of Beams and the Whirling of Shafts. 93> 



The terms o£ the second order are thus found to be 



a l a 2 . . .a n I 2# 2 iDr • r +iDn — 2y 2 iDr-i • r+iD» ■ — r 

 t ] 2 a>r-) 



n-l 



== 2x 2 (a x + . . . + a r )(a r+ i + ...+«„) 

 i 



«-i 

 — 2y 2 (a x + . . . + a»— i)( a »-+i + . . . + a„) 



2 



= -2/2 («!+... +a T ) 2 



1 



ra-1 



+ 2y 2 {(«i+ ... +<2^) 2 — (ai + ... +a r )a r } 



2 



71-1 



= — 20" — y) 2 («i + • • • + a r y 



1 



{n — 1 n-l -^ 



Sa/f 2 a r a s (all unequal suffixes) > 



ra-1 



= — 2(cotfi7r— eosech.7r) 2 {a x +... H-a r ) 2 



1 

 x 



— cosech ^(a^ + a 2 2 -f . . . + a n 2 ),. 



and this quantity is always negative. 

 Now let k tend to zero, then 



<t>(Kl r ) = l,Klr and ^(KZ r )= - ±Kl r nearly. 



The sign of /(K) will depend on the sign of the determinant 



h + k, -¥2, 



2^2? ^2 H~ hi 



0, J 



— 2^3? ; 









2^i — 15 'n— 1 ~f '«• 



With a notation similar to that adopted for D, the terms 

 of J^n' containing l r 2 or l r are 



fl^r-l • r+lDnV + (l^-l • A' + lD/ • r +lD 7l ')/ r . 



Hence, if all the determinants in this expression are 

 positive, jD n ' increases when l r increases. When all the /'> 

 are zero 1 D„' is zero. Hence JJJ is positive if ,.1\,' is- 

 positive for all values of r and s for which r DJ is of lower 

 order than jD/. 



Now ]D 2 ', iD 3 ', 2 D 3 ' are positive. 



Hence by induction iD n ' is positive, and y(K) is positive. 



