scattered waves calculated by the met hod I haveindicated Theagreement is good tot he accuracy of the experiments which was about 1%. There is no ustatle constant. and the patterns reprodue not merely the general features of the x-ray pattems but detalls due to special arranoements of the adjustable constant, and the patterns reproduce not merely the general features of the X-ray pattemns but details due to special a s or crystals in the films which were known to occur from previous investigation by X-rays. Later work has amply confirmed this conclusion, and many houeande ot nhet ooranhe have heen taken in my and ot her laberatoree withot any dicacreement with the therry hel found The thousands of phot ographs have been taken in m and ot her la boratories without any dis greement with the theory bei ng foun d. 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 pre of e in which he acy of I in 1,00 Before discussing the thegretic al mplic ations of theseres utsthere are two mad ncat ns ofthe ex peri ments w hich s hould be mentioned. In the one, the electrons after passing throughthe film are subject to a uniform magnetic field which defl ects them. It is found that the el ectrons whoseimpact onthe Mate forms thering patt ern are deflected equally with those which have passed through holes in the flm. Thusthe pattem is duetoelectrons which have plate f ms ering pattern are de flected equally with thos e which assed through holes in the us the patter is d oelectrons which preserved uncha nged the propert y of being def ected by a magnet. This distinguishes theeffect from anything produced by X-rays and shows that it is ate roety electre. the other noit k a maticat to awid the need foe menarln the verv thin flms whch are needed to transmit th electrons, an appar atus has been devised to w ork by reflection the electrons strikingt he diffr acting s urface at a sm all glancing angle. It appears par atu tingsurrace appears that in many cases the patterns so obtained are really due to electrons transmitted through small projections on the surface. In other cases, for example to a great variety of phenomena with success, but owling lar gely to mat hem atical afficulties there are not mamy cases in w hich an accurate com parison ety nenom nsuccess rgely to mat hemati ere are nany cases in wh an accurate com parison is possible betw eentheory and experiment. The diffraction of fast electrons by crystals is by farthe s everest numerical test which has been made and theratre Imootant to cee tist what cocie lone the evre a ent between theory and these experiments permits The calculations sofar are identical with those in the corresponding case of the diffraction of X-rays. The only assumption made in determining the drections of the difracted heame le that we hawe to deal with a traln of wave conelderahle denth and with nlane wave. etendin owera directions of the difracted beams is that e to deal with a train of wave of considerable depth and with a plan -tront ling over a considerable num ber of atoms. The minimum extension oft he wave system sideways and frontways can be found I fromthe s har pness oft he lines. Taking Frlecend's fouree t at leact waves from back to frot ower a froot of more than 2ach way But the real trouble comes when weconsider the physical mea ning of the waves. In, as we have seen, the electrons acke nthe photographic plate at those places w rong. Following Bohr, Born, and Schr inger, we can express this by saying that the intensity of the waves at any place measures the probability of an electron m ani festing itselfthere. This view is strengt hened by meas urements ofthe relative int ensnes ings, w hichagree well withalc dations by Mott based onsSche ingrs equation. Such a view, how ever sucessful as formal statement satvanlance with all ordnan ldess why shoud atile in cetain daces aseclated with of waves why should wavee odure efecte with all ordinary ideas. Why should a particle appear only in certain places associated with a set of waves? why should waves eftects only through the medium of particles? For it must be emphasized that in theseexperiments each electron only sensitizes the photographic plate inone minute he in that redlonit the reg of pe he energy is distri buted throughout the w aves as in a sound or wat er wave, the wave is only effective in the one place where the electron appears. The rest w, the wave is nroug aves as a sound of it is kind of Once the particle has appeared the wave disappears like dream when the sleeper wakes. Yet the motion of the electron, ulike that ofa Newtonian particle, is infuenced by what he ppers ov er the whole front of the wave, ashown by the effct of the size of the crystals in point of view is fundamental, and we have to face a break wit h ordi nary m echanical ideas. Particles have not a unique track, the energy in these waves is not continuously distributed. probebility not determinism governs nature. 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 aueral wave inwhich the hencmen the old on the s harpness of the diffr acted beams, which we have just mentioned. On the wave theory it is sim ply an example of the factthat a diff ractiongr atin themhertth the chmecetthe dutteted heame douhled ingp Double the num ber of the lines and the shar pness ofthe diffracted b is doubled also. However if there are already many lines, the angular change is small. But imagine a perticle acted on by the material which forms the slits of the grating, and sunse the forces surh asto deflect it into one of the difracted beams. The forces due to the material round the slits near the oe throuh whlch t e of the diffracted beams. The forces due to the material round the slits through which it sses will be the most im partant, an increase in the num ber ofslits will affect the motion but the angular ection due to add ing successive slits wil passe e most im portant, an rease in the num ber he motion but e angular der e dding successives ts 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. Similarty for the length of the wave train It this were limited by a shutter mowving so quickly as to let only a shart wave train pass through, the wave Similar or the gth of the wave tra n. this e limited by a shutter kly as to let only as ive train pass n, the w theory woud requ rethat the velocity ofthe particle would be uncertain over a range increasing with the s hort ness ofthe 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