SMALL OSCILLATIONS 349 



so that we can write the value of W in the form 



W=W + u # + 2 a lt fafa + - - + ,,<#, (188) 



in which rjbwers of fa, fa, higher than the second are left out 

 of account, because we are going to confine our attention to motions 

 in which fa, fa, are all small quantities. 



The kinetic energy, as before ( 270), is a quadratic function of 

 <i>4> >< Letus'say 



^ =6 U # + 2 & u fc4 + .-. + &..&. (189) 



The coefficients J n , 6 la , , ~b nn are, strictly speaking, functions of 

 ^i 02> ' ' ' 0n> but we mav regard their values as being equal to 

 the values in the configuration of equilibrium, and so may treat 

 them as constants. 



280. Now consider two quadratic functions of n variables x v 

 x v ' ' ') x n> defined by 



f(x lf x 2 , -.., x n ) = a u x\ + 2 

 FX x - aj = bx* + 2 



Since the function T defined by equation (189) is necessarily 

 positive, it follows that F(x lt x 2 , -, x n ) is positive for all values 

 of x lt x 2) - -, x n . Hence, by a known theorem in algebra, we can 

 find a transformation of the type 



\ / 



in which the coefficients /c n , etc., are real, which is such that / and 

 F transform into expressions of the type 



f(x lf x 2 , 



^(^, * 2 , 



and all the coefficients yS^ /S 2 , , @ n will be positive. 



