216 THE THEORY OF SCREWS. [211 



to the generators of a right circular cone. All the screws reciprocal to 77 

 form a cylindroid, and fy is the one screw of the system which is parallel 

 to the nodal line of the cylindroid. The virtual coefficient of ^ and 17 is 

 greater than that of 77 with any other screw. 



If 6 be a screw about which, when a body is twisting with a given 

 twist velocity it has a given kinetic energy, then we must have 



U *e? + uje* + u 3 2 3 2 - E(e? + e.? + e/) = o, 



where E is a constant proportional to the energy. It follows that the locus 

 of 6 must be a conic passing through the four points of intersection of the 

 two conies 



u*d? + u?e? + u 3 2 6 3 2 = 0, 



The four points in which these two conies intersect correspond to the screws 

 about which the body can twist with indefinite kinetic energy. These four 

 points A, B, C, D being known, the kinetic energy appropriate to every point 

 P can be readily ascertained. It is only necessary to measure the anharmonic 

 ratio subtended by P, at A, B, C, D, and to set off on a straight line 

 distances w x 2 , u, u 3 2 , h 2 , so that the anharmonic ratio of the four points 

 shall be equal to that subtended by P. This will determine h 2 , which is 

 proportional to the kinetic energy due to the unit twist velocity about the 

 screw corresponding to P. 



A quiescent rigid body of mass M receives an impulsive wrench of given 

 intensity on a given screw 77 ; we investigate the locus of the screw 6 belonging 

 to the three-system, such that if the body be constrained to twist about 6, 

 it shall acquire a given kinetic energy. 



It follows at once (91) that we must have 



where E is proportional to the kinetic energy. The required locus is there 

 fore a conic having double contact with the conic of inertia. 



It is easy to prove from this that E will be a maximum if 

 V#i : Ptfi = W 2 2 2 : p 2 i] 2 = u 3 2 3 : p s rj 3 ; 



whence again we have Euler s well-known theorem that if the body be 

 allowed to select the screw about which it will twist, the kinetic energy 

 acquired will be larger than when the body is constrained to a screw other 

 than that which it naturally chooses ( 94). 



A somewhat curious result arises when we seek the interpretation of 

 a tangent to the conic of infinite pitch. This tangent must, like any other 

 straight line, correspond to a cylindroid ; and since it is the polar of the 



