294 THEORY OF HEAT. [CHAP. VI. 



308. The equation 



m m 



92 



~*- ^ ~^~ 2 * ^ 



on being differentiated twice, gives the following relation 



We write, as follows, this equation and all those which may 

 be derived from it by differentiation, 



&c., 

 and in general 



Now if we write in the following order the algebraic equation 



JT = 0, and all those which may be derived from it by differentiation, 



dX d*X 



and if we suppose that every real root of any one of these equa 

 tions on being substituted in that which precedes and in that which 

 follows it gives two results of opposite sign ; it is certain that the 

 proposed equation X = has all its roots real, and that conse 

 quently the same is the case in all the subordinate equations 



-0 &c 



I &quot;* v 7 t&amp;gt; ~&quot; / i o &quot;~~&quot; V/j CX&amp;lt;V/ 



dx dx* dx* 



These propositions are founded on the theory of algebraic equa 

 tions, and have been proved long since. It is sufficient to prove 

 that the equations 



fulfil the preceding condition. Now this follows from the general 

 equation 



