128 LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [April 25, 



26. The first group of terms of (54) is the infinite series which 

 gives the solution of the four-term equation 



ay" -\- by'' -^ cy^ -\-I = o 

 obtained from the equation 



av" + hy'^x -{- cy^x + / = o 

 provided the conditions 



d" if c" 



^V^ '^ k\n-kY-'-' «V^' "^ ^^"' 

 are satisfied. 



The second group of terms has the common factor 



d 



en -^ ' ' 

 and the successive groups of terms respectively the common factors 

 d- ^ d^ ^ d' ^ 



-. 2m ., Am ^, im 



The convergency conditions of the successive groups of terms 

 are identical with the convergency conditions of the first group. It 

 follows that (54) may be written 



(55) j=y«+y.t-,}'«'+ ^vw-v'"+ i3A?-''»"' + •■•• 



where Y q, \\, Y^, Y^, Y^, • • •, represent the sums of convergent infinite 

 series. 



If Y denotes the largest of the numbers F,,, F^, Y^, Y^, •••, 



(56) 7 S F ( . + -"^ J- - + ^, jv'" + ^ ,t' +...). 

 The series (56) is convergent provided 



(57) 7^;JV'<'. 



If both members of the inequality (57) are affected by the exponent 

 n, condition (57) becomes 



(58) -rn;^<n\ 



ci t 



The conditions sufficient for the absolute convergence of (54) are 

 therefore 



