276 LAMBERT— ON THE SOLUTION OF [April 2^. 



Equating to zero the coefficients of the powers of 5" in this identity, 

 there result the following differential equations for the determina- 

 tion of yo,yi,y2>yz> •••• 



The equation in v,, is a homogeneous linear differential equation 

 and its solution is 



Substituting this value of y,, the equation for determining y^ 

 becomes 



ax'- ax ^ 



This equation becomes exact when multiplied by .t'"""\ The 

 resulting equation integrated gives a linear equation of the first order, 

 the solution of which is 



-^'1 ^ 2\n^) "^ 2\n- i) ■ 



Substituting this value of a'i in the equation for determining y^ 

 and proceeding in the same manner 



V = 1 



2*. 2 !(/^ -I- i)(« + 2) ^ 2'- 2 !(;/— i){n — 2) 



In like manner 



_ Ax''+^ 5;t'-"+« 



J', = 2^7! (;r+ i)(« + 2){n + 3) "^ 2''.3!(^/- i)(«- 2)(^z-3)' 



and so on. 



