191 1.] LINEAR DIFFERENTIAL EQUATIONS. 277 



Substituting these values of 3V,. Vi, jv. Ts, • • • in 



y = Vo + y\S + yS^ + V3^^ + v,^* + • • • 



and making 5" unity, 



A 

 + 2) 2*-T! 



I x^ ^ 



V I x"^ I x^ 



(;z+ i){n + 2){n+ 3) 2^3! 



+ 



^ I ;tr 



+ .5.ir-" I + -2 + 



2 T ,.4 



;/ — I 2^ (« — I )[n — 2) 2* • 2 ! 



I x^ 



+ (;,_ i)(;,_2)(;/-3)^^y!-^ 



]■ 



When n is not an integer the terms of both series in this value 

 of y continue indefinitely according to the law of formation which 

 inspection makes evident, both series are uniformly convergent ex- 

 cept when .r=:0, and both series are solutions of the given dififer- 

 ential equation. 



When n is a negative integer the law of formation of the terms 

 of the first series changes after the (?;)th term and when n is a 

 positive integer the law of formation of the terms of the second 

 series changes after the (;Oth term. The second case will be con- 

 sidered. 



W^hen ;/ is a positive integer the {n)xh. term of the second series is 



Bx^-^ 



^'"-' ^ 2-'"-^'(;/ _!)!(;/_ I)! • 



Substituting this value of 3'»^i in the differential equation for 

 determining 3',,, 



ax^ ax - " - " ' 



and solving for y„ by the method used in solvir.g for 3'i,3'2-3'3, •••, 



A= ^^-jiiijr^.y ['' log '^- - 27J • 



In determining 3'„xi.3'»,-o.3'».3, ••■, the second term in the bracket 



