equation (9.22) as 2s approaches infinity, Cor approaches 
1/2 from values less than 7/2, and the mesh of the net approaches 
zero. The rule for picking W(,0) must also be given before 
the partial sum is formed. For some forms of the integral 
W(,) can be a continuous function; for other forms of the 
integral, it must be defined in very special ways. The value 
under the square root sign is always positive by virtue of 
equation (9.13). 
An imporant property of the Lebesgue Stieltjes Power Integral 
for a short crested sea surface 
From the properties of E,(# ,@),it follows that E,(p ,0) 
has a definite limiting value when 6 = 7/2 as #L approaches 
infinity. This limiting value will be called Eo max? and it 
should be noted that the free surface considered as a function 
of time (when squared and averaged over time) for some fixed x 
and y may or may not be related to the Enax of Chapter 7. For 
this reason the difference between E and Eee must be kept 
2max 
in mind until their relationship is studied in Chapter 10. 
When the free surface defined by equation (9.1) is squared 
and averaged over the y,t plane, it can be proved that equation 
(9.24) is the result. The potential energy averaged over the 
y,t plane is then given by equation (9.25). 
Equation (9.24) can be proved by the procedures given in 
equation (9.26) and the steps which follow it. In equation (9.26) 
the definition of the integral given by equation (9.24) has been 
substituted for (Xt) By a correct application of the 
various limiting procedures, the result can be proved. In equation 
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