m- 1 n- 1 



S S (N., - aj - bk - c)j =0 

 k=0 j=0 ' Jk '' '-* 



(8.4) 



m- 1 n- 1 

 k=0 j=0 



m- 1 n- 1 



(N.y. - aj - bk - c)k = 



S S (N - aj - bk - c) = 



k=0 j=0 



jk 



The last equation simply states that the average of all the points in the 

 plane when truly leveled should be zero. Points on a tilted surface could still 

 average to zero, but V would not be a minimum; and thus the other two equa- 

 tions assure that V will be a minimumo 



The indicated summations can be carried out, and the result is three 

 simultaneous linear equations in the three unknowns, a, b, and c. 

 (8.26) 



mn(n - l)(Zn - 1) n(n - l)m(na - 1) mn(n - 1 ) 



n(n - l)m(m - 1) nirL(m.- l)(Zm- 1) nm(m- 1) 



mn(n - 1) 

 2 



nm(m - 1) 

 Z 



nm 



m- 1 n- 1 

 k=0 j=0 -^ 



m- 1 n- 1 

 2 E kN 



jk 



m- 1 n- 1 



S N 



k=0 j=0 



jk 



For m = 90j n = 60, the determinant is known, and the indicated sum- 

 mations on the right hand side, when performed on the data, then permit the 

 values of a, b and c to be found. 



Estimating the Directional Spectrum. 



In, theoretical discussions of wind generated gravity waves, it has been 



shown by Cox and Munk [1954] and by Pierson [1955] that 



60 



