SECT. II.] FORM OF THE SOLUTION. 231 



Retaining the two conditions a = a x and a n = a^, the value 

 of 2 contains the first power of h, the value of a 3 contains the 

 second power of h, and so on up to a B+1 , which contains the 

 n th power of li. This arranged, a a+l becoming equal to a n , we 

 have, to determine h, an equation of the n ih degree, and a t re 

 mains undetermined. 



It follows from this that we shall find n values for A, and in 

 accordance with the nature of linear equations, the general value 

 of a is composed of n terms, so that the quantities a, /5, 7, ... &c. 

 are determined by means of equations such as 



a = a/ + a/e* + a, V&quot; + &c., 

 = a/* + &amp;lt;e* &amp;lt; + a 8 V + &c, 

 7 = a/&amp;lt; + ay + a 8 V&quot; -f &c. 



to = a + &quot; &amp;lt; + a V* + &c. 



The values of h, ti, A&quot;, &c. are n in number, and are equal to 

 the n roots of the algebraical equation of the n ih degree in h, 

 which has, as we shall see further on, all its roots real. 



The coefficients of the first equation a lf a/, a&quot;, a&quot; , &c., are 

 arbitrary ; as for th&quot;e coefficients of the lower lines, they are deter 

 mined by a number n of systems of equations similar to the pre 

 ceding equations. The problem is now to form these equations. 



253. Writing the letter q instead of -j- , we have the fol- 



A/ 



lowing equations 



We see that these quantities belong to a recurrent series 

 whose scale of relation consists of two terms (q + 2) and - 1. We 



