105] HARMONIC SCREWS. 97 



Suppose the body to be in motion under the influence of the forces, and that 

 at any epoch t the co-ordinates of the twisting motion are 



dt &quot; dt~ &quot; 



when referred to the principal screws of inertia. Let / , ... &quot; be the 

 co-ordinates of a wrench which, had it acted upon the body at rest for the 

 small time e, would have communicated to the body a twisting motion 

 identical with that which the body actually has at the epoch t. The 

 co-ordinates of the impulsive wrench which would, in the time e, have pro 

 duced from rest the motion which the body actually has at the epoch t + e, 

 are : 



- 



1 dt &quot;&quot; n dt 



On the other hand, the motion at the epoch t + e may be considered to 

 arise from the influence of the wrench / , . . . f M &quot; for the time e, followed by 

 the influence of the evoked wrench for the time e. The final effect of the 

 two wrenches must, by the second law of motion, be the same as if they 

 acted simultaneously for the time e upon the body initially at rest. 



The co-ordinates of the evoked wrench being : 



I dV + J^ 



2p l ddi 2p n ddn 



we therefore have the equation : 



dt 2/&amp;gt;i 

 or 



dt 1 dV 



dt ~ 2 Pl d6J 

 and w1 similar equations ; but we see from 97 that 



ef -M^W 

 e ^- ^^ ~dt 



Differentiating this equation with respect to the time, and regarding e as 

 constant, we have 



dt Pl dt* 

 whence 



2 &amp;lt;9/ dV 



the same equation as that already found by Lagrange s method. 

 B. 



