145] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 133 



and thus obtain the construction used in 142. A similar construction is 

 obtained when fl is at infinity. 



The two points, A and A , will divide the arc cut off by XY in a constant 

 anharmonic ratio, for the pencil H (XQQ Y) always preserves the same 

 anharmonic ratio as H moves round the circle. 



145. A Special Case. 



If 77 be an impulsive screw, arid if a be the corresponding instantaneous 

 screw, it will not usually happen that when a. is the impulsive screw ij is the 

 corresponding instantaneous screw. If, however, in even a single case, it be 

 true that the impulsive screw and the instantaneous screw are interchange 

 able, then the relation will be universally true. 



Let fl and 1 (Fig. 26) be a pair of points belonging to the system 

 described in 144. Then A being given, A is found. If A is similarly to 



Fig. 26. 



determine A, then the figure shows that fl must lie on the polar of fl , 

 and, consequently, fl and fl are conjugate points with respect to the circle; 

 or, what comes to the same thing, they divide XY harmonically. The same 

 must be true of each pair of points H and fl , and therefore of and , and 

 we have the following theorem : 



If the points and be harmonic conjugates of the points where the homo- 

 graphic axis intersects the circle, then every pair of instantaneous and impulsive 

 screws on the cylindroid are interchangeable. 



We might, perhaps, speak of this condition of the system as one of 

 dynamical involution. In this remarkable case an impulsive wrench of unit 

 intensity applied to one of the principal screws of inertia will generate a 

 velocity equal and opposite to that which would have been produced if the 

 wrench had been applied to the other principal screw. The construction 



