138 THE THEORY OF SCREWS. [148- 



Any ray through P, the intersection of the axis of inertia with the tangent 

 at H, cuts the circle in two points, A and A&quot;, either of which will receive the 

 same kinetic energy from the given impulse. 



149. Euler s Theorem. 



If the body be permitted to select the screw about which it will 

 commence to twist, then, as already mentioned, 94, Euler s theorem states 

 that the body will commence to move with a greater kinetic energy than if 

 it be restricted to some other screw. By drawing the tangent from P (not, 

 however, shown in the figure) we obtain the point of contact B, where it is 

 obvious that the ratio of the perpendiculars on PH and PQ is a maximum, 

 and, consequently, the kinetic energy is greatest. It follows from Euler s 

 theorem that B will be the instantaneous screw corresponding to A as the 

 impulsive screw. The line BH is the .polar of P, and, consequently, BH 

 must contain , the pole of the axis of inertia. We are thus again led 

 to the construction ( 140) for the instantaneous screw 5; that is, draw 

 AOH, and then HO B. 



150. To determine a Screw that will acquire a given Twist 

 Velocity under a given Impulse. 



The impulsive screw being given, and the intensity of the impulsive 

 wrench being one unit, the acquired twist velocity ( 147) will vary as 

 (Fig. 30), 



AfH 

 A Q 



If, therefore, the twist velocity be given, this ratio is given. A must then lie 

 on a given ellipse, with H as the focus and the axis of inertia as the directrix. 

 This ellipse will intersect the circle in four points, any one of which gives a 

 screw which fulfils the condition proposed in the problem. 



The relation between the intensity of the impulsive wrench and the twist 

 velocity generated can be also investigated as follows : 



Let P, Q, R, S be points on the circle (Fig. 31) corresponding to four im 

 pulsive screws, and let P , Q , R , S be the four corresponding instantaneous 

 screws deduced by the construction already given. Let p, q, r, s denote the 

 intensities of the impulsive wrenches on P, Q, R, S, which will give the units 

 of twist velocity on P , Q , R, S . Supposing that impulsive wrenches on 

 P, Q, R neutralize, then the corresponding twist velocities generated on 

 P , Q , R must neutralize also. In the former case, the intensities must be 

 proportional to the sides of the triangle PQR ; in the latter, the twist 



