587 T. H. Havelock. 
where 
foo) 5 2 
(1 ae en (¥—i) tt? /2p dt 
P 
Ei nk 
) ra) egy 
The integrand in (17) is expanded in ascending powers of 1/p and then 
integrated term by term, using the formula 
72Q(y) (7) 
[. fe O- — D(r + HS He tH, 
0 
— (ly) cot —ty: (18) 
The expression was carried out completely to include all terms of order 
p 3, and the leading terms in py‘ were also determined so as to check the order 
of numerical approximation. Leaving out the intermediate expansions, it 
may suffice to record the final result ; we find 
> Serie 1 1i0 ald 380 rai) 
Q (y) = die + (4id8e 2 Bie = Ste : 
— ise Zl ep - 3 ) Bie? — 209 xt rig sav diet S 
eee ee be aL die Gel dite 9aY O° a 
inp 68 et 
(25 serio 1 (3 
a oy ee #15 
ae . — siete 4 OT 2 a) Steit9 
7 ey = FEE 64 
_ 16065 oxy ute 3465 any val 1 
art Tart Oe gto (19) 
Collecting these results, we have now reduced (4) to the form 
P 4 
R= aN , (8p) + 1s (Gp) + real part of 
afb wp 
(7/2p)* €” {Q (0) — 2e#°Q (28) + e-”Q (48)} |, (20) 
where the integrals I are defined in (13), (14) and graphed in fig. 1, and the 
asymptotic expansion of Q as a function of p is given in (19). 
5. Numerical calculations from (18) and (19) are tedious, and we have 
chosen the parameters so that we require the numerical values of the coefficients 
235 
