DYNAMICS OF A RIGID BODY. QI 



left-hand side, differentiating with respect to each co- 

 ordinate successively, and observing that the differen- 

 tial co-efficients of x must be zero, we have the n equa- 

 tions : 



(A n - U?X) ai + Ana* . . , + A ln a n = O. 



&c., &c. 

 A nl ai + A nz a z ---- (A nn - ujx) a n = o. 



We hence see that there are n screws belonging to 

 each screw complex of the n th order on which the time 

 of vibration is a maximum or minimum, and by com- 

 parison with 74 we deduce the very interesting result 

 that these n screws are also the harmonic screws. 



Taking the screw complex of the sixth order, which 

 of course includes every screw in space, we see that 

 if the body be permitted to twist about one of the six har- 

 monic screws the time of vibration will be a maximum 

 or minimum, as compared with the time of vibration on 

 any adjacent screw. 



If the six harmonic screws were taken as the screws 

 of reference, then u* and v* would each consist of the sum 

 of six square terms ( 59, 72). If the co-efficients in 

 these two expressions were proportional, so that u a z only 

 differed from z' a 2 by a numerical factor, we should then 

 find that every screw in space was an harmonic screw,. 

 and that the times of vibrations about all these screws 

 were equal. 



