4 T. H. Havelock. 
when we put d = 0, provided & is not zero. Hence we obtain, for all points 
other than the origin, 
2M KoM [” 6 ( 2 ene 
= ———_ } 2 6d | ————_—__———ck. (9 
‘ veLy TU [se 0 K — ky sec? 0 + iu sec 0 a) 
We shall limit consideration at present to the surface elevation along the line 
of motion, that is for y= 0; we have 
2M 4M 1/2, 3 [ exe cos 6 
= ic Og Wee se CRE Ne ean od ane Gt 
6 AIL TU I, cal ) K —Ky sec? § + ip sec 0 a“ cy) 
noting that we require the limiting value of the real part as u is made zero. 
The integration in « may be transformed by regarding « for the moment as 
a complex variable and considering a contour integral taken round a suitable 
path according as @ is positive or negative. In this process it is the residue 
at the pole of the mtegrand which gives the expression for the waves in the 
rear of the system. The result, when uw has been made zero, is 
| ” Cos (Ky ma sec 0) din 
0 1+m 
27 sin (kor sec 9) + 
forz> 0; 
(® cos (Kk gma sec 8) 
J0 1+m 
For the integration with respect to 0, we require the following results 
dm; tori << 0s) ((1i)) 
ie sec? 0 sin (gx sec 0) d0 = —= Y, (ko%), (12) 
0 hed 
|" sect 6 d6 I" cos («gma sec 0) Fp ia mace ia J, (koma) tes 
Jo 0 1+m 2Jo 1lt+tm 
el db ts Te i Jo (Kgma) cb 
Qype Aye Jo (1 + m)* 
2 2 
=— ae ale 7 ea (kot) — Y, (kx) — 2}. (13) 
In this J and Y denote Bessel functions, and H is Struve’s function, the 
notation being that of G. N. Watson’s “ Treatise on Bessel Functions.” Col- 
lecting these results and putting in (10) we obtain the surface elevation on the 
line y=0. To avoid any possible ambiguity in signs, we shall find it con- 
venient to write xz’ for —a and so restrict « and 2’ to positive values; 2 is 
thus distance in front, and «’ distance behind the moving system. We obtain 
cB LH (cae) — Ya ker) 21; a> 0 
: x) 
- 9 
c= Bea H oe!) — Ya oot) 2} FRE Va eee); > 0. (18 
350 
