HARMONIC SCREWS. 77 



Substituting for T we have : 



< dV 



with (n-i) similar equations. Finally, introducing the 

 expression for V ( 70), we obtain n linear differential 

 equations of the second order. 



The reader who is not acquainted with Lagrange's 

 magnificent equations of motion in generalized co-ordi- 

 nates will perhaps welcome reasoning by which the 

 equations which we require can be otherwise demon- 

 strated. 



Suppose the body to be in motion under the influ- 

 ence of the forces, and that at any epoch t the co-ordi- 



j/\ r 7/1 / 



nates of the twisting motion are -y)-, . . , . -= . , when 



it i- dt 



referred to the principal screws of inertia. Let ?/',...., n " 

 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 /. 

 The co-ordinates of the impulsive wrench which would, 

 in the time e, have produced from rest the motion which 

 the body actually has at the epoch t + e, are : 



, w ,....,n , 



On the other hand , the motion at the epoch t + c 

 may be considered to arise from the influence of the 

 wrench /', .... " for the time e y followed by the in- 

 fluence 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 : 



