( «4 ) 



tl r 



greatest value of x al which - = can still occur —■ x„ , then 



dx 



1 -J- £j = l / , // " (w — l) 2 fj = ,r, 7 2 (w — I) 2 — J , which value must be 



positive for s a . 



Now we niav proceed to demonstrate that the minimum of (</'") 



d{<p'") cPA 



cannot be given by the second factor ot = 0, viz. by =0; 



dx dx* 



dr 

 and at the same time furnish a proof for the theorem that -==0 



1 dm 



can only occur for volumes smaller than b 2 . The quantity 



x(l — x)c 

 A = - - begins with a value =0 at x = 0, and ends also with 



a 



at a? = l. So there is a maximum value and we find it from 



ilA c[a x (l— «)" — «,A' 8 l * | /a, 



— = - at = 1/ -. ror this maximum 

 d,r a- 1 — # y a 3 



. dM 



value ot A < 0, and we should be apt to suppose that this 



.dx* 



will be the case throughout the course from x = to x = \ . This, 



however, is not always the case. In some cases a point of inflection 



appears in the line representing A at certain value of x, and then 



d*A d*A 



is positive tor greater value of x. If we calculate , we may 



dx* dx* 



reduce it to the following form : 



d-A 2c I j 



— = a^ — c K (1 — l( -y + a t x A ] 



dx a ( ) 



And now it is the question whether it x <i., - [a, (I — x) 3 -\- a s ^' 3 ] 

 can be equal to 0. For x = this quantity is a x \a % — c], and as 



2 = ~ -. the value of ol -■ c will certainly be positive for 



c (n—l)' 2 J F 



positive £ a . Hence is negative tor x = 0. For x = l this quan- 



aj l+^i a, l+ f i 



tity is a. (a, — c) and as — = — , and 1= — - 1, we 



c (w — 1) c (n — 1) 



can get a negative value for it if the value of 8 : is small and that 



of n large. This case occurs when e , <^ ir' 2 — 2n. Then there is a 



d*A 



value of x for which — changes the negative sign into the positive 



one. Now we saw, however, above, that if e t <[ n* — 2n, the value 

 of (tp'") = Q is not real over the full extent from x = to x=A, 

 And now the question rises which value of x is greater: the value 



