P1 (cos 0) by Jo([ ] sin 0/2). While these approximations are very good. closed set that is not closed.

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1 62 PHYSICS: R. SERBER PROC. N. A. S. the collection { aj + E an }. Then every set is compact, but not every set is closed n =,m (e.g.! X - a is not closed). Thus X has Property a but contains a compactly closed set that is not closed. * This research was supported by a grant from the National Science Foundation. 1 Whyburn, G. T., "Open mappings on locally compact spaces," Am. Math. Soc. Memoirs, no. 1 (150). Also "Open and closed mappings," Duke Math. J., 17, 6-74 (150). 2 Halfar, E., "Compact mappings," Proc. Am. Math. Soc., 8, (157). 3 Gale, D., "Compact sets of functions and function rings," Proc. Am. Math. Soc., 1, (150). 4 Kelley, J. L., General Topology (New York: D. Van Nostrand, 155), p. 230 ff. 6 Halfar, E., "Conditions implying continuity of functions," Proc. Am. Math. Soc., 11, (160). 6 Whyburn, G. T., "Mappings on inverse sets," Duke Math. J., 23, (156). SHADOW SCATTERING AT LARGE ANGLES* BY R. SERBER COLUMBIA UNIVERSITY, AND BROOKHAVEN NATIONAL LABORATORY, UPTON, NEW YORK Communicated July 28, 165 As the energy of an elastically scattered particle is increased until its wavelength, 1/k, becomes small compared to the dimensions of the scatterer, it would be expected that the phase shift 5I should depend on the ratio p = (1 + '/2)/k, that is, that as k is increased, the scattering takes place at fixed impact parameter p, rather than at fixed angular momentum 1. This result is just what would be given by solution of a wave equation by the WKB approximation, appropriate for small wavelength (it is, of course, the WKB approximation which dictates the proportionality of p to 1 + l/2, rather than to 1 or [1(1 + 1) ]V/'). In these circumstances many terms contribute to the sum for the scattering amplitude, f _ CD ik - 2k2 g (21 + 1) a(l)p(z), (1) (where a(l) = 1 - e2iw and z = cos 0), and a frequently used approximation is to replace (1) by f co ik = J A(p)Jo(qp)pdp, (2) with q the momentum transfer, q = 2k sin 0/2, and A(p) = A([l + 1/2]/k) = a(l). (3) Equation (2) is obtained from (1) by replacing the sum by an integral, and approximating P1 (cos 0) by Jo([ ] sin 0/2). While these approximations are very good for small angles, at large angles the equality between the Legendre and Bessel functions is in error by terms of order sin2 0/2, and moreover the scattering given by (2) becomes so small that it may be comparable to the error in replacing the sum by an integral. In this note we shall investigate the relationship between the results

2 VOL. 54, 165 PHYSICS: R. SERBER 63 given by (1) and (2) for a class of problems for which (2) has an asymptotic expansion, for large q, in inverse powers of q. To obtain the asymptotic form of (2) we first divide Jo(qp) into terms behaving like e t"i for large qp, Jo(qp) = 1/2[Ho(')(qp) + Ho(2)(qp)]. Imagine, for the moment, that A(p) is such that the integral involving Ho(1) can be deformed to the positive imaginary axis, and the integral involving Ho(2) can be deformed to the negative imaginary axis. Writing p = iu/q in the first integral and p = -iu/q in the second, we obtain f_u ik= - iq2 fj [A( ') - A (- i)j Ko(u)udu. (4) As q -a oo, the contributions to the integral come from values of p = - closer and q closer to zero, and if A (p) has a power series expansion in p, substitution of this expansion in (4) will immediately give the asymptotic series in powers of 1/q. If A (p) is such that the contours of integration cannot be deformed to the imaginary axis, it will nevertheless be possible to deform them to the imaginary axis in the neighborhood of p = 0. If A is then expanded in a power series, and the integration is carried out term by term, the contours for the individual terms can be taken along the imaginary axis. Thus while (4) is not true as an equality, the correct asymptotic expansion is obtained from it by expanding A (p) and integrating term by term. A term asps in the expansion of A (p) contributes to (4) a term co fs _ 2a~sin(1/2r8) 'K(u)du = ik 7rq 2+s S 2 a. sin('/27rs)r(1 + 1/28)22(5) Equation (5) gives zero for s an even integer. Thus, if A (p) can be expanded in a power series in p2, the scattering for large momentum transfer falls off faster than a power of 1/q. For such a shadow, smooth as one crosses p = 0, the large momentum transfer scattering is not determined by the behavior of A (p) near the origin; in short, nothing interesting is happening at small distances. A parallel treatment can be given for the asymptotic form of (1) at large angles. For this purpose we use the Watson-Sommerfeld representation' of (1), _ = 2lrik2 'A)2 cos r ) P-'12+X(-z)xdx. (6) Using the asymptotic form of P (- cos 0) for large 1, we see that P-112+X (-z) contains terms behaving like both e4i-x(t) for large x. Thus P-1/2+X(-z)/cos -rx behaves like etxo in the first quadrant and like e-ix` in the fourth quadrant. Under the same conditions that led to (4), we can deform the contour to the imaginary axis, and, writing x = iu, obtain f 1 a(-'/2 + iu) ik 2irik2 J-x cosh u(ru 1 ('0 [a(-'/2 + iu) - a(-'/2 - iu)] -2irik2Jo cosh iru ~~P-1/2+iU(-z)udu, (7)

3 64 PHYSICS: R. SERBER PROC. N. A. S. the second form following from the fact that P,+ju(- + z) is an even function of u. If we use the connection (3), we have a (- 1/2 iu) = A (iu/k). Since P-/2+ii+(-z)/cosh 7ru behaves like e -u8 for large u, the contributions to (7) will come from arguments of A of order 1/kG for large kg, and the asymptotic form is again obtained by expanding A in a power series. In place of (5) we obtain fs a, sin(1/27s) Jet1b+' 8 _ a8sin(127rs) _'U~ P 1 /,+iu(-z)du. (8) ik 7rk+ Jocosh iru For s an even integer we again get zero. For s an odd integer the integral appearing in (8) can be evaluated by considering a particular example. Suppose in (1) we take a() = e -a(l+'/2), a choice of a, which permits the series to be summed without difficulty by using the familiar relation Putting h = e-a gives co (1 + h2-2hz)-'/2 -E h'p1(z). 1=0 (2 cosh a - 2 Z l=0 and on differentiating both sides with respect to a we find f 1 sinh a ik k2 [2 cosh a -2z Equation (7) now gives us the relationship sinh a 1 cosin au [2 cosh a - 2z]' = X Jo cosh rul P~ 1i+iu(-z)udu () The integral appearing in (8) can be evaluated, for odd integral values of s, by comparing coefficients of a' in the expansion of both sides of () in powers of a. Writing 1 CP us+1 Is= - I P-1/2+iu(-z)du, (10) or cosh xru we find I1 3/2, I3 = 6/2- I/ /2- /2 +,//2, where w = 2(1 -z).

4 VOL. 54, 165 PHYSICS: R. SERBER 65 A recursion relation for these integrals can be obtained by using the differential equation satisfied by the Legendre functions LIPI(z) = d diz (1 - Z2) d + 1( + dzy 1)} P1(z) = 0, from which we find L _ EP-1/2+ iu(-z) = u2p_ I/2+1(Z). Hence Is+2 = L-ij, or, written in terms of w, 1+2 = {Id [4w-. d 2dw 4j (11) Let us label the asymptotic term (8) by f,(')/ik, to indicate that it is the value obtained from (1), and the term (5) by f'(2)/ik to show that it is the one obtained from (2). Their ratio is f8(l) P = C8(w) (12) with C (W) T 8)2ls1 +(1+2SI1 (13) W 1+I8. CQ(w) is a polynomial of degree 1/2(8-1) in w, and C'(0) = 1. From the expressions already given for II, Is, and 1 we find G3 1- WY G51-w and the explicit expressions for f'(1), f3(m), f (1) are f,(l) _ ik f =l a q3 i k = - 1 Y) f w2 f5() 225a/ 1 WI(+ 1 2) ik q

5 66 ZOOLOGY: DAVIDSON ET AL. PROC. N. A. S. The asymptotic form of the scattering amplitude for large momentum transfer given by (2) is of course a function only of q. As we see from (12), the result given by (1) differs from that given by (2) by polynomials in w = 2(1 - cos 0) = q Ik, which approach unity as k2 oa for fixed q2. For example, a 1/q' term in the asymptotic form of the scattering amplitude is reduced by a factor 7 at 0 = 00, and a factor - at 0 = 1800, corrections which may be described as modest in view of the initial doubts which might have been entertained as to the relationship between (1) and (2) at large scattering angles. * This research was supported in part by the U.S. Atomic Energy Commission. 'Sommerfeld, A., Partial Differential Equations in Physics, (New York: Academic Press, 14) p. 27; Watson, G. N., Proc. Roy. Soc. (London), 5, 546 (118). EVIDENCE FOR PRELOCALIZATION OF CYTOPLASMIC FACTORS AFFECTINGjGENE ACTIVATION IN EARLY EMBRYOGENES1S* BY ERIc H. DAVIDSON, G. W. HASLETT, R. J. FINNEY, V. G. ALLFREY, AND A. E. MIRSKY THE ROCKEFELLER UNIVERSITY Communicated June 14, 165 Differentiation begins early in embryogenesis as different genes become active in different cells. Within the closed system of the early embryo, equal genomes thus direct the creation of diverse cell types. Though the nuclei of these cells contain complete copies of the same genome,1, 2 the nucleoplasmic and cytoplasmic environments of these genomes are not the same, as a result of the distribution of cleavage nuclei into diverse areas of egg cytoplasm early in the cleavage process. In some cases the fate of these nuclei, i.e., the type of differentiated cell to which they or their descendants give rise, has been seen to depend on the area of cytoplasm in which they come to lie. For example, it was demonstrated by Hegner in 11 that the determination of sex cells in chrysomelid beetles requires exposure of the appropriate cleavage nuclei to a clearly demarcated polar area of cytoplasm.3 Destruction of a small amount of the polar cytoplasm before it has been populated by the cleaving nuclei deprives those nuclei which normally would differentiate as sex cells of their special cytoplasmic environment. The result is the manifestation of other activities by these nuclei and their descendant lineage. Thus a complete insect is formed, but primary sexual tissue is absent. The first remarkable case of this kind is Boveri's famous 10 demonstration of sex-cell determination in Ascaris, in which only genomes exposed to certain polar cytoplasm are protected from chromatin diminution and thus preserved for their subsequent role as sexual stem cells.4 The pressure plate experiments carried out on a wide variety of cleavage stage embryos by Driesch and many others, beginning in 182,' 5, 6 suggest that qualitative cytoplasmic determination of nuclear differentiation is not confined to presumptive sexual tissue, but is common to all early cell types. In these