106 THE THEORY OF SCREWS. [115-117 



115. 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 the potential for every position in the neighbourhood of a position 

 of stable equilibrium, it is required to select from a given screw system 

 the screw or screws on which, if the body be constrained to twist, the time 

 of vibration will be a maximum or minimum, relatively to the time of 

 vibration on the neighbouring screws of the same screw system. 



Take the n principal screws of inertia belonging to the screw system, 

 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 or a 



V a 



minimum. 



Introducing the value of u a ( 97), and of v a ( 102), in terms of the 

 co-ordinates, we have to determine the maximum and minimum of the 

 function 



Multiplying this equation by the denominator of the left-hand side, 

 differentiating with respect to each co-ordinate successively, and observing 

 that the differential coefficients of a; must be zero, we have the n equations : 



(^ n - U^X) j + A 12 dz ... + A ln n = 0, 



&c., &c. 



AM&! + A H .,a., ... -I- (A nn ~ Unty H = 0. 



We hence see that there are n screws belonging to each screw of the 

 nth order on which the time of vibration is a maximum or minimum, and by 

 comparison with 104 we deduce the interesting result that these n screws 

 are also the harmonic screws. 



Taking the screw system 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 harmonic 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 a 2 and w a 2 would each consist of the sum of six square terms ( 89, 102). If 

 the coefficients in these two expressions were proportional, so that u a &quot; only 

 differed from v^ 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. 



