DIFFERENTIAL EQUATIONS OF THE ?i''' OrDER. 



If (3) be (litferentiated partially, we get 



'■ = " , = u -J 



i^ + S ^„-w,,- ^„-i_,-.M., =/'(*/)[ r + S^U_,,,-.\_w,,-+il 



which by elimination of /'('/') give 



i'=n-l i=n-l 



1 = i=0 



l=n-l in-l 



1=0 , = u 

 ~r \^ n—l—i_i "n—i.i \^ ^ „—\—i J ^n—\—i .i+\ 



(4), 



I =u 



(■=•1-1 



(5). 



yj n—\—i.i -n—i,i ^J^n—l—i.;~n-l-i,i+\ 



1=11 1=11 



= XY — XY 



In the products of two sums we put r instead of i in their 

 factors, whereby these terms become 



»-fi-l l'=n-l 



Q ^7 7 



Î — Ù r z= I j 

 î-rt-l r=n-l 



_ ^ Qz Z 



^^ k^^ R^l--i.t n — 1— r,i- ^11—1.1 ^« — 1— r,r-f 1 



last 



1=11 rm) 



l=n-l l=n-l 



or 



Ï — I 1 Ï I I L 



7=1) rrru 



-n— i, t *'«_i_r,r+i 



(6) 



The equation (5) becomes, as may easy be seen, 



(=n-l 

 1=1 



+ the terms in (G) = XY — XY 



Nova Acta Beg. Soc. Sc. Ups. Ser. III. 



^n-i,i ~t~ 



(')• 



