scattered waves calculated by the met hod I have indicated. The agreement is good to t he accuracy of the experiments which was about 1%6. There is ad ustable constant, and the patterns reproduce ely the general features of the X-ray pattems but details due to special arrangements of the crystals in the films which were known to occur from previous investigation by x-rays. Later work has amply confirmed this conclusion, and many agreement with ory bei ng foun d. the accur acy has increased with the improvement of the apparatus, perhaps the most accurate work being that of v. Friesen of Uppsala who has used the method in a precision ion of e in which he reaches an accuracy of I in 1, 000. Before discussing the theoretic al implic ations of thes e res ultsthere are two modi fications of the ex periments w hich s hould be me ntioned. In the one, the electrons after passing throughthe film are subject to a uniform m agnetic fiel d w hich del ects them. It is found that the el whase impact onthe plate forms the ring patt ern are deflected equally with thas e which have passed through holes in the film. Thus the pattem is due to electr ons w hich hav preserved unc ha nged the pro pert y of being deflected by a magnet. This disting uishes the effect from anything produced by X-rays and shows that it is a true property of electrons. The other point is a practical one, to avoid the need for preparing the very thin films which are needed to transmit el ectrons strik ng t he diffracton in many cases the patterns so obtained are really due to electrons transmitted through small projections on the surface. In other cases, for exampl when the cleavage surface of a The theory of de Broglie in the form given to it by Schr t inger is now known as w ave mec hanics and is the basis of at omic physics. It has been applied to a great variety of phenomena with s uccess, but owing lar gel y to mat hem atical difficulties there are not many cases in w hic h an accurate com parison is passi ble betw eentheory and experim ent. The diffraction of fast electrons by crystals is by far the s ever est numerical test w hich has been made and it is therefore important to see just what conclusions the excellent agreement between theory and these experiments permits us to draw a front of more than 200 ?each But the real trouble comes when we consi der the physical meani ng of the waves. In fact, as we have seen, the electrons bl acke n the photographic plat ose places w here the waves would be strong Following Bohr, Born, and schr t inger, we can express this by saying that the inte nsity of the waves at any place measures the probability of an electron m ani festing itself there. This view is strengt hened by meas urem ents of t he relative int ensities of ti rings, w hich agree well wit h calc uations by Mott based on Schr t s equation. such a view, how ever s uccessful as a formal statement is at variance with all ordinary ideas. Why should a particle appear only in certain places associated with a set of waves? Why should waves produce effects only hrough the medum of particles? For it must be emphasized that in t hese experime nts eac h el ectron only sensitizes the photog raphic plate in one mi nute is only effective in the of it is a kind of phantom. Once the particle has appeared the wave disappears like a dream when the sleeper wakes. Yet the motion of the electron, unlike that of a Newtonian particle, is influenced by what ha ppens over the w hole front of the wave, as is shown by the effect of the size of the crystals on the sharpness of the patterns. The difference in point of view is fundam ental, and we have to face a brea k wit h ordi nary m echanical ideas. Particles unique track, the energy in these waves is not continuously distributed, probability not determinism governs nature. But while emphasizing this fundamental change in outlook, which I believe to advance in physical conceptions, I should like to point out several ways in w hich the new phenome na fit the old framew ork better than is often realized. Take the case of the inn uence of the size of the crystals on the s harness of the dif fr acted beams, w hich we have just mentioned. o n the wave theory it is sim ply an ex ample of the factthat a di f raction gr ating with only a few lines has a poor resolving power. Double the num ber of the lines and the shar ness of the diffracted beams is doubled also. However if there are already many lines, the angular change is small. But imagine a particle acted on by the material which forms the slits of the grating, and suppose the forces such as to deflect it into one of the diffracted beams. The forces due to the material round the slits near the one through which it passes will be the m ast im portant, an increase in the number of slits will affect the moti on but the angular deflection due to addi ng successive slits will diminish as the numbers increase. The law is of a similar character, though no simple law of force would reproduce the wave ettect quantitativel Similarly for the length of the wave train. If this were limited by a shutter moving so quickly as to let only a short wave train pass through, the theory wou d requ re t hat the velocity of the particle would be uncertain over a range increasing with the s hort ness of the wave train, and correspondingscattered w av es calcul ated by the met hod I hav e i ndicat ed. The agreement is good to t he accur acy o f the experiments whic h was about 1%. Ther e is no adjustable constant, and the patterns reproduce not merely the general features of the X-ray patterns but details due to special arrangements of the crystals in the films which were known to occur from previous investigation by X-rays. Later work has amply confirmed this conclusion, and many thousands of phot ogr aphs hav e been tak en in my own and ot her labor atories without any disagreement with the theor y bei ng foun d. The accur acy has increased with the improvement of the apparatus, perhaps the most accurate work being that of v. Friesen of Uppsala who has used the method in a precision determination of e in which he reaches an accuracy of I in 1,000. Before discussing the theoretic al implic ations of t hes e res ults t her e ar e two modi fications o f the experiments w hich s houl d be mentioned. I n t he one, t he electrons a fter passing thr ough t he film are subject to a uniform m agnetic fiel d w hich defl ects them. It is found t hat the el ectrons whose impact on t he plate forms t he ri ng patt ern are de flected equally with t hos e whic h have passed through holes i n the film. Thus t he patte rn is due t o electr ons w hich have preser ved unc hanged the propert y o f bei ng defl ected by a m agnet. This distinguishes the e ffect from anyt hing produced by X - rays and shows that it is a true property of electrons. The other point is a practical one, to avoid the need for preparing the very thin films which are needed to transmit the electrons, an appar atus has been devised t o w ork by r efl ection, the el ectrons striki ng t he di ffr acting s urface at a sm all glancing angl e. It appears t hat in many cases the patterns so obtained are really due to electrons transmitted through small projections on the surface. In other cases, for example when the cleavage surface of a crystal is used, true reflection occurs from the Bragg planes. The theor y o f de B roglie i n t he form given t o it by Sc hr 鰀 inge r is now known as w ave mec hanics and is t he basis o f at omic physics. It has been applied to a great v ariety o f phenomena with s uccess, but owi ng lar gel y to mat hem atical difficulti es there are not many cases i n w hic h an accurate com parison is possi ble betw een t heory and experim ent. The di ffr action o f fast electr ons by crystals is by fa r t he s ever est num erical test w hich has been made and it is therefore important to see just what conclusions the excellent agreement between theory and these experiments permits us to draw. The calculations so far are identical with those in the corresponding case of the diffraction of X-rays. The only assumption made in determining the directions of the diffracted beams is that we have to deal with a train of wave of c onsiderable depth and with a plane wave-front extending over a considerable num be r of atoms. The mi nim um ext ensi on of t he wave system sideways and frontways can be found from t he s har pness of t he li nes. Taking v. Friesen's figures, it is at least 225 waves from back to front over a front of more than 200 ?each way. But the real trouble c omes w hen we c onsi der the physical meani ng of the waves. In fact, as w e hav e seen, t he electrons bl acke n t he photogr aphic pl ate at thos e pl aces w her e the waves woul d be stro ng. F ollowi ng Bohr, Born, and Schr 鰀 i nger, we can express t his by sayi ng that the i ntensity o f t he waves at any pl ace measur es the pr obability of an electr on m ani festi ng itsel f there. This vi ew is strengt hened by meas urem ents of t he r elativ e int ensities of t he rings, w hich agr ee well wit h calc ulations by Mott based on Sc hr 鰀 inge r's equati on. S uch a view, how ever s uccessful as a formal stat ement is at v ariance with all ordinary ideas. Why should a particle appear only in certain places associated with a set of waves? Why should waves produce effects only through the medium o f pa rticles? For it m ust be emphasized that in t hese experiments eac h el ectron only sensitizes t he photographic plate i n one mi nute regi on, but in that region it has the sam e pow ers o f penetr ation and photographic action as if it had neve r been diffract ed. W e cannot s uppos e that t he ene rgy is distri buted t hroughout t he w aves as in a sound or wat er wave, the wav e is only effectiv e i n the one place where t he electron appea rs. The rest of it is a kind of phantom. Once the particle has appeared the wave disappears like a dream when the sleeper wakes. Yet the motion of the electron, unli ke t hat o f a Newtonian particle, is i nfluenced by what happens ov er t he w hol e front of the wave, as is shown by t he e ffect of t he size of t he crystals on t he sharpness of the patter ns. The di fference in point o f view is fundam ental, and we have to face a break wit h ordi nar y m echanical ideas. Particles have not a unique track, the energy in these waves is not continuously distributed, probability not determinism governs nature. But while emphasizing this fundamental change in outlook, which I believe to represent an advance in physical conceptions, I should like to point out several ways i n w hich the new phenomena fit the ol d fr amew ork better t han is o ften r ealized. Tak e t he case o f t he infl uence of the size o f t he crystals on t he s har pness of t he diffr acted beams, w hich w e hav e just mentioned. O n the wav e theor y it is sim ply an ex ampl e o f the fac t t hat a di ffraction gr ating with only a few li nes has a poor r esolvi ng power. Double the num ber o f t he li nes and the shar pness o f t he diffr acted beams is doubled also. However i f there are already many lines, the angular change is small. But imagine a particle acted on by the material which forms the slits of the grating, and suppose the forces such as to deflect it into one of the diffracted beams. The forces due to the material round the slits near the one through which it passes will be the m ost im port ant, an incr ease in the num ber o f slits will a ffect the moti on but t he angular defl ection due to addi ng successiv e slits will diminish as the numbers increase. The law is of a similar character, though no simple law of force would reproduce the wave effect quantitatively. Similarly for the length of the wave train. If this were limited by a shutter moving so quickly as to let only a short wave train pass through, the wave theor y woul d requi re t hat the v elocity of t he particl e would be uncertai n over a range i ncreasing with t he s hort ness o f t he wave tr ain, and corr esponding