292 Dr. J. W. Nicholson on the Bending of 



to the same order of approximation, and similarly, it may be 

 shown that 



W^{4»»-I)^W«<<-!)}-<»>> 



For the purpose of the present problem, we require the 

 portion of dP n jd/jb which involves —cmd in the exponent, for 

 this alone leads to a series with a zero point. This portion 

 becomes 



and this may be substituted for dF n /d/ju in the summation. 



Second order values of R w , <f> n , </> nr , and % n are also re- 

 quired, and to these we proceed. General expressions for 

 R n and <jfr M were given in (21-24), and isolating the most 

 significant terms when m is of order z, and equal to zx, 



R„=(l-^)-l, (144) 



the next term being of relative order z~ 2 . Thus the value 

 of H n used in the first approximation is sufficient. The 

 second approximation to (/>„ is 



+ i{( 1 "-^)" i +i^( 1 -^)" f }- 



Three orders are here retained, for it must be remembered 

 that we are in this case dealing with an exponential. Let 



^ = 8z Ul-* 2 )* + 3(l-^ 2 )fJ ' " " (U5) 

 and similarly, c being ajr as before, 



Pnr ~ 8^1(1 -#x 2 )\ 3 (1— cVjtJ ' ' l ° ; 

 and the second order value of e™~ l ™ r becomes 



i(<l>n—<!>nr+Pn—Pnr) 



where, in the second expression, the </>'s denote the old first 

 order values. This may be written, since the /3's are of 

 order z~\ 



(l + ^ n -c/3 nr )e^n-Kr\ 



In a series involving the exponential of argument t(<f> n — (f> nr ) 

 we may therefore leave the <£'s unaltered from their first 



