102 Samuel Karp, Jack Kotik, and Jerome Lurye 
r 
a -v} ,cosh7u 
I,=vecoshu )| )) m, sin (vx, cosh u) e (A12) 
isl jel 
Z = wy -vy.cosh7u 
J, = 7v cosh u Ms Ds m, cos (vx, cosh u) e / (A13) 
i=l jz=1 
Passing to the continuous case with the y integration extended to infinity, we get, in- 
stead of (Al2) and (A13), 
L/2 @ a 2 
I = v cosh u { m(X) sin (vx cosh u) B per ead dydx (Al4) 
-L/2 “0 
L/2 © a 5 
J = -v cosh u | | m(X) cos (vx cosh u) eu tte Sard es dydx. (A15) 
-L/2 “0 
L/ 
Note that in Equations (A14) and (A15), we have assumed only the integrability of m(x) 
and not its boundedness. 
The y integration can be performed separately, so that (Al4) and (A15) become 
L/2 
ve 1 a x a a 
[= are J m(X) sin (vx cosh u) dx (A16) 
-L/2 
E72 
1 a a 
dh Fas: Sean | m(X) cos (vx cosh u) dx. (A17) 
-L/2 
Finally, we reintroduce the normalized variable x = x/L. Then 
1/2 
ces ale : 
I= Pantin i M(x) sin (Fx cosh u) dx (A18) 
-1/2 
L 1/2 
Oe me Rania: fi M(x) cos (Fx cosh u) dx. (A19) 
-1/2 
In these equations, M(x) is defined to be m(Lx), while F = L = gL/c? as before. 
