1905 - 6 .} Lord Kelvin on an Inifiational Form . 
401 
we see by (134) that the corresponding fluid motion of which F 
is the displacement-potential (132), has constant pressure over 
every surface (z + £) ; that is to say, every surface which was 
level when the water was undisturbed. Thus our problem of 
finding any possible infinitesimal irrotational motion of the fluid, 
in which the free surface is under any constant pressure, is 
solved by finding solutions of (133) and (135). 
§ 99. Now by differentiation we verify that, as found in § 3 
above, 
1 -g* 2 
F = , £ 4(z+ta;) (13 6 \ 
Jz + lX 
satisfies (133) and (135). By changing i into -i, and by 
integrations or differentiations performed on (136), according to 
the symbol - — r— r-^— = , where i,j, k are any integers positive or 
dtax^azr 
negative,, we can derive from (136) any number of imaginary 
solutions. And by addition of these, with constant coefficients, 
we can find any number of realised solutions. If, as in § 97, we 
regard any one of the formulas thus obtained as a displacement- 
it we find £, the vertical 
potential, then by taking — of 
component displacement, which we shall take as the most 
convenient expression in each case for the solutions with which 
we are concerned. Or we may, if we please, take any solution of 
(135) as representing, not a displacement-potential, but a velocity- 
potential, or a horizontal component of displacement or velocity, 
or a vertical component of displacement or velocity. 
§ 100. Thus it was that in § 12 we took 
/O ~ 9 & 
- £ = {RS}— — ^(Z+cX) 
sjz + LX 
= <t>(x,Z, t) 
/ 2 (qfx 1 \ 
Vpfb-^) 
-gt2 z 
e 4 P 2 
(137), 
where p= s /(z 2 + x 2 ) , and ^ = tan 1 (x/z), 
and in all of §§ 1-31, this notation </> and — £ was consistently 
used, with — £ to denote, when positive, upward displacement of 
the water (represented by upward ordinates in the drawings). 
In the two curves of § 4, fig. 1, that which has its maximum 
PROC. ROY. SOC. EDIN. — VOL. XXVI. 26 
