466 T. H. Havelock. 
Take O im the free surface, with Ow in the direction of motion and Oz verti- 
cally upwards; and let w be the velocity of the model. We consider first 
a distribution of horizontal doublets in the plane y = 0, extending from the 
free surface down to a great depth, and we take the moment M per unit area 
to be a function of x only. Further, we suppose that the distribution of M 
is confined to a finite range in 2, is continuous within this range, and is zero 
at the two limits of the range. 
The surface elevation along the median line y = 0 is given by 
CSE {IM eR] 2) + (MGR — Mai}, 
where the summation covers all points of sudden change in the gradient of M, 
and the integrals extend over the ranges of continuous variation of gradient. 
The function F is defined for positive and negative values of its argument by 
F (q) = — 5 Qo (xo) 
Ko 
; ) 
TT AT 
Ri 9) — Fe, Qo (Kog) + — Po (x09); 
Ko Ko 
with ¢ > 0, and ky = g/u?. 
We have also, for positive values of , 
x [P 
Pop) =—5 | Yon) ap 
4 Jo 
, (3) 
in the usual notation for Struve and Bessel functions. 
One of the approximations of the theory lies in the connection between the 
form of the ship and the equivalent distribution of doublets in the median 
plane y= 0. For a ship model, of infinite draught, whose horizontal half- 
section is given by y = f(z), the usual approximation amounts to taking 
M’ (0) = (u/2r) f’ (2). (4) 
With this relation, the surface elevation along y = 0 is given by 
2 , s ” 
C=SE IP OER —2)+ (sf MPe—Hail. — & 
Here x, and h, are positive in the direction from stern to bow, «,, being the 
position of any sharp corner in the form of the model. With this convention 
