38 WILSON 



Conditions for stability. Let z = /(.r, y). 



dz dz 



z = 2o + -^^x + —Ay 

 ax dy 



ART. C 



+ H^. ^^' + 2 ^ AxLy + ^, Ay^ + 



d^z ^ 



v^ ' dxdy dy"^ 



Tangent plane 



dz dz 



Zp = Zo + -- Ax -\- -- Ay, 

 dx dy 



^- ^P = Ht^, ^^' + 2£^Aa:-A2/ + ^,A2/2J + .. 



dH d^ 



dxdy dy^ 



Neglecting higher powers, the condition that z > Zp, except for 

 Ax = Ay = 0, is first 



dh , d'z 



^,>0 and ->0. 



and then by completing the square also 



dx"^ dy^ \dxdy/ 



> 0. 



For the limit of stability this last condition is zero. Re- 

 place 2 by e and x, yhy r],v and remembering de = idr] — pdv the 

 conditions are 



dh fdp\ dh (dt\ 



dv^ \dv/r, dti^ \dr] 



dh d^e 



dv^ df] 



2 



/ d^e Y _ _ (dp) (dt\ _ /dpV 



\dvdr]' \dv/„ \dri/^ xd-q/^ 



V 



The first condition means that when the change is adiabatic p 

 must decrease as v increases, and the second means that at con- 

 stant volume the temperature must rise if heat is supplied. The 

 third condition may be transformed. Note first that 



i!i - _ (^\ _ (^\ 



dr\dv \dr]/^ \dv/^ 



