296 THEORY OF HEAT. [CHAP. VI. 



real root between X and 3 and since the equation y = has all its 



roots real in infinite number, it follows that the equation A Q~ 



\j 



has the same property. In this manner we have achieved the 

 proof that the definite equation 



- -&c 

 2 



2 2 2 .4 2 2 2 .4 2 .6 2 



in which the unknown is #, has all its roots real and positive. We 

 A proceed to continue the investigation of the function u and of the 

 \ differential equation which it satisfies. 



310. From the equation y -f ^| -f 6 -~ = 0, we derive the general 



equation -jji + (i+ 1) J^TI + & ^r^ = 0, and if we suppose = we 

 have the equation 



d^y_ 1 y 

 dB i+l i + ldO if 



which serves to determine the coefficients of the different terms of 

 the development of the function/ (0), since these coefficients depend 

 on the values which the differential coefficients receive when the 

 variable in them is made to vanish. Supposing the first term to 

 be known and to be equal to 1, we have the series 



_ _^ __ ____ _.. 



If now in the equation proposed 



, d*u , 1 du - 

 gu + - r - z +-- r = Q 

 dor x dx 



x* 

 we make g^ = 0, and seek for the new equation in u and 0, re 



garding u as a function of 0, we find 



du d?u 



