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



Let n denote the value o£ n at the minimum point, and 



™ =?? + i. 



Writing p b =(«i.-ir)(6/*)* > p,o=(nio-*rX6/*r)*. (79) 

 Then € = B- 1 -R,7 1 -0 J 



R a =(27r)- 1 6i^-M/ 1 2 +A 2 ), 



sv = tan"Vi//4- tan^F^-flm, 



t0= _ e &* (kll n n iir /27rar 2 sin 0)*, 



Mr 



W,o = Wo<? Xo 



tan X o=-i3K»/3* = (3/w)C/i/i'+./4/ r A • • ■ (80) 



with the minimum point substituted in each. With these 

 values, 



«/ _co p — 1 



by the usual summation formula, where the limits have been 

 taken as infinite. Actually, they are not even of order z for 

 the harmonics of the type contributing mainly to the sum, 

 but owing to the rapid oscillation of the exponential they 

 may be regarded as infinite in the usual way. Again, to the 

 same order, the multiplier of the exponential in the integrand 

 may be taken as corresponding to the value of f making the 

 exponent a minimum, or £=0. Thus 



-^(wo + ^ioV*M #**+****'"*, . (81) 

 e — 1 J-« 



7 = 



and the integral is identical with 



dw cos (w 3 + aw), . . . (82) 



W") Jo 



where a=2€(6/zv /lf )i, 



and 7 is therefore proportional to an Airy's integral. The 

 transitional region therefore exhibits maxima and minima 

 after the manner customary in such problems when treated 

 by the ordinary methods, provided that this integral is of 



