257 



'*'" / — ï^ïï^T V" (■'') = ^>> "'A <■'') 



In this way we find 





,"+1 

 'A 



A' 



n— > 00 2/n '•''7 ^A n — ^ oo ^7 



Vn M 



which proves the proposition ; we see that 



k' 



Now, if in drawing the successive deflexions ?/„,?/,, y,, ... . it is 

 found that y„_|_i : y„ is sufficiently independent from -e, it will be 

 permitted occasionally to consider 



yn =yo + Vi + • ■ ■ + yn-i + 



1 f 



k' 



to he the deflexion y. For we have 



III . Vn "„ , ,, 1 -r-i "mWm\x) 



2/n 



1 + 



m=A i 



h 



1 + 





= j: 



Pm U'm {X 



n^h ^jn ~r ^ 



t:)\ / k' \>-i i 



' + 57 





and as 



!H=A ^m "T A tn=A V ^"i / Mm ~r "^ 



Aa 



^mih +/c') 



y = .Vt 



n=l ^m ~r "- 



we get 



yn ~ y=^ — 2 Pm W,, 



m=A+l 





+ k' K,{h+k')\' 



since m = h gives zero. If k' <^ A/,^i , the series has zero as a limit 

 for n —*■ CO, which is easily seen by writing it in the form 



.Vn — .V = — 



^ Pm Wm (.'■) I -^ I 



A+iy m=A+l 



Aa 



^m y ^^-fe' A^aA + 'i' 



