THE PRINCIPAL SCREWS OF INERTIA. 6 1 



principal screws of inertia, then the co-ordinates of the 

 reduced impulsive wrench are proportional to 



" ''*/,.< 



? 



Let T 1 denote the kinetic energy of the body of mass 

 M when animated by a twisting motion about the screw 

 a, with a twist velocity a. Let the twist velocities of the 

 components on any n conjugate screws of inertia be de- 

 noted by di', . . . d B '. (These screws will not be co-reci- 

 procal unless in the special case where they are the 

 principal screws of inertia.) It follows ( 58) that the 

 kinetic energy will be the sum of the n several kinetic 

 energies due to each component twisting motion. Hence 

 we have ( 59) 



and also 



u* = Ufa? + . . . + w n W. 



Let Q! , . . . a ra and ]3i , . . , j3 ra be the co-ordinates of 

 any two screws belonging to a screw complex of the n th 

 order, referred to any n conjugate screws of inertia, whe- 

 ther co-reciprocal or not, belonging to the same screw 

 complex, then the condition that a and /3 should be con- 

 jugate screws of inertia is 



To prove this, take the case of n = 4, and let A, B y C y D 



be the four screws of reference, and let A\, , A 6 be 



the co-ordinates of A with respect to the six principal 

 screws of inertia of the body when free ( 52). The unit 

 wrench on a is to be resolved into four wrenches of in- 

 tensities ai, . . . , a 4 on A, By Cy D: each of these compon 

 nents is again to be resolved into six wrenches on the 



