. The axis of inertia cannot cut the representative circle in real points, for