174 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



Changing the sign of t in T u and the order of the limits, we get 



J a 



Put t = pe*"^. Then we may integrate first from p m a to p = sM while remains equal to 0, 

 and afterwards from 0=0 to = f a while p remains equal to 3^c^. Let the two parts of the 

 integral be denoted by T, T". We shall evidently obtain a superior limit to T' by making the 

 following changes in the integral : first, replacing the quantity under the integral sign by its 

 modulus ; secondly, replacing f in the index by the product of f and the greatest value (namely 

 sM) which t receives in the integration ; thirdly, replacing a by the smallest quantity 

 (namely 0) to which it can be equal, and, fourthly, extending the superior limit to oo. Hence 



/m 

 e~% 1 dt, a quantity which 



vanishes in the limit, when a becomes infinite. 



We shall obtain a superior limit to the real or imaginary part of T" by reducing the quan- 

 tity under the integral sign to its modulus, and omitting \/ — 1 in the coefficient. Hence L 

 will be such a limit if 



J « 



L m Shi I e- c 'fWd9, where /(0) = 3 cos 20 - cos (30 - fa). 



We may evidently suppose a to be positive, if not equal to zero, since the case in which it is 

 negative may be reduced to the case in which it is positive by changing the signs of a and 0. 



If 



When 0= — , the first term in /(0) is equal to 5, which, being greater than 1, determines the 

 o 



sign of the whole, and therefore /(0) is positive ; and f(0) is evidently positive from — to 

 0= - , since for such values cos20>^. Also in general f{0) =- 6 sin 20 + 3 sin (30 - | a ), 



7T 7T 



which is evidently positive from = — to = — , and the latter is the largest value we need 

 consider, being the extreme value of when a has its extreme value — . When has its ex- 

 treme value | a, f(0) = 2 cos 3a, which is positive when a<-, and vanishes when a = -. 

 Hence f{0) is positive when 0<|| a; for it has been shewn to be positive when 0<~ , which 

 meets the case in which a < — or = - , and to be constantly decreasing from = — to = fa, 



which meets the case in which > — . Hence when a < — the limit of L for c = eo is zero, 



9 6 



inasmuch as the coefficient of c 3 in the index of e is negative and finite ; and when a = j; the 



same is true, for the same reason, if it be not for a range of integration lying as near as we 



