97] THE PRINCIPAL SCREWS OF INERTIA. 85 



97. Co-ordinates of Impulsive and Instantaneous Screws. 



Taking as screws of reference the n principal screws of inertia ( 84), we re 

 quire to ascertain the relation between the co-ordinates of a reduced impulsive 

 wrench and the co-ordinates of the corresponding instantaneous screw. If the 

 co-ordinates of the reduced impulsive wrench are v\&quot; , ...r} n &quot;, and those of the 

 twist velocity are a l} d 2&amp;gt; ... a n , then, remembering the property of a principal 

 screw of inertia ( 78), and denoting by u l} ... u n , the values of the magnitude 

 u ( 89) for the principal screws of inertia, we have, from 90, 



^ = M^&quot; 



whence observing that d l = dot 1 ; ... a n = da n we deduce the following theorem, 

 which is the generalization of 80. 



If a quiescent rigid body, which has freedom of the nth order, commence 

 to twist about a screw a, of which the co-ordinates, with respect to the 

 principal screws of inertia, are a lt ... a n and if p 1} ... p n be the pitches, and 

 iii, ... u n the constants defined, in 89, of the principal screws of inertia, 

 then the co-ordinates of the reduced impulsive wrench are proportional to 



U,* U n * 



!, ... a n . 



PI Pn 



Let T 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 

 denoted by a,, a 2 , ... d n . [These screws will not be co-reciprocal unless in the 

 special case where they are the principal screws of inertia.] It follows ( 88) 

 that the kinetic energy will be the sum of the n several kinetic energies due 

 to each component twisting motion. Hence we have ( 89) 



T = Mufa? + . . . + 

 and also u a * = u? a, 3 + . . . + u n 2 a 2 . 



Let i, ... a n and /3 1( ... /3 7i , be the co-ordinates of any two screws belong 

 ing to a screw system of the ?ith order, referred to any n conjugate screws of 

 inertia, whether co-reciprocal or not, belonging to the same screw system, then 

 the condition that a and /3 should be conjugate screws of inertia is 



Mi a a x ft + . . . + ujttnpn = 0. 



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

 reference, and let A l} ... A s be the co-ordinates of A with respect to the six 

 principal screws of inertia of the body when free ( 79). The unit wrench on 

 a is to be resolved into four wrenches of intensities cti, ... a 4 on A, B, C, D: 

 each of these components is again to be resolved into six wrenches on the 



