392 TRANSACTIONS OF SECTION A. 
and in general, the apes admit the following expansions: ? 
4m 
(2) Pm) = eat a =) F(—n,n+1,1-m,43(—p)) 1p] <2 
(2) Q,"(#) = 
n+m+1 
o( 2 ) o(-3) n+m+1 m—n1 
= 1, Qm— 1 ek(m—n)nt = es If i Ay Geel 4 eT —~—) =) id 
= 
wk: 
+ Qn ener ) ake =e 
(u2—1)™ jae ea 8: ) 
. —m— pameyO 
where //< 1 and pis slash e. If be negative, the exponential has argument 
3(3m + 2)ri. 
(c) If |n/>1, 
m = 2-7-1 ima @( + Mm) w( — 3 3) (n° abe. a +n+2 m+n+t 3 BS 
Q, (#) e o(n+4) “pment ( 2 4 2 : Soeur = 
-_ 
(a) If |p| >I, 
>m(,)— Sin(n+m)r _o(n+m) (w?-1)™ n+m+2 n+m+1 cage 
F (#) 9+. cos mr o(n +3) o(— ~4) pmenet F( ) 4 oy, Rat Dy =) 
“a 
+2 w(n—%) min—m p(mao-ntl m—n 1 _ a; 
rer pres yg cid Gear pomuaar r 
These define the functions of argument greater than unity. mis [ul<1, 
P,,(u) is defined as the common yalue of e+ '™” P ™(u4 20.7) and 
2 e'™™ Q™(n) = eH Q (wn + 0.7) + etm” Q™( —0.2), 
The expansions of the paper, when 7 is large, are as follows :— 
Let A,, A.» . . » be a series of coefficients satisfying 
k'(k? ~1)(r — 2)(r —4) (7 —6)A,_¢ + 4(2 — 3k*)(r —2)(r -—3)(" -4)Ay_4 
+ (1 +2) {K? + 8k?(7 — 2)? — (7 — 2)?}A,_2 + (r—1) {(2n + 1)*-(n—1)?}Ar =0, 
with A, =1, A, = — (4k? +1)/8((n + 3)?—-1°), 
Af (n+ $)?-27)A, = ~ 64?(2 — 3k”) + 3(4K? — 1)(28K? —9)/8{(n + 3)? - 1°} 
where k=m|(n+}), 
so that, if % is not of order greater than unity, A,, A,;, A, are of the same order, 
but A, of order n? less, and so for every third coefficient. Moreover, let D denote 
the operation 
mt d d \2 d\3 
= _ He Ms ( ay 
D1 + He (an) a a ian) 
where the p’s are given identically by 
Ltpyvtpot? + 2... S(L+AvtAge?74+ 2.7, 
and have the same property of convergence as the X’s. 
Then the following expansions exist, for a real argument, 7 being large:— 
Case 1: 
p>1, m<n+3, the latter not being an integer, 
zs eine sin (m+ sin (m+n) = a(n +m) Tp 
PAH) - 7 COS 7 Qum(e) = ae m). pe — i) é 
m = cima _ . @(n+M) | The-t 
Qa") se aa=aa ean 4 
' Hobson, Phil. Trans, Assoc., 1896, p. 4438 e¢ seq. 
