os ilvsbes 
resulting free surface is non-Gaussian it may be just as difficult to obtain 
results on the probability structure of the first and second order Lagrang - 
ian models as it would be to obtain results on a much higher order Eulerian 
model. If the problems prove intractable, numerical computation is a 
possibility that should yield some results. 
The real problem is thus to compare the six available random 
models with nature and to devise relevant experiments to see which model 
comes the closest to agreeing with that which is observed. Any linear non- 
Gaussian model that can be devised along with its higher order extensions 
should also be allowed to enter the competition. We suspect that such 
models will not do as well. 
Recapitulation 
So far in this paper, three new random sea surfaces have been obtain- 
ed. They are the short crested sea surface given by (17), (18), and (19) 
based on a linear superposition of the particle motions; the long crested 
sea surface given by (21), (22), and (23) based on a linear superposition of 
the particle motions; and the long crested sea surface given by (29) and 
(30) based on the second order correction to the long crested model. Each 
model is irrotational to the order to which it is carried out. The notation 
chosen to represent the second order long crested model is not as useful 
as the notation used by Tick [1959] in his study of the second order long 
crested model in the Eulerian system. These results, however, show the 
existence of such a model. 
Properties of the models 
The multivariate probability structures of these models have 
not yet been found. Only a few of the properties of the long crested 
linear model have been found. A possible realization of z = z(x) has 
been constructed, and the probability density of z(x) for a fixed x has 
been found. 
