134] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 123 



We are now led to a simple geometrical representation for u e 2 . Let A, B 

 (fig. 19) be the two canonical screws of reference. Bisect AB in , then 



= -2AO -+2XO -, 



= 2XY.XO . 



It is obvious that the point 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 

 X . X Y, where 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 . Both points, and , are of much 

 significance in the representative circle. We can easily prove the following 

 theorem, in which we call the polar of 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 X to the 

 axis of inertia. 



The geometrical construction for the pitch given in 51 can also be 

 applied to determine w fl 2 . This quantity is therefore proportional to the 

 perpendicular from on the tangent at X. It thus appears that the 

 representative circle gives a graphic illustration of the law of distribu 

 tion of u e - around the screws on a cylindroid. 



The axis of inertia cannot cut the representative circle in real points, for 



