1 10 DYNAMICS OF A RIGID BODY. 



The linear parameter u a on any screw of the cylin- 

 droid is inversely proportional to the parallel diameter 

 of a certain ellipse, and a pair of conjugate screws of 

 inertia on the cylindroid are parallel to a pair of conju- 

 gate diameters of the same ellipse. This ellipse may 

 be called the ellipse of inertia. 



The major and minor axes of the ellipse of inertia are 

 parallel to screws upon the cylindroid, which for a given 

 twist velocity correspond to a maximum and minimum 

 kinetic energy. 



An impulsive wrench on a screw j acts upon a quies- 

 cent rigid body which has freedom of the second order. 

 It is required to determine the screw 9 on the cylin- 

 droid expressing the freedom about which the body will 

 commence to twist. 



The ellipse of inertia enables us to solve this problem 

 with great facility. Determine that one screw on the 

 cylindroid which is reciprocal to TJ ( -8). Draw a 

 diameter D of the ellipse of inertia parallel to 0. Then 

 the required screw 9 is simply that screw on the cylin- 

 droid which is parallel to the diameter conjugate to D 

 in the ellipse of inertia. 



The converse problem, viz., to determine the screw rj, 

 an impulsive wrench on which would make the body 

 commence to twist about 9 is indeterminate. Any screw 

 in space which is reciprocal to would fulfil the required 

 condition (54). 



We have seen in 66 that an impulsive wrench on 

 any screw in space may always be replaced by a pre- 

 cisely equivalent wrench upon the cylindroid which 

 expresses the freedom. We are now going to deter- 

 mine the screw j, on the cylindroid of freedom, an impul- 

 sive wrench on which would make the body twist about 

 a given screw 9 on the same cylindroid. This can be 

 easily determined with the help of the pitch conic ; for 

 we have seen ( 42) that a pair of reciprocal screws on 



