420 T. H. HAVELOCK 
from rest; the solution is carried to the second order of approximation. 
It has been found possible in this case to reduce the expressions to forms 
which are not too difficult for numerical computation, and curves have 
been drawn to show the influence of the acceleration upon the resistance 
and upon the effective inertia coefficient. 
2. We shall construct the expressions by the method used in the 
previous paper. With the origin O at a depth f below the free surface, 
Ox horizontal and Oy upwards, suppose a singularity of order n created 
at the origin at time ¢ = 0, maintained for a short time 67 and then 
annihilated. To satisfy the condition at the free surface during this 
impulsive motion, we have for the complex potential function 
hy >S Tee Pz 2uf)-”, (1) 
where P, may be complex, and P* denotes the conjugate complex 
quantity. 
To obtain the surface elevation in a convenient form we write (1) as 
ioe) 
(=1) ye nN—1 pike (a)” * N—1p—tK2—2kKf 9 
Hh = Gail a iip) || ie ei ermee GPO ES aire. (2) 
0 
The result of the Saha vertical velocity acting for a time 67 is to leave 
the free surface with an elevation 7 given by 
oO 
2(2 yale ia Np—ikx—Kf 
N= ES = D! dr | Ke dk, (3) 
0 
where Re denotes the real part. 
The potential function for the subsequent fluid motion due to this 
initial surface elevation, without velocity, is 
ive} 
Qghn ; 
P* or | Kr —he—tk2—2KF sin (ght?) dk. (4) 
we (n—1)! 
We now consider this to be a continuous process occurring as the origin 
moves parallel to Ox with a velocity c, with c and P, functions of the time. 
Let s, be the distance travelled by the origin from the starting-point; then 
in (4) we replace ¢ by t—7, and z by z—(s,—s,), so that z is now referred 
to the moving origin. Integrating from the start up to the instant ¢, we 
obtain for the complex potential 
= Prt) Prt) ws ep A | Pe dr | Dente F,en-t dc; (5) 
BO (B= NP. (a 
with DL = en ilesr—sz)—gh el} __ gif xis, -s7) +27}, (6) 
546 
