90 DYNAMICS OF A RIGID BODY. 



u \M 



Regarding the rigid body and the forces as con- 

 stant, and comparing inter se the periods about different 

 screws a, on which the body might have been constrained 

 to twist, we see from the result just arrived at that the 



time for each screw a is proportional to -. 



85. Property of Harmonic Screws. As the time of 

 vibration is affected by the position of the screw to 

 which the motion is limited, it becomes of interest to- 

 consider how a screw is to be chosen so that the time 

 of vibration shall be a maximum or minimum. With 

 slightly increased generality we may state the problem 

 as follows : 



Given a rigid body, and the forces which act upon it, 

 it is required to select from all the screws of a given 

 screw complex the particular screw or screws on which, 

 if the body be constrained to twist, the time of vibra- 

 tion will be a maximum or minimum, relatively to the 

 time of vibration on the neighbouring screws of the same 

 screw complex. 



Take the n principal screws of inertia belonging 

 to the screw complex, as screws of reference, then we 

 have to determine the n co-ordinates of a screw a by 



the condition that the function ^ shall be a maximum 



Va. 



or a minimum. 



Introducing the value of u a ( 67), and of v a (72), 

 in terms of the co-ordinates, we have to determine the 

 maximum and minimum of the function 



U T**\ + ^ a * : - LL - = x, 



Multiplying this equation by the denominator of the 



