144J PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 131 



It also follows that 



is constant ; whence we have the following theorem : 



Draw through the impulsive screw A a ray AH parallel to the homographic 

 axis, then the ray from H to a fixed point fl on the homographic axis will 

 cut the circle in the instantaneous screw A , and the acquired twist velocity will 

 be inversely proportional to CIA . 



If the twist velocity to be acquired by A from a unit impulsive wrench 

 on A be assigned, then CIA is determined : there will be two screws A , and 

 two corresponding impulsive screws, either of which will solve the problem. 

 The diameter through Cl indicates the two screws about which the body will 

 acquire the greatest and the least velocities respectively with a given 

 intensity for the impulsive wrench. 



143. Twist Velocities on the Principal Screws. 



The quantities a and ft, which are the twist velocities acquired by unit 

 impulsive wrenches on the principal screws, can be expressed geometrically 

 as follows (Fig. 22) : 



Let to be the twist velocity acquired on A by the wrench on A, then, by 



the last article, 



aAY = (oA Y, 



/3AX=a,A X; 



A Y AY 

 whence : ft :: ^ : ^ . 



This ratio is the anharmonic ratio of the four points X, Y, A, A , that is, of 

 X, Y, 0, ; whence, finally, 



O Y OY 



* : P O X OX 



144. Another Investigation of the Twist Velocity acquired by 

 an Impulse. 



We have just seen that 



&amp;lt;*AY=a&amp;gt;A Y, 



whence aft AX . AY= tfA X . A Y. 



Let fall perpendiculars AP, A P , HQ on the homographic axis (Fig. 24). 

 Then, by the properties of the circle,. 



AX. AY : A X. A Y :: AP : A P ; 

 so that a/3 A P = a&amp;gt;-A P . 



92 



