of the Principle of Stationary Phase 53 



Cases of practical importance are those in which A = Ai and 

 t = fj> is a, true minimum or maximum of bt — tf{t), so that e and r) 

 are both + 1 or both — 1. The formula then is 



A cos {n/M'f'ifM) ± ^mTr} . (r - 1) ! (r !)'»-i T(m) 

 ~ 7r?i"* { I (^e-) (fi) I 1*'^ • 



If nV or ny tend to finite limits, the gamma functions have to 

 be replaced by incomplete gamma functions ; and if one or other 

 tends to zero, we modify the approximation by writing zero for 

 A or A^ respectively in the general formula. 



The general result reduces to Kelvin's formula when r = 2, 

 X = 0, 711= h, and e = ?; = 1, provided that (with Kelvin's notation) 

 x/t is constant. In that case, a sufficient condition for the validity 

 of the formula is that 



^ [{{ma^/t) - mf{m) - i^-f (;.)}*]-^ 



should have limited total fluctuation when m ^ 0. 



If X were a function of t, Bromwich's general theorem {loc. cit., 

 p. 443) would have to be used, and the enunciation of sufficient 

 conditions (even in their simplest form) for the validity of the 

 formula, would be exceedingly laborious. The reason for this is 

 that (with the notation employed in this paper) -^ and F (t) dt/dyfr 

 would both be functions of n. 



4. The problem of Riemann (see § 1 above) essentially consists 

 in obtaining an approximation for integrals of the type 



/■"" / ,x cos sin nt. 



when n is large and a-' (t)—>oo as t—> 0. 



These integrals are expressible by integrals of the type 



t-'pit)^"^^ {nt+ (7(t)]dt, 

 Jo sm ! ' 



so that the problem is, at first sight, very similar to that discussed 

 in % 2—3. 



There is however an essential difference, namely that, in the 

 problem we have discussed, ntf{t) owes its large rate of increase 

 (which balances the rate of increase of nbt at the stationary point) 

 to the large factor n, whereas, in the problem attacked by Fejer 

 and Hardy, the function a (t) owes its large rate of increase to the 

 infinity of a (t) at ^ = 0. In our problem fj, is fixed, whereas in 

 the other problem the stationary point of nt — (T{t) tends to zero 

 as ?i — * 00 . It seems to be this difterence which accounts for the 

 somewhat elaborate investigation given by Hardy and which 



