309] THE GEOMETRICAL THEORY. 333 



will, by the property of a principal screw of inertia, produce an instantaneous 

 twist velocity about the same screw. But the two twist velocities so 

 generated can, of course, only compound into a single twist velocity on some 

 other screw of the cylindroid. We have now to obtain the geometrical 

 relations characteristic of the pairs of impulsive and instantaneous screws on 

 such a cylindroid. 



In previous chapters we have discussed the relations between impulsive 

 screws and instantaneous screws, when the movements of the body are 

 confined, by geometrical constraint, to twists about the screws on a 

 cylindroid. The problem now before us is a special case, for though the 

 movements are no other than twists about the screws on a cylindroid, yet 

 this restriction, in the present case, is not the result of constraint. It arises 

 from the fact that two of the six principal screws of inertia of the rigid 

 body lie on the cylindroid, while the impulsive wrench is, by hypothesis, 

 limited to the same surface. 



To study the question we shall make use of the circular representation 

 of the cylindroid, 50. We have there shown how, when the several screws 

 on the cylindroid are represented by points on the circumference of a circle, 

 various dynamical problems can be solved with simplicity and convenience. 

 For example, when the impulsive screw is represented on the circle by 

 one point, and the instantaneous screw by another, we have seen how 

 these points are connected by geometrical construction ( 140). 



In the case of the unconstrained body, which is that now before us, it is 

 known that, whenever the pitch of an instantaneous screw is zero, the corre 

 sponding impulsive screw must be at right angles thereto ( 301). 



In the circular representation, the angle between any two screws is 

 equal to the angle subtended in the representative circle by the chord 

 whose extremities are the representatives of the two screws. Two screws, 

 at right angles, are consequently represented by the extremities of a 

 diameter of the representative circle. If, therefore, we take A, B, two 

 points on the circle, to represent the two screws of zero pitch, then the two 

 points, P and Q, diametrically opposite to them, are the points indicating 

 the corresponding impulsive screws. It is plain from 287 that AQ and BP 

 must intersect in the homographic axis, and hence the homographic axis 

 is parallel to AQ and EP, and as it must contain the pole of AB it follows 

 that the homographic axis XY must be the diameter perpendicular to AB. 



The two principal screws of the cylindroid X and Y are, in this case, the 

 principal screws of inertia. Each of them, when regarded as an impulsive 

 screw, coincides with its corresponding instantaneous screw. The diameter 

 XY bisects the angle between AP and BQ. 



