50 3Ir Watson, The limits of applicahility 



i 



-co 



x'"^-^f{x) sin nxdx ->/(+ 0) t"'-' sin tdt 



.' 



=/(+ O)r(w) sin I m-TT. 



[// 0<m< 1, the sines may be replaced throughout by cosines ; 

 and, if ny—^ a as n-^ oo , where a is finite, the infinity in the upper 

 limit of the integral must be replaced by o-.] 



As the formal analytical proof of a theorem* slightly more 

 general than Kelvin's theorem is quite simple, and as sufficient 

 general restrictions to be satisfied by the function' /(?n) are 

 apparent in the course of the investigation, it seems to be worth 

 while to place the theorem on record. It is applicable to all 

 kinds of stationary points, whereas Kelvin considered only cases 

 of true maxima or minima of the simplest type. 



2. The main theorem which will be proved in this paper is 

 as follows f: 



Let a, /3 be any numbers {infinity not excluded), possibly depending 

 on the variable n, such that the real function bt — tf(t) has only one 

 stationary value in the range a^t^ ^, at t = /m, b being independent 

 of n. Let the first r differential coefficients with regard to t o/i| 

 bt — tf(t), be continuousX in a range of values of t of which t = fxis^^ 

 an interior point, it being supposed that the last of them is the lowest 

 which does not vanish at t= /j,, so that r ^ 2. 



Let F (t) be a real function, continuous when a<t < /3, except 

 possibly at t = fi, and let 



Urn F{t).{t- ixY = A, Lim F (t) . {tju - 1)"^ := A„ 

 where A, Ay^ are not zero ; for brevity, let (1 — A,)/?- = m.. 

 Then, if the function 



F(t).\bt-tf{t)-,ji'^f{,,)Y-^-.\b-tf'{t)-f{t)\-^ 

 has limited total fiuctuation^ in the range a^i^/9, and if 



I nb^ - n^f(,8) - n/ii'f (/.) | , | nba - noif (a) - ntif (^) j 

 both tend to infinity luith n, the approximate value of the integral 



/ s ^ [ F{t) cos [bnt - ntf(t)} dt, 



* For the connexion between this theorem and a problem, due to Riemann 

 (Werke, p. 260), which has been discussed by Fej^r {Comptes Rendus, November 

 30, 1908, and a memoir published by the Academy of Budapest in 1909) and by 

 Hardy (Quarterly Journal, xliv. 1913, pp. 1—40 and 242 — 263), see §4 below. 



t It is convenient to modify Kelvin's notation. 



X It is necessary iox f{t) to have a continuous first differential coefficient when 



§ If the fluctuation depends on n, it must be a bounded function of n as 7i-».x . 



