262 



Mr. G. H. Darwin. 



[Mar. 18, 



values of h 2 being admissible) and X the abscissa ; it is shown in fig. 1. 

 Its equation is #. 2 — X(X 2 — 4). 



It is obvious that OA=OA' = 2. 



The maximum and minimum values of h 2 (viz., Bb, B'b') are given 

 by 3X 2 =4 or X=+2/3 . 



Then B&=B'&'= -2 3 /3t+4 . 2/3*= (4/3*)*. 



Since in the cubic, on which the solution of the biquadratic depends, 

 h 2 is necessarily positive, it follows that if h be greater than 4/3* the 

 cubic has one real positive root greater than OM, and if li be less than 

 4/3 , it has two real negative roots lying between and OA', and 

 one real positive root lying between OA and OM. 



To find OM we observe that since li 2 is equal fco (4/3*) 2 , and since 

 the root of X 3 — 4\— Ji 2 =0 which is equal to — 2/3* is repeated twice, 

 therefore, if e be the third root (or OM) we must have 



(x + | i )\x- 6 )=xa-4x-(*) 2 , 



whence (2/3*) 2 6=(4/3*) 2 , and e or OM=4/3*. 



Now OA =2 ; hence, if h be less than 4/3*, the cubic has a positive 

 root between 2 and 4/3% and if h be greater than 4/3*, the cubic has a 

 positive root between 4/3* and infinity. 



It will only be necessary to consider the positive root of the cubic. 



Now suppose h to be greater than 4/3*. 



Then it has just been shown that X is greater than 4/3*, and hence 

 (X being positive) 3X 3 is greater than 16X, or 4(\ 3 — 4X) greater than 

 X 3 , or 4/z 2 greater than X 3 , or 2h\~% greater than unity. 



Therefore {|(Xf+fc) \/l - 2K\1}* = - +A) ^2h\~i-l} 2 . 



Thus the biquadratic has two real roots, which we may call a and b. 



Then a=|(Xl+^)[l-f ^2£Xrf^l]; 



b=J-(Xl-f^)[l- ^jiivt^l]. 



It will now be proved that a is greater and b less than f A. 



Now a> or <%h, 



as (XI + h) [1 + </2KXr* - 1 ] > or <Sh, 



(XI + A) / 



as xi W2h-\i> or <2h-U ) 



as XI + Ji > or < X* /2ft-- X*, 

 as X 3 + 2ftXf + ft 2 > or <2ftX!-X 3 , 

 as 2X 3 + Ti 2 > or <0. 

 Since the left hand side is essentially positive, a is greater than f ft. 



