G4- Dr. J. W. Nicholson on the Bending of 



In determining the first approximation to this root, we 

 may shorten the equation to 



I 4 P '_ 3r(}) k f 2p'l 

 2! + 35!~ r(J) 1 33! J ' 



provided that the other terms are really negligible for the 

 ensuing value of p. This is found to be the case. Jf 

 p* = y) (substituting the values of the Gramma functions), w is 

 a root of 



^(l+£-) 3 = 27-895 (l+f)', • • (105) 



and this is a quintic with real coefficients, and therefore with 

 at least one real root. It is found by the usual method of 

 trial that there is only one real root, given by io=— 3"115. 

 With this value, higher approximations to the solution of the 

 actual equation may be deduced by the method of continued 

 approximation, but this is not necessary for the present 

 purpose, as an examination of the neglected terms indicates 

 at once. 

 Thus 



(m-z) B ~ = -3-115, 



z 



or 



m-£=-~*(l,o>, <» 2 )('5192)i, 



where w and &> 2 are cube roots of unity, given by 

 i(-l±*V3). 



But two of these values may be rejected, for, p 3 being real, 

 (104) indicates that p 2 must be proportional to — e^ 75 ", or 

 p proportional to e~ &*, and therefore to 1 — i \/3 with a 

 positive real part. The only root available is therefore 

 given by 



m-z = \z\ (l—« V3) (-5192)*, . . (106) 



the others being introduced in cubing (104). Thus for the 

 first root of ~dfoz . z* K m (z) = 0, the imaginary part is on 

 reduction 



-•696 izi (107) 



This imaginary part is negative, as stated in the previors 

 section, and /3 = '696, roughly equal to 2/3*. With this 

 value, the ratio of disturbed to undisturbed amplitude is given 

 by (101) as (8tt sin 6)\ (jto)* tan \Q /3~ l «-(*»>* 0* 



* This was inadvertently quoted as 1/3 in the previous section, but 

 no deduction was made from it. 



