DEPARTMENT OF PtJRE MATttEMAtlCS. 495 



1.2(8,r)- 172T3T4 (8.v)^ "* " ^^' 



where , . . , s . — ■ . r 



(8,r)'- 1.2.3. ij&^ 



and n.. 4»-'-P_ (4n^~l^) (4n^^y) (4>t^-5') 



8,f 1.2.3. (8.t)3 



(3) 



can, of course, be condensed into the form 



J: (■'•) = A /-.R cos (r + a-'^-n'^) 

 V I'-'f' 4 4 



where R^sP^ + Q^ I . . (4) 



P O I 



and cos a = J,, Ian a =^ 



If values of the function are required for large values of .c, beyond those given 

 by existing tables, it seems likely that the above condensed formula would be the 

 most suitable for calculation. The series for R- in inverse powers of .r is remark- 

 ably simple ; and, moreover, when .r is large it is very nearly constant. The value 

 of a can be obtained from the formula cos a = P-^R, when P and R are known, or 

 it could be independently obtained from a series given in the sequel. These series 

 were discovered by Mr. Walter Gregory (since unhappily deceased) and myself at 

 Coopers Hill many years ago, but were put on one side when Dr. Meissel's tables 

 of J„(.i) and J,(,i) were published, giving these functions up to ,r = 15-5. 



It is po.ssible that the series are already known, but I have not seen them else- 

 where, and they may be new. At any rate, the proof given in this paper is 

 independent. 



The series for R" is 



1 +^ 4«^-l 1^ (4n"-- l) (4?t^-3^) 

 2" (2,1)- 2.4" (2.V)* 

 ^ 1.3.5 (4w^-n (4w^-3-^)(4n' -50 

 2.4.0* (2()'^ " "^ ^''^ 



It is a remarkably simple series, considering the formiilse for P and Q, whose 

 numerators contain considerably more terms. It can be easily verified to a few 

 terms by actually squaring and adding the series (2) and (-3). 



It is, like them, a terminated series when }i is an odd multiple of J, and for 

 other values of ra it is a semi-convergent aeries — i.e., the terms decrease for a certain 

 range, and afterwards become alternately positive and negative. 



To Jind the Series for If- ;— 



Let 





and let M = a;-* 8* cos <^<, r , , . (G) 



where ■ S = R'J, and <^ = cr H- n-^ - «^. J 



'Jlien K satisfies the same differential equation aa J„(.(), viz., 



d"u Idu /■^_7i^\ _„ 

 3.1'^ X d.v \ x^J 



Taking the relation ,r'« = Si cos 0, and differentiating twice, we obtain 



1 d'^ic ,du, a 



a.i — , +.v-i -^ — |.r--^« 



-{*«-'SHS-'(i)'-s'(2)'}c,», 



