CLASS OF DEFINITE INTEGRALS AND INFINITE SERIES. 



183 



!* 1« Q2 , 12 Q2 /j2 . 



B ..2 rsV ^j : c- + ^ ( S-iyj;»--^^V-iYA.... 



Substituting in (48), reducing the result to the form A (P + v - 1 Q), adding another 

 solution of the form B (P - \/ - 1 Q), and changing the arbitrary constants, we get 



u = Ax~l(Rcosx+ S sinx) + Bx~% (Bsinx - S cosx), . (49) 

 where 



R = 1- 



I s . 3 2 1 2 .3 2 .5 2 . 7 2 



1.2(8,») 2 1 .2.3. 4(8#) 4 



S = 



I 2 . 3 2 . 5 a 



1.8.T7 1.2. 3 (8a.) 



+ ... . 



(50) 



21. It remains to determine the arbitrary constants A, B. In equation (46) let cos 0= 1 - n, 

 whence 



d6 = 



djx 



iln 



dfx 



sin 



9 (2 M - m")* (2/u)^ 



+ Md/n, 



where 



M =(2^- M 2 r h -(«/«)- *, 

 a quantity which does not become infinite between the limits of /i. Substituting in (46) we get 



M =* — f COS {(1 - Al)*} fl~^dn + - / COS {(1 - /(*)#} -Wdiu. • (51) 



By considering the series whose n th term is the part of the latter integral for which the 

 limits of fi. are nirx' 1 and (n + l)7ra? _1 respectively, it would be very easy to prove that the 

 integral has a superior limit of the form Hoc' 1 , where H is a. finite constant, and therefore this 

 integral does not furnish any part of the leading terms in u. Putting /ux = v in the first 

 integral in (51), so that 



fi~^dfi = Vtiridv, 

 observing that the limits of r are and m, of which the latter ultimately becomes 00 , and that 



cos v. V^dfi = Z f cos\ 2 d\ = \J — = 2 J sin \ 2 d\ = / sin v . iT^dv, 



we get ultimately for very large values of x 



u = (ttx)~1 (cos x + sin x). 

 Comparing with (49) we get 



whence 



A = B = v-l, 



- - £) * cos (• - 3 + (~) ^ sin {* - 3 •• • • < 52 > 



* Thig expression for «, or rather an expression differing 

 from it in nothing but notation and arrangement, has been 

 already obtained in a different manner by Sir William R. Ha- 



milton, in a memoir On Fluctuating Functions. See Trans- 

 actions of the Royal Irish Academy, Vol. xix. p. 313. 



