( 519 ) 



dx^ q d.i;* b.v dx\q 



flus limiting oase will evidently occur when 



-r— and -r — 



are at the same time. Now 



ö.Ui _ 7^T 2^ X dVi _ i^2' 2.4 l—2rx 



J^ ~~ ~ \^v ^~ V (1 + *••'■)' ' 0^ ~ ~ (l-.t')^ "^ V [i^-rxy ' 

 SO that for liiis ))oint of iiitlection we shall have the relations 

 .v{l—x)_RT _ {I- x)\l — 2r.v) _RT 

 {l-\-rxy ~~ 2^ ' ' (1 + nr)^ ~~ 2^ * 



On dividing, we find : 



.,; (1 _|_ ,. a^ = {1—x) (1 — 2 r .v), 

 or 



r.v' — 2 (1 4 r),r +1=0. 



When /' is either negative or positive, we tind from this : 



(«) 



when ,r^ indicates the value of ,*' at the point of inflection. .I'c may 

 run from \/.^ (if j' = 0) to 1 (if r = — 1), when r is negative. If, 

 however, r is po.sltwe, ,i\- runs from Ya (if '' = 0) to (if y = x). 

 The positive sign for t 1 -|- /■ -f- y •' would give in lK)tii cases impos- 

 sible values for .I'c- 



We now further obtain : 



Xc{l - X,) _ RT _R2\ 1 T_0 T 



(1+7.^ -- 2aq,~ ~^ ' 2ra' ¥,- 2ft' Y; 



T 

 tiiat is to say, when for— is substituted its value from (Sbis) : 



1 + 



■^•c(i-.t'c) = ^ , (i+>-.>-cr 



(l + r.re)='^2« ' 1-^%(1-.»;,)' 

 where the lower sign indicates conditions, where -^- <C 0, and which 



d.r 



are consequently stable. From this then follows : 



2 « ^^'^ (1 _ ^ Zo./ (1 -.tv) ) ^ ^ [(1 + ,. x^ + « .v/^] . 



l + r.Vc ^ 



Now, from the equation, from which iji) was deduced, we tind: 



