343 STABLE AND V 1 



is a position of MtaUibrinBk For in this case z is a 

 and therefore Bz always 0. 



If some of the bodies are rough the result will still hold it' 

 >n be such as to piv\vnt any sliding ; see case (6) 

 of Art. 256. 



262. Suppose a system in equilibrium, ami that an in- 

 definitely small displacement is given to it; it' it tli< i 

 to return t > it- original position, that position is said to be 

 one of stable equilibrium if the system tend to move further 

 from its original position, that position is said to be one of 

 'H'l'iinn. 



To determine in any case whether the erjuil Miriam of a 

 in is stable or unstable, is a question of dynamics on 

 which we do not enter. The reader may refer to Poisson, 

 Art. 570, or Duhamel, Tom. n. Art. 69; the best in 

 gation of the question, however, will he found in the Cours 

 CompUmentaire a" Analyse et de Mecanique Rationelle, par 

 J. Vleille, Paris, 1851. * 



The following general theorem is demonstrated. Suppose 

 the forces which act upon a system such that 



2 (Xdx + Ydy + Zdz) 



is the immediate differential of some function of the co-ordi- 

 nates, <f>; then, for every position of equilibrium, </> i>, in 

 general, a maximum or mini mum; in the former ca.se the 

 equilibrium is stable and in the latter unstable. 



An important particular case is that of th :n in 



Art. 'JC1. in which the equilibrium is stable when the centre 



of gravity has its lowest position, and unstable whm it has 

 its highest position. 



\Ve will now illustrate the principle contained in 

 the preceding Article by application to two examples. 



I. A uniform heavy beam is placed with its ends in 

 tact with a fixed smooth vrrtieal curve in the form 

 ellipse with its directrices horizontal : determine the p,. 

 of stable equilibrium, the length of the beam being supposed 

 not less than the latus rectum. 



