—* I —E—————— eer 
TRANSACTIONS OF SECTION A. 893 
Bie { i, Xe 
ope Bo teat aim: -} 
att) G+ RHI eB” PEEP 
fan 12h k) 
tm+3=D, { log. {u-+ (0? + 3°08} - Or ees Fk 
where 
ci to 
"log 
and y?=p?—]. This holds even when, m=n”+3, 
Case 2; 
p>1, m>n+4, which is not integral. 
The expansion in this case only differs from that of Case 1 in the form of t, which 
becomes 
kp+(v? +h)! k-1 
fs 21 72)8) _ be 
tne 3=D. Tog, {w+ (+h) }—5 1 iu? +h Bel 
If m and x are both integers, P,"(1) must be defined with another multiplier, and 
the expansion is readily obtained, 
Case 3: 
O<p<(1—k?*)i, kl. 
Pim) = - ae (2. 1—p?)? Ri sin p, 
Q,"(u) =m ene . (2. 1-p?)* Ri cos p, 
p=(m+n+1) > als Seth m) ree i ie ah Siegen 
where D’ is the operation D with the signs of p,,p,;... cena 
Casa 4; 
; k>1, p<, 
2 o(n+m : 
PMH) = ae agg) A= ty RB sin 
n+m a 
Q,"(#) = 7? ay (2.1—p?)"? R} cos Py 
1-p? A, Per Neae 
‘ak As Drie Bev (ey * i 
o* Kaito: dis) 2) eee ee ; 
~log ae + yy } 
Case 5: 
k<l, p>(1—2). 
The expansion is that of Case 3, with a change of p to 
Cay o*(” + 3) p{5 (P+ vb +h ahs is 
Spiga! 27 w(n+ m)o(n =r 2 loge . kp — (ke +7 a ea leg Gs i") 
—log p+ (i? 40°) by 
