Mil. G. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 287 



MV Inn-, t/n/s proved that fi (jr) f. t (x) possesses tin asymptotic expansion ofwhic/t /. , 

 M a yrade and /> is a radius, am/ n 



1 (2m+l+/ < " 1 )}, 



where m is the greatest integer such that ink,, < 1, and iji is defined as above. 



We now wish to determine an outer grade and an outer radius. 

 % We have seen that 



8. = m ._ ... 



-'' X 



so that* 



| S.*"* 1 | < A.B, 2 r (//-, + 1 ) r{( -/) /,+ 1 } PW+ B,r K+ 1) o-r (A, 



r = 



!.( us .suppose tliat / 8 S/j, so that /;,>: A-, ; then, since p,^o- , <rSor , we get, as in 

 the previous work, 



/, (2m'+ H-/- ! )}r (n/ + 1) CT O " 

 + A 2 B,i?", r (7^ + 1 ) o- " + B.B*'?", T (n/ + 1 ) y- V ", 



where IM' is the greatest integer, such that ?>i7 < 1, 



VK = ^o if 'o > ^i and / < 1 (otherwise 1/1 = 1), 

 ^"i = TJ if / u > ^i and / < 1 (otherwise 77", = 1). 



We consequently get always 



| S..T-" I < [(A A+ A,B,){ 1 +r, 3 (2m' + 1 +/.-)} + B.B^y- 1 ] <7 U T (n/.-f 1), 

 where 17, = iy if ' / < 1 , ^ a = 1 if / i 1 . 



t* to say, 1 is an outer grade, <T O an outer radius, il 



is a possible outer constant of the usi/mjifotir c.rjxtusion of f^ (jc) J\(x). 



We have thus oht.-iim-d tin- n-sults stated at the beginning of the section. 



We deduce that it' two functions /j (.<), j\ (./) lx>th have asymptotic expansions with 

 k, I, p, a- as a set of characteriatwa, and possible constants Ijeing AI, B, ; A 3 , B, 

 respectively, then the product /i (x) f 3 (x) has an asymptotic expansion with the same 

 rli.u-acteristics, the constants U'ing A u , BO, where 



A = A,A, (:>// + :> + *-'), a = ( 



f The quantity y is <liTmc<l in ciiinu-ctioii with equation (1). 



