134 -MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
The functions log p and }p*log p — 1p” are thus in a sense degenerate forms of the Bessel 
Functions, and any theorem relating to the G or Y functions will yield a corresponding 
theorem in these. 
Thus by equating coefficients of «° in the equation 
(V? + «°)(log J gxp — Yoxp) = 0 
we obtain 
Vv’ log p=0 
V"(4p? log p — tp”) =log p 
V*(4p" log p — Zp?) =0. 
and therefore 
We deduce at once 
v | ; (ZR? log R — 4R?)s(e’, y’\de'dy! = i | log Rfla’, y')da’dy! 
am v! | | (QRBlog R= IR) 2’, y’\de'dy' = Vv? | | log Ry(a,y')de’dy! 
= nfl (@ ? Y) : 
(e) Again, from the addition theorem 
Y «KR = Y xpd xp’ + 2 DYVVink Saxe" cos m(w-—w'); (p> p’) 
m=1 
we deduce 
oN , , 
log R = log p — ee cos m(w-w'); p>p 
and 
ZR? log R— ZR* = (Zp? log p — 4p") + gp log p 
Ses 
+ { (p= 2p log p)-& = ct cos (w — w’) 
4 8p 
1 ey" p> p ) A Ne ' 
7 ( m-1  m+l1 Berea eee: 
m=2 4m p 
(f) In the same way, from the results of (c), we may deduce the value of the 
integral 
= | : | ""(1R2 log B — LR2)J 8p" cos mu p'dp’du’, 
The form of the result varies in the cases m=0, m=1,m>1. 
n=: 
I, = ub ef al (a = F)(t98 apad,'Ba—J,Ba) 
+ au — log a)pad,(fa+ (2 log a - 1) Ja i » p<a 
= at ( iP? log p 7P*)( ~BadiBa) + os (Fe - “pala it TJ Ba) b p>a. 
m=1; 
2 2 ee 
Ue = =I, Bp cos w — e oe @ - P\(J,ba+ Bad, pa) 
2 2 
+ : € +2 log a) Ja + a — 2 log a)pas,'Ba > pa 
1 1 P 2 ' 
= Sree (1g Le Liss) (tale) + € (Sar sh80)}, woe 
