46 KARL PEARSON 



Here I x*e~ x dx is the incomplete r-function for an integer value of s. This 



can be found fairly easily from the above series, or may be determined from tables 

 of the incomplete r-function which it is hoped may be shortly published. 



Again: J,(2iJz) = S-^- 2> 



hence we have : 



f n (c) = NQ t e-' 1 J (2iJe^x) e~ x dx (lxxxii), 



J e 2 



a very concise form, which does not, however, simplify the calculations. Integrate 

 by parts and we have : 



F n (c) = NQ t e-<»-">S ~ R (2^)} 



aC 2 



But *J(2i'2)-S Z " - J 'W*) 



or : 



o { (We^f J 

 = NQ t e~ «"+•*) S (J 1 ^) Js (2*7^) (lxxxiii). 



This is the solution in Bessel's functions, and inside the cleared area, where 

 e 2 is greater than e 1( would give fairly good results if tables of the higher 

 Bessel's functions for imaginary values of the argument were available. 



We can also express the solution in terms of co-functions as follows : 



im-\; (^.)v^, 



Then F n (c) = NQ t S -. I s (a) E s (c). 



o s\ 



Now E. (r) = i e"* H ^ Q £J = 2*Ab w o> w , 



the 6's being undetermined constants, for dividing by the exponential factor we 

 have an integer algebraic expression in 7* jo* on both sides. Multiply both sides 

 by x^rdr and integrate between and oo , p being = or < s. Then : 



J e'^^x* (- y rdr = 2n<T 2 b 2p I x*P«>zp rdr > 

 2tto- 2 f» r" +1 dr 27 . , 



~7T jo "• (2a- 2 ) 5 = 27nr Mff ! )> b y ( X1X ) and ( XX1 )- 



