606 



RICE 



ART. L 



A/(a:i, X2, X3, 



9/ df df 



dxi dX2 dX3 



+ 1 



+ 2 



ay 



aa;i2 ^' ^ aa;2' 



32/ ^ 32/ 



+ 



32/ 



3a:i 3x2 



^1 $2 + 2 



32/ 



3xi 3X3 



^1^3 + 



+ 2 



92/ 



3rc2 3x3 



^2 $3 + 



]■ 



The values of df/dxi, df/dXi, etc, are zero when xi, xi, xt, ... are 

 the values of the coordinates for the configuration in question. 

 For convenience let us represent the values of d^f/dxi^, d'^f/dx^^, 

 . . . d^f/dxidXi, . . . for the same coordinates by the symbols 

 flu, 022, . . . ai2, . . . The symbol 021 would represent d^f/dXidxi, 

 but by the law of commutation for partial differentials this is 

 the same as a^. Now if the configuration is one of stable 

 equilibrium, the value of /(xi, X2, X3, . . .) is less at the equilibrium 

 configuration than for any neighboring configuration. Hence 

 if the equilibrium is stable the quadratic expression 



ail^l'* + ^22^2^ + «33^3^ 



+ 2ai2^i$2 + 2ai3^i6 



+ 2a23?2?3 + . . . 



is positive for any arbitrary values of ^1, ^2, ^3, ... In short 

 it is a "positive definite form."* The conditions which must be 

 satisfied by the coefficients an, 022, . . . an, . . . for this to be 

 the case are well-known and can be most readily expressed in 

 terms of the determinant 



dnl 



am 



anz 



Or 



* See the note on The Method of Variations, this volume, p. 5. 



