1903-4.] Lord Kelvin on Two-dimensional Waves. 
187 
1 
n / 2 ip 
[s/(P 
+ z) 
C°S0+ J(p-z) smfj] 
-gVz 
e 4 P 2 
where p 2 = z 2 + 
.r 
\/ — sin 
P \ 
where 0 = tan -1 ( 
p-zj 
(7), 
(8). 
The sign of ^/(p - 2 ) changes when a; passes through zero. 
Going back now to (5), and denoting by {RD} the difference of 
its values for ± 1 divided by 2t, we have another solution of our 
problem essentially different from (6), as follows 
1 
*) = {RD} 
-gt* 
;4(Z+lX) 
i;l> 
x . ut 2 x 
!)sm V 
V(2 + ta?) 
V(p^)cos|*] 
-gPz 
e 4 p 2 
= v 
(gf^ 
sn Tv +e -f- 
—gt 2 z 
(9), 
(10), 
( 11 ). 
§ 4. The annexed diagram, fig. 1, represents for t = 0 the solu- 
tions 2 <£ and ^ as functions of x, with 
the drawing. The formulas which we 
z — 1 for convenience in 
find by taking t = 0 
in (7) x J 2 and (10) x J2 are 
J(x 2 + z 2 ) 
Before passing to 
1 + 
2 <£ = 
VUA* 2 +**)-*] 
( 12 ). 
Jk + k 
the practical interpretation of our solutions, 
remark first that (12) contain full specifications of two distinct 
initiating disturbances ; in each of which <f> may he taken as a 
displacement-potential, or as a velocity-potential, or as a horizontal 
displacement-component or velocity, or as a vertical displacement- 
component or velocity. Thus we have really preparation for six dif- 
ferent cases of motion, of which we shall choose one, - £= ^2 x (7), 
for detailed examination. 
§ 5. Taking z = h= 1, for the water surface, let the two curves of 
figure 1 represent initial displacements, (12), of the water surface, 
left to itself with the water everywhere at rest. The displacements 
at any subsequent time t are expressed in real symbols by (7) (10) 
without the divisor J 2, and by (8) (11) with a factor J 2 intro- 
duced ; either of which may be chosen according to convenience 
in calculation. One set has thus been calculated from (8), with 
