PROPERTIES OF SINE WAVE PLUS NOISE 143 



where .4 is the area enclosed by the quadrilateral whose vertices are at the 

 points (r, s) given by (0, 0), (0, a), (/3, /3), (a, 0) and it is assumed that /S and 

 a are positive, w is a new variable and is not the one introduced in (7.13). 

 Setting a = QV{2bo) and ^ = QVi^h + 2g), using (7.18), and introduc- 

 ing the notation 



(7.19) 



(7.20) 



(7.21) 



. — /? 



where we have expressed X in terms of p and ^. 



The double integral defining y2 may be treated in the same way as (7.20) : 



p(X — ■M)e~"" du 

 y%- ' ' - • 



^ r r e~'-^' dr ds ^ r ue-^ du r p(X - u)e-"' 



J J ^ — r — s Jo ^ — u Jp (\ — p)(^ — 



Writing u= ^ — (^ — u) and X — 2/ = X — ^ + (^ — w) in the two numera- 

 tors leads to 



J p ^ — u \ — p J p 



(7.22) 



e 



du 



where we have used p(X — ^)/(X — p) = — ^ to simplify the coefficient of 

 the third integral. When the second and fourth integrals are evaluated, 

 their contribution to y2 is found to be equal to the terms independent of y i 

 on the right of (7.21). Hence, comparison of equations (7.21) and (7.22) 

 shows that 



Jo ^ — U Jp ^ — u 



