Studies in Wave Resistance. 575 
which we can compare the result by another inethod. In a previous paper* 
I have shown how to find the wave resistance of submerged bodies of various 
forms. Apart from the usual simplification of neglecting the square of the 
fluid velocity in the wave disturbance, the specific limitation in that analysis 
was that the dimensions of the submerged body should be small compared 
with the depth at which it moves; but on the other hand, the kinematical 
condition at the surface of the body was taken in its exact form. In 
particular, if the body is a prolate spheroid of semi-axis a, eccentricity e, 
moving with velocity ¢ in the direction of its axis and at a depth f, the wave 
resistance was found to be 
7/2 
R = 128 7°9pa%er A? | sec? per 2moF °° { T10 (xoae sec f) |? dd, (8) 
0 
where x = g/c? and A = [4e/(1—e”)—2 log {(1 + €)/(1—e)} J". 
The limitation in (8) is that @ is small compared with /, but there is no 
direct limitation on the form, for example the expression includes the case of 
the sphere with e=0. Now, if we apply Michell’s formula (7) to this case, 
we shall obtain a result in which there is no limitation of the ratio of a to /; 
but on the other hand the inclination of the surface must be small, so the 
expression will only hold in the limit as « approaches unity. 
The equation of the spheroid being 
Ee CS = 1, 
BY 
we have 
On [Oa =f’ (w, 2) = —Vafa {0 (0?) — a2 (2 PY”. (9) 
Thus from (7) 
b? — mc? wen eslg ] 
J= rane *f19 \ ea) ere sin ma da dé, (10) 
where we have put €=z—/, and the integration extends over the ellipse 
Pfla+?/P = 1. 
Integrating with respect to x first, we have 
| @ “sin mx da Ali AD 
ues a7 (ie ——— 
-» (@C—F)—Pape — PEE ("F JE=P) 
where p has been used for a(1—£?/?)1/”. 
Hence we obtain 
J = — 7b? efig | e(merb/)eos® J, (ma sin @) sin? 6 dé 
0 
pan, BPP IP @seS) eee 
= —I2arbhre~mreslg 2 
eee > GayrtiB (2n)! ( g 
* “Roy. Soc. Proc.,’ A, vol. 95, p. 354 (1919). 
I Jn+3/2 (ma). (11) 
203 
