358 GENERALIZED COORDINATES 



where T is the value which the kinetic energy would have if the solid were 

 removed. Suppose that the solid is acted on by external forces, besides the pres- 

 sure of the water. Let the sum of the moments of these forces about any axis 

 be , and let 6 be a coordinate which measures the angle turned through about 

 this axis. Then Lagrange's equation corresponding to the coordinate 6 is 



dt\s0 



If the external forces just suffice to hold the body at rest in the liquid, we 

 have ( - } = 0, so that 

 ' 



Hence the sum of the moments of the liquid pressure must be , or 



We can calculate a from the shape of the solid, and so can obtain a knowledge 

 of the couples acting on the solid. 



Illustration from electromagnetism. The energy required to establish the flow 

 of two steady currents of electricity of strengths i, i' in two given closed circuits 

 is known to be of the form 



where L and N depend on the shape of the first and second circuits respectively, 

 while M depends on the shape of both circuits, and also on their positions 

 relative to one another. 



Suppose that the second circuit is free to move along any line towards the 

 first circuit. Let x be a coordinate measured along this line, and let the force 

 required to hold the second circuit at rest be JT in the direction in which x is 

 measured. 



Let L denote the usual function T W, and let the second circuit be acted 

 on by an externally applied force X. Then Lagrange's equation for the coor- 

 dinate x is 



1/?A_^ = x 

 dt\dx) dx 



that, since there is no acceleration, 



X.-S5. 



dx 

 As a matter of experiment, it is found that 



dx 



