forward in the problem of understanding and forecasting ocean 
waves. 
The generalization which will be developed in this chapter 
will make it apparent that the finite irregular train of wave 
groups mentioned in the last chapter as given in eouation (6.5) 
involves too many special assumptions to permit its development 
to a completely realistic case. 
The Lebesgue Stieltjes Power Integral 
In order to extend the techniques of wave analysis, it is 
necessary to discuss a new type of integral which is well estab- 
lished in theoretical mathematics, but which is unfamiliar to 
Many people. The ordinary Riemann Integral is the one which 
is well known. “The concepts of the Lebesgue Integral and the 
Stieltjes Integral are employed in theoretical statistics, 
and Cramer [1946] is a reference for such a study. No attempt 
will be made for complete mathematical detail and for complete 
generality, but the derivation will be general enough to in- 
clude those properties which are needed for wave record analy- 
sis. The reader who is interested in greater detail is referred 
to Tukey and Hamming [1949], Tukey [1949], Levy [1948], Cramer 
[1946], and Wiener [1949]. The methods by which these con- 
cepts can be applied to wave analysis most directly are given 
by Tukey and Hamming [1949], and many of the arguments herein 
will be based upon quotations from and explanations by Tukey 
and Hamming [1949] and Tukey [1949]. 
Consider the Lebesgue Stieltjes Power Integral given by 
equation (7.1). 7(t) is the free surface as a function of 
OSTA me 
