If the primary wave be represented by = e-ikx the component rotations in the secondary wave are '1'3= P (- AN y) N r2 ' cwi= r x D y N 'y)' lw2=P (- AD + 6,N z2 - x2 ' D r N r2 where ik3T e-ikr _ P - 4 r The expression for the resultant rotation in the general case would be rather complicated, and is not needed for our purpose.
Accordingly, if E be the energy of the primary wave, dE 87-2n (D' - D) 2 T2 E dx 3 D2%4 ' whence E = Eoe-hx (II) where h = 8?r 2 n (D' - D)2T2 3 D2 x 4, (12) If we had a sufficiently complete expression for the scattered light, we might investigate (12) somewhat more directly by considering the resultant of the primary vibration and of the secondary vibrations which travel in the same direction.
Moreover, if OP = r, and AO=x, then r 2 =x 2 + p2, and pdp=rdr. The resultant at 0 of all the secondary vibrations which issue from the stratum dx is by (3), with sin ¢ equal to unity, ndx f ?
Cos(21r/X)(bt - x), and the resultant will then represent the whole actual disturbance at 0 as modified by the particles in the stratum dx.
Through the resultant scarcity of labor, much land fell out of cultivation.