— 204 — 

 and by substituting into the expression 2), 



so we get 



dy _ <p(x) V 



dx ki f 



or 



dy _ <fix ) dx 



y k, X ' 



Hence by intergrating and transforming we get 



1 P 



k, Jo 



rCx J o X 



y=ce 



where c is ihe integral constant. 



Thus if we know the form of expression for / ^^^■' dx, then we 



Jo X 



may easily determine the equation of the outline curve of the tree bole. 



According to my investigation, ^^^^ may be expressed empirically 



k, 

 as follows: 



dy 



X 



where a and b are the constants depending on the individual tree bole. 

 The characteristics of the expression ax+ are as follows: 



X 



(i) Lt,= y'.(x)=oo. Lt.=o*i^^=oo and Lt,=„--^^(-^'^ =00, 



dx dx^ 



(ii) U^=,Mx)=ah+-^= const, ^M-'^'^^^^^ =a~^ =const. 



h dx h" 



and Lt.=.^^-^(^> =+2_ft=const. 

 (iii) when x=+^~-, - \ "^ =o, j-iCx) =const. and 



dx- 

 So it is evident that ¥'i(a;) reaches a minimum at the corresponding 



ordinate to the abscissa x= + ./ . 



'^ a 



