348 GENERALIZED COORDINATES 



Making all these substitutions, equation (186) becomes 



which is Euler's third equation, and the other two equations 

 follow from symmetry. 



SMALL OSCILLATIONS 



279. Let 19 2) - , 6 n be generalized coordinates of any system, 

 and let it be supposed that these coordinates are all ^independent, 

 so that any set of values of lt 2 , -,# gives a possible configura- 

 tion of the system. 



Suppose that the configuration 



f\ f\ f\ /i f\ __ /j / -| orrv 



11> 22' > n n \/ 



is known to be a configuration of equilibrium. Then, if 



the quantities <j> lt (f> 2 , - , <f> n may be taken to be generalized coordi- 

 nates of the system, and will possess the property of all vanishing 

 in the position of equilibrium. 



Let W Q denote the value of the potential energy in the configu- 

 ration of equilibrium. The potential energy in any other configu- 

 ration may, by Taylor's theorem, be expanded in the form 







where all the differential coefficients are evaluated in the position 

 of equilibrium. In this position of equilibrium, however, we have, 

 by the theorem of 135, 



dfF = dJF = dW 



90, ~ M 2 = = dO n '' U ' 



