particular Differential Equation. 261 



Again, make ?^ = 5r~'Mj; then 



And therefore if2?i=0, we have 



As the two last values of p are diflPerent from the former, 

 the two solutions now obtained relate to different equations. 

 Now changey(?M) intoy(?j), and make ?^ = 7r-'?/, ; then 



Also ;^, = gives 



p=/(«+ 1) 4-/(n + i)-ai\vi-n - ^-^- )• 



Make u = 7r„_iM,, we have 



(i-„T;;, +/(«)x)'r„_i?/i +;?A7r„_,z^, = X, 



7r„7r;„7r„_iZ<i +7?i7r„X?/i = X, p, =p +/(«)> 

 and 



If by ? operations Pi = 0, we have 



p=-/(n)-/(7J-l)....-/(«-^ + l)=«/(«-m-i=i) 



As the two last values of/) are the same with the two former, 

 we have again two sets of particular solutions to the same two 

 equations, unless the complete solutions may be expresseil in 

 two different ways. 



Instead of the second condition of (2.) and (4'.), we might 

 make 



But then we shouhi have to integrate 



A 



