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fl Qp ?

00Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P, it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave.

00To find the former, we have, if OAQ=4), AOP=w, QP 2 =u 2 +4a 2 sin 2 2w - 4au sin la) sin (2w-4)) = (u +a sin 4) sin w) 2 -a 2 sin 2 4)sin 2 c0+4a sin 2 2w(a-u cos 0).

00But if we now suppose that Q lies on the circle u= a cos 0, the middle term vanishes, and we get, correct as far as w4, QP= (u+a sin 4) sin w) 1 ' 3 1 {- a sin2c?sin4w V 4u so that QP - u=asin0sinw -Ft asin¢tanOsin 4 w..

00By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.

00fl Qp ?

00Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P, it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave.

00To find the former, we have, if OAQ=4), AOP=w, QP 2 =u 2 +4a 2 sin 2 2w - 4au sin la) sin (2w-4)) = (u +a sin 4) sin w) 2 -a 2 sin 2 4)sin 2 c0+4a sin 2 2w(a-u cos 0).

00(8); 4 sin 2 2w=sin 2 cw+ sin4w, and thus to the same order QP 2 = (u-{-a sin 0 sin w)2 - a cos 0(u - a cos 4) sin 2 o+- a(a - u cos 0) sin 4(.0.

00But if we now suppose that Q lies on the circle u= a cos 0, the middle term vanishes, and we get, correct as far as w4, QP= (u+a sin 4) sin w) 1 ' 3 1 {- a sin2c?sin4w V 4u so that QP - u=asin0sinw -Ft asin¢tanOsin 4 w..

00By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.

00

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