167 



(247) [173] 



the constituents of which, by (44), are the same as the coeffi- 

 cients of the equations (245), (thus dt/d-q = d'^e/dif, dyL„/dmn = 

 dh/dnin^, etc.) and R^ is the determinant formed by erasing 

 the last row and column of Rn-\-\. Similarly, the determi- 

 nants Rn _ 1, /?„ _ 2, etc., are obtained by erasing successively the 

 last row and column of Rn, and 



/ dnn-i \ 

 \dmn - 1/, 



Rn 



r, /il, . . .Mn-2i lir 



Rn 



etc. (248) [176] 



Now according to (230) and (232) the phase is stable if the 

 differential coefficients (246), (248), etc. are all positive. 

 These conditions are satisfied if the determinant (247) and all 

 its minors, down to dh/dtf, are positive.* "Any phase for which 

 this condition is satisfied will be stable, and no phase will be 

 stable for which any of these quantities has a negative value." 

 Since the conditions (230) remain valid if we replace any of the 

 subscript /I's by m's, the order in which we erase the successive 

 columns with the corresponding rows in the determinant is 

 immaterial. 



For a body of invariable composition, it is only necessary to 

 use the terms which are common to the first two rows and 



* The differential coefficients in (246), (248), etc. would also be posi- 

 tive if all the determinants, Rn+\, Rn, etc. were negative. But the last 

 term d^e/dr]^, by (229), cannot be negative, so none of the others can be 

 negative. 



