Equal Order and Argument. 



235 



the expression (which occurs in the second line of the 

 ■analysis quoted above) 



by 



i r 00 



— I cos {n(w — sin w)\ dw 



l c* 



- I cos (nic z /Q) dw. 

 77 J 





To return to Dr. Airey's paper, his formula (9), which is 

 obtained by taking a new variable x? = n{iv — sin iv) in Bessel's 

 integral, becomes an exact result if the oo in the upper limit 

 of the integral is replaced by nir, so that the formula reads 



j "<"'=y."iH») ! '-' + foC)-iC)'"--}— 



r* oo 



Next Dr. Airey substitutes T(p)cos \pm for I x^cosxdx 



Jo 

 wherever it occurs, quite disregarding the fact that this 

 integral diverges except when 0<jt?<l, and that it is only 

 when 0</><l that the equation 



/"* 00 



l x p_1 cos xdx = Y(p) cos \pir 

 Jo 

 is true. 



In point of fact, by integrating by parts, it can be shown 



x p ~ 1 co^xdx differs from T (p) cos ^pir by an ex- 

 - o 

 pression whose asymptotic expansion, when n is an 

 integer *, is 



{(p — 1){)17t)P- 2 — (p—l){p — 2)(p — 3)(?i7T)P- 4 + . . .}C0S?17T, 



and so the integral of each term in the expansion given 

 above for J„(n) differs from the expression which Dr. Airey 

 substitutes for it by a term which may be written 



cos nir x 0(l/n 2 ) . 



Now it so happens that the aggregate of these terms 

 <cancel one another. But it seems to be impracticable to prove 

 this by employing BessePs integral alone. 



To prove the correctness of Dr. Airey's expansion (11), it 



* When n is not an integer, a second expansion of which sin??7r is a 

 factor has to be added to this. 



