252, 253] SMALL OSCILLATIONS. 287 



Now the velocity of each particle of the system can be ex 

 pressed in terms of and 0, and the kinetic energy T is thus of 

 the form %A6* where A may depend upon 6, but does not vanish 

 with 6. 



Also the potential energy V vanishes with 6, if the standard 

 position is the position of equilibrium. Thus Fis a function of 6 

 which may be expanded* in powers of 6 and the series contains 

 no term independent of 6. Again, the principle of virtual work 



shows that ^ vanishes with 0, or that the term of the first order 

 dc/ 



is missing from the series for F Thus F can be expressed as a 

 series beginning with the term in s , and more generally we may 

 say that, when 6 is sufficiently small, 



where C is a function of which is finite when 6 = 0. 

 The equation of energy accordingly is 



A6* + $C0* = const., 

 and on differentiating we have 



Omitting small quantities of an order higher than the first we 

 have 



A e + ce=o, 



where A and G have their values for 6 = 0. Thus, if these two 

 quantities have the same sign, the motion in 6 is simple harmonic 

 with period 2ir ^(A/C). 



Now A must be positive since otherwise the expression ^AO 2 

 could not represent an amount of kinetic energy. Hence there are 

 oscillations in a real period if C is positive. 



The value C (for 6 = 0) is the value of ^~ for 6 = 0, and thus 



the conditions for a real period of oscillation are the same as the 

 conditions that F may have a minimum value in the position of 

 equilibrium. 



* The possibility of the expansion is assumed. The function V is in fact 

 supposed known for any value of 0, and is assumed to be a function capable of 

 expansion in Maclaurin s series in powers of 6. We do not require to consider any 

 other case. 



