58 THE PRINCIPAL SCREWS OF INERTIA. 



Let ri be any screw on which the body receives an 

 impulsive wrench. Decompose this wrench into com- 

 ponents on a system of six screws consisting of any n 

 screws from P f y and any 6 - n screws from Q. The latter 

 are neutralised by the reactions of the constraints, and 

 may be omitted, while the former compound into one 

 wrench on a screw belonging to P '; we may therefore 

 replace the given wrench by a wrench on . Now, if 

 the body were perfectly free, an impulsive wrench on 

 must make the body twist about some screw a on P. In 

 the present case, although the body is not perfectly free, 

 yet it is free so far as twisting about a is concerned, and 

 we may therefore, with reference to this particular im- 

 pulse about , consider the body as being perfectly free. 

 It follows from 62 that there would be a loss of energy 

 if the body were compelled to twist about any other 

 screw than a, which is the one it naturally chooses. 

 This theorem is due to Euler.* 



65. Co-ordinates of a Screw belonging to a Screw com- 

 plex. It will now be necessary to make some extensions 

 of the conceptions of screw co-ordinates. Suppose that 

 a body have freedom of the n th order, we have shown that 

 it is always possible to choose n screws from the screw 

 complex expressing that freedom, such that each screw 

 is reciprocal to all the rest. As an example we shall give 

 the proof for the screw complex of the third order. Let 

 B^ B^ B z be three screws of the reciprocal screw com- 

 plex; then, if any screw A l be taken which is reciprocal 

 to BI, BZ, .Z? 3 , any screw A% which is reciprocal to 

 B ly BV, BZ, A ly and the screw A 3 , which is reciprocal to 

 J5i 9 Bo, B z , A ly A 2 ; then the group A ly A 2 , A 3 possess 

 the required property, and may be termed co-recipro- 

 cals. 



* Thomson & Tait : Natural Philosophy, vol. i. p. 216. 



