CHAP. VI.] THE EQUATION OF CONDITION. 293 



In the series 



l-* + |^ -^+&c. ( 



which expresses the value which u receives when x = X, we shall 

 replace *xy- by the quantity 0, and denoting this function of 



by / (0) or y, we have 



ffi /9 s 0* 



y =/ (0) = 1 - + * &quot; 



2*. 3 4 a 



the definite equation becomes 



6* O 3 6* 



JiX ~~ 2* ~^~ 32 3* &quot;&quot; 2* 3* 4* ~^~ 



^ * ff* fj* * 



1 ~^ + 5 ~ s + ia 2 ~~ &C 



/ (0) denoting the function - 



Each value of ^ furnishes a value for #, by means of the 

 equation 



and we thus obtain the quantities ^, ^r 2 , g z , &c, which enter in 

 infinite number into the solution required. 



The problem is then to prove that the equation 



must have all its roots real. We shall J&amp;gt;rove in fact that the 

 equation f(&) has all its roots real, that the same is the 

 case consequently with the equation f (0) =0, and that it follows 

 that the equation 







~) 



has also all its roots real, A representing the known number 



hX 



2 



