:c-pm ^^ ^a-:/ f^'ji'xi~.f] : (1 Ä t^ ft?? ^ ^ m, ^ m ^t mr » f3 3ê ft Jt THE JOURNAL OF THE COLLEGE OF SCIENCE IMPERIAL UNIVERSITY OF TOKYO. Vol. XLI. PUBLISHED BY THE UNIVEESITY. TOKYO. JAPAN. 1917—1921. TAISHO 6-10. Publishing Committee, Prof. S. Goto, Bigakuhakushi, Director of the College (ex officio). Prof. I. Ijima, Ph. D., Bigakuhakushi. Prof. F. Ömori, Bigakuhakushi. Prof. S. Watasé, Ph. D., Bigakuhakushi. CONTENTS. Art. 1. T. Terada, M. ]\iuti and J. Tukamoto : — On diural variation of barometric pressure. Publ. November 20th, 1917. Art. 2. Y. Shibata and T. Muraki : — Mesotomisation of diammine- dinitro-oxalo-cobalt complex and determination of the configura- tions of this complex and of diammine-tetranitro-cobalt complex. Publ. November 30th, 1917. Art. 3. K. Hirayama : — Researches on the distrib ition of the mean motions of the asteroids. WifJi 1 plate. Publ. March 30th, 1918. Art. 4. M. Kuntieda : — Asymptotic formulae for oscillating Dirichlet's integrals and coefificients of power series. Publ. April 30th, 1919. Art. 5. T. Terada, M. Ishimoto and M. Imamura : — On the effect of topography on the precipitation in Japan. Publ. June 13th, 1919. Art. 6. Y. Shibata : — Recherches sur les spectres d'absorption des ammine-complexes métalliques. III. Spectres d'absorption des sels complexes de nickel, de chrome et de cuivre. Avec 18 figures. Publ. March 20th, 1920. Art. 7. K. Yamada : — Magnetic separations of the lines of iron, nickel and zinc in different fields. With 20 plates. Publ. February 28th, 1921. Art. 8. Y. Takahashi :--Magnetic separations of iron lines in different fields. With l3 plates. Publ. February 28tb, 1921. Art. 9. T. Takagi :— Ueber eine Theorie des relativ Abel'schen Zahl- körpers. Publ. July 3 1st, 1920. Art. 10. K. Matsuno : — On the stereochemical configuration of the aquotriammine and diammine cobalt complex salts. Publ. March 31st, 1921. Art. 11. K. Matsuno : — The coagulation of arsenious sulphide sol by cobaltic complexes. Publ. March 31st, 1921. PRINTED BY THE TOKYO PRIN 1 ING CO., LTD. ~x^-^'^ JOUllNAL OF THE COLLEGE OF SClENeF, TuKYO IMPERIAL UNIVERSITY. VOL. XLT., ART. 1. On Diurnal Variation of Barometric Pressure. (Contribution II. from the Geophysical Seminary in the Physical Institute, Co!lege of Science}. By Torahiko TeRADA, Eg ikiihakusJii, Masazô KlUTl, , Jyun TUKAMOTO, Eigakmh f, Eigakushi. 1. While the amplitudes and phases of the semidiurnal wave of barometric pressure show a very regular distribution over the entire surface of the earth, those of the diurnal component depend remarkably on secondary local conditions, as was fully illustrated by the classical investigations of A. Angot'^ and J. Hann.^^ According to the recent aerological investigations/^ the diurnal component of the daily variation of temperature is conspicuous chiefly in the lowest kilometer of the atmosphere, and it seems quite natural that the corresponding diurnal component of the barometric pressure is influenced by the variety of the nature of the underlying earth's surface within comparatively narrow extent. Imagine for a moment the picture of the isobaric surface at about 2 km, over a land with irregular patches of water, desert etc., while the earth's surface is rapidly heated up by the solar radia- tion. The isobaric surface will be scattered over with numerous hills and dales, according to the nature of the substratum, and the resulting horizontal gradient of pressure cannot subsist without the flow of air tending to annul the gradient. When the heating 1.) A. Angot, Annales du Bureau Central Météorologique de France, 1887. 2.) J. Hann, Denkschriften d. kais. Akad. d. Wiss., uiath.-naturwis?. Kl., 55, pp. 49-121. 3.) Eeger, Arbeiten d. kon. preuss. aeronautischen Observatoriums bei Lindenberg, 8, p. 229. 2 Art. l.-T. Terîlda: is sufficiently rapid, the state can be maintained, since the flow is partly hindered by the friction of the underlying layer and also the deviating influence of the Coriolis's force. The irregularity of small scale due to the influence of the immediate neighbourhood will be gradually smoothed down as we proceed higher and higher, and the influence of the remoter substratum will gradually come into play. The above idea is not at all essentially new, being entertained by the authorities such as Hann and Angot among others.'^ It seems indeed otherwise impossible to account -for the irregular nature of the geographical distribution of the diurnal components as actually observed. As far as we are aware, there were however as yet no serious attempt made to consider these effects of local conditions a little more closely and to deduce anything in way of finding some rules or laws from among the apparently intractable chaos of materials. The present communication is the results of some trials dared in tliis direction. Though far from laying an^^ serious claim on the rigorousness of the method employed, nor on the exhaustiveness of the materials utilized, the essential features of some of the results given below may be of some interests for meteorologists. 2. The beginning of the present investigation dates back to several years ago when the attention of the one of the author was drawn })y the simple chart constructed by Buchan'\ showing roughly the geographical distribution of the daily amplitude of barometric pressure over the world. The dependency of the ampli- tude on the distriljution of land and water was so conspicuous that it seemed quite feasible to infer some quantitative relation between the amplitude and the proportion of land and water over a certain definite area. To carry out the comparison, the following procedure was taken. The amplitudes of the pressure variation at different points on the circles of latitude 20° and 40° respective!}' were estimated from the chart by interpolation and plotted in a diagram 1.) e. (/. Bürnstoin, Wiener Borichto, 113, 2a, 1904, p. 721 ; Met. Z.S., 7, p. 837. 2.) Buchan, Challen>fer Report, Phys. and Clieui. 2, Report on Atmospheric Circulation, p. 21, Fii,'. 2. On Diurual Variation < f Barountric Pressure. 3 with the longitudes as abscissa. On the other hand, the percentage of land area included within an area with the points in question Diurnal Component and Continentality. Fig. 1. Latitude -. O'-IO' 0 6 0-4- 0-2 1. Qnixeramobim. 2. Payta. 3. Borna. 4. S. Paul de Loan da, 5. Kibwezi. 6. Angola. 7. Gabun. 8. Batavia. 9. Quito. 10. Nauru. 11. Singapore. 12. Zanzibar. 13. Trevendrum. 14. Christiansborg. 15. Jaluit. 16. Indian and Pacific Ocean v8.'i''X.) 17. Bay of Bengal. 18. Ascension. 19. Pac. Ocean (GA'S.) 20. Costa Rica. 21. Atlantic Ocean (0--5'N.) 22. Atlantic Ocean (S'-IO^V.) Fig. 2. Latitude: 10''-20=' ov ? 1-2 V" i 1 y \ 0. -8 / / '- ß 5 ^ / 7 / 10 0'6 «,12 \ / / X 1 i X ; ■ 'US / 18/ ir X 0-^ •19 0 ()•- 1. Boroma. 2. Kartum. 3. Cuyaba. 4. Joal. 5. Mexico. 6. Acapulco. 7. Port Darwin, 8. Caetete. 9. Madras. 10. Puno. 0-6 0-B 10 K 11. Manila. 12. Port au Prince. 13. Bombay. 14. Kingston. 15. Pac. Ocean (16.3'S. 16. Samoa. 17. Dodabetta. 18. Tahiti. 19. St. Helena. Art, l.-T. Terada: Diurnal Component and Continentalily (Conthmedj. Fig. 3. Latitude : 20''-3ö° 1. Goalpara. 2. Patna. 3. Allahabad. 4. Asuncion. 5. Kimberley. 6. Calcutta. 7. San Paolo. 8. Galveston. 9. New Orleans. 10. Hazaribagh. 11. Hongkong. 12. Taihoku. 13. Mauritius. 14. Habana. 15. Mangewa I. Fig. 4. Latitude: 3ö°-40° 0-8 0-6 0-2 1. Cordoba. 2. Yarkand. 3. Leh. 4. Eocky. Mt. 5. Peking. 6. Eosario. 7. Memphis. 8. Yuma. P. Kairo. 10. St. Louis. 11. Cincinnati. 12. San Francisco. 13. San Diego. 14. Savannah. 15. Washington. 16. Hankow. 17. Philadelphia. 18. Dodge City. 19. Tokyo. 20. Denver. 21. Santa Fé, 22. Zikawei. 23. Melbourn. 24. San Fernando. 25. Pacific Ocean (33.3''6.) 26. Santiago. 27. Cape Town. 28. Lisbon. 29. Azores. 30. Cape North- umberland. at tlie centres and l;)oundu(l by two circles of latitude and two meri- dians, each 20° or 40' apart. Plotting the values of the percentage on the same diagram in a proper scale, a remarkable parallelism wag" observed, especially in the case of the land percentage in 40° Oil Diiirnal Variation of Barometric Pressure. Diurnal Compaaent and Continentality fContinuelJ. Fig. 5. Latitude : 40°-50' 0-ß 0-2 1. Tiflis. 2. St. Paul. 3. Madrid. 4. New York. 5. Toronto. 6. Duluth. 7. Cleveland. 8. Chicago. 9. Salt Lake City. 10. Sapporo. 11. Portland. 12. Milano. 13. Bucharest. 14. Xukuss. 15. Alpena. 16. Halifax. 17. Boston. 18. Gl neve. 19. Kremsmünster. 20. Bismarck. 21. "Winnipeg. 22. Wien. 23. Hobarton. 21. Albany. 25. Esquimault. 26. Paris. 27. Coimbra. 28. Lésina. 29. Triest. 30. St. Martin de Hinx. 31. München. 32. Pola. 33. Napoli. Fig. 6. Latitude : SO^-öO" 1. Irkutsk. 2. Prag-. 3. Leipzig. 4. Magdeburg. 5. Greenwich. 6. South Georgia I. 7. Upsala. 8. Bruxelles. 9. Petrograd. 10. Moscou. 11. Dorpat. square. Moreover it was clearly shown that the influence of the land distribution was remarkably less for the latitude 40° than for 20.° The result was communicated in a meeting of "Meteorologi- sches Colloquium" in the Meteorological Institute of Berlin. The publication was however refrained, since the Buchan's chart was far from ]:>eing up to date and moreover the amplitudes refers to the total amount of variation, but not to the diurnal component. Recently the subject wa^ resumed for a more detailed investigation, (5 Art. 1.— T. Terada: choosing as the materials those compiled by Angot and Hann in their classical papers above cited and also those given in the current numbers of the "Meteorologische Zeitschrift", from which the diurnal amplitudes «, were taken and their dependency on the distribution of land and Avater in a definite area enclosing the stations was to be examined. A circle with the radius of 10° was drawn, with each station in question a^ the centre, on a suitable globe, and the percentage of land area was determined by means of planimeter. The value of the percentage was called, the "continent- ality" of the station for simplicity's sake and denoted by K. It may be remarked that the area is roughly of the same order of magnitude as Australia. Different stations were then classified into groups according to the 10° zones of latitude to which the}^ belong. For each group, a diagram was constructed in which tlie values of «1 Avere plotted with the corresponding continentality as abscissa (Figs. 1-6). The points thus obtained, representing different stations, though rather capriciously scattered over the diagram, showed still an undeniable tendency to be arranged within a certain belt inclined to the axis of the continentality. Moreover, the in- clination of the belt to the axis of abscissa seems to become less as the latitude increases. The result shows at least that among the numerous factors determining «i, the continentality as defined above may play not quite an unimportant rôle. To proceed a little further, a straight line was drawn in each diagram, roughly representing the median line of the belt supposed. The ensemble (jf such straight lines for different latitudes taken together was adjusted among themselves, so as to show a regular transition according to latitude. From lines reduced (the full lines in Figs. ]-(*») a diagram was constructed in which the ordinate represents «i and the abscissa the latitude, and the system of curves Avas drawn representing the dependency of a^ on the latitude for different values of the continentality. It was found that the curves may be roughly repi-esented by an empirical formula of the form «/ = (a + 67i )cos Y — c, a=0.35, 6=0.80. c^0.15 in mm., Ou Diurnal Variation of Barometric Pressuro. 'J wiiero 'apoli -.323 Greenwich - .144 Coimlra -.148 Leipzig - .18(5 Lisbon -.2D3 Bruxelles -.235 B.'Ssekop + .009 Praj .133 Kla^vnfurt + .175 Paris -.133 Bozen + .5 47 Wien - .183 Tiflis + .113 München .253 Ma.lrid + .074 - The.se negative discrepancies are associated with the small values of the daily temperature variation, as will be shown in subsequent paragraplis. The abnormal values for Bozen and Klagenfurt are evidently due to the altitude of the stations. In Northern America, negative J prevails in the inland, while positive values are frequent in the coastal stations : Taiîi.i: II. Positive J. Negative J. Eastport .09G New Orleans .001 New York .115 Galveston -.027 Philalelphia .051 Sitka ■ .142 Washington .033 ^• Savannah .095 o Port au Prince .2Ü8 Kingston P. r; land San Francisco San Diego Yuuia .071 .038 .114 .139 . .171 Ou Diurnal Variation of Barometric Pressure. Positive J. Xegat ive J. St. Paul .008 Winnipeg -.078 Cincinnati .lOà Chioage -.134 Memphis .095 St. Louis Bismarck Saltlake City Denver Dodge City Santa Fé -.025 -.120 -.205 --.lui -.153 -.213 The same tendency is suspected also in South America, though the data availed of were too scanty. For Asian continental stations, the data are still more wanting; Nertinsk and Tomsk give negative, hut Irkutsk a positive J. While Japanese stations and also Peking give positive J, the Chinese stations Hankow and Shanghai as well as Hongkong and Tonkin show negative values. Again, it is interesting to ohserve that the positive anomaly prevails in the Pacific Ocean, while the negative values seem to he frequent in the Atlantic, though for the latter the data are scanty: Tabli: III. Pacific Stations Atlantic Stations Tahiti + .221 Ascension + .0S3 Mangarewa + .057 St. Helena -.019 Jaluit + .143 Ocean 5" — lO-^N. -.056 Nauru + .310 0"- 5°X. ■ .033 Manila + .2;3 Samoa + .2-Ö Batavia + .221 Ocean 6.1°S. + .067 „ 10.3° + .117 „ 33 3° + .071- From the ahove examples, we may realize that there remains still some important local factors determining the amount of the diurnal component, heside the "continuality'" a-^ artificially defined 10 Art. l.-T. Terada: above. The above discrepancies are quite systematic in character and cannot be generally accounted for by the local topography within a small extent. The last comparison of the two oceans, suggests a considerable influence of the continent l.ying far beyond tlie limit of the 10° circle here adopted for the determination of K. 3. The very unsatisfactory results of the above attempt to account for the variety of «i by the simple consideration of the "continentality", led us to consider the matter more closely under the light of some elementary theoretical considerations.'^ Take first of all, for simplicity's sake, only the two-dimensional problem, in which the earth's surface considered as plane is taken for the :ry-plane, and the isobaric as well as the isothermal lines are all straight and parallel to the y-axis. Supjiose now the isobaric surface originally plane at the heiglit z, Ije swelled up by C(^» ^) on account of the heating of tlie underlying strata. The horizontal pressure gradient at the level z will Ije Qi» '^^- . If we may assume the steady state soon established, this will be equilibriated by the friction, i.e. where u is the horizontal velocity toAvard x and /^ the coefficient of friction.'^ Now, C may be roughly given ]:)y wliere 6 is (he mean absolute temperature at z and assumed in- dependent of X, and d is the variable part of the temperature which is to be considered as a function of both x and z, and also of time L The above holds only on the assumption tliat the air columns are kept b}^ vertical partition walls from expanding horizontally: l»ut 1.) Prof, Okatla kiuilly dr.w our attention to a paper by Hill, Met. Zs, 5, 1888. p. 340, on the annual variation of barouiotri«.' pressure in ludiii, in wliicli a similar idea as fallowed in this section can be traced. 2.1 The influence of the rotation of earth is here provisionally put out of account. The effect will be referred to l.iti r. On Diurnal Variation of Birometric Pressure. 21 when tlie lieating is effected sufficiently rapid, we may take (2) as a rougli approximation. Next, let us assume that d is of tlie form where dX^) is ^^^^ variation of temperature at sea-level and is a function of x and t only, while A^^) is the function of z only. Tlie latter assumption amounts to saying that the variation of tempera- ture at z level is proportional to tlie variation at sea-level. Then we have To simplify the matter, we idealize the case still further and assume that the elevation C is nearly constant for a siuall finite altitude near the surface. Then, if we are considering only the initial stage of the motion where the pressure gradient near the earth' s surface is still insignificant, we may put u=o^ -^ — =o at z=o. Hence we have _g dd. tjyj?'d'M''- ^'^ In this integral, p{z) may for the present purpose be considered to ])e the mean value independent of the variable part of the temper- ature. It may therefore be regarded for the rough approximation, as the function of ,~ onl}^, say -^^), ""^'^ = IaO-'^IM''- ^'^ Then the total fiow of mass througli tlie infinite vertical plane parallel to yz, is given by U = l^updz = -^^---?^f^'oF(z)dz. (5) Jo /J- dxJo ^ The latter integral is a constant and may be denoted by /. The rate of cliange of the barometic pressure p at sea-level will be ap- proximately given by ]2 Art. 1.— T. Terada: dp ^ _ }'U ^ f^ ^% dt ^ dx ~ (l dx' ' (^^ The aljove holds of course only for the highly idealized case con- sidered; but, if the total amount of Uhe small and the variation of w sufficiently rapid, the essential feature of the yariation of pressure may roughl}^ be represented by (G), provided that the stationary state expressed by (1) is established instantaneously. In the above calculation, the effect of the poriolis's force was entirely put out of account. The comi^onent flow in y-direction due to this influence will become considerable compared Avith ii, especially for high latitudes and in high levels. Plowever, as long as the assumption of parallel isobars and the uniform deviating force is adhered, this component will bring no essential modifica- tion to the form of the expression (G), except the value of the constant factor. In applying the above theory to the actual problems, however, the nature of the abstractions made in the assumption must always be borne in mind. Again, in the above, we have tacitly neglected the influence of the topography in affecting the thickness of the air layer subject- ed to the daily yariation of temperature. In actual cases, ç will be generally diminished with the elevation of the earth's surface, if we provisionally assume that the temperature reduced to the sea- level is everywhere uniform and the variation takes place as in the case there were no elevation of land. In this case we must replace in the al)Ove equations l)y y . This makes ~~f. proportional to ^■^„ , j^ ^ dd^ Jh ^ dx' dx dx where A(.t) represent the elevation of the surface as a function of x. Ivemembering the nature of the assumptions made at the c-ut- set, let us simply take the form dp ^ ^ dti„ ^. ^ ^^^^,^^ • . ^,3,^ dt dx' ' ^ as at least (iiialitntively legitimate, neither entering upon the form On Diarnnl Variation of Earometic Pressure. 13 of ß(z) and thence the vakie of /, nor upon the influence of h(;x) wliicli we assume small, and carry through the application a little further on some appropriate actual problems, to see how far the formula may be utilized for the explanation of the phenomena in question. __ 4. Suppose a temperature wave proceeding toward the posi- tive direction of x, West say, over the earth's surface idealized as in the preceding paragraph, such that O-r 6, = aco^-^{t—^ + =2.t, we obtain Now suppose for very rougli approximation that the part of Western Europe, the Atlantic Ocean and Northern America can be compared with the ideal arrangement of land and water as above considered, if we take the meridians 0°, 60°W. and 120°VV. as the ideal coast lines, each strip of land or water having a breadth of G0° in longitude. The comparison ma}^ in some measure be justified, if Ave apply the above formula only to the narrow belt On Diurual Variation of Baromotric rrossure. 15 along the European coast. Take the origin of x at 30 E^., corres- pondmg to the assumption that tlie aniphtude of tlie temperature variation is minimum at tlie centre of Atlantic Ocean. Tn (12), we assume a=5°C. and 1=-^-. Then , \ f I >. \- i-x , , Carrying out the numerial calculation for each ten degree of longi- tude, we obtained the second line of the following Table, in which the observed value of «i, as the mean values for different stations included within successive strips with the breadth of 10^ are given in the third line. In the fourth line, the number of stations taken for evaluating the third line is given. Table IV. Longitude. 20° W. 10° w. o°w. 10°E. 2ö°E. 30°E. Calc. 2.71 2.18 2.03 2.65 3.3« 3.67 Mean a^. .227 .159 .245 .317 .203 Xo. of Stations. 4 1.5 21 1Ü 3 Some parallelism is to be observed between the second and the third lines, if we put range 20°-30°E. out of account, in which the number of stations is scant3\ G. For the next trial, we chose all stations falling within the zone of latitude 20°-60°N. and the mean value of the observed a, were calculated for successive areas with the breadth of 20' in longitude. To compare this with the result to be expected from the present theory, it will be most natural to take for « the mean value of the amplitude of daily temperature ^-ariation for each area ■J J'o mentioned and estimate the coefficients -^ and -^- from these 2(3 Art. 1. - T, Terada: -data. Since the necessary materials were, however, not at our disposal, we were compelled to resort to a provisory assumption : That the mean a for a given zone of latitude as the function of lonyitbde is proportional to the mean annual amplitude of the temperature for the same zone. The latter assumption may appear at first sight utterly un- justifiable, since the daily amplitude generally decreases with latitude, whereas the annual amplitude increases with the distance from the equator. If we, however, confine our attention to a given narrow belt of latitude and consider the dependency of the ampli- tude on the longitude, it will be rather plausible to assume for the purpose of rough approximation, as is aimed at throughout the present investigation, a parallelism between the two amplitudes, since both depend on the nature of the earth's surface in a similar way, in spite of the difference in the periods of the periodic heating and cooling.'^ ]) Avhen Ave consider the average daily fluctuation of temperature, we may consider the sun at the equator. Then the solar radiation per unit area of the surface will be proportion- al to cos , where (f is the latitude. The temperature amplitude of the earth's surface depends not only on the intensity of the radiation, but largely on the nature of the surface. Assume it to be proportional to F(X). Then the daily amplitude will be roughly of the form Fcos .. Now, consider F and F' expanded by Foi.ritr's series of the form CO CO y A„j sin //(). + ^ B,;, cos ih\. It is evident that A'„, and B'„, for F' converge more rapidly than the corresponding coelH- cients ft r F, since the daily variation of temj: eraturi\ being greater for the lower layer of atmosphere, will be influenced conspicuously by the minor irregularities of the surface, while the annual variation extending to a higher strata, will depend on the general condition of the wide area surrounding the station in question. Taking he we ver the mean value of the daily amplitude over a wide area, the terms depending on the large values of m will tend to cancel each other, and only the first few terms will remain as the leading terms. Our as- sumption actually amounts to assuming the proptrtionality of A„,, B,,, with A'„., B'„j for the small values of w, which will be in any case not very far from the truth. On Diurnal Variation of Barometric Preesure. ^'J The data used were the small chart giving the ^ lines of equal annual temperature variation" in Berghaus' s Physikalisches Atlas, III. Abt., Meteorologie, No. I., from which the mean annual amplitude for the zone of latitude 20° — 60° N. for every ten degrees of longitude was determined, from the mean values for the three latitudes 20°, 40° and 60°N. at the corresponding longitudes. The value a' thus obtained was plotted in a diagram with the longitude X as abscissa and the resulting curve was smoothed down by taking for each thé mean values of a"s at a; -20°, x- 10°, x, a;+10° and da' a: + 20°. From the smoothed curve thus obtained, the value of -5— was determined for every ten degrees of longitude by drawing the tangent at the corresponding point of the curve. The resulting da' curve for --j— was again smoothed down in a similar manner and d^a' then -j-ô- was determined^\ The values of the differential co- dx" efficients were substituted for -^ — and -r-r in (10) and a?^ and ^^ calculated for each 20° of longitude. The result is shown in the following table and also plotted in Fig. 8. , in which the calculated value proportional to a/ is given in an arbitrary scale. The agree- ment of the calculated and observed curves are rather remarkable, if we remember the nature of the assumptions made in the deriva- tion of the formula (10). 1) It may be remarked that the values of the differential coefficients thus obtained are actually given by dx' 1 / \ d2ix' 1 / \ dx^ 4didj ^ ^ - 1 -' ' - 1 J ' as far as we consider the part of the curve to be smoothed may be regarded as linear within the range 2fZi or 2^2- In our case 2di and 2i.j are equal to 50°. 2) Of course we may obtain only the value proportional to ax. 18 Art. 1.— T. Terada: Fig. 8. Observed and calculated values of the amplitude aj and the phase ^i of the diurnal component, as depending on the longitude. • • • observed. — X — X — X — calculated. S^l 360 180 V / m- -y^ jif^ y ^ .-•- ^ ;* Sfr /' ^' '-•— ^ •M- -»e ^ ^t^ / z' Iv y / / / 1 Ol ai 1 — 0-6 '*V *^ / \ \fr \ 0-5 /y r^ K \ 7 y / ; \ \ 1 0-4' j / \ V 1 : / \ \ ij > .\ J f V 0-3 jj V i y" '¥- m- > \ J \ f V f ^ » > 0-2 1 \ J ' •, ^ j 5^ J \ ^ /. ' •' \ Ol ^^ À f \^ '' t 0 . V 180W 180E On Diurnal Variation of Barometric Pressure. 19 Table V. Observed and calculated values of a^ and ^, as the function of the longitude. Longitude. Stations. Mean Mean Calc. Value ppl. «1. Calc. 180°— 160°E. - — 230 335° 160°— 140°E. Sapporo. 0.301 313° 394 301° 140° — 120°E. Tokyo Zikawei. 0.352 285° 528 272° 120°— 100°E. Irkutsk, Nertschinsk, Peking, Hong- kong, Tonkin. 0.519 284° 562 247° 100° — 80° E. Barnaul, Goalpara, Patna, Allahabad, Hazaribagh, Calcutta. 0.614 246° 452 229° 80° — 60° E. Ekaterinburg, Yarkand, Leh, Simla. 0.524 265° 347 219° 60° — 40°E. Nukuss, Tiflis. 0.465 265° 329 215° 40° — 20°E. Petrograd, Moscow, Tarnopol, Bucha- rest, Athen, Cairo. 0.199 261° 334 210° 20° — 0°E. Christiania, Upsala, Berlin, Magde- burg, Utrecht, Leipzig, Bruxelles, Prag, Paris, "Wien,:München, Krems- miinster, Budapest, Klagenfurt, Bozen, GenèTe, Aosta, Milano, Triest, Pavia, Pola, Lésina, Pelagosa, Napoli, Etna. 0.239 251° 307 155° 0°-20°W. Makerstown, Dublin, Oxford, Green- wich, Jersey Island, St. Martin de Hinx, Madrid, Coimbra, Lisbon, San Fernando. 0.121 220° 197 142° 20°— 40°W. Ponta Delgada. 0.046 92° 157 59° 40°-60°W. 241 334° 60°- 80°W. Eastport, Halifax, Toronto, Albany, Boston, New York, Philadelphia, Washington. 0.348 270° 407 267° 80°- 100° W, Winnipeg, Duluth, Alpena, St. Paul, Chicago, Cleveland, Cincinnati, St. Louis, Memphis, Savannah, New Orleans, Galveston, Havana. 0.418 250° 548 249° 100°— 120°W. Assiniboine, Bismarck, Salt Lake City, Denver, Pikes Peak, Colorado Spring, Rocky Mt., Dodge City, Santa Fe, Yuma, San Diego. 0.563 257° 493 218' 120''-140°W. Sitka, Esquimault, Portland, San Francisco. 0.300 213° 284 229° 140°— 160° W. - - 138 111° 160°-180°W. — — 127 23° • It must be remarked that a difficulty is often met with in taking the mean value of ^i, since the actual values for different stations in the same group vary widely and it becomes in such cases uncertain whether 'R, we have always IC=1. As far as the qualitative verifi- cation of our theory for a stations not very far from the coast is concerned, we must expect that the observed values of «i considered as the function of K will have a minimum near the coast, or for A''=0.5, and two maxima on both sides. To test the point, the data for 33 North American stations were chosen^\ The conti- nentality for each station was evaluated and compared with the corresponding amplitude au Grouping these according to the magnitudes of K, and taking the mean value for each group, we obtained the following result : Table VII. Range of K. 0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1.0 Mean K. 0.132 0.288 0.526 0.682 0.907 Mean di. 0.436 0.270 0.356 0.500 0.400 Number of Stations 2 2 12 5 13 In Fig. 9. the values of mean ai are plotted, with the mean K as abscissa^\ The result is qualitatively in accordance with the theory, in so far as there is a minimum of a^ near ^"=0.3 or 0.4 and two apparent maxima on both side of it. If the assumption (20) were actually the case, the values of da d^a and — - would be practically zero for greater value of x and dx dx the amplitude m in the interior of a large continent would be the same as in the ocean. In the actual case, this latter is never the 1) Hann, loc. cit.; also Met. Zs., 1899, p. 421. 2) The curve («i, K) may be considered as transformed from the curve (ai, x) by suitably varying the scale of the abscissa. The value of a^ for 7i =0 is taken from the mean value for Atlantic Ocean 0'-10°N. 24 Art. 1.— T. Terada: case, «1 increasing generally towards the interior of the continent as was evidently shown in the first part of the present communi- cation. Tlie I'eason is that the Fig. 9. 0-6 0-5 0-4- 0-3 0-2 01 fi -f>- >' 0-2 O-* 0-6 0-8 »1 : mean diurnal amplitude. K : continentality. assumption (20) roughly re- present the course of a only near the coast, while for the interior of the continent, there must be added another term on the right side of (20), which is to re- present the gradual increase of a towards inside, for an example such linear term as in (11). For the present, we are still at a loss, on account of the want of necessary data, how to determine the best expression of a appli- cable for the majority of cases. In actual cases, a will depend not only on the distance of the station from sea, but on many for examples, cloudiness, prevailing winds 1-0 R other conditions, etc., etc. 9. It must be remarked that the value defined by us as the continentality is a quantity of very arbitrary character. We could have better chosen the diameter of the circular area smaller or greater than 10°. Probably there may be found a more adequate magnitude of the radius for which the relation between the continentality Ä'and the amplitude ai may turn out more con- spicuous. Remembering that the irregularity in the daily variation of temperature must gradually disappear with the altitude, and that the higher the layer, the more conspicuous relatively will become the influence of the remoter area of the earth's surface, it will be more appropriate to consider a determined by a quantity, A say, instead of K, which is defined by On Diurnal Variation of Barometric Pressure. 25 A=Lt ^ftçi(r)dr, (23) B large ^-^ "g where L, is the area of land in a circular belt with the radius r and the breadth dr. 0(r) is to be considered as a function of r which continuously tends to zero for the large value of r. Let 0 be prac- tically zero for r>i?o, then ^ = -Î- fEr 0 (r) dr = M?M.fL, dr (24) where ^ is a fraction between 0 and 1. But the factor of 0(a, \ a^ \ may have two maxima at 0 and t: and two zero between them. Taking the equation for the general value oi p, compare two special points x=^0 and x=I/2, for which the amplitudes are a^ and a/ respectively. Then a 1— ^(i^' + l) a Now consider 6 as a periodic function of season, and put 6 = 6(1 + 6i cos 7i{t + (f), then we have 1 + io_(y + 1) + A_(p2 + 1) cos n(^ + ip) l_A.(y+l) — ^(/+l)cosw(^ + ^) a a a) If both bo(p^+l)/a and bj{p^ + l)/a be small compared with unity, On Diurnal Variation of Barometric Pressure. 27 and «1 and a/ will each attain one maximum and minimum during the course of a year. Moreover, the maximum of «j will cor- respond to the minimum of a^. b) If l-^{p^+l)^^{p^ + l), hxxi I + -^-^ if +l)>-^{f+l), a, a a a a will have one maximum and a/ two maxima. c) If l-_^(^2+i)<:i(p2+l), and 1 + A(_pHl)<-^ (i^'+l) , a, a a a a as well as a/ will have two maxima. The position of zero will vary wàth the value of a, b,p. Again consider the case when a varies only in the direction of meridian. Putting in § 7, (18) a=c + d cos- — y, (25) L we have a, oc c + d |l-(^ Y } cos -^ ?/ . (26) When we consider d an a periodic function of seasons, it will be evident that the annual course will be inverted at the two stations apart L/2 in the direction of y. In order that we may apply the above to actual examples, we must of course have a detailed knowledge about the geographical and seasonal distribution of the daily variation of temperature. Since we are not yet in possession of the sufficient data, we will merely mention here some facts with regard to the actual examples to which our theory may find application, and suggest a method of investigation convenient for the purpose. We wdll choose the data for British Isles, because the seasonal variation of a^ is very irregular, apparently owdng to the rather complicated distribution of land and water in this region. The seasonal variation of a^ was taken from Angot's paper. Our first procedure was to obtain the mean seasonal variation (Fig. 10) for the ten stations, and then to obtain for each station the deviation curve (Figs. 11-18). On comparing these deviation curves among themselves, several interesting facts may be noticed: Annual Variation of the Diurnal Components a^ (Fig. 11 to 18 show the deviation, from the mean value given in Fig. 10). Fig. 10. Mean of 10 stations. ^-^ Z xZ -y^ , / ^ \IX 'kj/ . .. \/_ \_...v > ^-^ -t^ ' 1 1 ... .... 1 ■ - -^- inniTTflïowKîsai Fig. 11. Aberdeen. •10 •05 0 -05' -•10 -.15 ..'■' i ■■■■•» , /" ""■■■> J \ 9 1 / ,/ \ .-^ 1 .' ..'•■ \ / \ / i ^ \ •••"" 1 Fig. 14. Liverpool (upper) and Stonyhurst (lower). 0 -05 y '-.., n \ / '•.. /'^ ■■■ /■ ■• ■■■•■• f / 1 1. ^ 1 Â1 '■■■■ / .■ \ \ / \ / \/ \ / 1 \J '^ ■■•■"- \ ./• Fig. 1.5. Greenwich. OS ~" n 0 ,•■ ...-■■• \ / ', •./• Q5 \ 1 \ ,."«. ..''' s «' \ f 10 VI «J- Fig. 12. Oxford. 4- ? •10 / \ i_ •fl'i .•••■ . . ' "\ / Q «Ï7P - \ ■A -: i f" "03, \ — T W Fig. 13. Armagh (upper curve) and Glasgow (lower). •10 ~ 05 » O ; '. i \ '4—1 t i \ / \ f \ 1 k \ / "•■■■-, — L •0<> / ', / ■-N 0 ,..-J ' •\ / - ^ii". -- \ ,.'■ ..- ù't '■••■■' " 1 7 I S I Fig. 16. Kew. •10 0 -05 / •-.^ / *..,„ / \ i 1 ) \ / \ . J / ~' \ \ ,../ Fig. 17. Falmouth. ■ / ■\ i I \ '\ / / ■•' ■■} ; / '-^ -■:. .^'' Fig. 18. Valencia. .. •^ - J 1 \ / \ 1 .. "\. \ ! 5 ■: y. ,.-•' \ i \ ^ — On Diurnal Variation of Barometric Pressure, dû a) How much the daily variation may be affected by the local influence, may be seen by the comparison of the two curves for Kew and Greenwich. b) Deviation curves for Aberdeen and Oxford have two annual maxima and show very nearly opposite course. Falmouth and Valencia resemble Aberdeen in some measure, while the mean of Kew and Greenwich is rather of the Oxford type. This may probably be accounted for, referring to the ideal case corresponding to the equation (26) above. c) Comparing the deviation curve for Armagh and Glasgow, we observe that they are reversed to each other between February and July, while they are parallel in the interval July to September. Moreover, taking the mean of Armagh and Glasgow, it is found to be nearly parallel to Stonyhurst curve in the interval January to September and also parallel to Liverpool curve between March and September. d) Shifting the deviation curve for the mean of Kew and Greenwich two months later, the curve becomes similar to Valencia curve. e) The mean of Aberdeen and Oxford deviation curves is inverted to the mean curve of the ten stations. It will be very interesting to study the actual origin of these complicated anomalies under the light of an elementary theory as propounded in the previous paragraphs. The results may in any case be instructive for elucidating the actual mechanisms of the daily exchange of the atmosphere over a region with various geographical conditions. We hope we will be able to resume the subject when the necessary data for the temperature variations are at our disposal. In conclusion, we wish to express our best thanks to Prof. T. Okada of the Central Meteorological Observatory for many valuable informations. 30 T. Terada SUMMARY. 1. The geographical distribution of the diurnal component a^ of daily barometric change is compared with the distribution of land and water, a quantity called ' ' continentality ' ' being in- troduced which is the percentage of land in a definite area sur- rounding each station. 2. A linear relation between the amplitude and the con- tinentality was assumed and a systematic discrepancy was dis- covered. 3. An elementary theory based on a number of simplifying assumptions is proposed and applied to actual examples. 4. The variation of the amplitudes and phases as functions of longitude could be explained in its essential features. 5. The inversion of phase near the pole is explained. 6. The minimum of amplitude near coast is pointed out and explained. 7. The influence of geographical conditions on the seasonal variation of amplitude is discussed and different possibilities point- ed out. 8. A method of investigating the complicated variety of the seasonal variation under the light of the above theory is suggested, referring to some examples. Published. November 20Lh, 1917. JOUBNAL OF THE COLLEuK OF SCIKNCK, TOKYn IMTEKIAL UNIVERSITY. VOL XLI., ART. 2. Mesotomisation of Diamniine=dinitro=oxalo=cobalt Complex and Determination of the Configurations of this Complex and of Diammine=tetranitro=cobalt Complex. by Yuji SHI BATA, Hi'i'ihiiluilaifihi. and Toshio MARUKI, Fii'jakiushi. •'( Chemical Laboratory, Science College, Tokyo Imperial University. In a recent communication [Yuji Sbibata, Recherches sur les spectres d\ibsorption des amrnlnes- complexes métalliques I. Spectres d^absorption des solutions aqueuses des ammines- complexes cobaltiques ; Journ. Coll. Seien. Imp. Univ. Tokio, vol. XXXVII, Art. 2, 1915.] one of the authors discussed the relation between the constructions of various cobalt ammine complexes and the absorption spectra of their aqueous solutions. Among those studied, some nitrammine cobalt complexes were also included, and it was found that their absorption spectra are strongly influenced by the difference of the space positions of nitro-groups in the complex ions. Thus, if at least two nitro- groups occupy the furthest corners of a regular octahedron, which is considered to represent the space model of such metal-ammine complexes, three distinct bands appears in their absorption spectra, the absorption maxima existing respectively at about 2000, 3000 and 4000 of frequencies. Another group of nitrammine cobalt complexes, which have their nitro-groups only in the adjacent positions, shows merely two absorption bands and lacks the one in the most refrangil)le region of the spectrum. Art. 2. -Y. Shibata and T. Maruki From tliesG diversities in the optical behavier of tliese com- plexes, tlie author was able to determine the spacial arrangements of the nitro-gronps in several nitrammine cobalt complexes. Thus, the result of the investigation of the absorption spectrum of diammine-dinitro-oxalo-cobalt complex [Co(NH3)2(N.02)2C20 J ' Me' easily led him to ascertain that its two nitro-groups are in the adjacent positions (cis-position), this complex showing only two bands at a^jout 2000 and 3000 of frequencies. Now, if we could determine the space positions of tlie two ammonia molecules in this complex, the configuration of tlie salt would become quite clear, the oxalic acid-residue necessarily occupying the cis-position [A. Werner, Ann., 386, 10, 1912]. Further, the configuration of diammine-dinitro-oxalo-cobalt com- plex having once been determined, that of its mother substance, diammine-tetranitro-cobalt complex [Co(NH3)2(N02)4] ' Me', will be easily ascertained, the one being readily obtainable from the other by the action of oxalic acid [Jörgensen, Zeitsch. anorg. Chem., 11, 440, 1896]. Thus: — [Co(NH3)2(N02\]Me + C.O.Ha = [Co(NH3)2(NO,)2C20,]Me + 2N0,H Now, owing to the fact that there are two possibilities in tlie spacial arrangements of the ammonia molecules in the complex ion, w^e may express the above chemical reaction in either of the following ways : — NH, NH, NO, NO« oxalic acid NO. NO, NO., NO. NH3 (Ammonia molecules in the trans-pcsition.) NH3 (I) Mesoto Ulisation of Diammine-dinitro-oxalo-cobalt Complex. 3 ^^H, NO "NO, NH. or oxalic acid NO., > NH, NO., NO. NO., (Auiuionia molecules in the cis-position.) CoO^ (II) The clioico between these two formulae for diammine-clinitro- oxalo-cobalt comjDlex offers little difficulty, when we trake their constructions into account. As may be seen, formula (I) is con- structed quite symmetrically, which allows no possibilit}^ of the existence of enanthiomorphs, while with formula (II), the con- struction of which is obviously as^'mmetric, the resolution of the complex into optically active isomers has to be expected. Accordingly, mesotomisation of diammine-dinitro-oxalo- cobalt complex has l)een tried, with the result that both of the optical isomers have actually been obtained. Details of our pro- ceedure will be described in the experimental part, but its essential points may here be outlined. The collait complex, which was initially prepared as an ammonium salt, Avas first transformed into the corresponding barium salt. To a concentrated solution of this salt, sulphate of an alkaloid, such as brucine, strychinine or cinchonine, was added, the precipitated barium sulphate was filtered off. and the alkaloid salt of the complex, which readily ciystallised out from the filtrate, was then fractionated in the usual manner. The most easily and difficultly .soluble portions of the alkaloid complexes, which were thus obtained by repeated fractionations, were then converted into the potassium salts by the action of potassium iodide. The two potassium diammine-dinitro-oxalo- cobaltiates [Co(NH3)2(N02)2C204]K thus obtained from the two NH, 4 Art. 2.— T. Shibata and T. Maruki : extreme fractions were both proved to be opticall}' active, as had been anticipated, and the one was found to be the optical antipode of the other, having the foUowing specific rotatory power: [«]p = ±115°. The possibility of the mesotomisation of this complex shows that it is asymmetrically constructed and that, therefore, its two ammonia molecules must be in the adjacent positions, correspond- ing to the space formula (I) (see page 2). Consequently the position of the two ammonia molecules in diammine-tetranitro- cobalt complex, the mother substance, must also be adjacent. The mesotomisation of diammine-dinitro-oxalo-cobalt com- plex not onl}" affords us a means of determining its configuration in the manner above described, but it also offers us some inter- esting facts concerning tlie stereocliemistry of metal complexes. In so far as this asymmetrically constructed complex ion is found to be an anion, this is virtually the first example of a mesotomised complex anion containing a cobalt atom^'\ the large number of optically active cobalt complexes Intherto obtained bj^ A. Werner and his pupils being all of them cations. Moreover the type of the optically active anion [Co(NH3)2(N02)2C204] ' is entirely new in respect to the molecular asymmetry of such metal complexes. When A. Werner found optical activity in some cobalt- ammine complexes, he drew attention to the fact that the activity is due sometimes to the asymmetric cobalt atom, sometimes to an asymmetric structure of the complex ion as a whole. Those in which optical activity is due to the asjmimetric structure of the complex ion as a whole were further classified by him into the two types [MeAsBy (or [MeA.C']) and [MeAg]^'', calling them respectively the type of molecular asymmetry I and the type of molecular asjnnmetry II. For example, cvVdinitro-diethylene- diamine-cobalt complex [Coen2(N02)2(2)]', and carbonato-diethylene- 1) A. Werner mesotomised formerly a chromium complex containing- a complex anion [Cr(C204)sJ". [Jier., 45, 30Ü1, 1912]. 2) In these general formulae, A means a molecule of organic amine ))ases which cor- responds to two molecules of ammonia, or a bivalent acid radical, while B and C represent respectively mono-and bivalent acid radicals. Mesotomisation. of Diammine-dinitrc-oxalo-cobalt Compley. 5 diamine-cobalt complex [CoeiiaCOa]' belong to the type of mole- cular asymmetry I, while triethylene-diamine-cobalt complex [Coeiis] " and trioxalo-chrominm complex [01(0204)3] " belong to that of molecular asymmetry II. Now, it will be noticed that the optical activity of diammine- dinitro-oxalo-cobalt complex is due not to the asymmetry of the cobalt atom but to a molecular asymmetry, the central cobalt atom in this complex ion which may be expressed by the general formula [CoEsFoD^^]' being by no means asymmetric, as is the case with [Coen.NOo-Cl]', [Coena-NHs-Br]" etc [compare A. Werner, Ber., 44, L'^S" and 2445, 1911]. This new type of molecular asjamnetry to which attention has just been called is, therefore, to be in- troduced into the stereocliemistry of inorganic complex compounds and, in extension of Werner's classification, it may be called the type of molecular asymmetry III. Experimental. 1) Preparation of Barium Diammine-dinitro-oxalo-coballiale. To start with, ammonium diamrnine-tetranitro-cabaltiate [Co(NH3)-2(N02)4]NH4 was first of all prepared according to the method described by Jörgensen \^Zeitsch. anorg. Chem., 17, 476, 1898], this substance having the advantage of being easily con- verted into the l)arium salt and the barium salt being required in later operations. The large yellowisli-brown coloured crystals of the ammonium salt thus obtained were carefully purified by repeated recrystallisa- tion, and subsequent!}" converted into ammonium diammine- 'dinitro-oxalo-cobaltiate [Co(NH3)2(N02)2C204]NH4 by the action of oxalic acid upon a concentrated aqueous solution of the original salt [Jörgensen, Zeitsch. anorg. Chem., 11, 440, 1896]. The oxalo-complex thus obtained was then dissolved in wate]', and finely pulverised barium chloride was added in portions. When one molecular proportion of barium chloride was added to two molecular proportions of the complex, barium diammine- dinitro-oxalo-cobaltiate began to crystallise out from the solution Q Art. 2.— Y". Shibata and T. Maruki : ill the form of fine brownish red needles containing 3 molecules of water [Jorgensen, ibid., 445], This important salt was also repeatedly recrystallised, until the estimation of cobalt in the complex gave a quite satisfactory value. 2) Mesotomisalion of Diammine-dinitro-oxalo-coball Complex as Brucine Salt. To a saturated aqueous solution of l)arium diaramine-dinitro- oxalo-cobaltiate, which was kept at about 40°C, finely pulverised brucine sulphate (C23H2cN.04)28<>4H2, THoO was added in small portions under constant agitation. The barium sulphate formed was filtered ofï after the whole of the calculated amount of brucine sulphate was added. As the filtrate became cooler, brown needles aggregating in a radial form gradually began to appear here and there on the walls of the vessel. After allowing the solution to stand over night at the ordinary temperature, these crystals were gathered on a filter as the first fraction. The mother liquor was then kept in a vacuum over sulphuric acid, and the crystals, which separated out from it, were gathered from time to time. In this way four fractions in all were obtained. An estimation of cobalt and water with one of these fractions gave the following results: calc. for obs [Co(NH,\(NO,\,C,0,]-H (CogKeN.O,), H^O Co =8-52% Co =8-59^ H20 = 2-72 " H.,0=2-63 » The four fractions gave the following values for their specific rotations in an aqueous solution (0.3%, 10 cm): I [«]-'= -21.;3° III [«]?r=+27° IV [«]-g° =+'2y.:r The first fraction which consisted of the least soluble crystals was found to be fairly unstable wlien dissolved in water, for on Mesotoinisation of Diamuiinc-diuitrc-oxalc-cobalt Complex. 7 standing the insoluble free alkaloid gradually sei:>arated out from the solution. However, by working rapidly but cautiously we were able to effect its further fractional crystallisation and purifica- tion, and ultimatel}^ a crop of crystals which had as high a specific rotation as [a]g-5»=-70-7° •was obtained, a \'alue which has not been increased by further recrystallisation. These crystals we have designated as fraction la.^ Fractions II, III and IV, having been ascertained to show comparatively small differences in their rotatory power, were now together dissolved in the mother liquor from fraction la, and this solution was again kept in a vacuum in order to repeat the fraction- ation. Two more fractions were thus obtained, fractions Y and VI, which had the following specific rotations: V [«?i>°=0' VI [a]l°=+45° Fraction VI was once more dissolved in a small quantity of water and refractionated by partial crystallisation in a vacuum (fractions VII and VIII), Fraction VIII, which was most easily soluble in water, had a specific rotation of [«]d°= + 68-3. As this value remained unchanged even after further recrystallisations, it was concluded that the fractions la and VIII were essentially the l- brucine salt of /-and <^-diammine-dinitro-oxalo-cobalt complex respectively. Elimination of the brucine molecule from these fractions was then proceeded with. For this purpose, the example set by A. Werner, who mesotomised trioxalo-chromium complex {Cr(C204)3)]"' Me'3 (loc. cU.) as str^^chinine salt and subsequently decomposed it by means of potassium iodide in order to eliminate str3''chinine as its insoluble iodide, was followed. The operation was carried out in tlie following manner: the brucine complex was first dissolved In a small quantity of water, and to this solution a calculated amount of solid potassium iodide was then added. On vigorous agitation the hardly soluble brucine hydroiodide completely separated out from the solution, and the g: Art. 2.— Y. Shibata and T. Maruki : precipitates thus formed were immediately filtered off. In order to avoid autoracemisation, which might occur in the dissolved active complex, absolute alcohol was added to the filtrate in small portions, until a slight but permanent turbidity was established. The walls of the vessel were then strongly rubbed with platinum spatula to accelarate crystallisation, when fine brownish red needles were readily formed. These crystals were collected and well sucked on a filter, washed first with a small quantity of water and then with absolute alcohol, until they had no more a bitter, but slightly sweet taste. An estimation of cobalt, potassium and Avater in the product thus obtained from fraction la gave the following numbers, proving that the desired potassium salt of the complex was obtained: calc. for obs. [Co(NH3),(NO,)2CA]K-liH20 . Co = 17-88% Co = 17-38% K=10-96" K=11.53" H20= 645 " H,,0= 7-96 " Its specific rotation was found to be : [a]^»=-115' (0.1%, 10 cm) The salt prepared from fraction VIII in the same manner gave the following analytical and optical data: obs. tlieor. Co=17-95% Co=17.38% K = 11.59» K=11.53" 11,0= 7 17 " H20= 7.9G » Sp. rot. : [«]!,*= + 1 15° (0.1 %, 10 cm) It is evident, therefore, that as already pointed out, the least soluble fraction consisted of /-brucine-/-diammine-dinitro-oxalo- cobalt complex, while the most easily soluble fraction consisted of ./-brucine-c?-diammine-dinitro-oxalo-cobalt complex. 3) Wesotomlsation of Diammine-dinilro-oxalo-coball Complex as Sirychinine Salt. Mesotomisation of diammine-dinitro-oxalo-cobalt complex with strychinine sulphate (C'sJissNoC )2)S04H2, GHgO was next tried. Mesotomisat.ion of Diainm)no-dinitro-oxalr-coh)alt Complex. 9 The mode of proceedure was quite the same as before; but in view of the fact that the solubility of strychinine sulphate is considerably less than that of brucine sulphate a slight modification was in- troduced in taking a less concentrated solution of the barium salt of the complex and in keeping its temperature at about 50°C, instead of at 40''. The filtrate from the precipitated barium sulphate soon yielded a quantity of pale brown needles, showing that the strychinine salt of the complex is likewise much less soluble than the corresponding brucine salt. An analysis of these crystals gave the following numbers: calc. for obs. [Co(NH,)2 (NO.,),CA]-H-(C2iH,2NAVH2Ü Co = 9-60% Co=9-4:\% . H20=H.62 •• H20 = 2.88 " In the course of the fractionations of the strychinine salt of the complex, it was observed that the least soluble fraction, which contained ^strychinine-^diammine-dinitro-oxalo-cobalt complex, was, in the state of a solution, even more unstable than the cor- responding brucine salt; consequently, an accurate measurement of its rotatory power in a dilute solution could not be made. The strychinine salt after two recrystallisations was, therefore, converted into the potassium salt in the manner already described, and the specific rotation of the potassium diammine-dinitro-oxalo-cobaltiate thus obtained was determined with the following result: [a]g»=_104*' (0-\%, 10 cm) Due perhaps to the small solubility of the strychinine salt the c^-variety of the complex could not be separated even by repeated fractionations. 4) Mesolomisalion of Diammine-dinitro-oxalo-cobalt Complex as Cinchonine Salt. As the alkaloids used in the two preceeding cases were them- selves laevorotatory, the optically active cobalt complex obtained as the first fraction by using a salt of these alkaloids was also, in each case, found to be laevorotatory. With the view, therefore, 10 Art 2.— y. Shi bâta and T. Maruki. : of obtaining a d-vaiiety of the optically active complex as the first fraction, mesotomisation of diammine-dinitro-oxalo-eobalt complex was next tried with cinchonine sulphate (Ci9H22NsO)2S04H2, which is itself dextroi^Dtatory . The method of proceedure was quite the same as in the previous cases. The cinchonine salt of the complex forms fine pale brown needles which are anhydrous. The results of the analysis were calc. for obs. [Co(NH3),.(NO.A-CA]-H-(Ci^^N,0) Co= 10.04% Co = 10.40% H20= 0 " H20= 0 » At first» the cinchonine salt was separated in three fractions; the first fraction consisted of the crystals obtained directly from the solution, while the second and third fractions consisted of those obtained by partial crystallisation in a vacuum. The three fractions had the following specific rotations: I [«]g°=+88.5'' (0.2%, 10 cm) II [«]j?°=+8S° (0-25%, „ ) III [«]!,•'= -28-8° ( „ , „ ) The first two of these were then mixed together and recrystallised from water. The least soluble portion of the crystals thus obtained gave the value for its specific rotation, which remained unaltered even after further recrystallisations. This fraction was then transformed into the potassium salt. It had the following specific rotation: [«]2p^»= + 111° (0-1 %, 10 cm) On account of tlie scantiness of the material, we were unable to obtain, from the more soluble portions, the corresponding /-variet}' of a sufficiently strong rotatory power. 5) Optically Active Ammonium Diammine-di- nitro-oxalo-cobaltiate. Optically active ammonium salts of this cobalt complex Mesotomisation of Diammine-dinitro-oxalo-cobalt Complex. 11 [Co(NH3)2(N02)2C204]NH4 were prepared from /-brucine-Z-diam- mine-dinitro-oxalo-cobalt complex and c?-cinchonine-cZ-diammine- dinitro-oxalo-cobalt complex, in quite the same manner as in the case of the potassium salts, only using ammonium iodide instead of potassium iodide. The /-and tZ-ammonium salts of the complex thus obtained had the following specific rotations: [aj^»=-l07° (0-1%, 10 cm) „ = + 116- („ , „ ) These values are practically the same as those of the potassium salts. This is probably due to the fact that the difïerence between the molecular weights of the potassium- (=302) and ammonium- (=291) salts is not large enough to exert any distinct influence upon their rotatory powers, which were always measured in so diluted a solution as 0-1%, because of a fairly intense colour of the solution. Sammary. 1. Diammine-dinitro-oxalo-cobalt complex [Co(NH3)2(N02)2 C2O4] ' Me* has been resolved into the optically active isomers by fractionation of the salts of brucine, strychinine and cinchonine. 2. In the fractionation of the alkaloid salts of the complex, either the /-alkaloid-/-cobalt complex or cZ-alkaloid-tZ-cobalt com- plex was always found to separate out as the least soluble fraction, while either the Z-alkaloid-cZ-cobalt complex or c?-alkaloid-/-cobalt complex always constituted the most easily soluble fraction. 3. The specific rotations of potassium and ammonium diammine-dinitro-oxalo-cobaltiates, which have been obtainq^l by replacing the alkaloid molecules with potassium-and am- monium ions respectively, were measured, and found to have the values of ca ±115", using sodium light. 4. As a results of the mesotomisation of diammine-dinitro- oxalo-cobalt complex, the configurations of this complex and also of its mother substance, diammine-tetranitro-cobalt complex, have been made clear, the possibility of mesotomisation indicating that 12 Art. 2 —Y. Shibata. and T. Maruki : the complex ion [Co(NH2)2(N02)l'C204] ' is constructed asymmetri=- cally, which, means that the two ammonia molecules in this complex ion occupy the adjacent spacial position. As to the two nitro-groups, their position was previously determined, by one of the authors (Shibata), by a spectroscopic study. 5. The molecular asymmetry of the above mentioned complex ion, which is the first example of a mesotomised complex anion containing a cobalt atom, belongs to a new type and has been introduced into the stereochemistry of the metal complexes, calling it the type of molecular asymmetry III, in extension of Werner's classification. The senior author, Yuji Shibata, has profound sorrow in re- cording here the death of his collaborator Mr. Toshio Maruki,. Rigahushi, which occurred during the progress of the investigation recorded in this paper. At the same time, he has much pleasure in expressing his hearty thanks to Messrs. K. Matsuno, Rigakushif, and S. Mitsukuri for the great assistance they have given him in completing this work. Published November 30, 1917. JOURNAL OF TirE COLLKGK OF SCIENCE. TOKYO IMl'lUflAL ITNIVEUSITY. VOL. XL I., ART. 3. Researches on the Distribution of the Mean Motions of the Asteroids. By KiyotSugu HlRAYAMA, Bi(iakuhakiishi, Assistant Professor of Astronomy, Science College, Tokyo Imperial University. With 1 plate. In accordance with the views of some astronomers the fall of meteors, the zodiacal light and the gegenschein suggest the possi- bility of some kind of resistance to the planetary and satellite mo- tions. It may be exceedingly small for the major planets. But for small bodies like asteroids or satellites'^ it may not be entirely negligible if the interval of time be sufficiently long, say hundreds or thousands of years. Last year while at Yale University I considered the theoretical effect of a resisting medium, supposedly motionless, on the libra- tion of asteroids, and tried to explain the gaps of the asteroid dis- tribution on that hypothesis. But I did not succeed. Recently I have worked on the supposition of another kind of resistance suggested by Prof. E. W. Brown. ^^ According to this, resisting materials having the size of ordinary meteors are supposed to move around the central body in circular orbits. The result of my study seems satisfactory to explain the gaps in the first approxi- mation. I shall present the course of my investigation in the following chapters. The numerical computations throughout this investigation were duplicated by Mr. S. Terada, to whom the writer desires to express his sincere thanks. 1) For comets see § 8. 2) Sir G. Darwin seems to have had similar id(-a. See A. N. 184, p. 263. 2 Art. 3.— K. Hirayama : Chapter I. Effect on the Elliptic Motion, of Resisting IMaterials supposed to move around the Central Body in Cir- cular Orbits. 1 . I shall assume that the 7'esisting particles and the asteroid move in one and the same plane. The components of the velocities of two bodies at a common point, referred to the center of the sun and a system of fixed axes, are Asteroid Particle d?- nae dt y>y\ — < sin IV 0 dw na?V\—é^ "•^ r ""■ where ?•, w^ n, a and e are respectively the common radius vector, the true anomaly, the mean motion, the semi-major axis and the eccentricity of the asteroid and 7io, the circular mean motion of the particle. We have <) 5 9 3 rfa^7iQ-r or V- Hence the components of the relative velocity of the asteroid are ITT nae T , = sill 20 V 1— e" y^^,,^^aVr^_Ja\^,,J + ecoä2ü-Vl + ecosw Assuming that the resistance is proportional to the hth power of the relative velocity, and denoting by S and T, the components of the resistance in the direction of the radius vector and of the per- pendicular to it, we may write (2)- S=-cpV''-^V, T=-cpV"-'V, Researches on the Distribution of the IVIean Motions of the Asteroids. 3 where c is a constant depending only on the size of the asteroid, />, tlie density of the particles and F, the resultant relative velocity of the asteroid. The constants c and p are essentially positive. 2. Let /> be a continuous ftmction of r, so that it may be devel- oped in the convergent series, Put 7 _ a- dp 7, _ 1 a' d'-p ^ Pq da' '^2 po dar' then ,=4n.,,i^+^r==(O0(1 — ÄTi^ cos ÎU— Ä;ie^sin^'iü + A-2e' cos- ÎU+ ) 3. The equations for the variations of the elements are^^ ^= ^==- [Se sin w + T{l + ecos ?o)1 de Vi — r Va ■ , rrf , cosîo + e \~1 L \ 1 + e cos ?o/J — >Scosîu + T( l+-i |sin?t' L \ l+eco&iv/ J dt na dw^ VT^ dt na ds __ 2(1—6^) çy e" dv3 dt 7ia(l + e cos iv) 1 + VI — e"^ dt where -iss and e are the mean longitude of perihelion and the mean longitude at the epoch. Since V contains the 1st power of e as factor, S and T will contain e'' as factor. Hence, if we confine ourselves to the order of e''^^ in the development of the differential coefficients, we may neglect e^ terms in the coefficients of S and T. Thus, 1) Tisserand, Mécanique Céleste, I p. 433 and IV p. 218. Art. 3. — K. Hirayama : -^r- = — ÏSe sin ic + 2Y1 + c cos w)'] dt n (4) •de _ 1 dt na \_S sin w + T (2 cos ?ü + e sin- w)\ e-^— = f — *S' cos ?r + ri2— (3C0S ^ü) sin ?rT rt?' na dt 2^/1 ^ 1 = .Oo( 1 — A:] e cos ?ü) 4. We have to change the independent variable from t to w in the equations (4). The relation between the differentials is dt _ {\-êf n{\ + e cos wj- -dw or neglecting e^ (7) dt= — ( I — 2e cos w ' dio n 5. Combining tlie equations (!2), (4), (5), (6) and (7), and pvitting 4 \ 16 1 — f cos^w/ Researches on the Distril.aition of the ÄTean INIotions of the Asteroids. I obtain da dio = Wa y^e sinhv + cos wll + -^ cos w j fl + e cos iv \ É^= W V^ui-iD + cos ni 1 + -|- cos uAi cos w + -^ ^iirw J -= W —sill ic cos /Ü + sill 10 cos yrf 1 +^cos }c\( 1 — ö"^*^* '^^j — — = —2 IF Sill tüfl — f cos «V + -;r- -^— dw 2 rf/r Or neglecting e'^'^ and simplifying TF=-w«-W-v(l--|cos»„.)'^'{l-(.2 + ^-,-Jj^j,^p|^>cos,.} ^^* TFa(cos?r + 2e-Apcos"';tO (8) dw -= ^7(1 + 4- cos w — ^ cos-^z^;) ; ^ 2 4 '^ e— — = — W -r sin w cos-iv div 4 fZS OTT7/ ■ • , , 6' dv3 — — = — 2lr (sm lü—e siii ic cos ?r) + -T; — ^ — diu 2 rtR' 6. Tu find the secular variations of the elements in a unit interval of time we have to evaluate r da ~l _ ju^ r '^"^ da -, V. dt A '2,nJ(\ dw etc. etc. Since the effect of the resistance is supposed to he very small, we may neglect cVo^ within a single revolution, so tliat the quantities n, a and e may be integrated as constants. Now if we put X—{\—q cos-zü)"2"siii'?r cos'/t^ where (/< 1 and .^, /, j are any positive integers, it may be easily seen that Art. 3.— K. Hirayaina ; r'^xdw=[\+{-\y][\ + {-\y]J -xdw Hence the integral vanishes when either l or j is odd, and becomes àf'^Xdw when both are even. 7. Applying this result to the equations (8), I obtain after some simplifications (9) where «t'tZtü Taking the first integral and expanding -, 2 /-I/t 7i-l/3\ .. , (/i-l)(/i-3V3 \-_ . b 2 Now — / COS ^oau'' 7: J 0 ,_l-3 (i-1) 2-4 i when i is even. Therefore T_,_l-1 1/3 \, (/t-l)(/i-3) l-3/^3\- ^'~ ~^'^\~\r 2^4 2tVtj _ (//-l)(/t-3)(/i-5) 1-3-5 / 3 Y^ 2 4-0 2-4-6 \ 4 / ■*■ Researches on the Distribution of the Mean Motions of the Asteroids. Similarly J ^^ /6-x l-3/3\ (A-l)(;t-3) l-8-5/3\- ' 2 2 2A\^y 2-4 2-4-6U/ J h-l 1-3/ 3 y (A-l)(;t-3) 1-3-5/3 2 2 2-4\4y'^ 1-3 /^-3 1-3-5 / 3 2-4 1-3-5 / 3 \ Qi-'d){h-b) l-3-5-7/^3 Y 2-4-6\4/"'" 2-4 2-4-ü-8\4y These series stop at a finite number of terms when h is odd, and continue infinitely when h is even. If we put (10) «, = 27,— ^7, + ^^Js a,=I, ß,=I, we get finally (11) 0 The numerical values of «„ «o, and ^i, for /(î=1, 2, o are computed as follows : — h «1 a.. ß. 1 0-62 0-50 1-00 2 0-69 0-32 0-77 3 0.70 0.22 0-G2 8. If we put h=2 in (11), as it seems most natural, -4r is Lda ~\ i- at A -57- J to e^ Hence the effect must be very small for the bodies whose orbits have small eccentricities. Or in other Avords if the effect be appreciable in the motion of asteroids which have small eccentricities in general, it must be remarkably great for the motion of the comets. This appears to be almost fatal to our assumption, supposing the resistance still to exist, because the effect of the resistance on the cometary motion, whatever may be its law, is known to be very small if it exists at all. But tliere is an answer to this objection. The comet, as far as we know, is not a single body rigidly bound like a planet. It 3 Art. 3.- K. Hirayauia : seems to be a loose aggregation of small bodies^^, perhaps composed of the same kind of material as meteors, with rare gaseous envelope. Most of the particles which are supposed to effect the resistance will pass freely through this meteoric swarm. Uarely it ma}' occur that some resisting particle strikes a cometary particle. Then the latter will be projected outside of the swcirm and take on an individual motion. Gradual degeneration will follow this action if repeated frequently but there is no effect on the motion of the main body. Yet one more thing is conceivable, viz. an indirect effect coming in through the gaseous envelope. This, however, would be very small owing to tlie tenuity of the latter. 9. The assumption that the particles move in circular orbits in a definite plane is nothing but an imaginary convention to make the problem simpler. Practically this may he said to be that at a point in or near the plane, the resultant composite velocity of the particles passing through that point is equal to the velocity of the circular motion. As for the density of the particles it is natural to assume a certain amount of decrease as the distance from the sun increases, that is, to assume a negative value for ^i. This is not all. for there is some reason to believe that the particles are not numerous near the path of the major planets. They cannot move in orbits of small eccentricity in the neighborhood of the planets. If they did, they would be disturbed a great deal by tlie action of the latter, or they might even be swept up. except those moving about the triangular equilibrium points. Chapter 11. Motion of the Asteroids wliose Mean Motions are nearly Connnensura1>le to the Mean Motion of Jupiter. 1) Yoi 11;^' compares it with " piJi-lu-uls s.-v.-ral hundred feet apart. Kesearches un the Üistrilmtion of the Meaii Motions of tho Asteroids. 9 10. In this chapter I shall develop a simple theory of planetary librations according to the method of Prof. Brown.'-* Assuming that the asteroid 7noves in the fdane of the orbit of Jvpiter and that Jupiter inoves m a circular orbit, let n. a, e, s and TIS be the elements of the asteroid as before, n', a, and s', the circular elements of Jupiter and R the perturbative function, as usual. Take the mass of the sun as unity and the unit of length, so as to make (1) 7m'=l, neglecting the mass of the asteroid, and also n'V' = l + w' m' being the mass of Jupiter \Yhich is about 1/1047. 11. Let Ho be the mean motion commensurable to n' and let (2) n=noil+x) or a=at(l-{-x)"^ The quantity x is supposed to be small, of the same order as e^ at most. Let also (3) jL=jL n„ s where s and s' are positive integers prime to each other. 12. As a consequence of the assumptions mentioned above, any argument of long period-terms, or critical argument as usually called, takes the form is'l—isl'+jiv where i and j are any integers positive or negative, and / and I' are mean longitudes. By the properties of R we have 'is' — is+j=0 or j=iÇs — s') Hence the critical argument is ll Month. Notices of the R. A. S., Ixxii, p. 609. 10 Art. 3. — K. Hirayaina (4) i{s'l-sV+{s-s')vs}=id The corresponding term will be factored by e'^' '\ so that for the principal term we have to put i=l. Now l—lndt + £ = Uf^t + n^^jxdt + e by (2) and V=n't + t' whence (5) 6 = s'nfjxdt + s's — St' + (s — s') qj 18. According to the theory of perturbations Ave have (6) in which da , N "hit -—-(a,£)-,— dt ?£ de ( ^tB , ^ xSiJ (7) I de r ^IB , / _-,lB dw ^ dt ^ Is. Ivo dt {a,t) = — ( £,a) = 2)10^ f^ \'^B ce (.,e)= -(£,.)= —!^^^^l:iil(l -VT3^ {e,m)——{w,e) = 7ia^/l—é^ Neglecting all shoï't period-terms in i?, let and T> m' ( a 2\ + Til e functions f, ^, 5^^,,. ..are deve1Bc = C oust . ii*o is a function of a and e^ and ê is a function of x containing an arbitrary constant. Hence developing ^»=W»H^lF+^f)/ + where the suffix 0 outside of the parentheses means the value for ^^=0. The terms with o^ and higher powers may be neglected, since they are multiplied by m . Now lda\__1 (dê\ _ 2 8- s' whence Substituting this expression in the integral we get or putting (10) .„=.«„=(^)^-2i^„„(^|^ the integral becomes (11) (.r-a-„)- + B«,.Rc=0 Resoarehes on tlig Distribution of tlio Mean Motions of the Asteroids. X3 where C'is an arbitrary constant. 16. By the last equation of (8) dt " la e ^e Neglecting m'x, me and higher powers we may write or introducing Uo by (10) (12) 4^=,^^^(,,_^J+(,_,yv^o ^Bc dt "' "' "■ ' e ?e The equation (9) may be written also (13) e'=E + ^-^^(x-x,) where E is an arbitrary constant. 17. The constant .ro is a quantity depending on -^ and F, multiplied by w'. Now JS is supposed to be small, of the same order as e^ Hence, neglecting m JE in Xq, the latter becomes a constant depending only on -J-. Since x is always diminished by Xo, if we change the origin by that amount, we may write x for x—Xo and Xo disappears in the equations (11), (12) and (13). Thus x- = C—6a(fRc •> ET , 2 s-s' ^" = ^ + T^^^ Wo at e ce (1^) 18. The expression of Xo in terms of the powers of-^may be obtained easily by the usual process. The result is J 4 ■'^î't' 3.— K. Hirayama : m' ""^ da^ 2 V da.^ 2 da^^ I = (4-3.)(i)V + 2(4-5v)(^)V+3(4-7v)(^)V + where '^o=-% and ^=-77- The numerical values are s/s' Xo X 10^ 2/1 -0.005 3/2 + 0.027 3/1 -0.012 which are too small to be considered in our problem. The equations (14) represent the motion in the librating region with remarkable simpUcity. This will be shown geometric- ally. 19. General Case of the First Order, s—s'=l. The principal term of Ec, becomes in this case aoEc=—pi''''* e cos d in which ^'" = ^(2^'"'+^o^) The three equations of (14) become (15) x'=G + 6p,^'^ecosd (16) e'=E + -^x (17) l_^^^,^_pl^çosd_ no at e Suppose the quantities x and e are two rectangular coordinates. The equation (16), then, represents a paraôo/a whose axis is the Kesearches on the Distribution of the Mt'im Motions of the Asteroids. 15 axis of 2-. Let this parabola be designated by .4. The equation (1')), also, if we put a definite value for d, represents a parabola whose axis is the axis of e. For the hmiting values of cos d, ±1, we have two limiting parabolas whose equations are (18) x'=C±Q>p,e ^> Let these limiting parabolas be designated by Bo and B^, respectivel3^ ^o »"ind B^ meet at the same points (±\/(7, 0) with the axis of x. Two straight lines passing through these points and parallel to the axis of e represent the equation (15) for 0=-^ and 2 • exm \ \ B. ^0 \ / A / \ / o Fig. 1 XXXQ exio xxio 1) Writing f\ ior p^^ and i^g.,- for "p^'lg, generally, for the sake of brevity. 26 Art. 3. — K. Hiiayama : The parts, or tlie p«irt as tlie case maybe, of A inside of Bo and outside of ^i are tlie real patlis of the point (x, e). In Fig. 1 two parts, one on the negative side of x and the other on the positive side, are separate. The argument 6 may take any value in each part. But when £ becomes larger or C becomes smaller, these two parts approach each other until A touches Bi, in which case they are connected. After this A does not intersect B^, as in Fig. 2, and the argument 6 never takes the value n or—-. It increases from 0 to the maxinmm value do and decreases, passing through 0 again, until it reaches the minimum value — ^o- Then it increases again and so forth. This is a kind of libration.^^ 20. We can distinguish six types of the 7noiio7i, namely: — 1 Revolution on the negative side (of .t), 2 ,, extending on both sides, 3 ,, on the positive side, 1 Libration on the negative side, 2 ,, extending on both sides, 3 ,, on the positive side. 21. In order to determine the limits of C and ^ for each type, we need to consider some singular cases. 1. Condition of the contact of A and ^o or ^i. Difïerentiat- ing the equations (16) and (18) with respect to a- and equating -T-we get the condition (19) ex=±^ The same equation results from (17) putting cos d=± 1 and -rr =0 in it. To find the relation between (J and E, eliminating x and e from the equations (IG), (1 3 >> + + j> -F, Fo -F, Fo -F, Fo Libr. 1 5, 5, — — >> impossible F2 Es E, +F, >> 2 -Bo Bo — + Fo -l-oo + F, +00 + F, 4-00 E, -l-oo j» 3 »> + + E, Fo E, -F, E, -F, E, -F, This will be very easily understood by the geometrical representa- tion of the motion. •7/3 23. The sign of -rr for different types will be discussed now. ■tu The point at which -^ changes its sign is determined by the equation s'ex—pi cos 0=0 combined with the equations (15) and (16). Eliminating e and cos 6, we get easily x^+2s'Ex+^=0 0 or (25) x==-s'E±Js'^E'~ The corresponding values of e are given by Eesearctes on the Distribution of the Mean Motions of the Asteroids. 19 Since e must be positive, two points, real or imaginary, will be determined by these equations. Now evidently only an odd number of zero-points in the librations and an even number in the revolutions, are possible. Accordingly, since there cannot be /Id more than two, one and only one zero-point of-rr will exist in the case of the librations. The conditions necessary in order that two zero-points may be real, are 3s'^ 9/ E>0, ^^E',<'>=^ [^(4s^_55)6^^'4-(4s-2)ao^+ao='^] Putting ^=^y 20 Art. 3.— K. Hirayama: we have by (14) x^=G—ßp2ycosd (27) \ dd , , A j- = s' X + 4:p^ COB Hq dt The curve A becomes a straight Ime in this case. The condi- tion of the contact of A and ^o or B^ is s' and the relation between C and E becomes simply S There is no contact of the second order. The condition that A and Bq or Bi may touch each other on the axis of x is C=^p,^ = C, 7;t . 16 25. The limits of C and JE for different types of the motion are as follows : — Limiting Curves Limits of X Limits of G Type -oo 0 0 G, G2 +00 Limits of E Kevol. 1 B, B|i — — impossible impossible +F, E, >> 2 ^0 ^l - + '» F, +F, F, E, )> 3 >» + + J» -F, F, -F, F, Libr. 1 )) impossible impossible » 2 B, B, - + F, +00 + Fi +00 F3 +00 )> 3 ■ '» + + E, F, E, -F, impossible Researches on the Distribution of the Mean Motions of the Asteroids. 21 The limits are simpler than in the case of the first order. The libration of Type 1 is impossible throughout o^nù the librations are restricted to that about n. lift 26. The point at which -^ changes its sign is given by the equations =-^«±y(^^;_c -±y^-(^T« Only one point is possible on the positive side of e. Hence, there is no zero-point in the revolutions and one and only one point, in the librations. 27. General Case of the Third or Higher Orders^ s—s* >2. We have in this case acßM-^y"'p,..'e'-' co^d where jo,_,/ is a function of «o with a positive value. Putting we have by (14) x'^=-C—{ — Vf'''&p,.,.yco^d (28) 1 dd g-a'-a — -^=^'x + {-^y~''{s-s'fp,.AJ '-"' cos 6 The curve A becomes a parabola of various degrees with the axis parallel to the axis of y, and the vertex on the axis of x. The constant C2 becomes 0. The table of the limits is much simplified by this change. The libration of Type 1 is always impossible and the librations are restricted to those about 0 when s—s is odd and those about n when s—s' is even. The libration of Type 3 22 A-rt, 3.— K. Hirayama : becomes impossible when C>0, in the cases of the fourth and higher orders. Chapter III. Theoretical Effect of the Resistance on the Motion of the Asteroids in the Librating Region. 28. Putting /i=2, the equations (11) of Chapter I become (1) ari^i=_«, \_^y-ße^ a \^ dt A where Now, by the equations (1) and (2) of Chapter II, we have J_ f— 1 = -— — T— 1 a Ldt J 3 n L dt J and rdx~[ ^ 1 r dn "] Since n and 7io are supposed to be very nearly equal 1 rda"] ^ _ 2 rdx~] a Ldt J 3 Ldt J Hence (2) [1]=^' We have, also, by (5) of Chapter II dd , , f de de' , . .äw and, since x is not affected directly by the resistance, we have by (1) Researches on the Distribution of the Mean Motions of the Asteroids. 23 (3) ra- The differential coefficients of x, e and 6 are composed of two parts, viz; the differential coefficients due to the perturbation and Ldx "1 I ax \ Hence, if we denote the former by [-jrj^ 6^^- dx _/ dx\, r dx~\ dt~\dtrLdtJ ®^^' Or, substituting the expressions of -jj- , etc. we get dx _( dx \ 3a 3 dd_ldd\ dt \ dt J (4) dt de dt Since x is supposed to be very small, we can write As it will be seen in Chapter I, ß is essentially ^oÄ'^e and a may OL be positive or negative according as ki is less or greater than — - which is positive. We shall consider the case in which kx is negative and consequently a is positive. 29. Oeneral Case of the First Order. Differentiating the equa- tions (15) and (16) of Chapter II completely with respect to t, we get 2 _^_.^;E , _2_ dx dt dt 'as' dt 24 . -A^rfc- 3. — K. Hirayama Differentiating the same equations with respect to t, supposing the effect of the resistance is null, 2.(^)=6„cos.(^)-6,,sin*(f) \dr)~ds'\dt ) Substituting the expressions of -jr, etc. by (4) and taking the differences between the two sets of equations, we obtain (5) -^= Sae^x + 6ßpie^ cos d which are the equations for the variations of the arbitrary con- stants. These equations show that JE always decreases while G increases or decreases depending on x and cos d. Now the signs of X and cos 6 are both negative in the libration of Type 1 and both positive in the libration of Type 3. Hence C always decreases in the libration of Type 1 and increases in that of Type 3. dC 30. There are three more cases in which the sign of -jr may be determined easily, namely; the libration of Type 2 when Cis negative, the revolution of Type 1, and the revolution of Type 3 when JS is negative. In the first of these cases the motion being a kind of libration about 0, 6 may increase from 0 to an angle Oq and decrease to— ^o- The limit ^o is not greater than -^ so long dC as C is negative and consequently the second term of -rr is always positive. Now, we have by the equation (17) of Chapt. II no/e*xdt = /—. 3- j J sex— Pi cos o and Eesearches on the Distribution of the Mean Motions of the Asteroids. 25 'K ^-^0 -^ôo ./o -^ -%/ s'ex-p, COS d where T is the period of the libration. Writing x and e for x and e in the part in which -jr is negative, we have nJ' e'xdt=2f '(-, ^^ 5— , , ,^"^' n]dd = 2f Xdd X is evidently positive when both % and x are positive. Also, Y_. s'{e^—e'^)ee'xx'—px oo^d{e*x—e''^x') (s'ex—piCosd)(s'e^x'—piCOsd) The denominator is negative, e^— e'^ is positive, cos 0 and e^a:— e'*:c' are also positive and therefore JT is positive when x' is negative. -nT^i is 0 dit positive. Thus C increases algebraically when negative. dC 31. The sign of the first term of -jr is exclusively negative in the revolution of Type 1 and positive in that of Type 3. For the second term we have fiol e-coQ ddt=±/ — ^dd /g ./y s'ex—pi cos a in which T is the period of the revolution and the double sign cor- responds to the positive and negative values of -^, i. e., to Type 3 and Type 1 respectively. We may write J g sex—p^ cos 0 ^0 sex—pi cos o sex +Pi cos o/ n _ = 2f^Ycosddd 26 Art. 3.— K. Hirayama: in which x and e are written x' and e' where cos d is negative, and y_ s'ee'(e^x'~e'^x)+p^ cos d(e^ + e'^) (s'ex—pi cos 6){s'e'x'+2^i ^^^ ^) _ s'ee'(x'—x)E+p^ cos 6(e^ + e'-) (s'ex—pi cos 0){s'e'x' -Vp^ cos ^) The denominator is positive in both cases. The quantity x —x is positive in Tj^pe 1 and negative in T^^pe 3. E is positive in Type 1 and may be positive or negative in Type 3. Hence, for the révolution of Type 1 taking the negative sign for the double sign, the integral becomes negative and consequently C decreases. For the revolution of Type 3, if E is 7iegative or less than a certain posi- tive value, Y is positive and, taking the positive sign for the double sign, the integral becomes positive and therefore C increases. 32. Since e cannot be zero permanently in the general case of the first order, E decreases without limit so that it will become negative after a certain epoch. Now, when E is negative, the type of the motion is limited to two kinds, namely; the revolution and the libration, of Type 3. Hence we see that the revolutions and the librations, of Type 1 and Type 2, are not permanent forms of the motion ; hut will change ultimately either to the revolution or the libration, of Type 3. Also, since C increases algebraically when negative, it will become positive if it ivas initially negative, and it cannot become negative if it ivas initially positive; but will decrease to some limiting value which is positive and then will increase without limit. 33. It becomes necessary before proceeding further to find the limiting values of x and e in the libration or the revolution of Type 2, in which "^=0 for d=7:, supposing the efïect of the resistance is null. Let Xi and e^ be the values of x and e at the zero-point of -jr, then x,'=C-6p,e, e,'=E + ^x, Besearches on the Distribution of the Mean Motions of the Asteroids. 27 and s'eiXi+Pi=0 Evidently, the limiting values of x and e will be attained when 6^0 or 7t. Denoting these values by x and ê, we have x^=G±6pie -é=E+p in which the upper sign corresdonds to ^=0 and the lower to (e^-ê7] at Au where s'ex—p-, cos d 35. Putting 0=7: and changing x, e and ^ to x^,, e^ and ^i respectively, apply the above result to the case of the revolutions of Type 1 and Type 2 very near to the libration of Type 2. In this case, (a) se^x^+p^ is very small, negative in Type 1 and positive in Type 2, (b) e^—e^, is always negative in Type 1 and positive in Type 2, (c) 31 is positive by (a), (d) the quantity ^s'e,' + x,= ^^"^]~^' s'e, is positive by (7) and therefore (e) N is negative in Type 1 and positive in Type 2. Since N is very great, we may write In the revolution of Type 2, 3f, iV and e^—e^, are all positive and therefore —jr- is always positive. In the revolution of Type 1 , N and e^—e^, are negative. Writing 30 Art. 3.— K. Hirayama : and denoting the smallest value of e by eo, ^/ is always positive if But and by (6) approximately. Hence in the revolution of Type 1, ,.^ is always positive if the condition X, =; — ,=ei- -/ 4pi ''-M^-fJ or ^0 s'/3 be fulf^Ue^l. 3 6. Since s'^ia^i+j^i is zero when the path A touches the curve j(5i, and negative or positive according as the revolution belongs to Type 1 or Type 2, the quantity H^ is equal to — ^^ when the path is in contact with B^ and is less or gi'eater algebraically than— -nr- according as the revolution belongs to Type 1 or Type 2. Now it has been proved that Hx increases in the revolution of Type 2, and in that of Type 1 if e^ is greater than a certain positive value. Hence the path A recedes from the position of the contact in the revolution of Type 2, and approaches to that position in the revolution of Type 1 if e^ is greater than a certain positive value. 37. To see how the librations are changed by the resistance, let us find the variation of the minimum or the maximum value u of cos(?. Let X and e be the values of x and e corresponding to cosd=u. We have then s'êx—piU=0 Researches ou the Distribution of the Mean Motions of the Asteroids. 31 _ 2 - and x^—x^=6pi(ecosd—êu) é^—e^=-^(x—x) Differentiating these equations and eliminating -^ i?,ivl ^^ , we get Or substituting the expressions of [_-^J and |_-^J by (4) and eHminating x and cos^ by the known relations, we get (10) -4p,ë^=s'Me{e' -e')+^s'C^a + s'ß)e{e'-eJ dt ^ where M=3ae^-2ßx as before. 38. The second term of the right hand member of (10) is positive for all values of e. The sign of the first term depends on the sign of if and e'—e'. Now, if we confine ourselves to the libration of Type 2 in which u is very nearly equal to— 1, M is positive since x is negative, and the integral dt may be proved to be positive supposing the quantities C and E are constants within a single revolution. To prove this I shall proceed as follows : — We have, using the symbol of § 28, and therefore nj(e'-e')dt= -f^'~'^\de Now, since d is very nearly equal to tt, go Art. 3.— K. Hirayama: a;2 — 5^ = Qp^e cos d + 6^ië e^ _ ^2 _ (a; — a;) and s'ex=—pi Accordingly p^sind s'\/l2p^ê—{x + xf in which the negative sign will be taken if e^—e' and sin^ have the same sign, and the positive sign if otherwise. Hence where eo and e'o are the limiting values of e determined by the equations Putting fie) = 12pé -(x + xf and e=e±B e>0 we have 3s' '^è Now Researches on the DistriVtutidU of the Mean 'MotionK of the Asteroids. 33 and whence .^(6,3_M^i^_3..)(^-3..) The quantity —^ — 3ê£^ is positive within the limits and 3ë^— -^ is also positive by (7). Hence the integral 0 is positive. Now I eiê-e^dt^-lj {e--e')dt^ [e-è){é'-ë')dt Hence / e{é'-e')dt>0 err and therefore /* au ^^^^^q J . dt This result shows that the quantity w, Avhen it is very nearl}^ equal to—], approaches to— 1. Hence we may conclude that the libra- tion of Type 2 in which the maximum value of 0 is very nearly equal to TT changes to the revolution of Type 1 or Type 2. Now it has been proved that the revolution of Type 1, when é"i is greater than a certain positive value, approaches the posi- tion of contact, and that the revolution of Type 2 recedes from that position. Hence it may be seen that the revolution of Type 1 changes to the libration of Type 2, when Ci is greater than a certain positive value, and immediately later to the revolution of Type 2. 34 Art. 3.— K. Hirayama : 39. The equation (8) becomes, when ê is very small and negligible, dH s'x dt 4^^ cos d [-2;9s+|-(2« + s'/3>^j — dTT Accordingly, if x be negative, the sign of —57- is determined by the sign of cos^. Applying this result to the limiting cases of the revolution and the libration, of Type 1, we see at once that the quantity II at the point very near to the axis of x, increases in the revolution and decreases in the libration. Now, the quantity 11 is very small and negative in both cases. Accordingly it approaches 0 in the revolution and recedes from 0 in the libration. Hence we conclude that the revolution of Type 1 in its limiting case changes to the libration of Type 1. The same result may be obtained in the limiting cases of the revolution and the libration, of Type 2, when C is positive and less than C2. Thus we see that the libration of Type 2 changes to the revolution of Type 2, in this case. In the limiting cases of the revolution and the libration, of Type 3, the sign of the quantity in the brackets is not definite. It becomes positive when ß is very small compared with a. It this case ^increases in the revolution and decreases in the libration, and, since H is very small and positive, it approaches 0 in the libration and recedes in the revolution. Hence, if ß be very small compared with a, the libration of Type 3 changes to the revolution of Type 3. 40. Let us next consider the extreme case of the libration of Type 1 in which -jr is very small for ^=;r. We have by (9) Now s'e,Zi+pi is negative, 3s'ei^ + Xi is positive by (7), and therefore N is negative. The quantity 6*— e^ is negative for all values of e. The lower limit of e is /J-^ — Ci hy (ß), and Researches on the Distribution of the Mean Motions of the Asteroids. 35 Hence -Tr^>0, and therefore the libration of Type 1 changes to the revolution of Type 2. 41. When the motion changes from the revokition or the libration, of Type 1, to the revolution of Type 2, the limits of x and e increase discontinuous^. Referring back to the table (6), it may be seen that the lower limits change from the second row to the third row, and the upper limits from the third row to the fourth rOAV. Hence the lower limits of x and e increase by the amounts respectively, and the upper limits by V12^ + ^ and J^^ respectively. The upper limits become minimum when os'^ei^=pu and the minimum values of x and e are sf^^j and 3(-Äf) respectively. The corresponding lower limits are Numerical values of these limits for different cases of the first order are as follows : — gß Art. 3. — K. Hi raya ma: sis' p^xW _(§^)X10^ 3(^yxiœ (|_yxl0 3(|^,)xlO 2/1 0.716 -1.15 3.4() 0.62 1.86 3/2 1.474 -1.48 4.45 0.50 1.49 4/8 2.237 -1.71 5.13 0.45 1.34 The variations of the constants, Cancl E, l:)econie rapid by this increase of the eccentricity. 42. The revolution of Type 2 will change to the revolution of Type 8 sooner or later, since E decreases and C cannot become neg-ative. It has been proved that 6' increases when E is less than a certain positive value. Hence C begins to increase without limit just as E decreases. 43, The equation (10) may be integi'ated by expanding iu series when 6 is small. Let us first take the integral I e{e- — e-yit where T is the period of the libration as before. Putting $=x—x '// = e—c 2 we have e'-e^—2êr^ + y = ^$ and a;^ — 5' = 25=1^ + ^ '" = ^Pt (^ cos ^ — è cos d) Eliminating $ and making use of the relation s'ex=Pi cos 0 we get whence, neglecting r/, ffi"', ffi"^ and higher powers, we obtain Now and Also Eesearches on the Distribution of the Mean Motions of the Asteroids. 37 Me- - è')dt = f-^-^^i^-Jl- dd 7 ^ ./ s'ex-p, cos d s'ex—p^ cos d = s'{^s'é--\-x)rj-\--^:r::^{Ç>s'é- — x,rj- + Hence e^(e--ê-) 2êH5è''îy + s'C^s'e' + x)\ 2 3s'è- + S êy neglecting rf. Accordingly we can write Since x is positive in this case the coefficients P and Q are finite and positive. The double sign Avill be taken the same as that of ■f}. Now ■I« <^* dd and the sign of jy is the same as that of -tt throughout. Hence 3g Art. 3. — K. Hirayama : Again J'e(e'-ejdt=f f(''-^')^ = g; ßd« J ^ ^ J sex— Pi COS a s{os'e^ + x}J ' neglecting r/. Hence n,Je{e'-eJdt^±QJ'^'W^W < in which Q' is a quantity finite and positive, and the double sign will be taken the same as that of ;y. Consequently fT ..ë noj e{e''-èJdt=2Qj _Vd'-d'dd=zQ'7:d' and • where S is a finite quantity, positive Avhen 31 is positive. Also A,_ /' edd _ ê I'dd p//" dd where P' is a quantity finite and positive, and n^T^yJ' dt^'AP'f /}^ =27tP' Hence '■'''^M=-^,ff'^~S'g' and 1 /. '1 1 /"dd y w'^^''" Kosoarches on the Distvibutiou of the Mean Motions of the Asteroids. 39 Consequently the variation of â is slow when â is small. The same proposition may be proved for the libration of Type 1 when 6 is very nearly equal to tt. Hence in general the amplitude of the libration varies very slowly lohen the amplitude is very small. 44. Now when the amplitude of the liljration is very small x and e are connected approximately by the relation s'ex=p^ in the libration of Type 3. Hence the point (x, e) moves in a rectangular hyperbola, and therefore when e is great the point moves down nearly parallel to the axis of e until e becomes small and consequent- ly the variation becomes very slow. In the case of the libration of Type 1 the point {x, e) will move in another rectangular hyperbola represented by the equation s'ex=—j)i. It will move nearly parallel to the axis of x with a small eccentricity until reaches at the point and the motion abruptly changes to the revolution of Type 2. 45. The conclusions for the general case of the first order may be stated as follows: — 1. The revolution of Type 1 will change to the revolu- tion of Type 2 either directly or indirectly and finally to that of Type 3. The limits of the mean motion and eccentricity increase discontinuously when the motion changes to the revolution of Type 2, whence the variations of the constants become rapid, and the asteroids of this class will not stay long near the critical point. 2. The libration of Type 2, when C is not negative and relatively great, changes to the revolution of Type 2 either directly or indirectly and finally to that of Type 3. 3. The lil)ration of Type 2, when C is negative and relative^ great, changes to the libration of Type 3 which may change to the revolution of Type 3 if the constant ß be 40 Ai't. 3.— K. ffiravama sufficiently small compared with «. Tlie amplitude of the libration varies very slowly when the amplitude is small, and the asteroids of this class .will stay long near the critical point with small eccentricities. 46. General Case of the Second Order. Putting 6=- + d' in the equations (27) of Chapter II, we have (11) x- = C + Q2hß'' cos, 6' 1 dd n„ df = s'x — 'ipoGOSd' The equations for the variations of the constants become dC dt = Sae'x + V2ßp,e' cos 6' -^' = -2(-^ + ;5J< The quantity JiJ always decreases as in the case of the first order. G increases in the libration of Type 3, since x and cos^' are always positive. We can pro\'e also that C increases if negative, that it decreases in the revohdlon of Type 1 and that it increases in the revolidion of Type 3 if E he less than a certain 'positive value. 47. The limits of the mean motion and eccentricity, wlien the path Ä is in contact with the limiting curve B^, may be obtained as follows: — d' Ap, on ^±vws i^+^P,±J§j,,E For the limiting case in whicli ro=^i=0 we have Reseai-ches on the Distri'mtion of the Meau IVIotions of the Astevoiils. 41 7t, 0 0 4j9.; s' 0 3s'- Numerical values of these limits for different cases of the second order become as follows : — sis' i),xiœ -^xiœ ^/),xlO-^ (^^,^,^,^'xlO 3/1 0-0275 -0-1 10 0-330 0-766 5/3 0-222 -0-296 0-888 0-725 48. The equation corresponding to (8) becomes ^'^■=^^re' + (^a + s'ß){e'-e^)y dt 8(V^-4îo,cos / 9 — .\ tf , S,5\ ,- 9 -2\2 -^P-i^'-jr = — s'ae-e{e- — e-) — s'ia + .^ je{e' — ey 42 Art. 3.— K. Hirayama : wliere y/=co-^' is the minimum value of cos^' and e is tlie corres- ponding value of e. Now ^o/ ip'—'^')ài-=-^ I (x—x)dt _ 8 ('^ ( X — X __ X' — X \ja, ~ Ss' J ^ Vs'ic — 4^2 coso, the order of x will be higher than tliat of e' by a unit order at least, so that e may be supposed to be a constant in the first approximation. In the second member of the third equation of (12), the order of p,é~\ being /+1, is higher than that of x by a unit order at least, if i>?). Hence we may write -— = SX — Q COS u 7lo dt -^ where y is a constant. 54. The e(iuation corresponding to (H) becomes in this case -|-(^ê-^-')= [^è^ + ^(r-r)+,9(.-2)^4-?^(r-ê^')] -^'^'e in which ^r_ s' Ss'e'—i(i — 2)x -,( ^ 7— ^^> — -•> T Zi7~^ I S X — l'Pfi' - COS 0 , îîosearchos ou tho Distrilnition of the Mean Motions of the Asteroids. 45 Or neglecting tlie quantities of higher orders we get at 2 4 ^ where 2V=-£l ^*'^' * s'x—î^p^e^'"^ cosd' If we put ^'=- in this equation, N(e^—e^) is always positive, and hence Accordingly the revolution of Type 1 in its limiting case approaches the position of contact, and the revolution of Type 2 recedes from that position. 55. The equation corresponding to (10) takes the form neglecting the quantities of higher orders. Now nJe{e'--e')dt=f , <^'-~^') dd' J J sx—gco%o' According^ we can prove that I e{e'-~e')dt>0 0 as in the case of the second order. Hence /, dt ^g Art. 3.— K. Hirayaina : and therefore, the librations ultimately change to revolutions. We can prove also that the amjylittide of the libration increases very slowly when it is very small, as in the previous case. Thus we obtain the same conckisions as for the general case of the second order. Chapter IV. Peculiarities in the Distribution of the Mean Motions of the Asteroids and their Possible Explanations. 56. We shall first examine the nature of the gaps. The ratios of the mean motions up to the seventh order and lying within the denser portion of the asteroids are as follows: — Order njn' Order 7ifjn' Order Wo/w' 1 2/1 5 9/4 7 13/6 2 3/1 8/3 12/5 3 5/2 7/2 11/4 4 7/3 6 11/5 10/3 The simplest way of determining the width of the gaps is to find the difference of the two mean motions nearest to n^ in both directions. But this will be very rough, especially when one or both of the mean motions is not reliable, as in the case of (132) of the class 3/1. The following method will answer for this defect. Denoting by n_i5, n_u, ?i_.j, n-i, Wj, 7U, n^, v-i^ the mean motions arranged in ascending order and interposing no between n.i and ni, the quantities Çi=-g-(ni + n.+ +W9— nis— Wu— »hs) — no Rosearches on the Distril>utioii of the Meau Motious of the Asteroids. 47 and Ç_i=no— -p-(n-i + ?i-2+ + »-9 — n_B— ^î-u — w.is) may represent the widtli in the positive and negative directions of X respectively. The sum of these two quantities becomes zero when the distribution is uniform, and becomes negative when there is some condensation near no. The number of the mean motions in each direction may be taken more or less. But if this number be too small the result will be inaccurate, while if too many it may be affected by other irregularities of the distribution. 57. The width and position of the gaps thus determined, according to the Berliner Jahrbuch for 1917, are as follows: — tioln' Order ",) Q-i to the ffth order seems established beyond doubt. It may also be observed that the width becomes narrower as the order advances and also as the denominator or the numerator of the ratio increases. 58. In the portion with smaller mean motions than 500", though the number of the asteroids is very small we see at once that the character is reversed (PLATE). No asteroid can be found 48 -^rt. 3.— K. Hirayauia : except near the commensurable points in wliicli the gaps are found in the otlier portion. The positions of the condensations and the numl>er of tlie asteroids are as follows: — Order Mo No. of Asteroids 1/1 0 299-13 4 4/3 1 398-84 1 3/2 1 448-70 6 If the distribution of the mean motions can be compared with the spectrum, the portion with greater mean motions than 600" will correspond to an absorption spectrum, and that with smaller mean motions than 500" to an emission spectrum. The portion inter- mediate between these portions may possibly belong to the former class, although the character is not distinct owing to the scarcity of the asteroids. 59. That the eccentricity of the asteroids near the gaps is smaller than its mean value was remarked by Prof. Brown.'-* To verify this I have taken ten asteroids on each side of the gap and computed the mean angle of eccentricity as follows: — «o/« Urder Oiîter Inner Mean Diff, 2/1 1 7-49 6*09 . 679 + i-\o 11/5 ■6 9-08 8-57 8-82 + 0-51 9/4 6 7-16 6-24 6-70 + 0-92 7/3 4 8-94 610 7-52 + 2-84 5/2 3 8-41 6-54 7-48 + 1-87 8/3 5 8-15 7-57 7-86 + 0-58 3/1 2 7-89 7-87 7-88 •+0-02 7/2 5 6-65 9-47 8-06 -2-82 Mean 7-97 7-31 7-64 + 0G6 1) Science, .Tan. 20, loll. 3 Eesearches on the Distribution of the Mean Motions of the Asteroids. 49 The mean value of the angle of eccentricity of all the asteroids, 791 in number, is 8 '50. Accordingly the mean value in the outer portion is less than the total mean by 0*53 and that in the inner portion, by 1°19. Thus it seems quite certain that the eccentricity in the portion near the gaps, especially on the inner side, is smaller than that in the other portions. 60. In order to determine the types of the motion near the commensurable points I have computed the quantities (7, E, Eo, F^ and E% for each asteroid by the formulae — x=—-\ d=sXl-l') + (s-s')(w-l') no E.=Fo + ^^^îfl, {-Q^f^ (approximately) which may be easily deduced from the equations in Chapter II. The constants ;)i2,. were computed by Leverrier's formulée, and the elements of the asteroids, n, e, vs and /, according to the data in the Berliner Jahrbuch for 1917. 61. For the class 2/1 taking 41 asteroids with n—n^ within the limits— 35" and + 25", the results are as follows: — 2/1 «0=598 26 p/^^xlO^'rrrO-Tie No. Epoch m I V n xXlQSeXlO 6 CXlO* -Exl02 F0XIO2 FjXlOS Type 790 1914 VII 10-5 235 6 274°0 319'-0 564-31 -5-68 1-48 281-6 +30-99 + 5-97 52-03 371 Ri 76 1911 VII 6-0 87-5 309-6 227-5 564-54 -5-64 1-73 3021 +27-86 + ^I'o 4206 352 Ei 50 Art. 3. — K. Hirayama: Xo, Epoch -c^ I i n .rXlOSeXlO 6 CXlO^ -EX 102 FqX 102 FjX 102 Type 713 1911 TV 28-5 349-4 209-6 221°9 56533 -5-50 1-59 115-2 +3317 + 6-20 59-61 733 1912 IX 19-5 152-6 85 264 3 56613 -537 059 352-5 +26-33 + 393 37-56 225 1903 XI 5-0 2985 27-2 35i-8 567-59 -513 2-64 3361 +1595 +10-40 1379 528 1913 IX 130 52-2 9'6 2940 567-84 -509 0-21 1938 +2678 + 344 38-86 566 1905 VI 1-5 17-0 2603 42-6 570-18 -469 136 1921 +27-70 + 4-99 41-58 692 1910 V 30-5 111-8 194-2 194-1 57082 -459 1-65 2778 +20-10 + 578 21-89 168 1899 V 290 23-8 242-2 220-2 571-59 -4-44 0-76 1856 +22-97 + 3-54 28-59 466 1915 V 26-0 198-1 246 5 345-6 57595 - 373 074 113-4 +1517 + 304 12-47 643 1907 IX 12-5 902 95 1118 577-58 -3-46 077 2361 +13-82 + 2-90 10-35 525 1904 III 18-5 47 4 116-7 6-0 581-34 -2-83 371 1521 +22-09 +15-66 26 44 401 1913 III 170 239-3 164-5 279-0 58439 -232 0-49 205-8 + 7-28 + 1-79 287 745 1913 III 7-5 129-2 152-6 278-2 606-78 +1-42 0-90 85-4 + 1-71 - 0-13 781 1914 I 25-5 2676 136-1 305-2 608-78 +1-76 0-86 153-3 + 6-40 - 0-43 175 1914 I 11-0 330-5 90-4 3039 609-57 +1-89 1-87 1731 +11-55 + 224 723 S30 1911 IX 3-5 323-0 3237 2325 61021 +2-00 177 181-7 +1160 + 180 7-29 777 1914 I 28-5 167-1 1218 305-5 611-31 +2-18 1-46 379 - 020 + 0-68 O'OO 756 1908 IV 26-5 19i-8 217-6 130-7 61232 +2-35 120 151-0 +10-03 - 0-12 758 1912 VI 9-5 53-2 2562 255-8 612-61 +2-40 112 160-8 +1031 - 0-35 122 1911 V 70 189-9 2144 2225 614-37 +2-69 0*56 319-3 + 5-42 - 1-48 581 1905 XII 24-5 63-5 92-1 597 61596 +296 0-44 36-2 + 7-24 - 1-78 300 1895 Vn 10-0 325-4 302-1 1023 617-27 +318 0-43 62-9 + 927 - 1-93 318 1912 IV 110 78-4 186-5 2507 61767 +324 059 1235 +1191 - 1-81 108 1911 IX 210 164-9 3245 231 1 617-91 +328 105 21-2 + 655 - 1.08 667 1908 VIII 24-5 98-4 3347 1406 618-03 +3-30 171 151-9 +17*37 + 0 73 16-35 325 1913 XII 2-0 604 69-9 300-6 618 24_;' + 3-34 165 249-1 +13-68 + 0-50 10-14 580 1906 II 12-5 549 86-8 638 61861 +340 1-33 14-1 + 6-01 - 0.50 755 1915 VIII 140 217-5 3162 352-2 61988 +361 1-27 1893 +18-42 - 0-79 595 1906 V 18-5 289-5 221-1 71-7 620-18 +3-66 0-75 7-2 +10-20 - 1-88 645 1907 IX 29 5 899 14-6 113-2 620-25 +368 155 2381 +17-06 - O'OS 491 1903 I 00 41-1 21-8 329-1 620-55 +3-73 065 124-7 +15-51 - 206 331 1906 III 14-0 268-4 1749 663 620-62 +374 126 310-7 +10-46 - 0 90 236 1905 VI 7-0 32-8 241-8 43-0 62063 +375 0-13 1916 +14-61 - 248 702 1910 VIII 4-5 315-3 3160 199-6 621-86 +395 015 2621 +1570 - 261 696 1910 II 1-5 37-9 926 181-4 621-91 +395 2-41 121-7 +21-04 + 3-18 23-99 618 1906 X 25-5 3166 197 85 0 62209 +398 060 1963 +18 32 - 229 436 1906 II 20 15-4 1)6-1 629 62210 +3-99 083 355-7 +1237 -- 197 181 1910 XII 18-0 191-0 75-6 2108 622-48 +4-05 0-60 205.0 +18-74 - 2-34 92 1904 II 13-0 323 4 1059 31 622-68 +4-08 091 631 +14-84 - 1-85 316 1912 V 1-0 75-4 229-0 2524 623*00 +4-14 1-29 159-6 +2233 - 1-09 3-84 Ri 3-42 Ri 2-67 Ri 3-45 Li 3-51 Ri 2-99 Ri 3-20 Ri 2-60 Ri 2-48 Ri 3-13 Ri 1-80 Li 0-87 Rb 1-69 Rs R3 R3 La 2-11 R3 214 R3 1-55 R3 1-79 R3 2-03 R3 2-30 R3 1-70 R3 R3 Rs 1-63 ïl/jï 2-86 R3 2-13 E3 2-76 R3 2-62 R; 216 Kg 2-55 R,i 2-64 E, Rg 2-85 R., 2-34 R3 2-89 R3 2-57 R3 315 R3 Researches on tlie Distribution of the Mean Motions of the Asteroids. 5I Two asteroids (401) and (528) seem to make the libration of Type 1, but as the difference of E and F^ is very small the motion may be changeable to the revolution of Type 1 depending on the amount of the smaller inequalities. The asteroid (777) has a negative value of C and the motion is the libration of Type 2. All the remaining 38 asteroids make the revolution of Type 1 or Type 3. 62. For the class 3/2 six asteroids only may be taken. The results of computation are as follows: — 8/2 ^0=448-70 pf^'>xW=V^l No. Epoch -üJ I r n xXlOSfXlO 6 CXlO* £ X 102 FqX 102 i^^lX 102 Type 153 1911 III 28-0 282'-6 207°9 219°2 449-46 +0-17 1-62 40°8 - 1079 +257 1*49 Lg" 748 1913 III 8-5 103-0 160-9 278 3 451-35 +059 1-36 309-9 - 734 +1-65 361 1914 XI 270 93-4 61-6 330 6 453-60 +1-09 208 304-8 - 9-26 +3-96 190 1910 XI 8-0 103-7 71-0 207-5 45369 +1-11 1-67 343-2 -1287 +2-42 499 1911 I 30-5 92-5 112-4 214-5 457-15 + 1-88 2.14 338 1215 +3-95 334 1913 IV 26-0 8-5 225-4 282-3 45951 +2-41 0-15 332-4 + 4-63 -0-78 Five asteroids out of six have negative values of C and make the libration of Type 2. The asteroid (334) makes the libration of Type 3 with a positive value of C 63. Only one asteroid may be taken near the point 4/3 with the following result: — 4/8 no=898-84 ^i<*>x 10^=2*24 No. Epoch -m I I' n aXlOS^XlO 6 CXlO* £X102 F0XIO2 F1XIO2 Type 279 1913 VI 17-5 296°1 294°7 2867 397-60 -0-31 0-64 33°-4 -708 +0-48 028 Lj The type of the motion is the same as those of the five asteroids of tlie class 3/2, i.e. the libration of Type 2 with a negative value of a 0-69 La 1-10 Lj 2-13 L,. 1-90 L.., 0-72 Ls 52 Art. 3.— K. Hirayama: 64. For the class 3/1, taking 25 asteroids with n—Ua within the limits— 25" and + 25", the following results are obtained: — 3/1 710=897-39 p.py.lQ- =0-021 b No. Epoch -o I l n a;Xl02eXlO 6 CxlO* £Xl02 i^jXlOS F1XIO2 Type 765 1913 X 3-5 367 2 1°3 295°7 874-04 -2-60 2-81 287°6 +7'16 +11-36 43-4 Ki 714 1911 V 25-5 102-7 2142 224-1 874-17 -2-59 0-45 107-3 +6-70 + 3-65 40-6 472 1908 III 23-0 62-2 177*7 127-8 875-74 -2-41 0-98 278-7 +5-83 + 4.17 35-3 787 1914 IV 22-5 309-5 230-8 312-4 876-73 -2-30 1-24 2726 +5-30 + 4-60 321 355 1905 I 2-5 86-9 99-3 30-1 877-28 -2-24 1.08 182-8 +4-82 + 4-15 292 695 1909 XI 7-5 353-4 40-6 177*2 877-30 -2-24 1-56 215-8 +4'69 + 542 28-4 660 1908 I 12-5 2640 126-0 121-9 877-99 -2-16 1-03 288-3 +4-72 + 3-94 28-6 421 1912 VIII 29-0 34-6 349-8 262-4 87856 -2-10 292 351-8 +5-79 +11-32 35-1 292 1902 IV 40 331-4 206-7 306-6 881-55 -1-77 0-29 309-7 +3-14 + 2-44 19-0 46 1910 XI 28-0 354-5 62.6 209-2 884-45 -1.44 1-67 144-0 +1-71 + 4-71 10-4 518 1903 X 20.5 322-5 10-2 353-5 885-77 -1-29 2-23 314-7 +2-24 + 6-56 13-6 619 1906 X 22-5 2-4 37-7 84-8 886-62 -1-20 0-75 148-1 +1-35 + 2-16 8-2 132 1895 XI 30 5 152-4 123-2 114-2 903-69 +0-70 3-31 85-4 +0-64 +10-02 3-9 495 1902 XI 21-5 26-5 47-4 325-9 910-12 +1-42 1*47 202-7 +1-68 + 0-27 10-2 329 1901 VIII 27*0 217-0 337-1 288-3 912-13 +1-64 0-28 266-2 +2-69 - 2-11 335 1906 II 2-0 288'8 134-3 62-9 912-66 +1*70 1-80 163-2 +2-38 + 0-97 14*4 17 1911 VII 26-0 263-0 290-0 229-1 913-55 +1-80 1-33 128-7 +3-06 - 063 248 1905 VIII 6-0 247-8 319-5 48-0 913-94 +1-84 0-64 311-1 +3-43 - 2-05 556 1905 I 16-5 101-0 116-6 31-2 915-85 +2-06 1-01 225*0 +4-13 - 1-73 752 1913 V 10-5 105-8 212-5 283-6 917-80 +2-27 0-74 293-3 +5-18 - 2-48 623 1907 II 5-5 71*7 1230 936 91832 +2-33 1-15 3456 +563 - 1-79 650 1907 X 4*5 31-7 34-8 113-6 918-48 +2-35 1-87 117-4 +5-25 + 0-36 31-8 732 1912 IV 24-5 236-9 212-8 252-0 919-07 +242 0-46 290-6 +5-87 - 3*02 178 1910 III 130 261-3 1781 187-6 919-41 +245 0-45 137-9 +5-98 - 3-07 198 1910 VII 31-0 356-4 3106 199-2. 920*05 +2-53 2-23 65 8 +675 + 1-82 40-9 ♦ £sX 102=4-0 All the asteroids except (132) make the revolution of Type 1 or Type 3. Tlie asteroid (132) seems to make the libration of Type 2. But this asteroid has not been observed since its discovery in 1873 and is known as one of the lost planets. ^^ So the existence 1) Mr. D. Alter corrected the mean motion of this asteroid to 883'. 47 on the assumption of its identity with the object observed at the Lowell Observatory in 1913. See the Lick Obe. Bull. Nos. 275 and 285. Rl El El El El El El El El El El L. E;-{ 2*2 E3 E3 2-3 Eg 2-5 R.i 2-7 E3 3-0 E3 3-2 E3 Ra 3-2 Eg 3-3 E, E, Researches on. the Distribution of the Mean Motions of the Asteroids. 53 of the librating asteroids near the point 3/1 is doubtful. 65. Only two asteroids will be taken near the point 5/3 with the following results : — 5/3 no=498'55 p.}'^xW =0-^222 No. Epoch -uT I I' n a;Xl02öXlO 8 C'XlO' JE X 102 FoX 102 FiX 102 Type 522 1913 IV 6-0 13 228°3 280°6 513'62 +3-02 078 4-5 + 9-93 -0-74 1-4 R3 721 1911 X 18-5 29-0 19-2 2362 526-85 +5-68 1-18 14-6 +34-05 -1-13 2-6 R3 Both make the revolution of Type 3. 66. In the cases of the third or higher orders, the effect of the neglected terms being relatively great, it becomes very hard to determine the types of the motion accurately. As a rough approximation however the same method was applied for the classes 5/2 and 7/4, arriving at the following results : — All the asteroids near the point 5/2 (?io=747"'82) make the revolution of Type 1 or Type 3, except (464) which seems to make the libration of Type 2. But this asteroid has not been observed since its discovery in 1901 and therefore full weight cannot be assigned to its result. For the class 7/4 (?io==523"'48) two asteroids (522) and (721) only will be taken. The former seems to make the revolution of Type 1 and the latter, the libration of Type 2 with a positive value of C. 67. It is a well known fact that all of the four asteroids of the class 1/1 make the libration about the triangular equilibrium points. Hence, including this case, the results will be summarized as follows: — 1. All the astreoids with smaller mean motions than 600" make the libration and form a series of the groups near the commensurable points Ijl, 3l2 and 4j3. 2. The asteroids with greater mean motions than 580" do not make the libration (except a few cases which are mostly doubtful) and form a series of the gaps at the commensurable points 2jlj Sjl, 512 etc. g4 -^rt. 3. — K. Hirayama : 68. The first of these remarkable peculiarities may be accounted for by a consideration of gravitation only. Taking the cases of the first order, we have d = s{l-l')-{l-w) If we put l'=l in this equation, then l=w-d which shows that the conjunction of the mean longitudes occurs at the point l=w—d. Now in the libration of Type 2 with a negative value of C and in that of Type 3, the argument 6 oscillates about 0 and the amplitude of the oscillation is less than —^ . Hence in these cases the conjunction takes place near the point l^w, that is, near the perihelion of the asteroid. Contrarily if the type of the motion be the libration of Type 1 or the revolution of any type, the conjunction maj" take place near the aphelion of the asteroid. 69. The linear distance of the asteroid from Jupiter when the conjunction occurs at the aphelion of the asteroid is a' —a{\ + e) This expression becomes zero when a \ n'l The values of the eccentricity satisfying this condition are n 350" 400" 450" 500' e 0-11 0-21 0-31 0-41 This shows how the asteroid with a moderate eccentricity approaches Jupiter when the conjunction occurs near the aphelion of the asteroid. The eccentricity of the asteroid is generally Researches on the Distribution of the Mean Motions of the Asteroids. 55 smaller than 0"20, but this will not remain always small if the orbit of the asteroid be sufficiently close to that of Jupiter. So the motion of the asteroids with smaller mean motions, something like 450" or less, if the conjunctions occur near the aphelia, will be disturbed a great deal by the attraction of Jupiter. In the cases of the asteroids which make the libration about TT zero with the amplitudes less than -0-, since the conjunctions always take place near the perihelia, they will not suffer large disturbances and the motions will be stable. Contrarily the asteroids librating about TT, or making the revolution will sufïer large disturbances, since the conjunctions may take place near the aphelia. So their motions will be unstable. Theoretically speaking the libration of Type 3 is possible for any positive value of x. But if the value of x be great the range of JE for the libration becomes very small. Consequently the libration may be changeable to the revolution by a slight variation of ^ due to the smaller inequalities. The libration of Type 3 is thus practically impossible when x is not small. The fact that the asteroids do not exist at the intermediate positions is thus explicable. 70. The asteroids of the class 1/1, in spite uf the proximity of their orbits to that of Jupiter, never approach very near to the latter and consequent!}^ their motions are stable. This fact is in perfect agreement with the above explanation. An instance analogous to the librating asteroids of the 3/2 and 4/3 classes may be found in the Saturnian system. Hyperion, the seventh satellite of Saturn, has a period of revolution very nearly commensurable to that of Titan, the sixth and the largest satellite of Saturn. It has been shown in theory and by observations that the argument 4/— 3/'— 0?'-' of Hyperion makes a libration about tt, so that the conjunctions always take place near the aposaturnium of Hyperion. Consequently these satellites, in spite of tlie 1) Here I, ur and l' denote the mean longitude, the mean longitude of perisaturnium, of Hyperion, and the mean longitude of Titan, respectively. 56 Art. 3. — K. Hirayama : proximity of their orbits, never approach very near to each other. 71. To explain the second character of the distribution of the mean motions, I shall introduce the hypothesis of resisting materials. Assuming the existence of the resisting materials moving around the sun in circular orbits, it was proved in Chapter III that, in the general cases of the second and higher orders, the librations ultimately change to revolutions, while the revolutions do not change to librations. In the general case of the first order it was proved that the revolutions do not change to librations except that the revolution of Type 1 ma}^ temporarily change to the libration of Type 1, that the librations ultimately change to the revolution or the libration, of Type 3, and also that, if the constant a be sufficiently great compared witn /S, the libration of Type 3 changes to the revolution of the same type. Now in the class 2/1, the unique case of the first order with greater mean motions, we may naturally suppose, as it was noticed in § 9, that the density of the resisting materials rapidly decreases as the distance from the sun increases, so that a becomes great in com- parison with /9. Hence in general, in the cases of gi'eater mean motions, the librations ultimately change to revolutions, while the revolutions do not change to librations, except the revolution of Type 1 in the general case of the first order, which may temporarily change to the libration of Type 1. In the cases of smaller mean motions, we have sufficient reason to believe that the density of the resisting materials is very rare, as was noticed in § 9, so that the effect on the librating motions is insignificant. The fact that the eccentricities of the asteroids near the gaps, especially on the inner side, are generally smaller than those in the other portions (§ 59), may be explained in the following manner: — On the negative (outer) side of a:, the eccentricity cannot be great so long as the type of the motion is confined to the revolutions. Even when the eccentricity is moderate, the motion will sometimes change to the libration on account of the smaller inequalities and the asteroid will remove to the positive side of X. Hence in order that the revolution of Type 1 may be Researches on the Distribution of the Mean Motions of the Asteroids. 57 stable, the eccentricity must always be very small. On the positive (inner) side of x, the asteroids which made the libration previously with negative value of C remain near the critical point with small eccentricities, as was shown in Chapter III, The asteroids which were removed from the negative side do not stay long near the critical point, and therefore will not affect the mean value of the eccentricity, although they may have unusually great values. In concluding the present paper I wish to express my gratitude to Prof. Brown of Yale University, and to Prof. Eichel- berger of the Naval Observatory at Washington, both of whom read the first part of this paper and gave valuable and encouraging suggestions in connection with my investigation. ' Astronomical Observatory, Tokyo, Nov. 6, 1917. 58 Art. 3. — K. Hirayaina: List of Notations Used. Numbers refer to tlie articles in which th e notations are introduced. The paren theses enclosir g numbers indicate that tlie notations are used temp orarily. A Bo 19 Q. 1 . 56 a 1 a' 10 n-i6 " a 28 «0 18 Br C 15 21 B Bo 1 B.\ 10 ■ 13 »0 11 6(U8, 19 c 1 71 , •m «J ß 28 C, 21,24 8 1,(43) e 1 Wl5 , ß. 7 E 16 eJ S' T V (48), (50) 1,30,31) e' (30), (31) 60 35 61 83 Ps-s' ' il9, 24, 27 (6) £ 8, (88) e' 10 rj (43) E, ■ F. 1 ë (33),34,37 r 1 /? 12 Fo 21, 60 V, , / (38) s (6), 11 <9' 46,52 F, W 5 g 53 s' 11 Oo 19 H 34 X (6), (30) h 1 t 1 ^ 84 H, 85 Y (31) ^ (6),(12),52 u 87 6' 48 I^ Z (49) j (^).(12) lu 1 ^ (18) I2 (7) X 11 e (43) I? x' (30), (31) w 8 M 84 N 84, 54 ;h Xo X, 15 33 ^ 1 Po 2 P ^ Wo 1, 11 X (38),34,37 ^ P' y (24), (27) ^1 ^2 (13) Q'\ 1 Published March 30Lh, 1918. Jour, Sei. Coll. Vol XL I, Art. 3, PI. I. Z!.\^ — I ^|s+l K. HiRAYAMA : Eesearches on the Distribution of the Mean Motions of the Asteroids. JOURNAL OF THE COLLEGE OF SCIENCE, TOKYO IMPERIAL UNIVERSITY. VOL. XLI., AST. 4. Asymptotic Formulae for oscillating Dirichlefs Integrals and Coefficients of Power Series* By Moloji KUNIYEDA, Eigalcashi. Introduction. 1- G. H. Hardy, in his interesting papers f with the title "Oscillating Dirichlefs Integrals", has discussed almost com- pletely the oscillating nature of an integral of the form f^ Ax)^^dx (e>0), when X tends to infinity. Hereby /(a^) is of the form where p{x) and <^(aj) are logarithmico-exponential functions (or L-functions) and (y{x) tends to infinity as a;->0. As he remarks, the problem is equivalent to that of investi- gating the convergence or divergence of the Fourier's series defined by a function which has a single oscillating discontinuity of the type specified by , \ cos / ^ ü[x) • o), where, f and 1* as ä-^0, ^ being a positive number chosen sufficiently small so as to ensure that p and l{\lx). The results arrived at concerning the sine-integral are designated as theorems A, B, C in his papers. As will be seen from these results, Hardy has principally considered the cases in which the sine-integral oscillates as X^oo ; but he did not went into a minute discussion of the cases in which the integral tends to zero. It will be interesting to find asymptotic formulae ior S{X) in the latter cases; and it appears quite natural that the formulae obtained by him are also available to a certain extent in such cases. I have succeeded in extending the range of validity of his formulae considerably — roughly speaking, to all cases in which the order of S{X) is greater than -r-. * Throughout this paper, I will entirely adopt the symbols and notations defined in Section II of " 0. D. I. i." Oscillating Dirichlet's Integrals. 3 As for the cosine-integral C(>^), it will be seen that, in the two cases (A) and (B), it alwa3^s tends to zero as ^i-^oo, when conver- gent, and, in Case (C), its behaviour is very similar to that of the sine-integral. I have found for it asymptotic formulae whose range of validity is the same as that of the formulae for S(X). That the formulae should cease to hold when the order of the integrals sinks as low as -j- is to be expected. For, it is easy to see that the parts of the integi'als away from x—0 are in general of the order -r-, so that in such cases the behaviour of S{^) or C{À) is no longer dominated by the parts near x=0. 3- The principal results arrived at are as follows: Writing oix) = x-'^e^x), x' = x-«6> {x), x' 1), C(/l) r-^ / (-a) cos (ia;r) p(lß)e"^'''^ (-1 *tZ p = x-''0(x), x*<0<(}/xy, a£\, then, as ^-^oD , we have {S{l) = 0{)r'^') {a£-l\ ^ ^ \S{X) <-> -l\-a-hi)^in {i(a + 6*>} / (-l>1). In Case (C), Hardy gave formulae which were shewn to be valid when x^/a"< p < xa' . I have succeeded in proving that they are valid for x^a"\a'<^ p < xa' , thus extending the range of validity of the formulae considerably. It will be seen that this lower limit of P (namely p^x^'y"l<^') cor- responds to our natural limit y of the order of tlie integrals S{^) and C{X). Combining these results with Theorem (7, we obtain the theorem : Oscillatino- Dirichlot's Integrals. g.. The integrals SX?^) and C{1) are convergent when K^M <oo , when x0,p>0 and ;öl determined respectively by the equations da La[^~a'{a)]J ^ ^ It will be proved that as i^^-^Qo . The proof of this relation plays an important rôle in the discussion of Case (C). The integral /S{À) is still convergent when xa'/(e7r) p being any real constant. 1 have considered still more general cases in which the func- tion/(^) has a singularity of the following types: * Journal de Math. Série 3, t. 4 (1878) pp. 5—57, 377— él7. t „ Série 6, t. 4 (1908) pp. 203-283. X Comptes JRendm, 30 Nov., 1908 ; and " Äfiympti)tihux értékek megatnrjznsârôl " (1909) Budapest. Oscillating Dirichlet's Integrals. (i) A-) = -(TZ^v^'"""' ' A being a certain constant of the form Ä=ae''\ My results in Case (i) are as follows: — Let a>0 and p be any real constant, then the behaviour of ci^. as w->x , is determined asymptotically as follotvs : jy q = ], a = 7r, (9) a„ <- J- a-^'^-^e-^" 11^'-^ sin [2ahii-{^p-^)7:]* ; if 01 as a-->0. As was already mentioned, these integrals will be called " tlie sine-integraP' and " the cosine-integral " respectively. It will be supposed that ^ is a positive number so small that the range of integration does not * Me.isent/er of Math. Vol. XL VI (1916) pp. 70 — 73. t A preliminary notice of this Part appeared in the (,>uarti'rli/ Journal, Vol. XLVIII (1918) pp. 113-133. Oscillating. Diriehlet's Integrals. 9' include any point at which the subject of integration possesses any irrelevant discontinuity or other singvilarity. We distinguish as in Hardy's papers, the following three cases: (A) a < l(l/x), (B) l(}lx). II. Lemma for Case {Ä). 6- The proofs of the theorems A and B of ''0. D. I. 1." are principally carried out by means of H-lemma* 29. By examining the proof of this lemma, we can easily extend the range of validity of the formula given there. In fact, the integrals ./ 0 \ U J there considered, are absolutely convergent also in the case — I<œs0. Hence the argument of " O. D. I. 1." for the case 0'■''<*>, ;r*<:6'<(l/a;)*, ip{x)l) ; if a=\, b=f=0, this result (14) still holds, provid- ed that the integral ./o dt is convergent ; and if a = ^, è=0^ and T{x) is still convergent, III. Discussion of Case (J) : ^(x) < l{Vx). 7. At first we shall consider the sine-integral S(/l) =/ %(;c)e '■'<*> -?^5_^ dx. ./o X As is given in the paper " 0. D. I. 1.", the necessary and suffici- ent condition for the convergence of this integral is p{x) < a'{x) as x^Q. As in the same paper, hy perfornn'ng integration by parts, we have (15) = 0(l//)4-J(4 Avliere jK, =p — X{>', B., = x fja' ; (17) Oscillating Dirichlets Integrals. H and hence (16) J(k)==J^'^k)-iJ^'\X), Avhere Now Ave can Avrite Avbere a^l and x'0*; if a.= -l, B, = -x-S' = -xS,, B, = x'e, Avhere 6i=x6' is a function of the same type as 6 and 8i<6. Hence Ave have: (i) Let a^ — 1 Then, applying Lemma ] to the integrals (17), Ave obtain hence, by (15) and (16), S (A) = Oi^-'^'). (ii) Let — lî«^(l/>l)e*'(VA) ; and since ö<6', we have and hence 5f(^) ^ _r(-a) sin {Un) ,o(l/;)e»'Wi) , for, in this case, KV^) > -^ as >^>->go. (ill) Let a = 0 and 6'<1. Then B,r^ p= e, B.

l) = O(>î-^+0 (a§-l), SiX) ^ -r(-a)8iu (ia;r)/)(l//l)e''W^> (-l0.* Then, if a^O and a=k—l, if a=-l, B, = -^-p' ^ (a + l)x-<''+^>^ = (a + l)-^ ; X X B,=: -xS' = -e„ where Si is a function of the same type as 6 and Oi< S. Hence applying Theorem I, we obtain ( jw (X) = 0 (1) (a<-l), (a=-l), (-lî) a'(l/yl) e''^'"^ (a = - 1), j^'\x) 1) e'-t^^*) . (iv) Let a=0. Since 0<é?, by H-lemma 10, we hav^e T(lß), is deter- mined asymptotically by the follounng formulae ;* C{X) = Oilß) (rt< -] or ft = 1, xS'<\, pa^l), (4) [ C{X) ^ ^{^7:ß)[{\IX)e'{\lX)^ip{\l)^a'{\ß)}e''^':'^ («= — 1, a;0'or/>l), C{X) r^ r(-a) COS (^aTt) p(\ß)e''^''"^ (-l-00 Oscillating^ Dirichlet's Integrals. ']JJ Theorem III. The Inteyral where =x-''0{x)^ where x^ ^^^^ dx. Jo X Since the function ^ has the form (21), tlie conchtion for the con- vergence of this integral G{}^) is as a? ^ 0. As in Case (A), we have c(/) = o(i//) + J(;,\ where /(/) = 4- ['{-R-i B,)e'' -'-^-^ dx, / ./ 0 X B,=-^-p', B. = pa'. X ' ' If we write, as before, p = x'^d (x), wh ere a^O, x'<0<(\ jxY,* we have B,-iB.,r^ (a+l + bi)x-("-^'^e(x). * Observe that, when a=0, x^ -< © ^ 1. 18 Art. 4. -M. Kuniyeda : Hence, applying Tlieorem III, we obtain: Jf a<-l^ or if a=-l and 01. or if -l\). V. Examples^ of the Cases {A) and (B). 11. As examples of the two cases (A) and (B), we shall give some discussion abont the behaviour of the integral J{1) =re''' x'-'flocr —Y'dx as /^oo , where B(r)>0, E(s)>0. At first, we consider the case in which * Observe that, here also always t (>,) = o(l) as X-^X . fin the followings, I have given examples and verifications, quite similar to those given in Hardy's papers, for the purpose of parallelism. Oecillatjno^ Dirichlet's Integrals. 29 \r = —a, —1 <;a<;0, (22) (<; = « + mi, 0<«<1, m=^0. Then = I(/) + iZ(/), where 1(A) = f\-Jlogiy~'e^''-s'og(i!^cosh^ ^^^ J 0 \ X J X J 0 \ X J X Now = !,(/) + 7,;/) say. JCviclently the integral I\J^) is convergent, if a<0. The integral ^é.^) may be written in the form LI?.) =y"\l_:r)-«Ylog-J-y^^^^°s^«s [1/(1--)} cos }il-x)dx, and, when :^ is small, we have log^ = a-[l + 0(a-)}. Hence the integi'al I-i^^) is convergent if r/>0. The integral I\{'^^) may be divided into the two parts IXX) = (/"+/^'*)a;-'(log-I)"".-^i««-i^^(i/-'-)^^^rc say, Ç being a sufficiently small positive number. Then ./ 0 X 20 -'^i'*- 4.— M. Kimiyeda : log— j , ^ = mloglog(l/a;). Applying Theorem II, we obtain I^'{À)^ r(-a) COS (ki tt) /"(log /)-^e"^i ^o- log x (~l7i" as >^^x . Thus we obtain 2i(;)= r{-a) cos {\an)X''{\og /)-ie"iilt'^lûg;(l + s^) = r{r) cos {\rn) A'^ (log Xy-\l + ^,), where lim£^=0. The integral I-i{^) may also be divided into the two parts =j/(;0+i./'(/) say. As in the case of -fî'(^), we easily see that J/(^) = 0(1/;). * Hobson, Theory of Functions of a Real Variable, p. 672. Oscillatmo- Dirichlet's Integrals. 21 In considering the integral I/W, introduce tlie relations (l-x)-''-' = l + 0(x), Then we have L' (X) = f'x^. - -i i«s- (V-) [1 + 0 (x)} ""^^-^^-^^ dx ./ 0 ^ ' ■' X ^^r, ) /"* a - mi log (l/x) f 1 , /a / \-) COS yJ.iC -, = COS / / a:"V ° ^ ' M 1 +0 (a;)] arc ./ 0 a; I „,-., } /'* t -milog(i/.r) f 1 , n/ >f sin /a- , + sin / / a^ e °^ ' ' {[+C)(x)\ ax / 0 a- say. Then, by Theorems III and IV, we have, for 0<«<1, fx-e-"^' ^"^^ (^/^^ ^^— f?a; r-' r(« + mi) cos [i(« + mi);r] ;.-«ö-n^i log).^ /"Ve-"^^^°"(^^''^— ^--^x <-> r(6: + mi)sin [i(« + mi)7r] /-^-milog). Hence, we can easily see tliat j {?.) l2' as x-^x . Thus we obtain Z, (;.) = r[a + mi) cos {/ — ^(a + mi)?:} /r^c - mi log lÇl-\. e',) = r(s) cos (/-is7r) /-»(I + e',\ where liin ^a' = 0- 22 Art. 4. — M. Knniyeda Hence we liave /(/) = r(r) cos {irTr)}."- (log ?.y-\i+e,) + r(.9)cos(/-i-s-);r^'(i+0- Similar!}^ we can prove that Z(;.) = r(r)sin{h-7T)?r^ilog?.y-'(l + sn + r(.s) sin (^-i5;r)A-^(l +£/")• Thus we obtain J"(/) = A"^.îc'-^ (log — )' \lx = Fi^r) eW^i/r' (log ;0"^ (1 + 0 where lira £ = 0, lim £' = 0, r and s having the values of (22). 12. This result may be verified as follows. Hard}^ proved* that, if B(r)>0 aiid B{^)>0^ then, for pure imaginary values of t, we have = r(r)(-0-'[iog(-/)}'-Ui+^.) where (-^)-'- = exp[-Hog(-0] =exp[-r[log|^|-k.T7}], V = ex\y[ — s\ogf} = exp [ — s{logl^| + ^e;ri} J, *Proc. London Math. Soc. Ser. 2, Vol. 2, pp. 401 ai x('0 or -v-<0. Herein put t = '/d (/>0), then (-O"" = e^^^'/-^ r' = e-h^^-^ l-\ Therefore (23) /'V^^r'-^Aogi-Y'W = r(,-)eèr^^/-'-(log/0^-\l + £) where hm e = 0, ]ini s' = 0. This formula quite agrees with our i-esult obtained for the case in which r = —a, — l<;a<0; s = a + mi, 0?.a; (^^ = l\(i)A"' cos ^a- /■CO / x"'^ sill >^.a; (7a: = Fi^a)/."- wo (0 X /^ X Here again we have ohtained the result which quite agrees with the formula (23) fur /,..<(//). Oscillating- Diriclilot's lutegrals. »25 14- Finally consider the case in which (r=—a—hi, — 1<«<0, b=kO, (24) { In this case we have say. Then, proceeding as before, we easil}' see that the discussion of tlie integral Jp.) may be carried out by means of the integrals I'x-' flocrXyV {^' log (1/-^)+'» log log (i/.f)}/ sin /g- ^^^^ J 0 \ '^ X J X ' /■ X-" flog L)"", ('' 10«- (1/.^) + '« log log (l/.r)]. ^2i^ax^ J 0 \ X / X and that of J^^X) by means of the integrals r^^ ^-mi log (Ijx) siu/iT ^^^ /•V,-^»nog(i/.r)cos/^^^^ ./o a; Thus by another application of Theorems III and IV, we obtain + r(s)/-v^'-^''"^'(i+s' , where liin e=(), lim e'=0, T and s having the values of (-4). This result is nothing but the formula (23) for the case (24). IV. Lemmas for Case (C). 15. Among the lemmas given in the paper "0. D. /. ^. ", the most important ones are H-lemmas 32 and 33. They give 2Q Art. 4.-M. Knniyeda : important properties concerning the variations of the functions P(x) p{x) x[X-a'{x)} ' œ{). + a'{x)} for .-^ufticiently large values of /, provided that l{^/x), and X < fj < xff'.^ I will give two more lemmas of a similar nature, concerning the variations of these two functions, for the case in Avhich l{\lx), and x' /(i/a;). (i) If f> = Ax{\-Jix)], where A>Q, p>0 and o<\, then the function > 0, a > 0. Wc can easily see that, liy this assuiiiptiou, no loss «if i^euerality will l)o introduced. Oscillating Dii-ichlet's Inte<;-rals. 27 X-a'{x) since «7'<0, (7'>1, Hence the first part of the lemma follows im- mediately. Now consider the second case (ii) of the lemma, in which p = Ax {l—i'ix)}, where A>0, p>0, p we obtain ï< — ^and, by differentia- tion, a'-r'0, /><], /.'>0, iind hence V<1- Therefore a" -> x-f/ X whence it follows that, for sufficiently large fixed values of >'., the equation (25) lias one, and only one, root. Thus the function

) a:wxär" and, by (25), ;,-.'(«) = ^i^. Hence ^<^"^ = ^^'- and .,/(.) =-_^(^) = ^:-^, which proves the equation (27). Since — 7->i> we see that f.«) tends to zero as /^oo r 'J'he ])roof of the lemma is thus completed. 17. Lemma 3. Let rr>i{llx). ( i ) //■ r = Ax[^ +7'(x)], where A is a positive constant and then the function ^ x{X + a') Oscillating Diriclilet's Integrals. 2 0 has, for sufficiently large fixed values of X, one and only one stationary value in the ranye 00, p^O and p0, if x>d, 6 being the root of the equation A + a'{d) = 0. ( i ) At first, consider the case in wliich !' < X, and write p = xy, so that r > 0> r < 1- Then , = ^, ^^'^ ' L = ^ gives dx (28) x=''":i-a'. Now o'<0, a">Q, r>0, ^'>0, -^>0, r so that we have 30 -^Tt. 4. - M. Kimiyeda ; r whence it follows that, for sufficiently large fixed values of ?., the equation (28) has one, and only one, root a. Thus the function (f has one stationary value. By (28), we have which proves that « > ^. Also we have (29) ç(o) = ^^^ = - f'(''^-"-f''(^-^ ^""^ ^^^ a" {a) a'-' a" (a) Since /' <^"> — a"? whence it folloAvs that ^-7r-\ < ^^ /''(*) co . Next consider the case in which where ^ > 0, J>>0, ]><\. and -^ — 0 gives (30) i^^^lSUflL-a'. I' Now o'0, J»0, //>0, Oscillating Dirichlet's Integrals. 3]^ SO that we have whence it follows that, for sufficiently large fixed values of z'., the equation (30) has one, and only one, root o.. Thus the function

6. Also w^e have (31) ^.->co . (ii) Let p = Ax{\—]> x)], where ^ > 0, p>Ç>, p< 1. Then f = ^^^^'') and -^ = 0 gives If we write ?. + <7' 7 p = a y, so that r <0» (^'r < 1, then we have 32 -^rt- 4. — M. Euniyecla (7" + n' (32) / = - / From the relation a'y<,\, we obtain Hence --j—^ "^ r > '^' since a" > 0 and o > 0. Tlierefore the right-hand side of (32) is ultimately negative and hence there is no stationary value of ^. A If /o = 0, then f = yip^' and evidently there is no stationary value of d from above, (p^ + ^ . Thus in the case (i), the function ^ is a steadily decreasing function of x in the interval 0f(y) > (Vyy> y >Uy) > (^lyY> My)-^f(y)>^^ as î/->cc . Jf y=6 and y='di are respectively the roots of the equations yfiv) = ^> yfiiy) = c^, for large indues of /, c being a, positive constaid independent of ^, then Oscillating^ DirichlctV Iiite«,'rals. 33 d,f^ cd US /->x . Proof. Evidently yfiu) and (jfi[y) are ultimately monotonie and tei:id to infinity as .?/->x). Hence each of the equations yf{y) = À yf:(y) = c/ has, for sufficiently large values of ?•, one and only one root which tends to infinity as À->:c . By hypothesis, we have Of {6)=^)., dj\{d,) = cX; and, since f{y) r^ f(y), we have f((fô=fi(fôO+'\ where 6->0 as Oi^-jo or À->-x: . Hence we obtain Let V be a function of / sncli that Then we have (33) c.f{e) = rjf{d,){\+t), where e^O as I-^od . We have to prove that Tj (^ C as )-^ 00 . Evidently ^ is positive and continuous for all sufficiently large values of I, and it might tend to infinity or zero, or might oscillate finitely or infinitely as X^x) . If we suppose that iy>l or f) oscillates in an infinite range of values, then, corresponding to any prescribed positive number P, however great, there will exist a sequence {E) of values of X tend- ing to infinity, namely, "34 Art. 4.— M. Kuniyeda : {E) : /;, I,, , >^,„ (lim/„ = oo), such that, for every À„, we liave all values of ;. iu (U) being greater than a certain positive number A which can be determined corresponding to each given P. Since ^ = y > 1 as ; tends to infinity, taking the values of the sequence (^\ we can always choose a number a such that 1 < a < /^ < dja. Easily we can see that H-lernma 24 is available in our case. Hence we have /W=m/-^)^l Therefore, by (33), we have or (34) \-^f{-^){^+^)P, --y/(:y) >//-*> P- -i and the value of P may be chosen as large as we please. Hence neither of the inequalities of (34) can be true. Thus ^ cannot take values which become indefinitely great as )^-^a:> . Next, if we suppose that 3y->x, then, corresponding to any prescribed positive number p, however small, there will exist a sequence {E) of values of )~ tending to infinity such tliat, for every value of / of this sequence, we have Oscillating Dirichlet's Integrals. 35 Ti ^->oo . Therefore there must exist two certain positive constants j? and P such that But in this case we have as ^->oo {or k^cc), since ''/>f(y)>{Vyy as //->x.* Hence, by (33), we obtain whence it follows that 7] r^ c as i^-^oo . Thus the lemma is proved. Let f{y), fx{y) and c be the same as in our Lemma 4, then we have the following corollary, n denoting any positive constant. Corollary. If y = d, y = d^ are respecthwli/ the roots of yf(y) = ^^' y%(y) = cÄ for large values of X, then * This can be easily shown by means of H-lemma 18. Qfi Art. 4. —M. Kuniyeda : 1 as X^ X . The trutli of this corollary can be inferred immediatel}?- by writing our equations in tlie forms 1 1 1 y [/(>/)] " = / " , y [/(y)} « = c " ;. n . 19. Lemma 5. Lef. fix) and fi(x) be L-functions such that />o, /;>o, />/>! as x^O. If x = d, x = Oi ((re respectively the roots of the equations f{x) = K Mx) = i for large values of X, then e > d, for every sufficiently large value of X. If we notice that/ and /i are ultimately monotonie and/ 7(l/a;). 20. We now pass to the discussion of the behaviour of the integrals* C(h = f l>U)r^''^''>-^-^^^dx, Jo X . s (/) = f'o(x)r:<'^^^^^^ dx, ./ 0 X as à-^:d, when / (1/a;) < ^ < (l/a;)^ It will in this case be conve- nient to separate the real and imaginary parts of the integrals. Thus we have to consider * Although the sine-integral S{\) has already been treated by Hardy, we shall discuss it again, reproducing briefly his analysis, because, for the purpose of this paper, it is necessary to modify his argument and to extend it to the case x'-^p^x, while the same argument applies to the discussion of the cosine-integral t'().). Oscillating Di ri chiefs Integrals. g^7 I Z, (/) = /' V.t) cos a(x) ^^^ dx, L (/) = f '(>{x) sin a{x) -^^^ cZa-, 7. r^l/) = / ('{x) cos ff (a-) '^ dx, IM) = / (>{x) sin a(x) ^"^ ^'^^ tZa-. ./ 0 a:" ./ 0 a:; All these integrals are convergent if (35) ' x'{x) y H-lemma 32 and Lemma 2. we have to separate the following cases. 38 Art. 4. — M. Kxiniveda: (i) Let o0, p^O, ^<1. Then

= ^Ja'(l-^o), where ^>0, /^>0, ô<]. Then eing the root of the equation dx \\q now write •^=« = (/-I + ./■' ) .{ÄW] "^ ' "■' = ^='^^=" say, where /9 = la—a{ri). Tlien, by another application of the Second Mean Value Theorem, Therefore J-! = ^^p^^^'^^^^^j • 0 (1) = 0 [^.(a)} . Similarly we obtain Hence we have The same argument applies to the integral J^i^^)- Hence, if p0, ,tj^0, /^ < 1, Oscillating Dirichlet's Integrals. 39' if x = Ax (I — f>), where ^4>0, ;f;>0, />0, tl. Since xû" ^ At' in our case (C). the last condition of (38) is equivalent to £->/V^'((?). Let ^'(^)= --^-, so that u(x)>l, and lei o{x) = xo\x)i^{x), so that v(cc)< 1. Then the above conditions (88) for e are equivalent to d (380 s>Ö, e), where ^>0, ^>0, /><1. In this case, by H-lemma 33 and Lemma 3, the function (39) \ where yl > 0, <, > 0, i> < 1. In this case, by Lemma 3, the function . ./ 0 ./ 9-: ,/ e c J a 23- Integrals J/'^ and J!'''\ As x increases from 0 to d-z, the function y—Xx + o{x) decreases from x {o k{d — t) + (T{d — t). which is large and positive when 0 is small and £ smaller. Also //'> = /"' r - , . '"^^T-l «*« '/ ^ '/• ' .//(»-c)f»(fl-c; L x[Ä + a'{;x)} A ■' ■' The factor— cJ.L o which multiplies cosy is i)ositive and x{k + o'(x)] ' -' ' monotonie, as we have already seen. Hence, by the Second Mean A'alue Theorem, we obtain Oscillating Diriohlet's Integrals. 41 / , < K r ^'^^-'' 1 wliere 0 < e, < e. Now p, a and all their derivatives satisfy the condition and so each of them satisfies the relation if <{> 1. Therefore, in both of the cases (i) and (ii), we have 7 (1) V l'{d^ J (3) . (>iP) 24- lo the case (ii), we have assumed the relation d + e < a, which may be proved as follows. 42 -^rt- 4.— M. Kuniytnla : When /' < .T, X = a is the positive root of tl^e equation where {> = xy{x). This equation lias, for sufficiently large fixed values of /, one and only one positive root a. Hence tjie function /(re) changes its sign when X passes through the value o.. Now, since 0, yip) > 0, y\d) > 0, a\d) + / = 0, we have --^^^^':^^I^^^^ -s. -(. + ,) (0<, 0, and therefore f{d + e) > 0. Thus f{d) and f(ß + ^) have ultimately tJie same sign. Therefore it follows that 0 <: d + e < a. Next, when ,■' = Ax{l—J)), r = a is the positive root of tjie equation /(.r, = {1 + v{^)Yy^^ - '^'(■^)- / = 0. Oscillât) n0. since o"{6) > 0, Y/(ä) > 0, J>(d) < 1, a'{d) + / = 0. And fie + e) = [ 1 + r(6 + s)] i^'l - 0, ;>'(/9) > 0, £ /7(^) < ^(^). Therefore it follows that 6 <:â + t< a. 25- Integral Ji'\ We now consider the integral J-/--') = /' '^ 'iM. cos .// clx = J/ + J"/', where f J' = /"^"''^''^'^'Hpixl cosy \ . \ ' Jß \ X ' J+(t\x) I ^' , ' .//Î \ X à + (t\x)} -^ ß = Ad + a{d). Now let us consider the following difference of integrals (42) i' = / ,/'-7 , cos 2/ ^y which may be Avritten in the form (43) J = ;■_,.;, cos ij dy + ^-^-^ / yXti) cos 7/ dy. + (y\x) 1 1 where r(^) = ^, Z(^) - ^^^ - -^^^^.-.^^j^^^. 44 -A.rt. 4. — M. Kuniyeda : By the analysis of §§ 33—35 of " 0. 1>- I. ,V.'', we see that /•!r=9+£. coaydy K where 0 s e^ s e^ s $. Hence we have If we put r=/-"'"iMz^cos,.,, then as hi " 0. 1>. I. 2/\ we obtain and hence l^'l- /,v//jn ' A' £ r(d) and, if x< p < xa', e r'((^) < rid). If x'tain Oscillating- Diriehlofs Int-e^'rals. 45 Hence, by (42), we have T,_ p{d) f /•^("^^H-'C''-) cosy , ^ ,,.1 Similarly we obtain ru_ m / /■^'^^--)^-C'>-0 cosy , ^ ,.A Now /(^ + e) + a{d + e) = ß -\- is-V"(^ + e,), where 0 < e^ < e, so tliat a"{d + e,) ^ o"{d) ; and, by (38), t-a"{d) > 1. Hence we have /•A(«+e),.(no cosy , f^ cosy , A v(^'-'^=.A v(F^'^^^^^^- Therefore we obtain Similarly Hence, in both of the cases (i) and (ii), we obtain 26- Integral J'/^\ Finally we consider the integral Ju X '^ J x=a. x[X + a'{x)] "^ -^ In this integral, ^o^^y//^)-) i^ a steadily increasing function of x in the interval a+i::V^ + o(l)]; in the case (ii) Similarly we obtain : In tlie case (i) "^■■'^'^ = öJ{'2%-)Y ^'"' ^'^^'^""^ ^""^ '^^^^ ' in the case (ii) I Now we can state Theorem V. //" ,/■' < /> < a- or i> = Ax{l-\- jix) ] , where /i > 0, ()>Q and /><1, ^Ae?2 '^-^^^ = dJito'Xd)} ^"^^ 0?+l^)v/^+o(i)] + 0(1/;.), /,(/) = o(i/>î), '^'^'^ = dj[2a''(d)} ^'"^ C/^ + i^) V:r + o(1)] +0(1//;), /,(/) = 0(1/^); //* x-0, J>>0 avd /7< 1, then "^"^'•^ = lV0m ^"" ^''+ '"^ V-+ »(')} . '^A-^) = ^;,^ ■ 0(1); Oscillatiu'^- Diriohlet's Integrals. 47 finally, if (> = Ax, ivhere A>0, then a?ic?, i/i these formulae, 6 and a are functions of X deter mined respec- tively by the equations da La[À — (T\a)\ J and ,'i = Àd + a{a). Corollary 1. If x^<(>{x)], where .-I > 0, p^O and /><1, then if x<^x<7' or (I = Ax[\ — J){x)}, ivhere ^ > 0, /7>0 and J> <\, then «« = ev\Sw\ i--^"V^+o(i)] + -^^£(^ . 0(1) ; d, a, ß being the same as in the theorem. By H-lemma 32 and Lemma 2, we know that a[k — a\a)\ ^ 1 { fi\ as A-^oo, and evidently y^O as /^x. But o/\i2,a"(d)\ ^^ver tends to zero as A^oo, if f'>^x^a". Hence we liave Corollary 2. If x.ja" <_i,1/;. and S(Ä)>lß as /î-^oo. 28- For the purpose of this paper, it is necessary to compare the order of magnitude of J^ p{a) p{d) as >^v->oo, when p<^x^a". At first, we consider the first and the last of these functions. Now, since X + a'{d) = 0, we liave Hence, if /' < x^a"la', then if py- X's/o"l(r', then P(d) ^^ 1 pm vi We know that, if o > l(l/x), then n' > ^cr". [H-lemma 31] Hence, if i' > a:, tlien pW >1. Therefore we obtain : Oscillating Dirichlet's Integrals. ^4.0 If P(); if x^a"la' •< ^> < X or if p = Ax{\ + ''p{x)] , where A>0, p ^0 and p <1, then the formulae (45) hold, i.e., C{?) S{X) p[d) d^['la"{d)] f>m ,yH')i^^ e<^-i''^V", (A-^od). 29- It now remains to compare the order of magnitude of f(«) = - ''("^ a{k-a'{a)\ 4.ß) = P{0) d^\:2a"{d)] as -^-^00, when x0, p>0 and p < 1. When xl, if a = — 1. Under the supposition that 1{\Ix)l, if 0 = 0, since a' >■ 1/x. From the condition pKX'y/o", we obtain a<^h, or a = ^h, e-l. At first we consider the case in which >0 or & = 0, -l0 and 0,>O, which follow from the assumption p >0 and t >0. Oscillatinor Dirich let's Integrals. 51 and ^« = -""-"1^) since ß(a) r^ e{cd) ^ 6>(^), Hence we have K«)--f±ic'-"rW = A-/J (ß) Therefore K^^^^ Vl2a^mx . Next consider the case in which h = Ç>, a = 0. Then we have f p = ß(x), a'= - X-' e,(x), \ e < ^e„ a, > i. In this case 6 a'r x(Ä-ff') l-a' ' Hence r = — -^, ocf (T"(^—x'^a'. ' -' a' — '— -<^ a aY + a"r 52 -A^ft. 4. — M. Kuniyeda : and the equation (48) takes the form where t «^ — — < 1 r as x->0. As X = a is the root of this equation, we have -i- = _ t{a) < 1, < V^i» ^-^ > 1 and a < 6 ; and Oscillating Dirichlefs Integrals. if Or^ A, %1 ^ 1 . 0(0) 8(a) ^ • Hence it follows that in all cases we liave Thus we obtain ^(«) ^ ^,cß^ as -^->oo , 31. (ii) Let a= -1 and & > 1. In this case we have 53 ^^"^-^=^ gi^-e^^ / — ^»6»'. Hence the equation for x = a takes the form -a'(œ)t(œ) = ;., where ^-^ - (l + ô)-|^ > l. Hence we have >^ " t(a) whence <1, '{a) an<^l a>d by Lemma 5. Now é(d) = i^(^) - ^K&) 54 Art. 4. — M. Kuniyeda : and hence 6. Thus we obtain f («) < (/>{()) as /-^ X . 32- (iii) Let fj = Axll—J^x)}, where A > 0, Ji > 0 and /^ < 1, In this case and # = 0 gives fZa- ^ - -^^'- ' In § 16, w^e have seen that — ^^r-^ + ^ '^ —r . xr' < 1. p Hence, observing that xn" ro _ (^i + ö)^', we have p Xft' Therefore the equation for x = a takes the form - 1 ; and hence XfJ J t{a) ^ ' whence f («") = —4 — /5 , ^ f^ A^ Oscillatin<( Dirichlet's Integrals. 55 Now m = tf^f2%)j - vpi^wr ' and hence, observing that ;. = — (T'(d), we have y(«) V{^rr-(d)} __ V{2 =^ Ax[\—{>{x)], ivhere A>0, p>0 cuid |ö < 1. Then a{l-a'{a)] ^ d^{1o"{d)] as À->cD , a and d being functions of À determined respectively by the equations da L a[À — (T\a)\ J 33- In the above arguments, except in a few special cases, the whole thing depends on the fundamental Lemma 4. We can also prove the same proposition, with exception of a few special cases, by a more direct method without recourse to this lemma. The principal object of the method is to find such asymptotic expressions in terms of À for the functions (d), which are of convenient forms for the purpose of comparing their order of magnitude as >^^oo . The analysis is not very difficult and I content myself with giving only the following results. Du Bois-Reymond proved^ that, // y be the root of the equation * Math. Annalen ßd. VIII (1875), pp. 394 et neq. Du Bois-Eeymond does not state clearly the conditions to which his functions are subjected. 56 Art. 4.— M. Kuniyeda : yfijj) = ^ where f {y) Is mi. L-f unction such that if ^ f{y) > {VvY ^-^ ^-^^ > then, for large values of ?^, we have ivhere v is a certain function of ?., tending to zero as /^x . Easily we can prove that >' > {f {?)}-'"> {W)' as ;.->oo . Hence we liave Lemnia 6. Let f{y) he an L-f unction such that //>/(?y)>0/?/)* as y-^oD . If y = 0 is the root of the equation yf(u) = ^^ for large values of ?,, then t) can be expressed in tlieform where g is a certain function of the same type as /, namely, 'f>g{y)>{m' as y-^cc . As before, write f, = rr-" ('Kx), n' = - œ'^'^"^ ('J,(x). Then, applying Lemma 6, we arrive at tlie following result. Let X < p < xy/a" or (> — Ax[l-Ji{x)}, where ^ > 0, J> > 0 and ^ < I. Then, if b > 0, ive have, for large values of /, ^C) = 4;.%)i = '^-'■^('•^■ Oscillating Dirichlet's Integrals 57 1 where /^i = ?^ ^^^ , and y^ h are certain function î< satisfy ing the condition ?/>f(y)>(Vuy . i as y-^ao . ' Hence we have ! as ?.i^cc , since 6 > 0. Thus we obtain as À->oo . 34- We can now state Theorem VI. The integrak SU) = r p(x) è"^'^ ^^^^ dx, Jo œ Ca) = f ' o(x) e''^"> _S2?^ dx, Jo X ivhere l(lfx)<(T <(i/xy' and poo , is determined asymptotically as follows. If x^(x) ßi'M^}^^:^ dx, ./ 0 X where are L-functions such that (51) iQ/x) IIX; (ii) a^AUAlx), p!/;.; (iii) (T > /(I /a-), Xy/(T"la' < p < xo'. In the followings it will be seen that the same relation holds also in our case (51), except in a few special cases. Hence the above assumption seems very likely to be admissible. 36. With the above assumption, Ave can prove the lemma. Lemma 7. If S(ä), S(?^) are the integrals of {50), then m < si^) as ^->oo. Proof It is convenient to separate our integrals into the real and imaginary parts; the same methods appl}^ to both parts. Thus we consider 60 A-rt. 4. — M. Kuniyeda : lU) = I t/^x) cos fr(x) ^—âx, Jo X I(^X) = I J>{x) cos(t[x) — - dx. The?>e integrals are convergent, if f> < n'. Put K^) = 44 • Then t{x) is ultimately monotonie and tends to zero as x^O ; and we assume that ç is chosen sufficiently small to ensure that e(a;) is monotonie in the interval 0 < a- < c. We may write ^(''0 = / K^) K^) *2C>s (t(x) dx = I ^(x)f(x) sin ÀX dx, ./ 0 where j,.^ ^ p(x) cos ajx) ^ SO that f(/) = / /(«) sin ?.x dx. ./ 0 Now, corresponding to any prescrihed positive number d, however small, there always exists a positive number ?', independent of /, such that 0 < £(ç') < O, (0 < ç' < ç). We have /(;.) = (^ r + / '\ e(x'f(x) sin Xx dx say. Then, in the integral J"(2)(A) =y ' e{x)/{x) sin Ax dx, Oscillating Dirichlet's Integrals. Q\ the coefficient of sin ax m the subject of integration is absolutely integrable in the range of integration $'^x^^. Hence, by a well-known theorem,^ we have J^--')(/) = 0(1) as ?^->:c . In the integral J^). Hence y(;0 = i(/i) + 0(1). As to the integral ;'(A) = /' V(^) sin /a- ^a; (0 < e, < ç' < ç), ./ 0 we observe that the upper limit ç^ of integration is a function of ^, and it may be inferred that |/(/l)iK\i{x)\, and I j-(i)(/i) 1 = I e(ç')[y(/)-/(/)] I > K\m\. Therefore it follows that \s{r)\> K\s{X)\, contrar}'^ to our assumption (52). Thus w^e have |/(A)i(/Î)| < oZ|7(/)i +0(1), and \î{k)\oo . 37. Now we consider the integral S(?) = f\o{x)e''^- / 0 )^in^^^. Performing integration by parts, we have ! a X 0 ./ 0 ax K(T X ) Since p < (t', we obtain (53) Si?) = 0(1) + C,X?) + i S.i?), wiiere ( c,i?:) = i?rJ^e-^^^^dx, J 0 (T X (54) I S,il)=fj.,é''^^ dx, ^ ^' dxXxa') xo' ^ a {a J ' Then, in the integral Ci(/), we have X < -i-.- <. XG , — a since a, /> satisfy the first two conditions of (51). Hence Theorem VI may be applied to this integral. Thus we have CI?) = i? m _-^[.^-^^)^V:^ + o(l)], -da'id)^[^Äa"id)] where 1 as -^->x, since xo'^p. Now take the integral Jo X where xa' ^ a' {aj If we write (56) I ;, = x-^ 6, a^O, x' 0. And we observe that, if b = 0, since a > IÇl/x). From (56), we liave x(t" ^ _ (i + h)a', and ( xp' ^ - ap (a > 0), I x(/ < p (a = 0). Hence iC 1. Oscillating Dirichlet's Integrals. ß5 Now S{?.) cannot tend to zero as >^->x. For, if S(À)<^1, then, by the relation (52),'^ we have S,(^) < 1, and, by (53), CiU) + 0(1) -< 1, contradictory to the above result Ci{?.) > 1. Thus S(À) does not tend to zero as ?>->x>. Hence, by Lemma 7, we have (58) SiÀ) < >S(/l) as iî->x. Hence, h'om (53) and (55), we obtain which is nothing but the formula (7) in our case. Thus we may state, by combining this result with theorem VI, Theorem VII. The integral Jo X where i{l/x) < o < {VxY and p <, a', is convergent. The behaviour of S{^), as ^->30, is determined asymptotically as follows : If x'< p < x^a"lo'^ then if x^o"la' < p < a', then (7) SU) r^ ^ é^''^"^' ^t: where ß z= Xd + a{d), and d is determined as a function of X by the equation o'{d) + ; = 0. * Here S'](),) is replaced for S{\). 66 Art. 4. — M. Kuniyeda : 38- We now pass to another proof wliicli is quite independent of the assumption (52). We have to prove tlie relation (58), or as ^->oo . At first we consider the case in which h > 0. Now in the integral ./ 0 X we have f'l pï < (ci-h)^ {a^h). xa XCF (a = h). By (56), xy = -x-''6„ Hence, if a — b < b, we have xV^" f — _ a-- (■"-''> xo 6», • l'y < -^ < xa'. xa and Theorem VI may be applied to tlie integral S^{}). Thus we have and, as f>i < f», we obtain s,'/) < c,(;o. II a — b^b, then, by performing integration by parts, we ol)tain S,{^) = 0,\) + C,().) + iS,(?.), where Oscillating Dirichlet's Intt^urrals. 67 Sj)^) ^ f - sin Ix J 0 ^>,e' - dx. Now ^' xa' ^ a' {a'f ' fK.^-(a-'2b)^ (a > 26), X(T h xa' {a = 26), and If «-—26 < 6, then (a-..)_^ -^^(a-l).- ^. ,0, -• -^ < xa\ ■ — a-ff and, by another apphcation of Theorem VI, we obtain since S,(?^) < C,iÀ) < Ci(/Î) as ?.-^-c . If a— 26 >6, then repeat a similar process. Since 6 > 0 and b ^ a ^ l+b by (57), there exists a positive integer n such tliat (?^ — 1) b ^ a < nb. Hence, after repeating n times the above process, we are led to the equation wdiere Jo X /'. < ^^ ^ -ia-b) (a -26) •• -(a -71-16) a'-^"-"''^^ ^ — xa u{' eg Art. 4. — M. Kuniyeda : Since a < nb, Theorem VI may he applied to the integral S„(^), and we ohtain Since fj > pi> ■•• > fn-i, we have CiW > a(/l) > ••• > C',.(/), and hence we ohtain Thus, in the case b > 0, always we have Thus the proof is completed for the case b > 0. 39- Next we consider the case in which 0 = 0. Thus a'=-x''H,, ('A>1, p = x'" 0. We observe that, if a = 1, 6» < Ö^ ; ifa = 0, 6/>0i. In this case xa' = — 0i. (i) If a > 0, we have n xa T + —V ^7^^ <^ — « a; Kf ^ ^'^ W and, for any positive integer n, Hence, if a>0, then the method of the last paragraph fails, (ii) Now consider the case in which a = 0. Oscillating- Dirichlet's Integrals. ß9 First, let 6 '-^ Aß,. Then 0 = Ad/^l + elx)}, rv where e(a-) is an L-function such that e(.j;) < 1 as .r^O; and we have sm?..x 7 , A f^.n ia sin^a? S(^) = Af d^é'^^^^^dx + Af eR, / 0 X / 0 e^'^^^^^^dx X say. Then, performing integration ])y parts, we have /,(/.) = 0(1) -U r é' cos Ix dx J 0 ^ ^^^)-'^ Vi2^"(^)] '^^''''^''^^ + ^^^)] [by Theorem VI], In the integral LC/'O» we have Hence, by Theorem VI, or I^f.^) = 0(1/;,) ; and we have UX) < u?). Therefore since ^ i-' ^ 0i. Thns, in the case when a = 0 and 6 r-^ Ad^, the truth of the formula (7) is proved. Next let 8 > ßi. 70 Art. 4.— M. Kuniyecla: Then in the integral ./ 0 ax \ x(7 j we have _/^ _ _ _^ v. i Let UP write Six, 1) ^ _ d f 6 \ Ô»! "^ dx\ äj' SO that ^ (^' 1) = ^ {-^ - -^} < ^ since x6' < S and xO^' < 6*1. We observe that, since -s- > !■ we have Ö(a?, 1) > 0, and 6{x, 1) is a function of t]ie same type as 0, namely x^ < 8(x, 1) < {l/xf. Thus we have ./ 0 .X wJiere ,oi = — ^ ' ^ • If 0(a;, 1) < (9j, then /^i < xa' and, as before, applying Theorem VI, Ave see tliat If 6(x, 1) <-j A6i-, then, by proceeding as in the case 6 f^ AS^, we easily arrive at the same result. If 6{x, 1) > &i-, tlien repeat a similar process. Thus we have to consider successively the functions 0(x, 1), 6{x,2,), , 6{x,n) defined l)y the equations Oscillating Dirichlet's Inteijrals. 71 f 6(x, 1) _ (59) e(x, 2) _ d [ 6{x, 1) \ (^(x, n) ^ _ d fd(x,n-l)\ ■ V 6»!" * dx I öl" J ' where &{x,n-l)> 6{\ We easily see that 6 > &(x, 1) > 6'(a:, 2) > > 0{x, n). There are two different ca?es. (a) For a certain integer n, we have o(x, 7i) < ^r'- In this case, applying Theorem VI, we ohtain Ä„(/) 6/i"+\ In this case the above method again fails. We have thus proved that the formula (7) holds also when xa' < ,» < a', with the exception of the following special cases. . (i) 6=0, 0 < rt s 1 ; (ii) 6 = 0, a = 0, e{x,n)>0i, for any integer ii, 6yX,n) being the function defined by the equations (59). I have ah'eady got a certain proof for some of these special cases, but not yet completed it. Perhaps I may return to this problem on another occasion. 40- Here I will give another lemma which will be useful in Part II. 72 Art. 4. — M. Kuniyeda : Lemma 8. Ld f>{x) ^>^arHi) an L-fu)iction, but contlniioua and different ruble in the interval (0, ?) save for x = 0, satisfi/ing the relation w{x) r^ p{x) in mich a leay tlad vi{x) = j>{x) [I + t[xy. , where b{x) is tdtimateh/ monotonie and tends to zero as x-^Q. If tJtere exists an L-fanctioti y{x) such tliat then, under the conditions the same as those of Theorem VI, the same asymptotic formulae (7). {8) and {49) hold respectively for the inteyrfds obtained by replaciiaj w{x) for (>[x) in S{?<), G{?), 7i(/), I^i)-\ I-l?-) and UX). Proof. If t(x) be an L-function, then the lemma follows immediately from Theorem \'I and its corollary. If e(a?) is not an L-function, still it behaves like an L-function under our hypothesis and hence the truth of the lemma may l)e conjectured from Theorem VI. Take the integral /(•;.) = / ' ti?(.T) cos a(x) ^^1^ dx / 0 ,r /"^ r \ ^^r, f \ COS Ix 7 , C^ r \ I \ f \ COS Ix. J f)[x) COS fT(a-) dx + I t{x) Mx) cos a{x) dx. 0 X ./ 0 ■ X Let y(x) be an L-function such that lix) < rix) < 1* ' ■ as x^O : and write Pix)=f>{x)r(x), e(x) = 6^, SO that /' ■ Oscillating Dirichlet's Integrals. Y3 J-(/l) = /,(;.) + / ' e(.^) i,{a:) cos a{x) _?2!A dx, ' / 0 X where, l),y the eoroUary to Tlieorem VI, we have /,(;) = 0(1//) {p < a-^r' ; and since, b,y hypothesis, TS is differentiable, ê is also differentiable and -3— has a constant sign in the interval (0, c), ? being chosen sufficiently small. Hence by the Second Mean Value Tlieorem, we obtain J(À) = I e(a:) f>{x) cos a[x) cos }.x X cos Xx dx = e(c) / fjyx) cos aix) ^ dx (0 < ç^ < ?) J S^ ' ' X say. Then, l)y the corollary to Theurem VI, we have J 7) = 0(1/;.). if 7' < Xy/o"la' ; and, if x^a"la' < xa', 6, ß being the same as those in the al)Ove formula for Ii{}). Therefore, from the relation J> < (>, it follows that Y4 Art. 4. — M. Kuuiyeda : as /^x , wlien X's/n"lrf' •< (> < xa'. In the integral /(/) = I'' 7>(a-) cos o{x) ^^^^ dx (0 < Çi < ç), / 0 3? we ol)serve tliat ci varies with I; still, if Ave examine the proof of Theorem VI. we can see without difficulty that j'{À) < x(t'. Therefore we have: If (> < xV^"/a', then ^(/) = 0(1//) ; if x^r7"/(r' < ;, < xa', then /(/) -^ I^{?^). The same argument applies to the otlier integrals. Tluis the lemma is completely proved. VIII Km.mpks of Case (6') 41- Let us consider the case in which -a '>n wJiere m is positive, so that ni \ cos Xx /o X ./o a <.jc< ff' give respectively a ^ — h (* < 1' « < 2. The equation / + ^'(0) = 0 gives " = h- ) ■ and we have ß = U) + n{6) = 2(w;.)l Hence we have: If _| < a < 1, then IiW "^ i -*?>r i -U^"-^ cos (2w* ;.^ + ^tt), !,(;.) .^ i;r^wr i«- i /i'- * sin (2m* ?i + i-) , CC/^O ^ i7r4m-i'*-^-/>-^exp [(2?/i*;.i + i;r)i] ; if — -I < a < 2, then J.,(/) f-» ^-*m-*"-*>^*''-^'^cos(2m*/*-^;r), J,(/) --^ i^;r*w-*''"T/i'-^sin(2w4/* — Itt), We observe that all these integrals tend to zero as ;>^x if a < h and they oscillate if a ^ i- 42- These results may l)e verified as follows. Hardy proved* that /"* / 3-\ du I cos II COS I -'— -T— r ./o \ U J It =: ^ ^^"^ [ - J\'2/3) + J-''(2y9)-e-4-- J ^(2'i/9) + e*"'^ J -"(2^/9)] . 4 Sin ^VTT * Messenger of mathematics Vol. XL, ^jjj. Ji et seq. 7Q Art. 4. -M. Kuniyeda From this we can easily deduce I cos n sill - -^r- ./ 0 \ // / // 4 cos ^V- '- ^ ' ^ ^ ' ^ \ I y \ > / } These tormulae hold respectively for -1 < V < 1, and for — 1 <; v <; 2, it being understood that, in certain special cases, the expression of the rigiit-hand side must be replaced b^^ its limits. For v = ^ thej^ assume the forms ' / "^cos u COS f^-^ — = iv^a-) (-sin 2,9 + cos 2,? + e-^^), ./ 0 \ U J 's/u - - /"*cos n sin { ^'\ ^-'' = hVßr:) (sm 2,9 + cos 2ß-e--^) ; ./ 0 \ u J V ' and their values are expressible in terms of elementar}" functions also when ^ = - i- 4 (the last value, of course, only in the second integral). Now write ^w for ß\ and put u = kr in the integrals. We obtain /"cos Kx COS (-'"-) 4^^ = 5 , ( ™ f' r_/'(2v'(W.)} + J-''[2V(m;0] -e-^''''V''[2/V(m;0] +ei'''='V-''[2V(w/'0]1 /^os^^sinf^'L)-^!, = ,--V-(Tf r^^i^VlW.)] ./o \ a- / a:. 4 cos |-vn- \ -^. / L Now, when /J is large, Oscillating Diriohlet's Integrals. 77 JW) =3^^°' [2,9-i:]+2.);:]: where I ^^1 < —5- • Hence, when ß is large, -J-''(2/3) + J"-'(2;5) and J\%-i) + J-\%^) behave respectively like 2 sin |v- • /o Q 1 \ and — 77:^^^ cos (2^9-1-). On the otlier liand tends e.xpo}icnthdlii to zero as /9->3o i.e., as ;.-^co, so that this term is negligible in comparing witli the remaining terms. Hence we obtain the results: — The integral is convergent if — l^.a^ dx and /'W = -(1 +a.) :r-^'+'-^> cos (-^' j + ,n.T-<^+''> sin Ç^^ wlience / /'C^) sin /ia? <^a- ./ e since a>—l. Therefore we obtain /,(>() = 0(1//), and evidently J-i(/) < />- whence it follows that Similarly B(/) ^ Z, /) Therefore we obtain Ii(/) ^ -i-^?Ar4'-^;>-^cos(2;/i*/i + J;r) Z.,(/) '-^ J-i7/i-i-Ui<'-^sin(27;i^/i + i-) C(/î) -^ ^zi,ii-^"-i/}'^-i exp [(2wi*;.è + i-)/] (-l-^ exp {(2w-^;i-i-)/] (- 1 < a < 2\ which also agree witli the results obtained from Theorem \IJ, only the difference being that the lower limit of ^^ is -1 instead of -i. Thus our theorems YI and YII are verified. PART II Coefficients of Power Series /. Preliminaries. 44- Consider a power series m la.,'. n = 0 whose radius of convergence is unity, representing a function f{z) which has, on the circle of convergence, one singular point only at - = 1, being regular at every other point on it. Let ABCB be the circle of convergence of the series (60), A being the point z = 1 and 0 the centre. Draw a circular arc BPD inside the circle of convergence, cutting it at the points B and D, with the centre at A and the C radius ri < 1. \ Let J(;-i) denote the integral (61) I{>\) l-i J DPB where the path of integration is the arc DPB, starting at the point D, turning round the point A in the clock-wise direction along this arc and ending at the point B. gQ Art. 4. —M. Knniyeda : Lemma 9. // (62) lmiJ(rO = 0, then «n = -o_y 1,^ ^rr ^z, ^>^l ./ (C) ^ provided thd thh udegnd is convergent, the contour (C) of integration being the circle of convergence. Proof. Siiice the function f{z) is regular at ever}- point on tlie circle of convergence except only at the point A, the integral J BCD ^"+1 dz. where the path of integration is the arc BCD, is convergent and so also is the integral /(n) for every rj such that 0 < ri < 1. Hence, by means of Cauchy's Theorem, we have ''" ~ 2-i Jbcdpb ^"+'^ ^' Now let vi tend to zero. Then the arc BCD tends to the whole circle of convergence and we have a„ = since, by hypothesis, lini 7(/-i) = 0 and the last mtegral is convergent. 45. Ijemm.a 10. f-<'t ^ be d small positive constant and (63) lin) = -j- ( /■ ' + /''^ ) fie'') e-""' dd (0 < c < ;r). Then, if tiie conditions of Lemma 9 are satisfied, the behaviour of the coefficient a„ of the poiver scries (60), as n-^:D , is asymptotically determined as follows : Oscillating- Diriohlet's Integrals. g] //• I{n) < — , then », = 0 (1/n) ; if 1(71) > — , the?i a„ --^ I (71). Proof. By Lemma 9, we have = I{n) + r{7i) say. Then I\n) = J- /"''" /(e«^) e-^'^dd = -_L /' '"\u+iV) (cos 7id-i sin nd) dd, where f{e'^) = C7+iF, fZand F denoting real functions of 6. Since /(^) is regular on the circle of convergence, except at z = }, the functions U, V, ^^^ ^^ dd ' (Z/? have no singularities and are integrable in the interval (ç, 2.rr—$). Hence / (J COS 710 do ^=\ — — / -^^^ sm 710 do J s \ 71 \s 71 J ? do = 0(l/w). Similarly the integrals 27:-? r-^-^ TJ^innddd, / V iid dd g2 Art. 4.— M. Kuniyeda liave values of tlie same type. Therefore we olitain r{n) = on/n). Thus we 11 ave a„ = I(n) + 0{l/7i). Hence, if ^(^0 < — , then 11/ «« = 0(1/«); if I(n) > ^ , then a„ (^ lin}. 46- Now wo may write (64) I(7i) = J-(n) + J{n), wliere ml V Ä7Z J '27c~s At: J 0 n being a positive integer. If, in the neighbourhood of ^ = 0, l)oth of the functions /(e*0 and /[e(2»-*^*] take the form where , =, <0, a > 0, q > 0, 0 s r/ < 2-. It is to be understood that, when p and q are not integers, (l — zY and (l—z)'' assume respectively the values where log(l— ^^) assumes its principal value. At first we consider the integral ^w=iys?i.7^*- Let F be any point z = re"' on the arc DPB and let f denote the angle between the radius OA and the straight line AP, namely (f = ^ OAF. Then \ — z = 1 — r cos d—ir sin 6 = y^ cos ç- — i)\ siu 0 Now, if cos(« + g^)>0 in any part of the range then e""''^ " cos { 0 as e > 0. Now, by (66), I /•/--r ^ I 0. Hence re-"''"'''''dt <:Kr{', and we obtain Qg Alt. 4. — M. Kuniyeda : as ri->0, provided tliat p < 1+ '] =^((9)e'-''W, where (72) so tliat we liave, l»y (<')4) ;ind (65), Jin) = } r/(y')e-''dd = i- /■' <;-rals. 87 (()4) I{v) = J(n)+J[n). 49- Integral J(n). First of all, we shall consider the integral J(n). Observe that, when d is very small, we have COS [a + Uj[7: — âj] = cos (a + hj') [1 + 0(6'^)} + sin (a + hl7T).hlO{l + 0(6')], sm [a + hqi- - 6)] = sin (a + ^q7r)[l + 0(6')} - cos (a + l-q7t). h2d[l + 0(^)1 . Hence the equations (70) may be written in the forms r(-ß\ = i a + 0(d-) ^ e*^"' ^'^"^ <«+*'/") + ^'^"^ «i^ («+ è'7^)] + o(e-"0^ (70') .-}--l. \ 24 / lU2 2 / j Hence ei and £.j may be expressed as power series of .c, which are uniformly converuent for sufficiently small values of x. Thus the first terms of these series may respectively be taken as Yi and fo, and ^t immediately follows that ---^- and — -^ have ultimately constant signs. ti.r < xa' as x->0. Introducing the above expressing of {> and ö-, this con- dition becomes which is nothing but our h3'^pot]iesis. Thus, El and e,, having the above properties, Lemma (S of Part I may be applied to our integrals, and we obtain Ji{n) r^ I X ^cos('Ha:' + — J dx, JJn) (O / X'" sin ( nx -\- — \ dx (74) as M.->oo , and hence we l^ave J^{n) r^ A-*^+-M*-^ COS {1ahi^-+\-) (-^<^j<:2), Jin) r^ -*a-^^+M^-^ sin (2aM + i-) {-^(d) = -|L COS qr: + ^p- + Oiä'-') ; 90 Art. 4. — M. Kuniyeda : if q = h ( çr(^)--^6-««'\n+0(^^-)], I {()) = hp'+ 0(6^). Hence we have Avli ere /(n) = A eèi'^' / ' W{x) e- {'"■'■ + ''(■'■)]'> rix, io(x) = ax"' cos q7T, rjSi,x) = x-Pe-^''-''''''"^''{l + t^x) + is^x)}, ej and e- ])eing real, continuoas and differential )le function^^ of x in tlie interval (0, ç), such that lim e^ix) = 0, lim e,.-,(_x) = 0. .r^-O .!;->o ' Since sinqz>0, uj(x) tends ex]:>onentially to zero as x^O. There- fore the integral J{n) is ahsolutely convergent. Now (75) / ü3(x)e~ ^" '■ + '^(•'')^ ' clx = / ÜJ (x) cos a(x) cos nx dx ,/ 0 ./ 0 - / Tjj(x)^\no(x)^\y\iix dx — */ vs {x) co^ a{x) '&\n nx dx I 0 — ■i / w(«) sill 0. Hence the last integral is convergent. Thei-efore we have / üj{x) cos (t(x) cos nx clx = 0(1/;?,). ./ 0 Similarly the other three integrals on the right-hand side of (~~)) assume values of the same type. Thus we obtain (76) Jin) = 0(l/w). We observe that, in this case, p may tak« any value, positive or negative. 52- (iii) The case in which 0<:q<'[, « = (3 — g)-^-, p, 1+'/ exp [-(A: n^+'-i/^rr + ir)/}, * The exact forms of rj and t . are ^ 2 sin è.<- / I ,r* (2 sin ^.r)" '^ / ^.(.r) = /-—-^L __f .-" sin è '/■'■/ >2 sin è.r)'' ,5^ ( » ,_ ^tcosè?JL_w\ . V 2sinir .' I .r/ (2 sin ^r)'' J These functions are continuous and differential ik- when x is suttioiently small, including the value .T=0. And we can easily obtain the said result. Oscillatini; Dirichlet's Integrals. 93 53. (iv) The case in which 0 1///, as //.->oo . But, by proceeding as follows, it will be seen tliat this restriction al»out the value of p niay be removed. Now, in a certain region near the point z — 1 and interior to the circle of convergence, we may put — ^ — .m-^f ^2a,^2-, and we may differentiate this equation with respect to z, since our series ^cinz" is uniformly convergent in the said region. Thus we ol;)tain Observing that a„ is a function of ii and 2^, Ave write «„ = a{)i, p). Then we liave p ^ a[n, p + l)z'^ + qA ^ a(n, p + q + 1) ,~^ = };] (//• +l)a {n +l,p) z'\ n = 0 « = 0 )i = 0 * Our method fails to determine the asymptotic formulae in other cases. Qg Art. 4. —M. Kuniyeda : whence (83) p a{n, i^ + 1 ) + qAa(n, p + q + ^) = (n + l) a{n + 1, p) (h = 0,1,'2, ). Hence (84) a{n, p + q + A) = ^^^ a[n +\,p)- y^- a{n, p + l). For instance, take the case 0 a{ih p+V. Thus we obtain, hy (84), ain,p + q + '\) ^ . {n+\) a(n+\,p) . qA Oscillating Dirichlet's Integrals. 97 Now qA = qae^^-^''^'^\ and r~> e 9/(1 + ï) Hence, writing p^ =p + q+l^ we obtain where /)i 0 and q, a are certain constants, then the hehaiùour of the coefficient a„, ax n^x>, is determined asymptoticallij as f(jU()ir.< : (i) If q = \, <'■ — -, or tJie .sinyulfwiti/ is of the type then (9) a„r^ —a-iP^ie-^SiiP^^sml^akii-ity-î)-} (ii) If 0 <:q <\, a = (\ +q)'^, or the singularity is of the type then (10) "n^V^-]^,:7]^}"'('^*)''^'"'' ''' exp |^[^-y,i+'^-(i^j-^J-]iJ, (iii) //' 0 < g <; 1, ^/ = (;^ — g}-^^ ^y^. ^/,g ninyalarity is of tJic type f(y\ — 1 .^"(sin è'm + '■ COS ir77r)/(l-z)'^ / ^ > . = , < 0 ; \ z*^-^ - {\-zY U > 0, 0 < g < 1 ; ' then A" heiiuj the xarne as in the case {ii). Ill Case in ivhich the Singularity is of the Type _J: //(I- -f^ /lo^ 1 V 57- ^\ ^' now pass to llie case in wliieli f{z) luis a singularity of the type ^^ ^ (1-^f \ ^^ 1-J ' Oscillatiag Dirichlet's Integrals. 99 where A = a e"', a>0, 0 ir ^/ < 27r, q > 0, and p, r denote arhitraiy re;il constants. First of all, in considering the integral I[ri), we observe tlvat log T = log ( - — «"'"M = loo- — + 0 Next, when 6 is small, we have log ,. = log ■ ■ ,,y+(i"-|^)^ 1 —e^ 2 sin U' = logi- + i-/+0(^) = log|{l+o(llog|)), '<>B-x^.)"=(<'4n'+«("°4)}- whence Similarly (,og^_U.y = (,o,]7(i+o(i/io4)}. Hence the dicussion may he carried out quite similnrly as in the preceding case, the presence of the logarithmic factor producing no great change in the an;dysis, and \ve content ourselves with giving only the following results. CO Theorem IX. Let Z^a,^z" he a power serie.^, whose radms 71 = 0 of convergence is vriity, 7'epresenttn(/ a function f{z) n-hich has, on the circle of convergence, one singvlnr point only at z — \, being regular at every other 'point on it. If the singularity is of the type loo Art. 4— M. Kuniyeda where a > 0, q ami a denote certain condanti^, fi/ul p, r i, then the beh(wioiir of the coefficient a„, as 71-^00, is deteinnined asi/mptotic 0, q and a de- note certain constants, and p and all o"' s arbitrary real constants, then the behaviour of the coefficient an. as n->x, is determined asymptotically as follows. (i) If q = \, a = 71, or the singidarity is of the type then (ii) If 0(ga)'^+5 n ^^''' gin) exp j^- [kn^^'-{hp-^)-]i~\^ , k being the same as in the case {ii). Published April 20, 1919. .TOUl'.NAT. OF THE COLLEGE OF SCIEKCE, TOKYO IMPERIAL UNIVERSITY. VOL. XLI., ART. 5. On the Effect of Topography on the Precipitation in Japan. (Coutributiou III. from the Geopbysical Seminary iu the Physical Institute, College of Science). By Torahiko TERADA, RifjakuliakusM. Misio ISIMOTO, Bigakiishi and Masuziro IMAMURA, Rigakiishi. 1. As to the general distribution of precipitation in Japan, there is an earl}^ investigation of Prof. Kiyoo Nakamura^^ He pointed out a marked difference of the annual course of the precipitation on the Pacific and Japan Sea ^ides. According to his results, Japan Sea side has abundunt precipitation in autumn and winter compared with spring and summer; the maximum falls in December and the minimum in May; but the seasonal fluctuation is generally small on this side. On the contrary, the Pacific side is characterized by abundunt precipitation in summer and autumn and also by a large fluctuation. He divided the Pacific side into five districts and described the peculiarities of each district in some details. Besides, he alluded to the remark- able effect of topography in some examples. Recently, a Decennial Report of Precipitation,^^ 1901-1910, was published by the Central Meteorological Observatory, in which monthly records of observations in 1570 stations are given. In a note appended to the Report, Prof. Fujiwhara gave a brief account of the general distribution of precipitation in entire Japan, and confirmed in the main the results obtained by Prof. Nakamura. He also discussed the dependency of precipitation on the latitude 1) K. Nakamura, Dai-Nippon Hûdohen :^0^®L±^ (Climatology of Japan), 1897, Chapter VI. 2) Uryô-Zyîmenhô MÄ+¥^ 1914- 2 Art. 5. — T. Terada, M. Isimoto and M. luiauiura: and pointed out a peculiar fact that the decrease with the latitude is comparatively small in Southern Japan, but remarkable in the Northern, the dependency being quite different from those obtained by Murray and Supan.'^ He compared the relation with that of the temperature and precipitation and suggested an intimate physical connection between the latter elements. Moreover, he gave brief but suggestive discussions on the influences of the shielding mountains, altitude of the station, the slope of land, the distance from the sea etc. In a previous communication,^^ we have shown a remarkable influence of topographical condition on the distribution of rain accompanying cyclone. The present note which gives a resume of some statistical investigations on the relation of the geographical distribution of the yearly and mean monthly amounts of precipita- tion with the prevailing barometric gradients, may ]^e re'garded as a supplement to the previous note. It may be remarked that the subject in question is not without interests also from the seismological point of view, since as already shown by Prof. Omori,^^ there exists a correlation between tlie yearly seismic frequency of some localities and tlie annual amount of precipitation in some other districts. It seems, liowever, still an open question whether the precipitation is the direct agent acting as a secondary cause of earthquakes, or it is rather the barometric gradient which affects the seismic origin and the precipitation at the same time. We will add later a few remark on this latter point, though unfortunately we were not yet able to trace the relation in any conclusive manner, on account of the want of data. '2. The data used for the yearly mean barometric pressure and precipitation were taken from Kisyôyôran of the Central Meteorological Observatory, the epoch ranging from 1900 to 1017, while for the monthly means, the materials were taken from the 1) Hann. Lehrbuch, 2. Aufl. p. 295. 2) Terada, Yokota and Otuki, Journ. Coll. Soi. 37, Art. 4, (1916) 3) F. Omori, Bull. I.E.I.C. Vol. II. No. 2. On the Effect of Topography on the Precipitation in Japan. 3 Climatological Table of Japaii'^ recently issued by the Central Meteorological Observatoiy. Since it was our main purpose to study the effect of the discontinuity of wind velocity on land and sea, only those stations were chosen which are situated not far from the open coast facing either to the Pacific or to Japan Sea. The stations of which the yearly courses of rainfall were to be studied were grouped into the following six regions: I. Akita and Niigata, representing the Northern Japan Sea coast. II. Miyako, Isinomaki and Tyôsi, the Northern Pacific coast. III. Husiki, Kanazawa and Hukui, the central Japan Sea coast. Fig. 1. 1) Kisy.j-Zassan (UM^M) Vol. I. No. 4, (1918). 4 Art. 5. — T. Terada; M. Isimoto And M. Imamnra : lY. Nuniaclu, Hamamatu and Nagoya, the central Pacifie coast. V. Sakai, Hamacla and Hakuoka, the Southern Japan Sea coast. VI. Kôti, Mij'^azaki and Kagosima, the Southern Pacific coast. The distribution of the stations are shown in Fig. 1. The choice of the stations may seem somewhat arbitrary, but we were led to it by different subsidiary considerations. In some stations, the observations were interrupted once or twice during the interval of eighteen years taken. In order to fill up the gap, the following procedure was taken. When the amount of precipitation is wanting for a certain year at a given station, the ratio of the mean amount for the remaining seventeen years for the other stations in the same group to that of the station in question was taken and multiplied to the mean value of the year concerned of the other stations and the result was assumed as the reduced value to Ije replaced for the wanting data. Though the annual amounts of precipitation recorded in Kisyô-3'ôran are given in mm. and its fractions so that the numbers are made up of five figures, it was considered convenient and rather reasonable for the present purpose to cut the figures to only three, stopping at the place of cm. The data thus reduced are given in Table I. Ou the Effect of 'i'opography on the Precipitaticu in Japau. 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CD CD !> oq Ol Ol in m CD o Ol rH o o CD LO 00 O 1-1 1-1 rH CO Ci o CO 1> rH O Cl O CO CD Ci OC- CD CO IM 1 00 'S* o rH in CO 01 Ol oq t}< in CO Cl Ol Ol CD 1 t^ CO CD i> Cl in 10 CJl t~ 00 i>- Ci 1 Ol -* o 1> -* 00 rH rH -H lO -# CD Ol 00 Ol 1 O CO Ol CO oq rH rH Ci in Ol oq Oi 1 1 1 2.1 5 o Ö 8 5 o a s tu ^1 5 o a o o 3 o H d c« ^ 2 '5 1^ -3 S c 3 ^1 ce 3 o d a o d < rä )^ O c3 3 -y cß d a d 3 o d <1 o o g o d < d 1 'S' ;3 o i-H ."^ 1^ 3 o d < S rH S o d < o -^3 -H = = HH r^ 1— 1 > iE D d-^ =3 t! a> d d i (!>•" CJ rd d ö -1-3 aj '75 -HH > 2 bD Ol V -d d m ni n> o bo a a • rH ^ ,_^_^ d Ol a a drd fl S '1 in o.i:; • d -"-^ .22 d a O g c3 '-> d 2 § 2 d -S =» d 'S *" Q P c5 3 2d cä 9 !ï> o ^ m d a d s-^ a H Srâ 2 £ =^® ?. H >^ H * ^ 3 o 3.» O t» ^ -H^ .H Ol lî Art. 5. — T. Terada, M. Isiuioto and M. luiauiura : 3. Referring to the above Ta])Ie, it will be remarked that the mean annual precipitation is most abuncliint in the region VI and least in II. In the middle Japan, the Japan Sea side III has decidedly larger precipitation than the Pacific side IV. On the contrary, in the SW Japan, the Pacific side VI has 1.44 times more precipitation than the Jnpan Sea side. In the NE part of Japan, it is again the Japan Sea side which has more precipitation, though the contrast is comparatively more pronounced than in the case of III and IV. The mean values of I-II, III-IV and A^-VI are respectively 101, 222 and 214, while the difference, Japan Sea side minus Pacific side are respectively 51, 50 and -78. According to our previous theory, this result might have been explained at least qualitatively, if we could assume an area of inland high pressure near the junction of the central and SW parts of Japan. That this is realty the case may be seen from the annual isobar map. In the map, we see a general fall of pressure from the Asiatic continent toward the Pacific. The general slope is, however, disturbed by a remarkable tong or promontory protruding from Corea toward the central part of Japan along the axial line of the land. Such a distribution may conveniently be interpreted as due to the superposition of an elongated area of high pressure having its ridge near the -western part of Honsiu, upon the nearly uniform slope from the continent to the ocean. Hence, according to our rule, the Japan Sea side has more and tlie Pacific less precipitation on the NE side of high area, compared with the ideal case where no such high existed. The reverse may be said with regard to the SW side of the high. The very marked contrast between III-IV and V-VI, shows that the above efïect of the " tong " is rather prominent. 4. In discussing the matter more minutely, it is desirable to take the seasonal distribution of precipitation instead of the annual. In Table II. the mean monthly amounts of precipitation are given for the six regions concerned: (Jn the Effect of Topography on the Precipitation in Japan. Table IL* Group \ Month I II III IV V VI VII VIII IX X XI XII M --Mean S = sum Akita 122 99 106 111 113 146 198 187 191 162 184 171 S = 1789 Niig-atii 193 127 108 109 92 132 166 130 183 154 184 235 S = 1813 Mean 158 113 107 110 103 139 182 159 187 158 184 203 M = 150 % 105 75 71 73 69 93 121 106 125 105 123 135 M.a. = 19.9 100 Anoui. Mean Anoui. 25 -126 -146 -136 -156 -35 106 30 126 25 116 176 Miyako 71 66 88 99 124 135 143 179 206 161 69 63 S = 1404 Isinomaki 47 49 76 95 121 124 148 122 156 119 56 44 S = 1157 Mito 61 59 119 141 161 163 150 165 211 159 67 55 S =1509 II Mean 60 58 94 112 135 141 147 155 191 146 64 54 M = 113 % 53 51 83 99 119 125 130 137 169 129 57 48 M.a. = 34.8 100 Anoui. Wean Anoni. -135 -141 -49 - 3 55 72 86 106 198 83 -124 -149 Husiki 239 164 137 134 114 151 193 160 204 151 200 323 S = 2172 Kanazawa 269 186 164 170 147 181 202 180 226 194 259 357 S = 2534 Hukni 249 218 163 159 151 192 190 168 188 166 213 335 S = 2391 III Mean 252 189 155 154 137 175 195 169 206 170 224 338 M = 197 Yo 128 96 79 78 70 89 99 86 105 86 114 172 M .a. = 19.7 100 A nom. Mean Anoui. 142 -20 -107 -112 -152 -56 - 5 -71 25 -71 71 366 Nuuiadn 83 77 150 200 187 220 234 229 280 172 104 75 S = 2012 Hauiamatu 63 68 147 203 212 240 214 205 271 157 100 69 S = 1949 Xag-oya Äfean 60 64 129 170 166 225 192 181 243 148 81 52 S = 1712 lY 69 70 142 191 188 228 213 205 265 159 95 65 M = 158 ."/ 44 44 90 121 119 144 135 130 168 101 60 41 M.a. = 36.6 100 Anom. Alean Anoui. -153 -153 -27 57 52 120 96 82 186 3 -109 161 Sakai 203 143 144 136 113 166 163 131 221 162 157 195 S=1931 Hamada 115 98 112 138 123 193 164 128 177 121 100 114 S = 1592 Hnkuoka 68 81 112 136 125 249 245 140 189 101 72 76 S = 1583 V Mean 129 74 123 136 120 203 191 133 196 128 110 128 M = 142 % 91 52 87 96 85 143 135 93 138 90 77 90 M.a. = 21.3 100 Anom. Mean Anoui. -42 -226 -61 -19 -70 202 164 -33 178 -47 -108 -« Kuti 69 98 186 300 292 355 328 279 419 225 125 79 S = 2754 Miyazaki 81 101 187 245 271 383 270 255 349 226 125 74 S = 2566 Kagosiuui 92 92 158 232 229 415 291 175 240 139 91 84 2240 VI Mean 81 97 177 259 264 384 296 236 336 197 114 79 M = 210 % 39 46 84 123 126 183 141 112 160 94 54 38 M.a. = 40.8 100 Anom. xvlean Anom. -150 -132 -39 56 64 203 100 29 147 -15 -113 -152 * Precipitation is givtn in mm. The line %■ gives the monthly mean of different stations expressed in % of the mean for all mmth. The next line gives the deviation of the % values from 100 expressed in % of the mean value of diviation (m.a.) :Art. 5. — T. Terada, M. Isimoto and M. Imauiura : Table III. Mean&Diff. I II i 1 III IV V VI VII VIII IX X XI XII è(i+n) 109 86 101 111 119 140 165 157 189 152 124 129 i(III+IV) 161 130 149 173 163 202 204 187 236 195 160 202 èiV + YI) 105 86 1 150 198 192 294 244 185 266 163 112 104 I— II 98 55 ■ 13 -2 -32 -2 35 4 -4 12 120 149 III— IV 183 119 13 -37 -51 -53 -18 -36 -59 11 129 273 V— VI 48 -23 -54 -123 144 -181 -105 -103 -140 -69 -4 49 In Table IIT. , we give the mean values and the difference of the Japan Sea and Pacific sides of the NE, Central and SW parts of Japan respectively. Referring to the latter, we may remark several interesting facts. Firstly, the mean values of the both sides, are greatest in the Middle part III-IV, during the colder months Oct. -Feb., while it is so in the SW part during the warmer months, Mar. -July and Sept. While during the colder months, the NE and SW regions have nearly same amounts of precipitation, contrasted w^ith the large amount in the Middle Japan, the amount of precipitation during the warmer season continously falls off from the SW towards the NE districts. Secondly, the difference, Japan Sea side minus Pacific side show quite parallel course, but the mean value successively falls off from the Northern part to the Southern. Thirdly, among different possible combinations of cases, I>II, III > IV, V>VI occurs in January and December, occurs in February, jMarch, October and November, occurs in April, May, June, and September, occurs in July and August. I>II, III>IV, VII, IIIVI, and IIV, V>VI do not occur actually. On the Effect of Topograiiby on the Precipitation in Japan. 9 A full discussion of these characteristic distributions will be given in the later part of this paper. Here, it may suffice to draw the attention of the readers to the remarkable mode of distribution in which the influence of topography plays an important rôle. 5. Turning our attention to the amount of the yearly fluctu- ation, the most prominent feature is the difïerence of the mean anomalies on the both sides of the land, the Japan Sea side showing generally less fluctuation than the Pacific. On the latter side the amount is least in II and greatest in VI, while on the Japan Sea side it is the least in V. In Table I, we have given the yearly amount of precipitation also in percentages of the mean value and besides, the percentage fluctuation of the yearly percentage values are given in the next line. We reproduce here in Table IV. the mean anomalies of the percentage lvalues for the six regions. On the Pacific side the Table IV. XE Middle SW INIean Japan Sea Side I 8.8 III 8.1 V 6.1 7.7 Pacific Side II 8.9 IV 11.3 VI 10.6 10.3 Mean 8.85 9.7 8.35 9.0 Difference 0.1 3.2 4.5 2.6 anomaly is the least on the northern part of Japan whereas in the Japan Sea side it is least in the Southern. While the mean values of the anomalies of the both sides are not very different for the different parts, the contrast between the both sides is most prominent in the southern part. G. The mean of the amount of precipitation fur the Japan Sea side I, III, V is 203 and for the Pacific side II, IV, VI 195. The difference is not remarkable. On the other hand, the mean anomaly for the Japan Sea side is 7.7 while is it 10.3 for the Pacific. Thus it appears that the Pacific side is generally more "sensitive" to the main causes of the fluctuation, among which JO Art. 5. — T. Terada, M. Isiuioto and M. linamura : the barometric gradiet must be one uf the most important. If such be the case, due consideration must be paid to this point, if we are to attempt to compare the amount of precipitation on botli sides of the land and thereby deduce some topographical relation with respect to the barometric gradient. If this precaution is not made, the difference l>etween the l)()th sides will be largely determined by the side on which the amount is decidedl}^ larger. Hence the deviation of the yearly percentage value from the mean, i. e. 100, Avas divided by the mean anomaly. These quotients (Table I.) were adopted for the final data to be used in the com- parison with the barometric gradient. 7. The barometric gradients of the different parts of Japan, to be used fe)r the comparison with the precipitation may be obtained in difïerent ways. Referring to the mean annual isobar chart, the isobaric surface over the land is far from being nearly plane, chiefly due to the protruding area of high pressure lying along the axial line of the land. On account of the latter, the actual gradients on both sides in different parts of Japan may differ considerabl}^ from each other and may even have nearly opposite directions. In the present investigation, however, we will at first put the axial high out of consideration, which is in all probability very shallow phenomena, brought out by tlie reduction to sea level, and take the gradient obtained from tlie coastal stations. The general procedure taken is as follows. For the Northern Japan I II, foi* example, we choose four coastal stations, say Akita, Kanazawa on the Japan Sea side and jMiyako, Tyosi on the Pacific side, forming the angular points of a quadrilateral, as nearly i"ectangular as possible, including the region in question. The difference of the pressure on both sides divided by the mean distance may be taken as the measure of the gradient in the direction combining the centre of the (opposite sides. ^-^ If the quadrilateral l)e nearly rectangular, we obtain thus the two rectangular components of the gradient. The stations chosen for the purpose are as follows (see Fig. 1) : 1) In the case when the isoViaric surface is nearly plane, it will he plausible to take a trianu;le for the determination of irradient. But in such a case as is here concerned, the procedure uieutioned almve seems more advisal)le as o'ivini' a kind of mean gradient. On the Effect of Topo<^raphy on the Precipitation in Japan. \'\_ Akita. Kanazawa, Miyako and Tyusi for Nortlierii Japan MI. Niigata, Sakai, Tyosi and Koti for Middle Japan III-IV, Sakai. Hukuoka, Wakayania and Miyazaki for Bonthern Japan V-\'I. The difference of the means, Japan Sea pair minus the Pacific, reduced to the gradient corresponding to 111 km. distance was taken as the :i:-c«)mponent. For the 7/-c(»mponent, the difference was taken between the mean vahie on tlie opposite sides of the rectangle running transverse to the axis of the land, the positive sign ce)rresponding to the case when the SW side is higher than the NE side. The mean distance between the opposite side was roughly estimated on a map, scale 1 :1(),000,000. Thus the component gradients :i-i, y/i, a-2 2/2, a's .'/s, for the three regions I-II, III-IV, V-VI, I'espectivelv were obtained according to the following schema: /Akita 4- Kanazawa Mivako + Tvosi \ r. 00/. ^^ = ( 2 — " 2-^ )^^-^^^ / Kanazawa + Ty ôsi Akita + Mivako \ r. oao y-i^ 2 ~ 2~^ jxO.248 _ / Niigata + Sakai Ty ôsi + Kôti \ ^ Aor x-2-y ^ ^ jxu.^ôo ( Sakai + Kôti Niigata + Tvôsi \ n^,-t^ 2 2 • ) ^^-^^^ (Sakai + Hukuoka Wakayama + Miyazaki \ ^ ,,.,, —2 2 jxu.4bd (Hukuoka + Miyazaki Sakai + Wakayama \ n oc 7 2 2 '~ )^ It must be remarked that in the case of Xi 2/1, the two components are not even nearly rectangular, so that this point must be remembered Avhen a quantitative relation is concerned. But for the most purpose in the present investigation, the general qualitative relations are not seriously modified by treating the components as rectangular. The values of the components thus obtained aie contained in Table V. 12 Art. 5. — T. Terada. M. Isimoto and M. Iinamura : 00 CD rH q O ^ L-î Cl rH q 2 S 1 1 ce o Ci CD q ■"! o î^ q (M J>- (N q -* o rH q et o ■^ q (M X> N q ■* CO -! Ö O 1-1 MS ^ I-l i-H q q CO o o Cl q o 00 CO rH CO 00 Cl q i-H O eo q CO (M CO rH o LO rH q O 0 o 01 q 00 CD oj q ■>J( CO rH q O o w q 00 1-1 ai r-J CD ■* rH q o o o r-; q o 1> w q -* o i-H q 00 o o 1> CD i> o rH t-H a CO >-; q 1 2 S rH q o o rH q CD 2 o o r-j i-< sa o C^l rH 2 3 1 g O '-' rH (M rH q 05 lO q q 1 o ^ q q ^ CD 'S« rH q CO q t- o rH q 1 (M 8 rH q 1> 00 rH q Ci CO q q o 1-; q IM (M Cl rH Ci o o o o o o O o q '-: CD rH .-; q o / / '^ / 5^ / ^ / s / ^ H S ■~' =^ ■? S 2 'te — < H^ d ' • '¥ ' , rH 00 rj m ^^ r-l S + 1 + I> ■ CO Cl 1> CD • ■ -^ CO ■ o ' CO -e rH ^ CD ^ ^ m N T—{ t- O ,-t rH rH 1 LO t- 00 Ci -# 01 O t-< 01 1 1 T? 'Jt Cl CO ■* O Cl r-< CO -* Cl ^ CO o 00 Ci Cl 1—1 oq -* ■Tl' Ci 03 CD CO o r-H rH ! i ^ o CO Ci ~^ Cl I> o ^ o o CO ^ CO o 00 C5 rH r-^ rH rH ■ 1 O rH CD lO O 00 ^ CO O rH 1 00 oo Ci ^ (3j Ift 00 Cl O ^ 1 x> -* 1> r-< o Cl Ci •^ o OQ 1 CD ^ CO _, O Ol o O '"' \a Cl CO US o Ci Cl CO Ci r-l •^ Cl o CO O o CD CO O rH CO rH CO 00 Ol o O 1> sq Cl O T-l r^ 1 , Cl o Cl ^ O CO Ci -H o rH i-i -H 00 Ci lO O CD Cl ■# o rH '"' 1 1 o O UO O o 1> CO 00 o Cl r-i u 1 C6 / i> / 1— 1 >l / l-H 1 1 / >-' '"' r^ >■ / '^ On the Effect of Topography on the Precipitation in Japan. 13 8. In comparing the jx'arly barometric gradient with the difference of j^early precipitation on both sides of the land, several procedures were tried among which the following seems to be most convenient for demonstrating the topographical influence. To take, for example, the case of the Northern Japan I-II, a vector diagram (Fig. 2) of the yearly barometric gradient was plotted on a sheet of coordinate paper, the end point of the vector for each year being marked and numbered according to the number of the year. On the other hand the difference of the values: Yearly anomaly of the percentage value of precipitation Mean anomal v ,, „ „ „ 10' -05 0 Oi 10 -IS 10 10 -OS 0 OS 10 15 20 2S 30 jy Fig. 2. ]^^ Art. fj. — T. Torada,, M. Isimoto and M. luiauaura : of the region I and II, was taken (Table VI) for successive years. Then the points on the vector diagram were classified according as the above difference of the ]H)th sides is positive or negative. Carrying out the procedure, it was found that the points belonging to the two classes may be separated by a boundary line dividing the area of the diagram into two halves. Though the line of demarcation is more or less sinuous, it may be compared for very rough approximation with a straight line making an angle of about 20° or 30° with the axis of x. This latter result is exactly what may be expected from our previous elementary theory, except that the angle expected from the calculation w^as IS'' instead of 20 or 30°. The fact that the actual angle is decidedly larger than the calculated, points to the conclusion that the difference of the angles between the Avind and the barometric gradient on land and sea must be assumed much larger than in the previous paper. This discrepancy may probably be explained if we consider that the angle between the gradient and the wind increases generally with the height and also that in calculating the vertical displacement of air caused by the disconti- nuity of the horizontal flow on the boundary of land and sea, it is reasonable to take the data of calculation from a certain mean or ' ' effective ' ' height froih the sea level. If the years of observation be multiplied several times, we may obtain sufficient numl)er of points on the vector diagram for drawing a more definite line of demarcation and thence deduce something about the effective height above mentioned. IMoreover, it seems quite possible that the sinuosity of the boundary curve may have a real significance, since the " effective height" may l>e a function of the velocity of wind as well as the direction of wind determined by the irregularity of topography. These latter points of interests nuist be postponed to a future when a sufficient materials would have accumulated. 9. In passing, it may be observed that vector diagram above described is very convenient U)y investigating the sensibility of different stations with respect to the barometric gradient, or in other worlds the mode and degree in -which the topographic conditions affect the precipitation of the stations. Taking the On the Eiïect of Topo^-raphy on the Procipitation in .Japan. 15 diagram for a given region, we classify different years according as the precipitation of a certain station situated in the said region, is above or below the normal value, and mark the corresponding points with proper signs. Then if the station is especially sensitive to the gradient, the points belonging to the two classes will be separated by a definite line of demarcation on the area of the dia- gram. Carrying out the procedure for the different stations chosen above, it w^as found that some Pacific stations such as Numadu, Hamamatu, Koti etc. show striking dependency on the topographi- cal rain, whereas most of the Japan Sea stations show rather irregular aspects. It was suspected that the latter fact could be explained by the influence of the high pressure zone along the axial line of the land. Choosing a number of stations near the axial lines, we have calculated the gradient between these axial stations and the coastal stations on both sides of the land res- pectively. It was found that the gradient between the axial part and the Pacific coast, say s, is generally parallel or at least have the same sign with the a;- component above obtained, whereas the pressure difference between the axial and the Japan Sea regions, say n, has frequently different signs with x. On plotting n-y diagram instead of x-y diagram, however, the distribution of the years belonging to the above two classes for the case of tlie Japan Sea stations is still very irregular. We may therefore infer that on this side of Japan some other terms predominates over the effect above considered, as far as the annual amount of precipitation is concerned. We will resume the question in a later section in connection with the seasonal distribution. 10. A similar comparison can be made with respect to the Table VII. Region ^"^^^^ Month Oouipt.^^^^-^ I II III IV V VI VII VIII XI X XI XII Year I, IT. .66 .27 .66 .27 .50 .14 .17 .06 .06 .04 -.17 -.04 -.14 -.06 -.27 -.10 -.15 -.10 .17 -.06 .42 .14 .61 .33 .23 .07 Ill, IV. a-_, 2/j .43 .36 .54 .34 .52 .15 .20 -.02 .02 -.01 -.15 -.06 -.15 -.04 -.15 -.11 .11 -.11 .41 .01 .39 .18 .30 .40 .17 .10 V, VI. 2/3 .21 .20 .32 .16 .25 .01 .09 -.03 .00 -.03 -.14 -.03 -.21 -.04 -.12 -.09 .19 .37 -.13-. 05 .23 .08 .21 .31 .09 .04 16 Art. 5.— T. Teracla, M. Isimoto and M. Imaumra ; Table VIII. Diff. ^^~\ I II III IV V VI VII Vlli IX X XI XII I— II 160 15 -97 -133 -211 -107 ■ 20 -76 -72 -58 240 325 III— IV 295 133 -80 -167 -204 -176 -101 -153 -161 -74 180 527 V— VI 108 -94 -22 -75 -134 - 1 64 -62 31 -32 5 105 - 3 - 2 -• / 0 \ -2 • 3 .1 \ 'S .6 .7 3 •2 % x/. E>-. ^ 1^ }ii" 0 A ^' ';f-' ^ y -f «/ IK 4 •3 •2 % X/l / ■0-.. II ,/^ ^ x/t y _*/// ^ ^ ÎK ^■- "v \ «1 .vii - h 3 2 % L' /■ \ / 4(1 0 X ■■-*. -^ '■■■if 'h ^3 w ^"'•\- /I -2 -/ 0 •/ 2 -3 •♦ -5 -6 7 Fig. 8. On the Effect of Topography on the Precipitation in Japan. J J monthly value of precipitation and the barometric gradient. Table VII gives the mean monthly gradient to be compared with the precipitation given above in Table II. In Table VIII, the differences of the percentage anomaly, obtained in similar manner as in Table VI are given. The vector diagrams plotted in exactly same manner are given in Fig. 3. Here also the months with more than usual excess of precipitation on the Japan Sea side, are distributed on the one side of a line of demarcation, similar as in the case of the yearly amount. From the results thus far obtained, it seems certainly worth while to apply the vector diagram method to the relation of the precipitation and barometric gradient of each month of each year, instead of taking the monthly or the yearly means, since as is well known the fluctuation of the monthly means of the both elements are considerable. At present, we must refrain from the task on account of the want of time, with a hope that in some days we will be able to resume the investigation. 11. In section 3, we have already alluded to the peculiar distribution of the precipitation of the different regions in different seasons. We will now make an attempt to interprète or analyse the actual distribution in terms of the elementary theory proposed. Let us put at first for a trial : i\ = B2 + C2, Vi = Ä2 — C'L (A-) where c,, C2, Ci etc represent the effects of the topography which will be considered as some functions of the barometric gradients, varying from month to month, i^i, Ro, R^ represent the principal terms which will prevail in absence of the topographical influence. It seems plausible to assume that the c's are proportional to c''s, if not equal, so that we will put The factor a must vary with season, but it must be generally positive if our theory be valid. Further, on the basis of the approximate equality of the annual amounts of n + rs and n+^e, we 18 Art. 5. — T. Terada, M. Isimoto and M. Imamura : will assume provisionally that R2=Bz=R' Putting it for i?,, the above expressions (A) assume the following forms: r^—It +Ci, r.2 = B — aci, r^=B' + c„ r\=B'-ac., (B) r5=B' + c^, r^=B' — ac^, from which we may determine the values of R, R\ a, Ci, d and c^: Cs r?-i\ _ n—r^i B = ry-ci, 1 + a ' _ n - Ti l + a ' B' =--}■■ -c.. _ n—Te (C) 1 + rt The results of calculation are siven in Table IX. and Fig. 4. Tap.le IX. Month (( ^"1 '■_ '^3 Px R I 0.098 89.2 166.7 43.7 69 85 II 0.235 44.5 96.3 -18.6 68 93 III 1.094 6.2 6.2 -25.8 101 149 IV 3.780 -0.4 -7.7 -25.7 110 162 V 4.471 -6.0 -9.3 -26.3 109 146 VI -5.570 0.4 11.6 39.6 139 163 VII 20.75 1.6 -0.8 -4.8 180 196 VIII 0.861 2.2 -19.3 -55.4 157 188 IX 7.100 -0.5 7.3 -17.3 188 213 X 0.905 6.3 5.8 -36.2 152 176 XI 0.167 102.8 110.5 -3.4 81 113 XII 0.067 139.6 255.9 45.9 63 82 It may be remarked that a comes out generally positive, except in June. It is generally less than unit}^ in the colder season and greater than unity in warmer nfbnths. Each of Ci, c., C3, On the Effect of_TopogTaphy on the Precipitation in Japan. 19 which run nearly parallel to each other, shows a maxi- mum in winter and a secon- dary maximum in summer. We may also remark a paral- lelism of R and R'. The fact that a for June becomes negative is no serious objection to our tlieory, but rather due to the inadequacy of the simple assumption (B). This might have been easily avoided if for an instance we distinguish R2 and R3 instead of putting them equal to each other. Assuming «=12.6 which is the mean values of «'s for May and July, and putting rc = R' + R"—(ic^ we obtain i?" = 39, c, = — .15, c.= -3.90 and C3=-13.30. Thus the positive values of CS disappears. Substituting these values of c"s, however, the general feature of the c-curves is not altered. According to our pre- vious theory, the values of c's must chiefly depend on the component of the barometi'ic gradient parallel to the ^'-axis making an angle of about 20° with the /y-axis. Calculating the x- and 2/ -components by 20 Art. 5. — T. Terada, M. Isimoto and M. Imamiira x'=x COS lS° + ;y sin 18° y'=y cos 18^— a: sin IS"" it was found that the general course of the //'-component (Table X and Fig. 4) is quite similar to that of c. Table X. Kegion ^■""■^--„^^^^ Month Compt. ^~~~~~-~-~^ I II III IV V VI VII VIII TX X XI XII Vear I, n. y\' .71 .05 .71 .05 .52 -.02 .18 .00 .07 .02 -.17 .02 -.15 -.01 -.29 -.01 -.17 -.05 .14 -.11 .44 .00 .59 .13 .24 .01 Ill, IV. y -2 .52 .22 .62 .16 .54 -.02 .18 -.08 .02 -.07 -.16 -.01 -.16 .01 -.18 -.06 .07 -.14 .39 -.12 .43 .05 .41 .29 .19 .02 V, VI. Xo >J3 .26 .13 .35 .05 .24 -.07 .08 -.06 -.01 -.03 -.14 .01 -.22 .03 -.14 -.05 .14 -.18 .34 -.16 .24 .01 .30 .23 .10 -.01 Mean X .50 .13 .56 .09 .43 -.04 .15 -.05 .03 -.03 -.16 .01 -.18 .01 -.24 -.04 .01 -.12 .29 -.13 .37 .02 .43 .22 .18 .01 As to the terms R and R' whicli represent the parts of the precipitation independent of the local topography, we may remark that the general annual variation is such as to be easily explained by yearly course of temperature and humidity, though the high values in September and October may j^robably be due to the cyclones prevalent in this season. From the above results, it may be seen that on the Pacific side, the seasonal fluctuations of the general term R and R' are generally enhanced by the topographical efïect in warmer as well as in the colder seasons, whereas on the Japan Sea side the seasonal variation is partially compensated by the topographical influence. This is the formal explanation based on the present theory why the seasonal variation is small on the latter side. In a previous paragraph we have mentioned that on the Japan Sea side, the yearly fluctuation of the precipitation shows no regular relation with the gradient components x and ?/, and inferred that on this side some other agents must predominate over the above gradient. It is interesting to observe that here in the seasonal fluctuation, the effect of topography appears conspicu- Ou the Effect of Topography on the Precipitation in Japan. 21 oLis, the effects of the otlier agents being apparently eliminated by taking average of different years. The formnla (B) fails in the case of the yearly variations, the values of c's obtained showing no regular relation with the com- ponents of the barometric gradient. Besides, their values are generally a small fractions of the general terms R and R' . Tlie simple assumption is therefore inadequate in this case. One point of interest is that the values of a thus obtained are generally positive and their mean value is very nearly equal to unity, being, 1.002, which points to the inference that on an average r-i—r^ =re—ri. The latter fact could only be explained by the presence of the high pressure area on the axial line of the land, as already mentioned in § 3. From the mere mathematical points of view, the above method of analysis is nothing more than the substitution of the six given amounts of precipitation by the new six quantities R, R', a and c's. Neither is the substitution the unique one possible. The chief physical interests, however, consist in the fact that the apparently irregular distributions of the precipitation may thus be explained at least in its main feature by the combination of the topographical effects on the basis of the simple elementary theory. These results may turn out useful for the practical pui'pose, as soon as the long period forcast of the barometric conditions becomes a matter of practice, the possibility of which is nowaday anything more than the dream of the modern meteorologists. 12. As already cited at the beginning of this note. Prof. Omori shew that there exists a remarkable parallelism between the yearly amount of precipitation at Niigata, a Japan Sea station, and that of the earthquake at Tokyo. In the present inv^esigation, the earthquakes originating in the submarine zone extending from Kinkwazan to Idu were chosen from the Kisyô-yôran. The yearly number of occurrence is given in the first line of the Table XI and plotted in Fig. 5. Comparing the graph with that of precipitation in different stations or regions, it was found that the correlation is rather remarkable in the case of the mean fluctuation of précipita- 22 Art. 5. — T. Terada, M. Isimoto and M. Imamura : Tatîle XL Year 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 Earth- quake Xo. — 6 24 26 40 48 42 28 23 42 -.43 -.02 37 -.18 .86 44 -.25 1.51 31 24 29 64 26 22 Bar. Diff. -.20 -.53 -.50 -.43 -.20 -.38 -.23 -.28 -.40 -.18 -.05 -.35 -l.CI -.53 -.81 -.33 .57 -.23 .54 -.38 Precip. (lI+IV+VI)/3. -1.14 -.19 .52 1.48 1.35 1.50 -.74 -.11 -.25 -.73 0 12 m- |6 18 On the Effect of Topography on the Precipitation in Japan. 23 tion of the entire Pacific regions, i.e. (II4-IV + VI)/3,'^ which is given in the third hne of the Table. The Japan Sea side, (I + III + V)/3 shows nearly similar yearly variation as the Pacific, except m a few years. Since the intimate relation of the earthquake frequency and the barometric gradient has been already fully established by the investigations^^ of one of the authors and also of K. Hasegawa and Saemontaro Nakamura, it was suspected that it may be also the barometric gradient that directly determines the rainfall on the one hand and the earthquake on the other hand. Hence the earth- quake frequency was compared with the different components of the gradient prevailing in different parts of the land. After a series of trials, it was found that the difïerence of the barometric pressure, the Japan Sea stations minus the inland stations, shows a parallel course with the earthquake frequency. Taking for the inland stations Matumoto, Takayama, Hikone and Osaka, and for the Japan Sea coastal stations Niigata, and Sakai, the difference is given in the second line of the Table. XL Comparing this wdth the earthquake frequency (Fig. 5), it will be seen that the earthquake curve shifted about one year earlier, shows a rather remarkable parallelism with the gradient curve. Whether the very curious correlation is merly accidental or not, cannot be ascertained at present from such a scanty materiah Though the apparent relation may appear rather absurd, the possibility of such a coincidence cannot be excluded by a superfical consideration, if we consider for an instance that the precipitation of the last year may affect the barometric pressure of the year concerned. A further investigation in this line is now in progress and we hope will be able to clear up the apparent mystery in a near future. 1) Here the values of (anomaly)/(mean anomaly) was averaged for the three regions. 2) T. Terada, Proc. Tokyo Math.-Phys. Soc, vol. IV (1908) p. 454 ; K. Hasegawa, ibid. Vol. VII (1913), p. 181 ; S. Xakaumra. Ihil. Vol. VIII (1915), p. 69. 24 -Ä-rt. 5. — T. Teracfe, M. Isimoto and M. Imamura : Summary. 1. The remarkable influence of topography on the distribu- tion of precipitation is demonstrated with respect to the yearly as well as the monthly amounts. 2. A vector diagram method of investigating the different relations of the precipitation with the barometric gradient is illust- rated. 3. The neccesity of taking the percentage values of the precipitation is emphasized. 4. The rainfalls in different districts are analysed in terms of the topographical effects. It is shown that the effect is largely determined by the component of the barometric gradient taken in a direction a little inclined to the axial line of the land, as was expected from the elementary theory proposed. 5. A peculiar relation between the earthquake frequency and the barometric gradient is pointed out. Published June iSth, 1919. JOURKAL OF TIIE COLLEGE OF SCIENCE, TOKYO IMPERIAL UNIVERSITY. VOL. XLI., ART. 6. Recherches sur les spectres d'absorption des amtnine=complexes métalliques. ; III. Spectres d'absorption des sels complexes de nickel» de chrome et de cuivre. Par Yuji SHIBATA, BigalciüiakusM. Professevir de Chimie minérale à l'Université Impériale de Tokio. Avec la collaboration de K. Matsuno, EigakusM. Avec 18 figures. Nous avons récemment publié quelques notes sur les spectres d'absorption des solutions aqueuses des ammine-complexes cobal- tiques (Journ. Coll. Seien. Imp. Univ. Tokio, Vol. 37, Art. 2 et 8.). On peut résumer les résultats de ces travaux dans les quel- ques règles suivantes: I) L'absorption de rayons d'un sel complexe est exclusive- ment effectuée par son ion complexe, ou pour parler exactement, c'est le point de la connection de l'atome métallique central avec les radicaux acidiques, ou les molécules de l'eau ou de l'am- moniaque etc. dans l'ion complexe, qui est le vrai siège de l'absorption. II) Les ions complexes semblablement construits absorbent semblablement les uns et les autres. "^ III) Si les atomes métalloïdes, qui s'enchâinent directement avec un atome métallique central dans un ion complexe, sont égaux, la constitution des groupes des métalloides n'a pas d'importance pour l'absorption. 2 Art. 6.— Y. Shibata : IV) Dans ces deux derniers cas, ni la. différence des signes de l'ion, ni la différence des valences, n'influence jamais l'absorption. V) L'ion simple, qui n'absorbe pas lui-même et s'accouple avec un ion complexe, n'influence guère l'absorption. VI) Le remplacement d'un groupe atomique dans un ion complexe par un autre exerce une influence bien caractéristique d'après les substituants sur l'absorption. VII) Les radicaux fortement acidiques dans un ion com- plexe sont généralement remplacés par les molécules d'eau, quand on dissout le sel dans ce même liquide. Le but du travail présent est donc de rechercher si les conclusions cités ci-dessus peuvent être appliquées aussi aux sels complexes des autres métaux, comme le chrome, le nickel ou le cuivre. Or, nous avons étudié spectroscopiquement les solutions des 48 sels complexes de chrome, de nickel et de cuivre, qui sont donnés ci-dessous: Sels complexes Sels complexes Sels complexes chromiques. nickeleux. cuivriques. [Cr(NH3)e]Cl, [Ni(NH3)e]Cl, [Cii(Py)6](N03), [Cr endCL, [Ni(NH3)e]S0, [Cu(Py)J(N03)2 [Cr end(SCN)3 [Ni(N,H,)3]Cl [CufPy)3](N03), • [Cr(NH3\Cl]Cl2 [Ni en3]CL-2H,0 [Cu(Py)o](N03)2 [Cr(NH3)5H.O]Cl3 . (NHJ,Ni(SO,),-6NH3-H20 Cu(NH3)2(C.H30,X, [Cr(NH3)5(SCN)](SCN)2 aNiSO/SNH.-TH.O Cui(NH3)3l(C„H3Ö2') [Cr(NH3),Cl-H,0]CL, [Ni(NH3),]SO,-2H,0 Cu(NH3)3Cl(CoH3Ô.)-H20 [Cr(NH3),Cl-H,0]S0, [Ni(N,H,)2]Cl2 NH,Cu.(CoH3Ô2)5-K,0 [Cr en,Cl]Cl [Ni en,]Cb-H,,0 Cu(C,H30,)o-H,b [Cr(NH3)'3CVH,0]N03 [Ni(NH3)3]CL-3HoO Cu(NH3)5- CA Cr(NH3)3(SCN)3 NiS0,-7H,0 Cu(NH3),-CA-2H.O [Cr(NH3)2Br2(H20)2]Br NiCL-6H,0 CuCNHjjo'CA [Cr(NH3)!i(SCN),]NH, CuNH3-C,0, Cr,,(SO,)3-18H,0(Sel violet) Cu(NH3\S0, Cr2sO,,)3-6H,0;Sel vert) Cu(NH3),S04 [Cr(CA)3]K3 CuS0/5H„0 [Cr(SCNJ6]K3 CuCl/2H,6 CrO/SNHg CrO/3KCN Reclierclies sur les spectres d."absorptioii des ammine-complexes métallique III. 3 Parmi ces complexes, ceux de chrome sont les plus stables et montrent une ressemblance remarquable avec les complexes cobaltiques. En conséquence, toutes les conclusions citées plus haut sont, sans exception, appliquables aux sels chromiques, tandis que les complexes de nickel diffèrent très sensiblement des deux précédents. L'atome de nickel central des ions complexes nickeleux est toujours bivalent, et le nombre de coordination n'est pas nécessairement six, comme dans le cas du cobalt et du chrome. Surtout, leurs solutions aqueuses ne sont pas du tout stables, car elles montrent la tendance de l'hydrolyse. Toutefois, on se trouve, dans ce cas môme, en présence d'une bonne concordance avec tout ce qu'on a observé dans les expériences spectroscopique sur les complexes cobaltiques. Quant aux sels complexes cuivriques, ils manifestent une instabilité plus avancée que ceux de nickel, et de même leurs absorptions sont distinctement influencées par les anions, qui s'accouplent avec les cations complexes. Nous discuterons sur la propriété optique spéciale aux complexes cuivriques, jdIus bas, dans la partie expérimentale. Les solutions aqueuses des complexes chromiques, qui sont bien stables et satisfont complètement la loi de Beer sur l'absorp- tion de rayons, donnent deux ou trois bandes d'absorption très nettes dans l'échelle spectrale visible et ultraviolette. Dans le cas du cobalt où l'on a aussi observé deux ou trois bandes bien distinctes, on a remarqué que l'une de ces bandes qui se place toujours à 2000 de fréquence n'est pas influencée sensiblement par une substitution quelconque dans les ions complexes (Y. Shibata: .Journ. Coll. Seien. Imp. Univ. Tokio, Vol. 37, Art. 2, P. 5). Il n'en est cependant pas de même dans le cas du chrome; les bandes d'absorption des complexes chromiques sont bien sensibles et mobiles d'après les substitutions dans l'ion complexe. Comparé au cobalt, les sels de chrome possèdent généralement beaucoup moins d'abilité d'absorption et quelques complexes chromiques ne montrent leurs absorptions que dans les solutions assez concen- trées, par exemple, décinormale (en employant un tube de 10 cm de longueur), tandis que les complexes cobaltiques donnent leurs Art. 6. -Y. Shibata: absorptions clans les solutions très étendues, quelquefois en dix- millième normale. Les solutions des complexes de nickel absorbent aussi faible- ment, et leurs bandes ne paraissent qu'à la concentration de décinormale, tandis que les sels cuivriques, soit les complexes, soit les simples, ne donnent aucune bande d'absorption; ils absorbent pourtant continuellement dans les deux parties du rouge et du violet. Partie Spéciale. I. Absorption de rayons des sels complexes chromiques. 1) Complexes héxammoniés chromiques. Les trois sels de cette catégorie [Cr(NH3)6]Cl3*H20 [Jörgensen: J. prakt. Chem., 1884, [2], 30, 12], [Cr enslCla'S^rH^O [Pfeifïer: Z. anorg. Chem., 1900, 24, 286] et [Cr eus] (SCN^'H^O [Pfeiffer: Ibid., 294] sont des corps bien cristallisés avec les couleurs jaunes ou jaunes rougeâtres. Ils sont facilement solubles dans l'eau et donnent des solutions bien stables. Comme on le voit dans la figure I, les abilités d'absorption de rayons du complexe héxammonié proprement dit et des complexes, dont une partie des m.olécules d'ammoniaque est rempli par les molécules d'éthylènediamine, sont considérablement différentes; le premier absorbe beaucoup moins fortement que les derniers, c'est-à-dire que, bien que leurs deux bandes d'absorption se placent respectivement aux positions des mêmes logueurs d'onde (2150 et 2870 de fréquence), les concentrations, dans lesquelles on peut avantageusement observer les bandes, ne sont pas de même. Les deux bandes du corps [Cr(NH3)6]Cl3 ne loarâissent bien nettement que dans la solution de décinormale, tandis qu'on les observe, dans le cas des complexes contenant d'éthylènediamine, déjà dans la concentration de centinormale. Recherches sur. les spectres d'absorption ties aminine-CDmplexes motalliquesvIII.]: 5 Fig. I. Fréquence. 1500 2000 2500 3000 3500 4000 4500 5000 4-0 35 2'0 1-0 0-5 00 \ / \ f \ y v> 1 \ I.'\ .•<-•' ^'<^ r \ \ \ k *. V *, \ X V •. \ •. \ *. \ *. [Cr(NH3)e]CV [Cr eDgJCL - - \ 10/ 1(— ■Juoo ■ 10000 1000 100 10 t?j [Cr en.] (SCN)3 |\100/ Nous nous rappelons que, dans le cas des complexes cobaltiques, la différence des intensités d'absorption entre le com- plexe héxammonié proprement dit et celui qui contient de l'éthylènediamine à la place de l'ammoniaque n'existait pas. Ainsi, on remarquera deux faits importants; premièrement les oomplexes chromiques sont influencés optiquement bien délicate- ment par les substituants dans leurs ions complexes et deuxième- ment, conformément aux complexes cobaltiques, l'absorption de Art. 6.— T. Shibata : ceux de chrome est pratiquement indifférente aux ions simples qui s'accouplent avec les ions complexes. 2) Complexes péntammoniés chromiques. Fig. II. Fréquence. 2000 2500 3000 3500 4000 4500 1500 4-0 3-5 - 3-0 2-5 20 1-5 1-0 0-5 0-0 V::: / vV7 "i W *< \ \ \ \ '* \ * \ \ \ V \ \ \ \\ \ \ \\ II 5000 10000 1000 100 10 w [Cr(NH3)5H,0]C], J ' 100/ Le chlorure du péntammine-monocliloro-chrome, [Cr(NI[3)5 CljCls [Christensen: J. prakt. Chem., 188J, [2], 23, 54], est un corps cristallin coloré ronge, tandis que, le chlorure du péntam- mine-monoaquo-chrcme, [Cr(NH )5H20] CI3 [Christensen: Ibid., 28], est composé de cristaux rouges jaunâtres. Ces deux corps Recherches sur les spectres d'absorption, des ammine-complexes métalliques IIL sont solubles dans Teau et donnent des solutions ayant la même couleur que les corps solides. Comme la figure II l'indique, les deux sels absorbent pareillement Tun et l'autre; leurs premières bandes se trouvent à 2000 de fréquence, tandis que les deuxièmes se placent à 2650. Par les courbes, on remarquera que ce qui était vrai pour les complexes cobaltiques, l'est aussi pour ceux de chrome, c'est- à-dire que les chloro- et aquo-complexes absorbent semblablement dans les solutions aqueuses. Dans l'eau, les deux corps sont, sans doute, dans l'état d'équilibre suivant: [Cr(NH3)5Cl] CL + H,0 -^-» [Cr(NH3)5H,0] CL Dans une solution assez étendue, comme centinormale, il faut considérer que c'est l'aquo-complexe qui prédomine. Or, les covirbes qui sont données dans la figure II représentent très probablement celles des aquo-Kîomplexes. En comprant ces cour- bes d'absorption avec celles des complexes héxammoniés, on peut facilement apercevoir que le remplacement d'une molécule d'am- moniaque par une molécule d'eau exerce une influence bathochro- mique, c'est-à-dire que les deux bandes sont déplacées bien sensiblement vers le rouge. Il nous semble que ce phenomème est bien général, car nous avons toujours remarqué que l'importa- tion des molécules d'eau dans un ion complexe cobaltique exerçait aussi une influence dans le même sens. 3) Complexes lélrammoniés chromiques. Le chlorure et le sulfate du tétrammine-monoaquo-mono- chloro-chrome, qui sont des cristaux colorés en pourpre, ont été préparés par des voies tout à fait différentes l'une de l'autre. Le chlorure a été synthétisé d'après Pfeifïer (Ber. deutch. chem. Gesell., 1905, 38, 3594) qui a donné le procédé préparatif suivant: [CrClo(HoO),]Cl + 2C5H5N ^ [Cr(Py),(H.O)2(OH)JCl + 2ClH Chlorure vert [Cr(Py),(H,QX,(OH)JCl ^^HsetHCl [Cr(NH3),Cl H,0]C1 Cependant Jörgensen (Journ. prakt. Chem., 1899, [2,] 20, 110) a préparé son sulfate en opérant la réduction d'un mélange de 8 Art. 6. -Y. Shibata; bichromate d'ammoniaque, d'acide chlorohydrique et de chlorm-e d'ammonium par l'alcool. Quoi que l'écart des manières preparatives de ces deux corps soit assez grand, leurs absorptions, qui sont traduites graphique- ment dans la figure III ne diffèrent presque pas l'une de l'autre. Voici l'équilibre qui s'établit: [C^NHsXCl-HoOjCl, H,o [Cr(NH3%(H20)2]Cl3 Fig. III. Fréq^uence. 1500 2000 2500 3000 3500 4000 4500 4-U ■^ i \ / \ /'•A / ' \ \ 35 1 / i 3'0 \ y .' A 2-5 V/ \ \ \L '\ x \^ 2-0 — -V 1.5 10 0-5 nn 5000 — , 10000 [Cr(NH3),Cl-H,0]Cl,,- (Pfeiffer) [Cr(NH3),Cl-H,0]S0,- (Jörgensen) ^^ N^\ lu 100/ 1000 100 10 H •Ö Reclierolus sur los spectres d'absorption des auimlne-complexes métalliques III. 9 Conformément au cas précc'dent, il faut que les courbes représentent T absorption du diaquo-complexe chromique. Les courbes indiquent que les bandes sont déplacées davantage vers le rouge que dans le cas du monaquo-complexe. 1500 4-0 3-5 3-0 2-5 2'0 1-5 1-0 0-5 0-0 2000 2500 Fig. IV. Fréquence. 3000 3500 4000 4500 \ \ \ \ \ \ / \ J \ \ \ \ \ \ \ \ \ » * ^ \ l \\ 5000 10000 1000 100 10 w w [Cr en,,CloJCl, la solution fraiche (violet). la solution laissée (rouge). Le chlorure du diéthylènediamine-dichloro-chrome, [Cr eng CLgaJCrHsGLPfeifïer u. Lando: Ber., 1904, 37, 4278], dont les 10 Art. ß.- Y. Shi bâta: doux atomes de chlore dans F ion complexe se placent à la position de ois, est un corps cristallin violet. Il donne une solution bien labile et, quand on la laisse, le cliangement de la couleur a lieu de telle sorte que sa coloration violette initiale disparaît peu à peu, et en même temps la coloration rouge prend naissance. Cette transformation de tonalité continue jusqu'à ce qu'après cinq ou six heures, elle soit devenue tout à fait rouge. La courbe, tracée par la ligne noire dans la figure IV, représente l'absorption de la solution nouvelle (violette), tandis que celle de la ligne brisée représente la solution qui est laissée pendant une nuit (rouge). La raison, pour laquelle cette dernière courbe montre une bonne concordance avec celle du diaquo-com- plexe (fîg. III.), c'est qu'il y a eu aussi la substitution de deux molécules d'eau dans l'ion complexe diaquo-chromique: [Cr en.Cip;jCl 2H^ [Cr en,(H20)2g]]Cl3 4) Complexes Iri- el diammoniés chromiques. Le nitrate du triammine-dichloro-monoaquo- chrome, [Cr (NH3)3Cl2H,0]N03 [Werner: Ber., 1906, 39, 2663], est un corps ciistallin gris verdâtre, presque insoluble dans l'eau. Mais si on le laisse assez longtemps en contact avec de l'eau, ou si on le chauffe un peu, il se dissout et donne une solution violette qui est vraisemblablement formée par la réaction suivante: [Cr(NH3)3CUK,0]NO,ja^[Cr(NH3)3Cl(Hp)J^iQ H^[Cr(NH3)3( On sait déjà que le triammine-triaquo-complexe cobaltique, dont trois molécules d'eau dans l'ion complexe se placent dans la position consécutive, donne une solution violette, tandis que son isomère, dont deux molécules d'eau parmi trois se trouvent dans la position diagonale (ou la position de trans) change la couleur de sa solution spontanément du violet verdatre jusqu'au violet clair dans l'espace de quelques instants (Comparer K. Matsuno: Journ. Tokio Chem. Soc, 1917, Vol. 38, 664). Recherches sur les spectres d'absorption des ammine-complexcs métalliques III. W De même, il est bien probable que la solution violette du triammine-triaquo-complexe chromique prend la configuration, qui contient les trois molécules d' eau dans la position consécutive : NH3 H,0 'NH, H,0^ Cr NH, H,0 -J Le bromure du diammine-dibromo-diaquo-chrome, [Cr(NH3)2Br2 (H20)2]BrnYerneru. Dubsky: Ber., 1907, 40, 4090], est un corps vert brillant. On peut légitimement penser, à cause de cette couleur, que les deux atomes de brome dans l'ion complexe de ce corps se trouvent à la position de trans. Il est facilement soluble dans l'eau et donne une solution verte, qui change pourtant peu à peu et devient enfin tout-à-fait rouge après quelques heures. Une pareille combinaison en cobalt, [Co(NH3)2(H20)iCl2]X, est aussi un corps vert, mais sa solution verte nouvelle, après quel- ques instants, prend la couleur bleue verdâtre; puis le changement s'arrête et ne marche plus (Matsuno: ibid.). Cette transformation intermédiare de couleur est, sans doute, provoquée par le rem- placement d'un seul atome de chlore: H,0 NH, H.O H.,0 ^NH, ' NH, H,0 N,H, Cl Il faut, de là, conclure que, dans le cas de chrome, où l'on obtient facilement une solution rouge, le remplacement des atomes d'halogène dans son ion complexe par les molécules d'eau est complète, et donne tétraquo-complexe : [Cr(NH3)2(H,0)2Br2]Br 2H,o [Cr(NH3)2(H20,\]Br, 12 Art. 6.— Y. Shibata Les courbes dans la figure V représentent les absorptions des solutions aqueuses des deux complexes chromiques triammonié et diammonié, dont les atomes d'iialogène sont très probablement tout remplacés par les molécules d'eau. De ce point de vue, on peut considérer que les courbes données ci-dessous sont di- et triaquo-complexes chromiques. Fig. V. Fréquence. 1500 2000 2500 SOOO 3500 4000 4500 4-U f'\ 'A \ 3-5 30 / » / 1 / \ 1 / . V \ V \ % \ \ 2-0 15 \ \ \ \ \ \ \ N \ \ \ \ \ 10 \ ^ 0-5 an 1 1 ,Br,(K,OX,]Br J '"ÏÔ" ÏOô) [Cr(NH3)3CL,H,OJNO [Cr(NH3) 5000 10000 1000 I 100 10 Reclierclics sur les spectres d'absorption des auiminc-complexes métalliiuics III. ]^3 Or, on trouve ici encore une fois l'influence bathochromique des molécules d'eau dans un ion complexe, parce que le tétraquo- complexe possède sa deuxième band à 2490 de fréquence, tandis que le triaquo-complexe Ta à 2550, et encore, si l'on compare leurs absorptions d'extrémité, on remarquera facilement que l'in- fluence s'y manifeste davantage en ce même sens. 5) Complexes ammoniés chromiques qui contiennent les radicaux de sulfocyanate. Nous avons étudié spectroscopiquement les cinq sels, donnés ci-dessous: a) [Cr(NH3)5(SCN)](SCN)o[Wern.er n. v. Halbaii : Ber., 1906, 39, 2670] b) [Cr en.(SCN),gf] SON [Pfeiffer : Ber., 1901, 34, 4307.] c) [Cr(NH3)3(SCN)3] [Werner u. v. Halbaii : Ber., 1906, 39, 2G68] d) [Cr(NH3),(SCN) JNH/H,0 [S. Christensen : J. prakt. Chem., 1892, [2], 45, 216] ' e) [Cr(SCN)JK3-3H,0 [Joseph Eoesler : Ami. d. Chem., 1867, 141, 18-5] Le corps (a) est formé de cristaux rouges violets; (b) orange carmin fade; (c) rouge fade; (d) rose foncé et (e) pourpre rougeâtre. Ils sont facilement solubles dans l'eau, sauf (c) qui est pourtant soluble dans l'acétone. Les solutions ont la même couleur rouge. Il a déjà été observé par l'un de nous (Y. Shibata: Journ. Coll. Seien Imp. Univ. Tokio, Vol. 37, Art. 2, P. 23) que le sulfocyano-ion dissocié dans l'eau n'absorbe pas du tout, tandis que, s'il se trouve dans un ion complexe, il donne toujours une bande bien caractéristique à 3400 de fréquence. On en trouvera encore quelques unes dans les fig. VI et fig. VIL De même, on remar- quera l'influence extraordinaire du radical de sulfocyanate dans la direction bathochromique, si l'on examine les courbes d'un com- plexe héxammonié (fig. I.) et du sulfocyanate de péntammine- monosulfocyanato-chrome (fig. VI.). La première bande à 2150 de fréquence de complexes héxam- moniés chromiques est alors déplacée jusqu'à 1750 vers le rouge, par la substitution d'un seul radical du sulfocyanate au lieu d'une 14 Art. 6.— Y. Slübata : molécule d'ammoniaque dans l'ion complexe, tandis que la deuxième est poussée de 2870 à 2400 pour la même raison. Quant à la troisième bande du pôntammine-monosulfocyanato- complexe cliromique, elle est, comme il a déjà été indiqué plus haut, pour le radical du sulfocyanate dans l'ion complexe; c'est ce qu'on a aussi observé pour la même sel complexe cobaltique. Faisons attention seulement, concernant cette dernière bande, que le radical de sulfocyanate dans l'ion complexe chromique montre une propriété hyperchromique bien remarquable: la bande en question du sulfocyanate complexe chromique parait déjà dans l'épaisseur de 5 mm de la solution centinormale, tandis que celle du complexe cobaltique ne commence à paraître que dans l'épais- seur de 40 mm de la solution, dans la même concentration. La courbe d'absorption du diéthylènediamine-disulfocyanato- complexe, dont les deux radicaux de sulfocyanate sont à la position de trans, est tracée, dans la figure VI, avec la ligne pointillée. On y remarquera un fait bien curieux; c'est que l'influence du deuxième radical de sulfocyanate est plutôt hypsochromique, c'est-à-dire que sa première bande est retrogradée jusqu'à 2100 de fréquence (vers violet), quoi que l'absorption du disulfocyanate- complexe (trans) soit distinctement bathochromique comparée à celle du héxammine-complexe. De même, sa deuxième bande est très insignifiante. Il faut donc considérer que cette anomalie apparente est provoquée pro- bablement par l'influence stéreochimique de deux radicaux de sulfocyanate, qui se placent, dans un ion complexe, à la position de trans l'un et l'autre. Ce qui est ainsi observé dans le cas du complexe chromique, c'est ce que nous avons déjà remarqué en certains complexes cobaltiques, bien que le phénomène dans les deux cas ne soit pas tout à fait identique. Nous nous rappellerons que les deux isomères de tétrammine-dinitro-complexes cobaltiques, crocôo (trans)- et flavo (eis) complexes, n'absorbent pas identique- ment l'un et l'autre; le crocéo-complexe donne trois bandes dans toute la region de l'échelle spectrale, tandis que le flavo-complexe n'en possède que deux. Cette différence de la manière d'absorp- tion doit être aussi due à F influence stéreochimique. A notre Kecherclies sur los spectres d'absorption des ammine^couiplexes métalliques III. 15 grand regret, nous n'avons pu étudier, faute des matière nécessaire, le cis-disulfocyano-complexe chromique, qui nous mettra en évidence évidemment des choses bien intéressantes. Le triammine-trisulfocyano-chrome a été étudié en une solu- tion acétonique, à cause de son insolubilité dans Teau. Comme 1500 2000 2500 Fig. VI. Fréquence. 3000 3500 4000 4500 4-0 3-5 3-0 2-5 2-0 1-5 10 0-5 0-0 '- Cr^^NH3)3(SCN)s (Solution dans] l'acétone) — L ^ -^ [Cr(NH3)a(SCN)](SCN), (\ iW ~100ö) (Cr en.(SCN),Pc]]SCN 5000 10000 t i '^-•. I^ ••••••^v. % ■■■} H. fl 1 ■•■À /Xi i 1 ' ' sN \ 1000 100 10 .^ 16 Art. 6. -T. Shibata : on le voit dans la fignre VI, ses deux bandes, qui se trouvent respectivement à 1900 et à 2500 de fréquence, sont encore sensiblement hypsochromiques relativement à celles du monosul- focjano-complexe, bien qu'elles soient logiquement bathochromi- ques, comparées aux complexes héxammoniés chromiques. Sa troisième bande, qui est due au radical de sulfocyanate dans l'ion complexe, ne parait pas, ou elle est plutôt couverte par une absorption continuelle d'extrémité, causée par le solvant (acétone). La similarité des positions des premières bandes du trans- disulfocyano-complexe et du trisulfocyano-triammine-complexe nous fait penser que la configuration I, dans laquelle deux radicaux de SON parmi trois se trouvent dans la position de trans, est préférable pour le trisulfocyano-complexe, à la configuration II, dont les trois radicaux de SCN sont arrangés cosécutivement: NH, SCN SCN NH, SCN SCN SCN SCN NH, NH, NH3 NH, I. II. Les absorptions du diammine-térasulfocyano-complexe et de l'héxasulfocyano-complexe sont données dans la figure VIL Ces deux sels donnent aussi trois bandes très nettes. Leurs premières et deuxièmes bandes mettent en évidence bien clairement la nature bathocbromique des radicaux de sulfocyanate dans l'ion complexe. Pourtant, par l'examen plus profond de ces deux courbes, nous pouvons facilement remarquer que le degré de l'influence bathocbromique est sensiblement moins important dans le tétrasulfocyano-complexe que dans le cas de l'héxacyano- complexe; en effet, la première et la deuxième bandes du tétrasul- focyano-complexe qui se placent respectivement à 1850 et à 2500 de fréquence n'atteignent pas même aux positions, où se trouvent les deux bandes du monosulfoc^^ano-complexe (1750 et 2400). C'est peut-Otre la raison, pour laquelle le tétrasulfocyano-complexe Eecherchcs sur les spectres d'absorption des ammine-complexes métalliques m. X7 absorbe moins bathocliromiquenient que le monosalfocyano-com- plexe, qu'au moins deux radicaux de SCN dans le premier corps doivent être nécessairement placés à la position de trans, qui, comme on a remarqué plus haut, exerce l'influence moins bathochromique. 1500 200O 2500 Fig. VII. Fréquence. 8000 3500 4000 4500 4-0 3-5 30 2-0 1-5 1-0 0-5 0-0 Il ^ A \ / ^ v / \ > - \\ 1 / \ \ \ \ « \ /y '> [Cr(NH3)2(SCN)JNH,-H.O- [Cr(SCN)e]K3 — V N _ J\ 100 NN 1000/ 5000 "■ 10000 1000 100 10 < Quant aux troisième bandes des deux complexes tétra- et héxasulfocyaniques, il suffira d'indiquer qu'elles montrent une IQ Art. 6.— T. Shibata : bonne concordance avec celles d'autres sels de cette catégorie comme les bandes caractéristiques au radical de SCN. 6) Sulfates et oxalate chromiques. Depuis long-temps il est bien connu que presque tout les sels chromiques existent en deux séries de composés: l'une représente des corps colorés violets tandis que l'autre contient ceux qui ont la couleur verte. Leur constitution a été long-temps le sujet des recherches de beaucoup de cliimistes; entre autres M. A. Werner et M. Recoura doivent être spécialement nommés, car nous devons beaucoup à ces savants à propos du développement de la chimie des sels chromiques dans cette direction. Donnons quelques exemples des sels appartenant aux deux séries : Sels verts. Sels violets. [^'fn^O) J ^^ *^^^^ [Ci-(H,0),] (N03)3-3H,0 [^^^XHX))e] °'' L^^^^HX))«] "^^^^ [Cr,(H,0),,](SOj3-5 on 6H,0 Nous avons remis, dans cette occasion, l' étude spectroscopique des membres de ces deux séries et, comme objets de nos recherches, nous avons choisi deux sulfates chromiques violet et vert. L'exam.en comparatif des courbes d'absorption des deux corps, qui sont données dans la figure VIII, nous a permis de mettre en évidence des choses bien intéressantes. Approximative- ment, ils absorbent très semblablement l'un et l'autre (exactement, les deux Ijandes des sels verts et violet se trouvent respectivement à 1720 et à 2250 pour le premier et à 1750 et à 2400 pour le dernier), 1)ien qu'ils fassent les solutions avec des couleurs très Recherches sur les spectres d'absorption des ammine-complexes métalliques III. 19 différentes. Toutefois, leurs abilités d'absorption ne sont pas du tout les mêmes, c'est-à-dire que le sel vert commence à absorber sélectivement à 28 mm de l'épaisseur de la solution pour sa première bande et à 33 mm pour la deuxième dans la concentra- tion centinormale, tandis que le sel violet commence à absorber sélectivement respectivement à 71 mm et à 130 mm de l'épaisseur. Cette absorption fortement hj^pochromique du sel violet est 1500 4-0 3-5 2000 2500 Fig. VIII. Fréquence. 3000 3500 4000 4500 5000 2-0 1-0 0-5 00 — I / ^.^ ■\"\ — % — /—^.îe^ r: — T-rr ' N • \ I \%, 10000 1000 100 10 Cro(S04)3-6H,0(vert) [Cr(CA)3]K3 Cr,,(S04V18H,0 (violet) f— 10/ H ►Ö su ta 20 Art. 6.— Y. Shibata: provoquée, il nous semble, par les molécules d'eau, qui occupent tous les nombres de coordination dans l'ion complexe, comme on le voit par sa formule dans la table donnée plus haut. Quant aux propriétés à la fois batliochromique et hypochromique des molécules d'eau dans un ion complexe, nous les avons déjà observé très souvent dans plusieurs aquo-complexes cobaltiques et chromiques. Il est pourtant bien naturel, que dans le sel vert, l'influence hypochromique ne soit pas très développée, parce que les molécules d'eau, dans ce cas, n'occupe qu'une partie des nombres de coordination de son ion complexe. Leurs constitutions expliqueront davantage la relation absorptive de ces deux sulfates. Sulfate Tert. (H,0)3 (OH^ -O. .0. 0,S Cr. Xr O. .0' SO., O, Sulfate violet. ILO H,0 OH, ^-Cr: OHo ■OHo OH, (S0,)3 G H,0 L'absorption de 1' oxalate chromique, [Cr(C204)6]K3, qui est indiquée dans la figure VIII avec une ligne pointillée, montre aussi une similarité étonnante avec celles des sulfates, spécialement celle du sulfate vert. Comme la constitution de 1' oxalate l'indique, les radicaux d' oxalate s'enchâinent directement à l'atome chromique central avec les atomes d'oxygène (Comparer Y. Shibata: Ibid. P«. 13): O' 0=0-0 ... t 0=0-0' ;cr. 0. ^c=o C>— 0=0 ^0—0=0 / ^0=0 K, Recliei'ches siir les spectres d'absorption, des ammine-complexes métalliques III. 21 En revenant aux sulfates encore une fois, nous remarquerons que les radicaux du sulfate et les molécules d'eau dans le sel vert se sont liés à T atome chromique central avec leurs atomes d'oxygène, tandis que l'atome chromique du sulfate violet, à son tour, s'enchâine directement aussi avec les atomes d'oxygène des molécules d'eau. C'est donc une raison pour que ces trois corps absorbent bien pareillement les uns et les autres. 7) Complexes de peroxydes chromiques. On connaît une série de complexes chromiques bien curieux: Cr04-3C5H5N, CrO^'SNHs et CrO^'SKCNCK. A. Hof mann u. H. Hiendlmaier: Ber., 1905, 38,3059]. Parmi eux, celui qui contient trois molécules de pyridine, étant insoluble dans l'eau, n'a pas été étudié, tandis que les deux autres sont bien solubles, donnant des solution jaunes. Fig. IX. Fréqtience. 1500 2000 2500 3000 3500 4000 4500 3-0 2-5 2-0 1-5 ro 0-5 0-0 \ \ \ \ \ \ \ \ \ ^ \ ^ V /\'* \ / ' V \ / \ ^y "V ^*°~~; n \ 1 V À CrO,-3KCN CrO^-SNHa- -J\ 100 1000/ 5000 1000 100 10 ^ ►Ö p 22 Art. 6.— Y. ShiLata: Comme on le voit dans la figure IX, les absorptions de ces deux complexes sont presque les mêmes l'une et l'autre, mais elles sont tout-à-fait différentes des autres complexes chromiques, que nous venons de décrire. Ils montrent deux bandes bien distinctes, comme les autres complexes, toutefois on n'ajDercoit point de similarité entre les courbes d'absorption en question et celles des autres complexes. Les absorptions de ces deux: complexes do peroxydes chromiques donnent leurs bandes à 2700 et à 3900 de fréquence, et dans l' allure tout le long de la courbe, on trouve une ressemblance parfaite entre celle-ci et celle des solutions aqueuses de chromâtes métalliques, qui a été observée par M. A. Hantzsch (Zeit, physik. Chem., 1910, 72, 363). Ce dernier fait nous indique que les chromâtes de potassium et d'ammonium sont faits dans les solutions des deux corps, d'après les formules suivantes: CrO^-SN H3 + 3H2O = CrO, + 3NH,0H Cr 0, + H,0 = Cr 0,Ho + O CrO.K, + 2NH,0H = CrO,(NH J, + 2H,0 et CrO,-3KCN + 4H2O = CrO.K, + 3K0H + 3CKH + O CrOÄ + 2K0H = CrO J\., + 2H,0 II. Absorption de rayons des sels de nickel. 1) Complexes héxammoniés nickeleux. Les quatre sels complexes donnés ci-dessous ont été étudiés: [Ni(NH3)6]Cl2 [Sörensen : Z. anorg. Chem., 1894, 5, 3G3], un corps bleu violacé fade. [Ni(KH3)6]S04 (Ibid.), un corps bleu violacé. [Ni(N2ll4)3]Cl, [H. Franzen u. O. v. Mayer : Z. anorg. Chem., 1908, ÖO, 262], un corps violet bleuâtre. [Ni enJCL (^Yerner et Spruck : Zeit, anorg. Chem., 1899, 21, 212), un corps violet bleuâtre. Ils sont solubles dans Teau, sauf celui qui contient trois molécules d' hydrazine. Ce dernier corps même, est bien soluble dans l'eau ammoniacale. Recherches sur les spectres d'absorption des ammine-complexes métalliques m. 23 Leurs absori:)tions sont étroitement analogues les unes et les autres, comme elles sont montrées dans la figure X; les trois premiers corps donnent leurs deux bandes à 1700 et à 2750 de fréquence, tandis que seulement le triéthylènediamine-complexe, étant un peu hypsochromique, en donne respectivement à 1800 et à 2900 de fréquence. Or, on se trouve encore en présence d'exemples des- propositions II, III et V, données en tête de cette note, qui annoncent que des complexes construits d'une façon analogue les uns et les autres absorbent semblablement et ne sont pas influencés par des ions simples qui s'accouplent avec les premiers. Fig. X. o "ii; 2 s Fréquence. 1500 2000 2500 3000 3500 4000 4500 5000 2-0 1-5 1-0 0-5 0.0 l II ,•/■ \ \*^'' ß ' !/ V • • t ■>\v \ 1 \ \ \ 100 10 H o « 3 o fe:^ [Ni(N2Hj3]Cl„-.- [Ni(NH3)JCl ^ [Ni(NH3)e]S0, - [Xi en3]Clo2H,0 ^(Solutions ammoniacales, ) 2) Ammine-complexes insalurés. Dans les complexes nickeleux, il y en a quelques uns, dont les nombres de coordination ne sont pas saturés. Comme exemples des sels de cette catégorie, nous avons étudié les quatre sels suivants: 24 Art. 6.— Y. Shibata : [Ni(NH3),]SO/2H20 [O. L. Erdmann : J. prakt. Chem., 1836, 7, 264], bleu fade. [Ni eiiojCL'HsO [Grossmann u. Schuck : Zeit, anorg. Chem., 1906, 50, 9], violet rougeâtre. [Ni(N2H4)2]Cl2 [H. Franzen u. O. v. Mayer : Zeit, anorg. Chem., 1908, 60, 262], gris violacé. [Ni(NH3)3]CV3H20 [Andrée: Compt. rend., 1888, 106, 937], bleu violacé. Ils sont très bien solubles dans V eau, mais ces solutions ne sont pas du tout stables; elles se décomposent assez rapidement et donnent des précipités d' hydro xyde nickeleux. Dans ce cas, quelques gouttes d'ammoniaque, ajoutées à la solution, suffiront pour arrêter cette décomposition hydrolytique. Comme on le voit dans la figure XI, les absorptions de ces quatre corps ne diffèrent guère les unes et les autres. De même, ces absorptions montrent une analogie bien notable avec celles des complexes précédents. La ressemblance de l'aspect des courbes des complexes appartenants à ces deux groupes nous indique que ceux qui sont insuffisamment ammoniés prennent quelques molécules d'ammoniaque dans la solution ammoniacale et deviennent des complexes héxammoniés en saturant des nom- bres de coordination. 3) Complexes nickeleux compliqués. 2NiS04-5NH3-7H20[Andrée: Compt. rend., 1888, 106, 937] est composés de cristaux bleus fades, qui se décomposent dans une solution aqueuse et donnent des précipités d'hydroxyde nickeleux. Ces précipités sont cependant solubles par l'addition de quelques quantités d'ammoniaque. L'autre corps [NiS04(NH4)2SOi"6NH3* 3H2O [Andrée: Ibid. 938] est composés de cristaux de la même couleur que le complexe précédent; il est toutefois bien soluble dans l'eau avec une couleur bleue et ne montre aucune tendance de décomposition. Dans la figure XII, on voit les courbes d'absorption de ces deux complexes qui montrent encore une fois une analogie étroite non seulement entre elles, mais aussi avec celles des auti'es com- Eecherches sur les spsctros d'absorption des ainmine-complexes métalliques III. 25 plexes nickeleux, dont les absorptions ont été discutées par nous jusqu'ici, c'est-à-dire que leurs bandes se placent à 1750 et à 2750 de fréquence. Fig. XI. Fréquence. 1500 2000 2500 3000 3500 4000 4500 5000 1-5 10 0-5 0-0 V /'^ 100 t?j o . — . 0 g ci- &• ^ ta O on [Ni(NH3)3]Cl/3H,0 [NienJCVHoO [Ni(NH3),]SO,-2H,0 [Ni(N,H,)2]Clo ^(Solutions ammoniacales, — — ) Quant à la constitution du corps 2NiS04'5NH3'7H20, il faut tout d'abord déterminer, s'il prend la forme, dans la solution, d'un sel double entre le sulfate nickeleux avec 7 molécules de l'eau de cristallisation et le sulfate de péntammine-complexe nickeleux, [Ni(NH3)5]SOi, ou s'il est devenu deux molécules d'un complexe [Ni(NH3)6]S04, en prenant encore 7 molécules d'ammoniaque dans une solution ammoniacale. S'il est vrai que ce complexe prend la formule d'un sel double contenant une molécule de sulfate nickeleux comme un composant, il devra donner l'absorption caractéristique à ce sel simple, dont l'absorp- tion va être décrite tout de suite (la figure XIII.). Mais un coup d'œil jeté sur la figure suffit pour remarquer que ce n'est pas le 26 Art. 6.— Y. Shibata : Fig. XII. FréqvTence. 1500 2000 2500 3000 3500 4000 4500 0-2 1-0 0-5 0-0 \ \ /\ 1 * f* \ * f • \ V -J LJ. --\w--- --1CX3(4) \J 1 \ • \ 000 100 10 Pc 11 et- p_, 2NiS04-5]SlH3-7H20 (Solution ammoniacales, ) (NH,)2SO/NiSO,-6NH3-3H,,0 (Solution dans l'eau P^re,-^ ) cas; l'aspect de sa courbes nous indique bien clairement qu'il est plutôt de la forme du complexe héxammonié. Le sel avec une formule brute (NHOsSO/NiSO.-ßNHs-SH.O est un corps bien stable dans l'eau, et sa courbe d'absorption ne diffère pas beaucoup de celle du complexe précédent. Dans la solution, le sel se dissocie évidemment dans les ions NH.l, SOI' et peut-être [Ni(NH3)G]," ce dernier donnant l'absorption caractéristique aux complexes ammoniés nickeleux. Le sel est bien stable dans Teau contrairement à autres complexes, parce qu'il est protégé contre la décomposition h^^droly tique par l'excès des ions SO4" et NH4, c'est-à-dire que les réactions hydrolytiques réversibles formulées ci-dessous sont poussées, d'après la loi d'action de masse, de droite à gauche respectivement par SO4" et nh;: [Ni(NH3)„]S04 + 2H,0::;:±:[îsi(NH3)„](HO), + SO/ + 2H- [Ni(NH3)„](HO)2-hnH20::;ztNi(HO),+ nNH^-HO, ?i<6. Kecherclies sur Its spectres d'absorption dos amminc-complcxes métalliques III. 27 4) Sulfate et chlorure de nickel. Ils se cristallisent avec 7 molecules d'eau en sulfate et avec G molécules d'eau en chlorure, leur couleur étant vert foncé. La figure XIII montre les "courbes d'absorption de ces deux sels. Bien que lem's anions soient tout-à-fait différents, les deux sels absorbent d'une manière analogue. Leur absorption se caractérise par sa très faible abilité. Son unique bande à 2550 de fréquence ne parait que dans la solution d'une concentration demi-normale. On observe encore des absorptions continuelles du côté du rouge et de l'ultraviolet; il nous semble que la première doit être un fragment d'une bande dans l'infrarouge, dont il est impossible de faire l'observation complète avec une plaque panchromatique même. Si cette considération est correcte, elles devront cor- respondre aux deux bandes des complexes ammoniés nickeleux, qui se placent à 1750 et à 2750 de fréquence. Dans ce cas, elles *sont alors déplacées considérablement vers le rouge, évidemment par l'influence bathochromique .des molécules d'eau qui sont liées avec l'atome nickeleux central en donnant un ion complexe Ni(H,0)„, Fig. XIII. Fréquence. 1500 2000 2-0 1-5 I 1-0 !^L 2500 3000 3500 4000 4500 5000 0-0 0-5 lu—^ i 100 fi ;i 10 6 5 et- pj O tn NiCb-GHoO XiS0/7K,0 28 -A^rt. 6.— Y. Shibata : III. Absorption de rayons des sels cuivriques. 1) Complexes pyridines cuivriques. Les quatre sels complexes pyridines donnés ci-dessous ont été étudiés : [Cu(C5H6N)6](N03),-3HoO [Pfeiffer u. Pimmer : Z. anorg. Cham., 1906, 48, 107]. [Cu(C5H5N),,](N03)o [Pfeiffer u. Pimmer : Ibid. 101]. [Cu(C5H5N)3](N03), [Pfeiffer u. Pimmer : Ibid. 103]. [Cu(C5H5]S[),]S04 [S. M. Jörgensen: J. prakt. Cham., 1886, [2], 33, 502]. Tous ces corps sont formés de cristaux bleus foncés et solubles dans l'eau sans montrer aucune tendance de décomposition. La figure XIV représente les courbes d'absorption d'un groupe de ces substances. C'est leur caractère bien notable qu'il n'y a aucune bande essentielle à ces corps, mais elles absorbent continuellement du coté du rouge et du violet. L'examen des courbes nous indique, que le complexe liéxapyridiné est, en outre, le i^lus stable dans l'eau et la solution satisfait parfaitement la loi de Beer sur l'absorption. De même, ce corps montre l'abilité d'absorption la plus forte; cela est bien naturel, car c' est un complexe, dont les •nombres de coordination sont complètement saturés par G molé- cules de pyridine. Le nitrate du complexe tétrapyridiné cuivrique est aussi assez stable dans l'eau, lorsque la solution n'est pas encore très étendue, mais la courbe perd sa continuité déjà dans les concentrations entre centinormale et millinormale. Il est bien intéressant qu'on observe une bande, dans la solution millinormale, à 4000 de fréquence, ce qui est caractéristique de la solution aqueuse de pyridine (Hartley: Journ. Chem. Soc, 1885, 47, 685). La naissance d'une bande caratéristique à pyridine n'indique pas autre chose que ce qui a paru dans la solution assez étendue, comme millinormale, par la décomposition partielle de l'ion complexe tétrapyriné [Cu(Py)4]. Kecherclics sur les spectres d'absorption des amminc-complexcs métalliques III. 29 1500 2000 2500 Fig. XIV. Fréquence. 3000 3500 4500 [Cu(CÄN)3](N03),- [Cu(CÄN) J(N03), U^ [Cu(CÄN)JSO, (VlO [Cu(CÄN)e](N03), 1000/ S3 Quant à la stabilité du complexes tripyriné, il nous semble qu'elle est peut-être moins avancée, parce que la bande de pyridine dans la solution de la concentration de millinormale, est beau- coup plus nette que celle du complexe précédent. Le sulfate du complexe tétrapyiidiné n'est plus inaltérable dans la solution de la 30 Art. 6.— T. Shibata : concentration de centinormale; si on la laisse, elle donne des précipités d'hydroxyde cuivrique^, qui nous ont empêché d'étudier les solutions plus diluées. 2) Acétates des complexes ammoniés cuivriques. Fig. XV. Fréquence. 1500 3-0 ^ [NH;Cu.,(aH.,0.,)5lK.O f Solution dans iL__^ N N V \ moniacales, 1^ 100 / 2^ ' '' ' " -'"-" -' \ l'eau pure, 10 100 Cu(NH3)o(C2H30),- Cu(NH3)3(C Jl30,)Cl-K,0 }( Solutions am Cu(NH3)3(CJ-l30,)I Comme exemples des complexes de ce genre, nous avons étudié les quatre sels suivants: Keclierclies sur los spectres d'absorption des amminc-complexes métalliques III. 31 Cu(NH3)2(CoH30o)o [D. W. Horn : Am. Chem. J., 1908, 39, 206]. Cu(NH3)3Cl(C,H30o)'H,0 [Th. W. Richards and Show : Am. Chem. J., 1893, 15,645]. Cu(NH3)3l(CoH30o) [Th. W. Eichards and Oenslager : Am. Chem. J., 1895, 17,298]. NH4Cuo(CoH302)5'H,0 [Th. W. Eichards and Oenslager : Ibid., 304]. Les trois premiers complexes ont presque la même couleur de bleu violacé brillant, tandis que le dernier est coloré en vert bleuâtre. Les trois complexes propres sont hydrolyses dans les solutions aqueuses, sinon un peu d'ammoniaque y est ajouté, tandis que le sel double vert bleuâtre est bien stable dans l'eau. Par l'examen comparatif des courbes, on remarque que les trois acétates des complexes ammoniés absorbent bi,en fortement, tandis que le sel double se montre sensiblement inférieur en cette abilité. Comme nous l'avons déjà remarqué dans l'introduction, les absorption des complexes ammoniés cuivriques sont curieusement influencées par des anions qui s'accouplent avec les cations com- plexes, et ce phenomème est le plus remarquable dans le cas des acétates. Prenant en considération ce fait, nous voulons donner les constitutions suivantes aux complexes en question: § NH3 N.H3 NH3 NH3 . 0„CoH, NH3 , Cl NH3.. ■ ■/! Cu Cu^ CuT NH, ^O.CoHj NH3 0.,C.,H3 ISÎH3 ■ O.CM^ NPI3 NH3 NH3 (I) ai) (III) Quant aux (II) et (III), ils subiront très probablement les substitutions de l'atome de cblrore ou d'iode respective- 32 Art. 6.— T. Shibata , ment par une molécule d'eau, lorsqu'ils seront dissolus dans l'eau: NH. NH, L NH, .Cu' NH, H.,0 ^0,C,H3 Cl ou I Le sel double vert bleuâtre se dissocie, sans doute, dans l'eau, d'après les formules suivants: NH.Cu^CCHsOOs li;: NH4(C.,H302) + 2Cu(C2H302)2 Z^ NH^ ' + 2Cu" + 5(C2H302), ' mais les ions de cuivre et d'acétat'e reconstrui- ront un nouvau sel complexe: Cu-- + 2(CoH30.,)'+4HoO±:;:Cu(H,0)4(C2H30.,)2, dont la con- stitution sera OH. H.,0 ... : " O0C.H3 H,0 :• OH., O.C^H, parce que ce sel double absorbe non seulement dans la region du violet, mais aussi à côté du rouge; cette dernière absorption ne paraît jamais dans l'acétate cuivrique simple, dont l'absorption sera traitée plus tard (la figure XVIII.). 3) Oxalates des complexes ammoniés cuivriques. Les quatre sels étudiés sont Cu(NH,)CA [D- W. Horn u. M. A. Graham : Am. Chem. J., 1908, 39, 508] Cu(NH,)-,CA[ " " J Cu(NH3),CA-2H,0 [ " " ] Cu(NH.,)5CA [D. W. Horn : Am. Chem. J., 1906, 35, 279] Il sont colorés en bleu très vif, et soluble dans l'eau très difficilement. Si Ton emploie de l'eau ammoniacale comme solvant, les oxalates s'y dissolvent bien facilement. Recherches sur les spectres cVabsorption. des amtaine-complexes métalliques III. 33 1500 2000 2500 3-0 2-0 1-0 0-5 00 Cu(NH3)CA- Cu(NH3),C,0, Fig. XA^I. Fré equence. 3000 3500 4000 4500 5000 \\ • \ \\ \ V il Il t 4 • 1000 100 10 Cu(NH3)4CA-2H.O Gu(NH3),C,0, Solutions am- N N moniacales, ^^ ^^^ ' tel Comme la figure XVI T indique, ces quatre corps absorbent bien semblal^lement les uns et les autres, et leur abilité est sen- siblement inférieure vis-à-vis de tous les complexes, dont les absorptions ont été discutées jusqu'ici. Le fait que les quatre complexes montrent des absorptions bien analogues les uns et les autres, nous indique qu'ils prennent une constitution similaire. On ne sait pourtant s'ils sont de la forme (I) en recevant les molécules d'ammoniaque des solutions, ou s'ils cessent d'être un complexe péntammonié et prennent la forme (II): 34 Art. 6.— T. Shibata ; NH NIL 3 . \ ^ NH, NH., NH, (I) ■NH, c^,o. mais par la raison que des complexes cuivriques sont toujours optiquement influencés par les anions, nous voulons accepter la formule (II). 4) Sulfates des complexes ammoniés cuivriques. Fig. XVII. Fréquence. 1500 2000 2500 3000 3500 4000 4500 30 2-0 1-5 1-0 0-5 0-0 "- -lr 1 t t ./ » 1 1 // k._ /."' \ \ \ ' ': \ Y \ Cu(NH3),S0, Cu(NH,)5S0, ••• Cu(C5H5N),SO, •f Solutions am- ~ ^ moniacalcs, ^^ 1^0 ( Solution dans — -j l'eau iDure, I^ ' 5000 1000 100 10 K Recherclies sxir les spectres d'absorption des ammine-complexes mûtallirpies HT, 35 Les deux sels Cu(NH3)4S04 [Bouzat: Compt. rend., 1902, 134, 1218] et CuCNPDôSO, [D. W. Horn: Am. Cliem. J., 1908, 39, 194] sont formés de cristaux bleus très foncés et bien solubles dans l'eau; leurs solutions deviennent cependant instables par dilution et donnent enfin des précipités d'hydroxyde cuivrique, qui sont encore dissous, quand on y ajoute quelques quantités d'ammonia- que. Les deux corps absorbent tout pareillement l'un et l'autre, comme on le remarque dans la figure XVII; l'absorption conti- nuelle à côté du violet montre une allure très particulière, qui est peut-être caractéristique aux sulfates des complexes ammoniés. Ce qui n'a pas lieu dans le cas du sulfate du complexe pyridine, dont la courbe est donnée aussi dans la figure XVII pour que l'on puisse faire la comparaison, c'est parce qu'il a une autre constitution que celles des sulfates des complexes ammoniés. Pour la même raison, que nous avons déjà annoncé dans le cas des acétates et des oxalates, nous donnons la constitution suivante au sulfate du complexe ammonié: NH3-. : -NHj Cn' NH3-- T'~-~^o^ O SO, 5) Sels cuivriques simples— Acétate, sulfate et chlorure. Pour faire une étude comparative, nous avons choisi trois sels simples CuCC^HsOOs'HsO, CuCl2'2H,0 et CuSO^'SHjO, dont les absorptions sont données dans la figure XV III. 36 Art. 6. ~Y. Shibata; • Fig. XVIII. Fréquence. ■1500 2000 2500 '3000 3500 4000 4500 COOO 100 1000 Cu(C,,H30,),K,0 CUCV2H3O CuSO/öHsO i>(Solntioiis aqueuses, — -) w SU Chacun de ces trois sels, contrairement aux autres sels com- plexes cuivriques, ne montre, clans la solution décinormale, qu'une absorption continuelle dans la région du violet. S'il est vrai, comme on le croit généralement, que des sels dans les solutions aqueuses se dissocient eléctrol3^tiquement dans un degré presque parfait, les trois sels étudiés ici doivent donner la même absorption l'un et l'autre, parce que, comme nous T avons déjà très souvent indiqué, les anions, qui n'absorbent pas eux-mêmes, n'exercent pas d'influence quelconque sur les absorj^tions de ces sels. Dans le cas actuel pourtant les trois sels absorbent bien différemment, c'est-à-dire que, l'acétate, qui n'a qu'une molécule d'eau, absorbe Kocherchcs sur les spectres d'absorption des ammine-complexes métalliques III. 37 le plus fortement, le chlorure avec deux molécules d'eau montre une abilité d'absorption intermédiaire, tandis que le sulfate, qui possède autant de molécules d'eau que cinq, absorbe le moins fortement. Comme on le connaît bien, la solution aqueuse de sulfate cuivrique réagit neutralement, tandis que les deux autres montrent une réaction acidique dans la solution aqueuse. On croit générale- ment que le chlorure et l'acétate de cuivre se décomposent hydrolytiquement dans l'eau, d'après les formules suivantes: CuCL/2ILO:;^[CiiCl(OH),,]" + 2H- CuCh-2H,0:^Z±[Cu(PL0),,]- + 1CV Cu(C,HÂVH,o::;z!:Cu(aH30,),,(HO) + h- " + wH,0:;z^[Cu(H20j„] + (CoH.,0,)' Ce sont certainement les ions complexes contenant les radi- caux d'acétate ou de chlorure qui f^ ^rbent le plus forteinent, tandis que le cation complexe qui ne diit que l'atome cuivri- que et quelques molécules d'eau absorbe bien inférieurement. RESUME. 1) Les complexes chromiques sont toujours bien stables dans les solutions aqueuses et donnent généralement deux ou trois bandes d'absorption très nettes. Tout ce qui est vrai pour les complexes cobaltiques l'est aussi pour ceux de chrome. 2) Les complexes nickeleux sont généralement instables dans l'eau, par conséquent, il nous fallait les étudier dans les solutions ammoniacales. Ils montrent toujours deux bandes d'absorption et leurs abilités sont beaucoup moins fortes que celles des complexes précédents. Néanmoins les absorptions des complexes nickeleux obéissent à la même loi que le chrome et le cobalt. 3) Les complexes cuivriques sont les plus labiles dans les solutions; de même, les anions exercent quelque influence sur l'absorption dans ce cas. Ils ne donnent aucune bande dans l'échelle spectrale entière, mais ils absorbent continuellement à côté du rouge et du violet. Published March 20th, 1920. JOUKNAL OF THE COLLEGE OF SCIENCl::, TOKYO IRirEmAL TJNIVEKSITY, VOL. XLI., ARTICLE 7. Magnetic Separation of the Lines of Iron, Nickel and Zinc in Different Fields. By Kogoro YAMADA, lii'iakKshi. With 20 Plates. I. The Object of the Experiment. Zeeman,^-* taking the " molecular ciii'ient " of Ampere into consideration, inferred that the outer components of triplets of iron lines may differ in intensity. Experimentally, however, he confirmed that there is no evidence of a directing influence of the magnetic field on the orbits of the light-ions as Preston'^ believed. Becquerel and Deslandres,'^ Reese, ^^ Hartmann, ^^ van Bilderbeek- van Meurs, ^-* Arthur King'^ and Graftdijk^^ studied the magnetic separations of iron lines and examined their characters with care; but all these authors assumed that the separations of iron lines are proportional to the strength of the magnetic fields. It is a question whether we maj^ assume this proportionality in spite of the disagreement of ~ of these lines with the value obtained b}^ Lorentz's elementary theory. Professor Nagaoka, from a theore- tical point of view, suggested that in many-lined spectra such as 1) Zeeman, Astrophys. Journ., 9 (1899), p. 47. 2) Preston, Phil. Mag., 45 (1898), p. 333. 3) Becquerel et Deslandres, C. E., 126 (1898), p. 997. 4) Reese, Astrophys. Journ., 12 (1900), p. 120. 5) Hartmann, Dissertation, Halle (1907). 6) Van Bilderbeek-van Meurs, Arch. Neerl., (2) 15 (1911), p. 353. 7) Arthur King, Astrophys. Jour.. 31 (1910), p. 433 ; 34 (1911), p. 225 ; Carnegie Institute Papers of the Mount Wilson Solar Observatory, 2 (1912), Part I. 8) Graftdijk, Arch. Néerl., (3) 2 (1912) p. 192. •2 . Art. 7.— K. Yamada: iron and tungsten, there ma}' be numerous lines showing abnormal Zeeman effect. It was with the object of testing this point that the present investigation was undertaken. On the other hand, the results obtained by various experi- menters differ so greatly from one another tliat I attributed these disagreements to the different values of -^ (here JA denotes the separation of two outer components of a triplet and H the magnetic field ap^Dlied) of one and the same line, and the discrepancy of many results is chiefly due to the fact that magnetic separations were studied by applying different magnetic fields. In order to decide this question, magnetic fields of different strengths must be applied to test the magnetic separations. Kent^^ in studying this problem found the drooping of the magnetic separations in fields stronger than 30,000 gauss. In reviewing his result, I found that the separation of zinc line v'^: 4680" 138 is not linearly proportional to the magnetic fields applied, while it was confirmed that this line is separated lineary proportional to the fields by Cotton and Weiss, ^-^ Stettenheimer^' and others. And I concluded that his determination of magnetic fields by the ballistic method bofore and after photographing the spectra may be different. Stettenheimer, •comparing her result upon the separation of X: 4680*138 of zinc with that of Kent, remarked tliat the value of Kent was smaller by 13*2 %. Hartmann also remarked that the value given by Kent was smaller by 8% as regards the iron lines. Van Bilderbeek-van Meurs^^ and Arthur King^^ are al>o of the same opinion. For the solution of this problem, we must have recourse to new experi- ments, and determine as accurately as possible the field at which the magnetic separation takes 'place. 1) Kent., Astropliys. Juurn., 13 (1901), p. 289. 2) Weiss et Cooton, Journ. d. Phys., (4) 6 (1907), p. 427. 3) Stettenheimer, Ann. d. Phys., (4) 24 (1907), p. 384. 4) Van Bilderbeek-van Meurs, loc. cit., p. 391. 5) Arthur King, Carnegie Institution Papers etc., p. 7. Ma>;nctic Separations of the Lines of Iron, Nickel and Zinc in Different Fields, -3 II. The Method of Experiment. 1. Light Source. A spark discharge between nickel steel and zinc wires by an induction coil was used as the source of light throughout the experiment. The primaiy current of the coil was supphed at 100 volts and 50 cycles from the secondary of the pole transformer of the laboratory, run at 3500 volts of the city main. The current strength was usually from 3 to 4 amperes, The spark discharge, as the source of hght, was placed in the middle between the two poles of the electromagnet and the spark gap made as small or large as the case required. In Fig. 1. the spark can be displaced by means of screw A. With a view of changing the spark gap, C can be moved up and down by means of D, and a small adjustment made it possible to keep the ends at a constant distance from each other, when the ends of the electrodes move away during the sparking. B and C are insulated by pieces of ebonite e and e'. As the terminals for the spark must be put in high magnetic fields, it was necessary to use pieces which were made non-magnetic if possible. A non- magnetic nickle-steel alloy containing about 25^ nickel'^ is most recommendable for this purpose as a portion of the photograph of -^lovn .tfxc -inOsiciloit Colt. JoJ;hc -InDtvct tC4 1 Colt, 1) Nagaoka and Honda, Journ. Coll. Sc, Imperial University, Tokyo, 19 (1903-1904) Art. 11. 4' - Art. 7.— K. Yainada: iron and nickel spectra can be taken on the same plate. More- over, I soldered a piece of the above mentioned nickel-steel, about 10 mm in length and 2 mm in diameter, to a brass rod. When strong fields were required, the proximity of the magnet poles necessitated the filing of the terminals into flat ends as shown in Fig. 2. Even when the magnetic field was raised to 35000 ^ gauss, these pieces were not found to be attracted by the 1 magnet poles. In this investigation, as the zinc spectrum for determining the field was photographed simultane- ousty, o?ie of the terminals ivas always a z'mc rod of the same diameter. '•* Similary as mentioned in man}'- published papers, condensers were connected parallel in the spark circuit, which also contained selfinduction coils. In my experi- l^ig-2 • nient, three Leyden jars, each of which was about 0'00404 micro-farad, were connected in parallel and the whole was inserted parallel to the spark circuit. Afterwards, on tlie advice of Professor Nagaoka, I introduced another spark gap into the spark circuit in addition to the selfinduction coils. This is recommen- dable, especially in the case of long exposures. Very small self- induction was sufficient in vay experiment, the sound of the spark retained its shrillness, especially after the auxiliar}" spark gap was introduced in the circuit. The spark gap of the light source varied from 1 to 3 mm. as the pole-pieces of the electromagnet were changed. From a long spark gap, I could get a brilliant light and a reduction in time of exposure was rendered possible. 1) When I read this paper at the ordinary meeting- of the Physico-uiathematil Society of Japan on the 8th of March, Professor Nagaoka put the followin»- question to me : "The non-magnetic nickel-steel can positively have no effect upon the strength of the magnetic field in which the spark is discharged. But the iron dnst grains scattered from the end of the spark electrodes are attracted uy the magnet poles. How have you disj^oscd of the influence of this dust upon the magnetic fields ? " I answered : " In order to get rid of this influence, I broke the magnetizing current and completely removed the dust several times during the photographing. Moreover, in my experiment, as the /ine line 4680"138 was simultaneously photograiihcd to deterenine the field, the change of the field due to these powders, if any, also afl:ected the separation of this zinc line at the same time ; in this manner the real value of th<} magnetic field in which the centre of vibration of iron and nickel spectra is found, was obtained. To determine the field accurately is one of the chief purposes of my experiuient." Magnetic Separations of tlio Linos of Iron, Nickel and Zinc in Different Fields. 5 2. Electromagnet. The magnetic fiele, in which the light source was placed, was excited by an electromagnet of Du-Bois half-ring type. Various fields were obtained by changing either the pole-gaps or the magnetizing current or both. I used four kinds of iron tips: 4 mm., 5'5 mm., G mm. and 0'5 mm. in end diameters according to the kind of spark desired as well as the width of the magnetic gap used. The length of the spark was always smaller than the diameter of the pole tips, and it was always placed in uniform fields. The gaps l:)etween the poles Averel'omm., 2"0mm., 2"5mm., 3*1 mm., 3'7mm, and 4"0 mm. The magnetizing current varied from 2 to 13 amperes. The vertical angle of the conical pole-pieces was about 120" to attain the maximum field. The highest magnetic field thus attained was 34120 gauss. The electromagnet can be excited for as long as six hours at 10 amperes, but it must be cooled down after three continuous hours when a current of 13 amperes is used. The current was obtained from the secondar}^ batteries in the laboratory. It was easily controlled when allowance was made for the increased resistance as the coil became warmer, and there was no difficulty in keeping the current constant as shown by the ammeter. The magnet was arranged broadside on, that is, the lines of force between the poles were in a direction perpendicular to the line joining the slit and the grating. 3. Spectrograph. The spectroscope used in this investigation was a Rowland concave grating whose focal length is 10| feet, the breadth and the height of the ruled surface are 3i and 1| inches respectively. Tlie number of ruled lines per inch is 14,438. The resolving power -g— ^i^^^ the linear distance in mm. on the photographic plate, corresponding to a change of wave-length of o one Angstrom unit, was calculated by tlie formulae inserted in Baly's "Spectroscopy" (p. 171, 1905). Let ;. be the wave-length in A. U., and g the linear distance in mm. from the spectral line of wave-length À to the slit, then we have the following data for the. ^grating: Art. 7. — K. Yaiuada : Order of Spectrimi dg d). d). do- d>. 1 ■ 0-182 5-50 50,000 2 0-364 2-75 101,000 3 0-546 1-83 151,000 4 0-728 1-37 202,000 5 0-910 1-10 ■ 252,000 The mountiiig of the grating was of the usual Rowland type, in wliich the grating and the photographic camera ran on rectan- gular iron rails. The source of light in most cases was focussed on the slit. by a lens. A quartz lens was used for the photographing of the ultra-violet region. The focal length of the lens was about 12 cm. and its diameter 5 cm. For the visible region, it was found profitable to use a short focus cylindrical glass lens together with a spherical one, in order to condense the light. To analyse the nature of the vibration in the magnetic field, a Wollaston or Rochon quartz prism was introduced between the lens and the slit. The optical arrangement is shown in Fig. 3. The spectra produced Ijy the Rowland grating almost always give rise to ''ghosts," to whicli 1 liave paid special attention. The distance A of the ghost from the main line divided by its wave-length X, i. e. -y- is constant for different orders of spectra'-* with respect to the instrument. Hence this constant is ser- viceable to decide the wave-length when its SLi C\tU.nJUicaJL oC*»ui. O TMaJ^^ order is known. In our graring -y- =2*568 x In this experiment, I succeeded in plioto- graphing as far as the fifth order spectra, by using a glass and a quartz lens separately to take the same region of the spectra. To distinguish between iron and nickel lines, the 1) Itowlaud, Physical Papers, p. 536 (1902) Baltimore. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. J /^ I* « iron (completely free -from nickel) spark was photographed and the spectra compared to dis- tinguish the line due to these metals. 4. Arrangement. The room in which the experiment was undertaken was on a concrete hasement. The iron rails on wdiich the rod supporting grating and the photographic camera were placed were fixed on stone piers. This ~^"' ■ firm setting enabled an expcsure of many hours duration to be undertaken without any risk from external shocks. The temperature did not vary so much as to disturb the sharpness of the lines. The arrangement is show^i in Fig. 5. OvicaveLmJ jy^^ß^ A/nmefer InducéCon ooti CapariLy S'eoonJQry _^, Ammefc.7- „ ^ , ^aitery /^^ Hcec-tronti Fig. 5. 5. Photography. Many kinds of photographic dry plates were tested to find whicli would be the best for my purpose. Those tested were (1) Lion 20, (2) Lion 225, (3) Lion, Ortho- chrome Plates, non-filter, (4) ditto, backed, (5) Paget Prize Plate, Q Art. 7. — K. Yamada : (G) Paget process Plate, (7) Chromo Isolar, (8) Wratten and Wainwright panchromatic plates, (9) Seed 27, (10) Agfa-Trocken- Platten, (11) Schleussnerplatten, (12) Cramer, (13) X Ray plate and (14) llford Process Plate. Of these fom-teen kinds, I found the "llford Process Plate" the most recommendahle for such accurate purposes. The peculiarity of this plate is its fine grain, though it has the drawback of requiring long exposure. The developer Avas in almost all cases " Agfa Rodinal " and afterwards " Azol," with the addition of a few drops of potassium bromide. A hydroquinone developer w^as also used, but it had no advantage over those above mentioned. 6. Identifications of Lines. Tlie identification of lines was not so easily worked out when three orders, the 3rd, 4tli and 5th overlapped on a photographic plate. At first I photographed iron spectra together with À : 4358*58, 4078*05 and 4046*78 of mercury lines. By the aid of this plate, I found to what region these spectra belong and by the help of Buisson and Fabry's charts inserted in "Recueil de Constantes Physiques (1913)", I could easily identify the lines and from the tables of line spectra'^ and those of iron lines'^ the values of weve-length -in international unit are easily known. When two or three orders overlap on a plate, a mere calculation with the help of the tables enables us to detect the value of wave-length. 7. Determination of the Magnetic Field. The magnetic separa- tion of zinc line >?.: 4680' 138 has been accurately examined and many experimenters have used it to determine the magnetic field. One of the chief purposes of this experiment was the simultaneous photography of zinc and iron lines. The groups " g," " h," "i" and "j" were taken in this manner. But the simultaneous photograph}^ thus obtained limited the number of lines, and I was obliged to photograph tlie zinc lines before and after taking the iron lines. The group "e" and "f" are those just mentioned. The ranges and orders pliotographed on these plates are given in Plates 1. and 11. 1) Kayser, Handbuch der Spectroscopie, VI (1912) pp. 935-1033. 2) Kayser, ditto, pp. 896-926. Maguetic Sépai'ations of the Tjiues of Iron, Xickel and Zinc in Different Fields. 9 8. Measurement and Reduction. To measure the magnetic separation of photographed lines, I used a dividing machine No. O030'\made by the Société Genevoise. Taking the tracelet off, 1 put a supporter of the photographic plate on the platform. Under this supporter, a plane mirror, inclined 45° to the platform, was fixed and by reflecting the light from an electric lamp, the plate was ilkiminated. The platform, and consequently the photo- graphic plate were moved by the rotation of the handle, and the spectral line under examination was placed under a single straight wire of a fixed microscope whose magnification was 20. But when the line was very intense, it was put between two parallel wires. As the magnification was moderately large, the former method gave better results. The drum of the dividing machine was divided into 0'005 mm. and by the aid of the vernier it could be read to 0*001 mm. A device was made so as to move the plate in two directions at right angle to each other. Hence the line w^as measured along its length as many times as possible. In the case of the fourth order spectra, as the line is long, measurement can be made on twenty different parts. As this work is very im.portant for finding accurate values the measurements were made b,y inj^self. The result obtained b}' the measurements of one and the same line on different days agreed well with each other. III. The Character of the Separation. After the publication of Voigt' s"^ asymmetrj^ theory with regard to the magnetic separation, Zeeman^'-* remarked that the asymmetrical intensity of the outer components of a triplet was also due to secondary circumstances, L e. to the reflecting grating and the focussing quartz lens. As, in my case, these two secon- dary circumstances are related to each other I could make out the character distinctly, but it may be of interest to report that on one 1) Société Genevoise, Price List (1912), Fascicle 1, General Instruments of Measurements, p. 13. 2) W. Voigt, Ann. d. Phys. (4) 1 (1900), 376. 3) Zeeman, Versl. Amsterd. Acad., 26 oct. (1907). 10 Art. 7. — K. Yamada : and the same plate there are various features of intensity of separated hnes. As to the triplet, I classify the following three- cases: (rt) the three components have equal intensities, (b) the inner component is stronger than the outer and (c) the outer components are stronger than the inner. I CL b c Fig. 6. IV. The Magnetic Separation of Iron Lines. The wave-lengths are represented in international units and the intensities of lines when the}^ are not affected are taken from the Gtli volume of Kayser's Spectroscopic. (1) A: 4415-13. Intensity 10. King took this for a septuplet. According to my measurements, it has many peculiarities; a remarkable character is the decrease of intensity in high fields, the cause of which may be the broadening of components, but further separations can neither he observed in the parallel nor in the normal component. The broadening of the central component is larger than the outer two.'-* The table contains the distance between two outer com- ponents. The fields were determined by the separation of À: 4680'] 38 of zinc, photographed on the same plate. Plato I, Fig. 1; Plate III, Fig. 1 and Plate VII, Fig. 1. 1) Mr. Y. Takahashi jirivatcly told uie that eacli component of this lino was found to be further separated in his research with an echelon spectroscope of greater resolving power than the concave grating used by me. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. XI Tap.le I. A). ^ XlOO mî X""' Deviation fi-om 10-88. H ^'' X1013 Deviation 10-88 9325 0-204±0-0026 21-88 11-22 + 0 34 + 3-1 10850 0-235±00037 21-64 11-10 + 0-23 + 2-0 12330 0-259±:0-0037 2100 10-77 -0-11 -1-0 13900 0 292±:0-0028 21-00 10-77 -0-11 -1-0 14140 0-298±:0-0018 2108 10-82 -0 06 -0-5 16600 0-353±0-0029 21-26 10-91 + 0-03 + 0-3 17800 0-374dz0-0035 2101 10-78 -0-10 -0-9 19000 0-399drO-0021 21-00 10-77 -0-11 - 1-0 20100 0-416±0-C063 20-68 10-62 -0-26 -2-4 21540 0-459±0-0042 21-35 10-95 + 007 + 0-6 23680 0-528±00073 22-30 11-44 ■ +0-56 -5-1 23860 0-506±0-0015 21-20 1088 0-00 0-0 26100 0-536rh0-0022 20-52 1053 -0-35 -3-1 26800 0-536dz00079 21-00 10-77 -0-11 -1-0 28500 0-637=tO-0020 22-36 11-46 + 0-58 + 5-3 29920 0-666±0-0053 22-28 11-43 + 0-55 + 5-1 30780 0-704dz0-0090 22-88 11-74 + 0-86 + 7-9 31460 0-720±0-0058 22-90 11-75 + 0-87 + 8-0 32300 0-752i±:0-0075 23-30 11-95 + 1-07 + 9-8 33800 0-767d:0-0041 22-70 11-65 + 0-77 + 7-1 34120 0-792±0-0005 23-20 11-90 + 1-02 + 9-4 From the table we see that JA increases as the fields become higher. The feature is seen from Fig. 1, Plate VII. The mean value of the first fourteen ^^-xlO^^ is 10*88 and the deviations from this value are given in Table I. For comparison, I insert the results of former observers: 12 Art. 7. — K. Yamada : OliservLT Character of Separation H A>. ^-- King Septuplet 16000 0-338 21-12 10-84 Kent Triplet 28000 0-540 20-4 10-48 Van Biklerbeek- van Meurs •. 32040 0-66 20-6 10-6 Graftdijk " 32040 0-684 21-38 10-95 Hartmann 20-4 10-48 (2) ;.:4404-75. Intensity 15; sharp triplet; character h. Plate I, Fig. 1; Plate III, Fig. J. The determination of the fields is tiie same as in the case of /: 441 5* 13. Tap.le II. Deviation from tlie Mean A), ^[\ xioc Ha2 H Deviation Mean Value 9325 0190±00023 20-38 10-50 -0-14 -1-3 10800 0-220±0-0018 20-38 10-50 -0-14 -1-3 10850 0-227±0-0005 20-92 10-78 + 0-14 + 1-3 12330 0-252dr0-0030 20-44 10-54 ■ 0-10 -0-9 13900 0-289±0-0035 20-80 10-72 + 0-08 + 0-8 14140 0-289±0-0004 20-46 10-55 -0-09 -0-8 16600 0-346±0 0015 20-84 10-75 + 011 + 1-0 17800 0-370±:0-0023 20-78 10-70 + 0-06 + 0-6 19000 0-396±00018 20-86 10-76 • +0-12 + 1-1 20100 0 -416^0-0017 20-68 10-67 + 0-03 + 0-3 21540 0-450zt00007 20-90 10-77 + 0-13 + 1-2 2*3680 0-497±0-0019 20-94 1079 + 0-15 + 1-4 23760 0-493d:0-0025 20-76 10-72 + 0-08 + 0-8 23860 0-496±0-0009 20-76 10-72 + 0-08 + 0-8 25300 0-523dr00000 20-68 10-67 + 0-03 + 0-3 26100 0-538zfc0-0015 20-64 10-64 0-00 0-0 26200 0-545rt0-0013 20-80 10-73 + 0-09 + 0-8 26800 0-549±:0-0036 20-48 10-57 -0-07 -0-7 27400 0-57ldbO-0022 20-84 10-74 + 0-10 +0-9 28320 0-589dz0-0014 20-80 10-73 + 0-09 + 0-8 .28500 0-594±0-0023 20-84 10-74 + 0-10 +0-9 Magnetic Separations of the Lines of Iron, Nickel and Zinc in different Fiekl? 13 AX ^'' XlOG H r;^'.; X1013 ri/r Deviation from the Mean H il X1013 Deviation Mean Value 29120 0-600±0-0013 20-62 10-64 000 0-0 29160 0-598±0-0036 20-52 10-58 - 006 -0-6 29920 0-615d=00021 20-58 10-61 -0-03 -0-3 30000 0-619±0-0015 20-64 10-64 0-00 0-0 30200 0-6a4=fcO-0030 20-76 10-70 + 0-06 + 0-6 30340 0-620±0-0013 20-44 10-54 -0-10 -0-9 30780 0-633±0-0009 20-60 10-62 -0-02 -0-2 31100 0-637±0-0024 20-48 lC-56 -008 -0-8 31300 0-646dr0-0024 20-66 10-65 + 0-01 + 0-1 31460 0-647±:0 0009 20-58 10-61 -0-03 -0-3 31650 0-644±00023 20-37 10-50 -0-14 -1-3 32300 0-660±0-0014 20-44 10-54 -0-10 -0-9 32300 0-661±0-00l7 20-48 10-57 -0-07 -0-7 32400 0-663±0-0016 20-48 10-57 -0-07 -0-7 32880 0-675rb0-0022 20-59 10-61 -0-03 -0-3 32900 0-675dr0-0015 20-52 10-58 -0-06 -0-6 33100 0-681zb00012 20-58 10-60 -0-04 -0-4 33340 0-681±:00038 20-44 10-54 -0-10 -0-9 33350 0-697±:0-0022 20-90 10-78 + 0-14 + 0-3 33500 0-688±:0-0030 20-56 10-60 - 0-04 -0-4 33800 0-696±0-0021 20-60 10-63 -0-01 + Q-1 34120 0-701±0-0025 20-58 10-61 -0-03 -0-3 Mean 20-62 10-64 J/ is linearly proportional to H (Plate VII, Fig. 2). results of former investigators are: The Observer Character of Ceparation H 1 A). 4-X106 ±1),-' King Triplet 16000 0-334 20-88 10-74 Kent Van ßilderbeek- van Meurs Graftdijk " 28000 ! 32040 32040 0-512 0-6S0 0-678 18-30 21-22 21-18 9-42 10-93 10-91 Hartmann " — — 20-20 10-26 Within 1%, the results obtained by King and by me are in good agreement with each other. (3) ;.: 4383-55. Intensity 20; sharp triplet; character Z». Plate I, Fig. 1; Plate III, Fig. 1. The determination of the fields is the same as in the case of. >l:4415-13. 14 Art. 7. — K. Yamada ; Table III. AX t^ X106 ^xio. Deviation from the Mean H ^^ X1013 Ma- Deviation Mean Value 9325 0-193±0-0014 2070 10-76 -0-03 -0-3 10800 0'220±0-0012 20-38 10-60 -0-19 -1-8 10850 0-224±00009 20-64 10-73 -0-06 -0-6 12330 0-256±0-0006 20-74 10-78 -001 -0-1 13900 0-287±0-0004 20-64 10-73 -0-06 -0-6 14140 0296±0-0007 20-92 10-88 + 0-09 + 0-8 16600 0-344±0-0014 20-72 10-78 -0-01 -0-1 17800 0 371±0-0011 20-84 10-85 + 0-06 + 0-6 19000 0-393±0-0006 20-68 10-75 -0-04 -0-4 20100 0-4l7±0-0009 20-72 10-78 -0-01 -0-1 21540 0-449±0-0035 20-88 10-86 + 0-07 + 0-7 23680 0-487±:0-0010 20-56 10-70 -009 -0-8 23760 0.498dz0-0016 20-98 10-91 + 0-12 + 1-1 23860 0.494dr0-0018 20-70 10-76 -0-03 -0-3 25300 0.523±0-0016 20-68 10-75 -0-04 -0-4 26100 0 544±0'0014 20-86 10-85 + 0-06 + 0-6 26200 0-547±0-0013 20-88 10-86 + 0-07 + 0-7 26800 0-556±0-0016 20-74 10-78 -0-01 -0-1 27400 0-561dr0-0018 20-50 10-66 -0-13 -1-2 28320 0-589±:0-0015 20-80 10-82 + 0-03 + 0-3 28500 0-588±0-0018 20-64 10-73 -0-06 • -0-6 29120 0-604±0-0009 20-74 10-78 -0-01 -0-1 29160 0-599±00018 20-58 10-65 -0-14 -1-3 29920 0-622±0-0005 20-80 10-82 + 0-03 + 0-3 30000 0-621di0-0010 20-70 10-76 -0 03 -0-3 30200 0-638dr00023 21-12 10-98 + 0-19 + 1-8 30340 0-628d=0-0013 20-72 10-78 -0-01 -0-1 30780 0-637±0-0003 ' 20-70 10-76 -0-03 -0-3 31100 0-642±0-00l7 2066 10-74 -0-05 -0-5 31300 O-653±OO011 20-88 10-86 + 0-07 + 0-7 31460 0-651dzO-0012 20-72 10-78 -0-01 -0-1 31650 0-653dr0-0023 21-64 10-73 -0-06 -0-6 32300 0-667±:00011 20-65 10-74 -0-05 -0-5 32300 0-673±00018 20-84 10-85 + 0-06 + 0-6 32400 0-67l±0-0006 20-72 10-78 -0-01 -0-1 32880 0-678dr0-0010 20-64 10-73 --0-06 -0-6 32900 0-687±00013 20-88 10-86 + 0-07 •4-0-7 33100 O-680rt0-0014 20-54 10-68 -0-11 -1-0 33340 a-69o±:0-0023 20-88 10-86 + 0-07 +0-7 33350 0-694dr')-0p09 20-84 10-85 + 0-06 + 0-6 33500 0-702±0-obl3 20-96 10-90 + 0-11 + 1-0 33800 0-701±:00020 20'74 10-78 -0-01 -0-1 34120 0-7l2±0-0006 20-88 10-86 + 0-07 + 0-7 Mean 20-74 10-79 Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. ^5 From the table we see that ^^ increases proportionally to the fields applied. Plate VIII, Fig. 1. The results of former investigations are: Observer Character of Separation H M ^} XlOG ^xio- King Kent Van Bilderbeek- van Meurs Graftdijk Hartmann Triplet 16000 28000 32040 32040 0-332 0-514 0-684 0-674 20-74 18-36 21-32 21-04 19-2 10-78 9-55 11-10 10-95 9-98 The value obtained by me is in full accord with King' s result. (4) X'A32ö-7Sy Intensity 15; seems to be a triplet, but in high fields re- sembles more or less :4415"13. Character b, Plate I, Fig. 1; Plate III, Fig. 1. The determination of the fields is the same as in the case of k : 44 15' 13. Table IV. Deviation from the Mean AX ^ XIOC j^XlO.» H .'^^-XlOis HX- Deviation Mean Value 9325 0-156±:0-0008 16-76 8-95 + 0-20 + 2-3 10800 0-l7l±0-0025 15-83 8-47 -0-28 -3-2 10850 0-l74dz0-0011 16-03 8-57 -0-18 -2-1 12330 0-199drG-0013 16-14 8-63 -0-12 -1-4 12900 0-219±0-0011 15-76 8-42 -0-33 -3-8 14140 0-231±0-0026 16-33 8-73 -0-02 -0-2 16600 0-266±0-0013 16-03 8-57 -0-18 -2-1 17800 0-280±0-00ll 15-73 8-41 -0-34 -3-9 19000 0-308±0-0009 16-22 8-67 -Ü-08 -0-9 20100 0-328±00013 16-33 8-73 -0-02 -0-2 21540 0-348±0-0015 16-17 8-64 -0-11 -1-3 23760 0-387±0-0024 16-29 8-70 -005 -0-6 23860 0-386dr0-0015 16-17 8-64 -0.11 -1-3 25300 0-402zt00016 15-89 8-49 -0.26 -3-0 26100 0-421±00024 16-13 8-62 -0.13 -1-5 26200 0-432±0-0014 16-50 8-82 . + 0-07 + 0-8 26800 0-429±0-0001 16-03 8-57 -0-18 -2-1 27400 0 447dzO-0033 16-30 8-71 -0-04 -0-5 1) In Mr. Y. Takahashi's experiment, this line also show complex separation. 16 Art. 7. — K. Yauiada ; Deviation from the Mean A>. ■^' X106 11 ^txio- H ^xio.= Deviation Mean Value 28320 0-474±0-0016 16-75 8-95 + 0-20 + 2-3 285C0 0-459±:0-0026 16-21 8-61 -0-14 -re 29120 0-484±0-0007 16-63 8-89 + 0-14 + 1-6 29160 0-485±0-0027 16-58 8-86 + 0-11 + 1-3 30000 0-495±0-0023 16-50 8-82 + 0-07 + 0-8 30200 0-500rb0-0038 16-57 8-85 + 0-10 + 1-1 30340 0-493±0-0012 16-27 8-70 -0-05 -0-6 30780 0-506±:0-0007 16-45 8-79 + 0-04 + 0-5 31100 0-508±0-0021 16-34 8-73 -0-02 -0-2 31300 0-519±:0-0023 16-59 8-86 + 0-11 + 1-3 31460 0-523±0-0013 16-63 8-88 + 013 + 1-5 31650 0-534±0-0025 16-87 9-01 + 0-26 + 3-0 32300 0-541±0-0012 16-76 8-95 + 0-20 + 2-3 32300 0-531±0-0007 16-45 8-79 + 0-04 + 0-5 32400 0-528dz0-0015 16-32 8-72 - 0-03 -0-3 32880 0-538±0-0032 16-36 8-74 -0-01 -0-1 32900 0-o48dr0-0047 16-65 8-90 ■ +0-15 + 1-7 33100 0-554±0-0022 16-74 8-95 + 0-20 + 2-3 33340 0-559±0.0047 16-78 8-97 + 0-22 + 2-5 33350 0-555±0.0027 16-65 8.90 + 0-15 + 1-7 • 33500 0-565±:0.0039 16-87 901 + 0-26 + 3-0 33800 0-567±0.0014 16-79 8-97 + 0-22 + 2-5 34120 0-583±0-0030 17-10 9.14 +0-39 + 4-5 Mean 8-75 From the table we see that J/ shghtly increases as the fields become higher, as in the case of X : 4415*13. Plate VIII, Fig. 2. The results of the former investigators are: Observer Character of Separation H A), 4-xio« H ->"''''■ King Triplet 16000 0-245 15.31 8-18 Kent Van Bilderbeek- van Meurs Graftdijk „ 28000 32040 32040 0-390 0-524 0-524 13-93 16-37 16-37 7-44 8-74 8-74 Hartmann " — — 16-0 8-62 (5) ;.:4307'92. Intensity 15; sharp triplet; character h. Plate I, Fig. 1; Plate III, Fig. 1. This line resembles ;:4404-75. The deter- mination of the fields is the same as in the case of yl:4415'13. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 17 Table V. Deviation from the Mean A>. ^xio. m, ^^°" H Deviation Mean VaUie 9325 0-188±0-0010 20-15 10-86 + 0-28 + 2-6 10800 0-215±:0 0014 19-90 10-73 + 0-15 + 1-4 10850 0214±0-0007 19-73 10-63 + 0-05 + 0-5 12330 0-244±0-0004 19-78 10-66 + 0-08 + 0-8 13900 0-273±0-0011 19-65 10-59 + 0-01 + 0-1 14140 0-277±0-0013 19-59 10-55 -0-03 -0-3 16600 0-330±000l7 1987 10-71 + 0-13 + 1-2 17800 0-346±0-0010 19-44 10-48 - 0-10 -0-9 19000 0 376dz0-0011 19-79 10-66 + 0-08 + 0-8 20100 0-397±0-0015 19-76 10-65 + 007 + 0-7 21540 0-427±0-0007 19-84 10-69 + 0-11 + 1-Q 23760 0-468±00014 19-71 10-63 + 005 + 0-5 23860 0-468±0-0009 19-62 10-57 -001 -0-1 25300 0-494±0-0005 19-54 10-53 -0-05 -0-5 26100 0-510±0-0005 19-54 10-53 -0-05 -0-5 26200 0-513±00011 19-59 10-55 -0-03 -0-3 26800 0-527±0-0023 19-67 10-60 + 0-02 + 0-2 27400 O-53O=t0-OO16 19-35 10-43 -0-15 -1-4 28820 0-550±:0-0027 19-43 10-47 -0-11 -1-0 28500 0-561±0-0026 19-70 10-62 + 0-04 + 0-4 29160 0-563±:0-0014 19-32 10-42 -0-16 -1-5 30000 0-579±0-0007 1930 10-41 -0-17 -1-6 30200 0-607±0-0026 20-10 10-84 + 0-26 + 2-5 30340 0-591r±:0-0029 19-47 10-50 -0-08 -0-8 30780 0-603=bO-0021 19-60 10-56 -0-02 -0-2 31100 0-601±:0-0036 1933 10-42 -0-16 -1-5 31300 0-608db00037 19-43 10-47 -0-11 -1-0 31460 0-604dr0-0012 19-20 10-35 -0-23 -2-2 31650 0-628dzO-0012 19-84 10-69 + 0-11 + 1-0 32300 0-627±:0-0025 19-42 10-46 -0-12 -1-1 32300 0-623±0-0016 19-30 10-41 -0-17 -1-6 32400 0-629dz0-0026 19-40 10-45 -0-13 -1-2 32880 0-642±:0-0026 19-54 10-53 -0-05 -0-5 32900 0-643±:0-0023 19-54 10-53 -0-05 -0-5 33100 0-643±0'0031 19-43 10-47 -0-11 -1-0 33340 0-654d=00025 19-63 10-58 0-00 0-0 33350 0-67'2 ±0 0012 20-14 10-84 + 0-26 + 2-5 33500 0-661±0-0026 19-75 10-64 + 0-06 + 0-6 33800 O-673±0-0021 19-91 10-73 + 0-15 + 1-4 34120 0-673±0-0042 19-74 10-64 + 0-06 + 0-6 Mean 19-63 10-58 18 Art. 7. — K. Yamada From the table we see that J^ increases proportionally to the fields applied, as in the case of 4404'75. Plate IX, Fig. 1. The results of the former investigations are: Observer Character of Separation H AX H ^^<^ ^) X1013 H King Triplet 16000 0-320 20-00 10-77 Kent Van Bilderbeek- van Meurs Graftdijk " 28000 32040 32040 0-480 0-643 0643 17-14 20-08 20-08 9-24 10-81 10-81 Hartmann " — — 18-8 10-12 The nearest value to my result is that found by King, whose value is in agreement with mine with a discrepancy of 2%. (6) ; : 3886-29. Intensit}^ 5; sharp triplet; character a. Plate I, Fig. 4; Plate IV, Fig. 4. The fields were determined before and after photographing this line by the aid of i^: 4680* 138 of zinc. But the fields marked with an asterisk were determined by À : 3856'38, taking -g- =20*94, which we shall discuss hereafter. (Table IX and its discussion). Table VI. Deviation from the Mean A/ ^^ X106 ^^ XlOls H ^l X1013 Deviation Mean Value 8220 0-174±0-0009 21-16 14-01 + 0-09 + 0-6 9815 0-200±0-0005 20-80 13-77 -0-15 -1-1 10880 0-228±0-0005 20-94 13-87 -0-05 -0-4 13430 0-282zb0-0010 20-98 13-90 -0-02 -0-1 14500 0-300±:000l7 20-70 13-70 -0-22 -1-6 16370 0-344±0-0009 21-00 13-91 -0-01 -0-1 17700 0-373drO-0010 21-08 13-95 + 0-03 + 0-2 19300 0-406±0-0005 21-04 13-93 + 0 01 + 01 20000 0-421dr0-0008 21-05 13-94 + 0-02 + 0-1 21740 0-462±0-0014 21-26 14-09 + 0-17 + 1-2 *25020 0-530dr00005 21-04 13-94 + 0-02 + 01 25250 0-532db0-0009 21-08 13-95 +0 03 + 0-2 *25540 0-538±:0-0008 21-08 13-95 + 0-03 + 0-2 *26100 0-547±:0-0006 20-94 13-87 -0-05 -0-4 26520 0-556rt0 0008 20-98 13-90 -002 -0-2 27240 0-569dr00007 20-90 13-84 -0-08 -0-6 28680 0-611±0-0021 21-30 14-10 + 1-8 + 1-3 Mean 21-02 13-92 Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. ]^9 From the table Ave see that JX is proportional to H. Plate IX, Fig. 2. The results of the previous investigations are: Investigator Character of Separation H A>. ^ XlOG JtL à\ X1013 King Kent Reese Van Bilderbeek- van Menrs Triplet 16000 28000 28300 32040 0-348 0-523 0-618 0-698 21-75 18-68 21-84 21-80 14-40 12-37 ■ 14-46 14-43 My result is smaller than those of King, Reese and van Bilderbeek-van Meurs, while the latter three are in good agreement with one another, (7) ;.: 3878-78. Intensity 5; sharp triplet; character a. Plate II, Fig. 1; Plate IV, Fig. 4. The determination of the fields is the same as in the case of A: 3886-29. Table VII. A>. ^Lxio. H^X'»" Deviation from tlie Mean H Hr^ x'0'= Deviation Mean Value 82* 0-l74±0-0020 21-18 14-07 + 0-09 + 0-6 9615 0-199dzO-0006 20-70 13-76 -0-22 -1-6 10880 0-231±:0-0006 20-22 14-12 + 0-14 + 1-0 13430 0-281±0-0009 20-90 13-90 -0-08 -0-6 16370 0-342±0-0007 20-80 13-83 -0-15 -11 17700 0-372dr0-0009 21-00 13-95 -0-03 -0-2 19300 0-401±0-0014 20-74 13-78 -0-20 -1-4 20000 0-421±G-0020 21-05 13-98 C-00 00 21740 0-462dr0-0018 21-26 14-14 + 0-16 + 1-1 *25020 0-538dr0-0012 21-52 14-31 + 0-33 + 2-4 25250 0-531±0-00I2 21-05 13-98 0-00 0-0 *25540 0-540zt0-0012 21-14 14-05 + 0-08 + 0-6 *26100 0-545±0-0010 20-88 13-8Ö -0-09 -0-6 26520 0-556±0-0011 20-96 13-94 -0-04 -0-3 27240 0-568±00022 . 20-86 13-86 -0-12 -0-9 28680 0-612dr0-0013 21-34 14-19 + 0-21 + 1-5 Mean 21-05 13-98 20 Alt. 7. — K. Yamada : AX is nearly proportional to H. Plate V, Fig. 4. The results of former investigations are: Investigators Character of Separation H A), ■^' XlOG Ja ^'' X1013 Kino- Van Bilderbeek- van Meurs Triplet 16000 32040 0-346 0-698 21-64 21-80 14-37 14-49 The discrepancy between my value and that of King is some- what large, amounting to o% . (8) ; : 3859-90. Intensity 6; sharp triplet; character a. Plate II, Fig. 1; Plate IV, Fig. 4. The determination of the fields is the same as in the case of X : 3886-29. Table VIII. A>. -^xio. m= x^°" Deviation from the Mean H J^}- X1013 JÜA- Deviation Mean Value 8220 0-l72±0-0005 20-90 14-04 + 0-06 + 0-4 9615 0-199it0-0009 20-70 13-89 -0-10 -0-7 10880 0-229db0-0006 21-04 14-13 + 0-14 + 1-0 13430 0-281±0-0009 20-90 14-04 + 0-06 + 0-4 14500 0-302±0-0009 20-84 13-98 -0-01 -0-1 16370 0-341 drO-0007 20-84 13-98 -0-01 -0-1 17700 0-370±0-0012 20-90 14-04 + 0-06 + 0-4 19300 0-392±:0-0007 20-30 13-63 -0-36 -2-6 20000 0-4l7±0-0008 20-84 13-98 -0-01 -0-1 21740 0-468±:0-0010 21-06 14-14 + 0-15 + 1-1 *25020 0-522±0-0006 20-86 14-02 + 0 03 + 0-2 25250 0-526±0-0003 20 84 13-98 - 0-01 -0-1 *25540 0-532db0-0005 20-82 13-98 -0 01 -0-1 *26100 0-543±00006 20-80 13-97 -0-02 -0-r 26520 0-550±0-0009 20-74 13-93 -0-06 -0-4 27240 0-568±0-0005 20-86 14-02 + 0-03 + 0-2 28680 0-598±0-0013 20-88 14-03 +0-04 + 0-3 Mean 2083 13-99 AX is nearly proportional tc II. Plate V of former investigations are: Fig. 5. The results Magnetic S îparations of tlic Lines of Iron, Nickel and Zinc in Different Fields. 21 Investigatoi* Character of tt Separation A). ^'' X105 n Hi:.xiox=^. Kino- Van Bilderbeek- van Meurs Reese Triple 16000 32040 28300 0-341 0-681 0-622 21-30 21-26 21-98 14-30 14-26 14-76 (9) ;: 3856-38. Iiitensit}^ 5; sharp triplet; character a. Plate IL Fig. 1; Plate IV, Fig. 4. The determination of the fields is the same as in the case of ; : 3886-29. Table IX. )A ^} X106 Deviation from the Mean H ^XIOIB Deviation Mean Value 6700 0-144d=0-0010 21-50 14-46 + 0-36 + 2-6 8220 0l73±:0-0007 21-08 14-16 + 0-06 + 0-4 10880 0-227±0-0007 20-88 14-04 -O-Oii -0-4 13430 0-280±:0-0013 20-84 14-03 -0-07 -0-5 16370 0-340±:0-0009 20-78 13-98 -012 -0-9 17700 0-370±0-0019 20-90 14-0Ö -0-04 -0-3 19300 0-401it0-0009 20-75 13-95 -0-15 -1-1 20000 0-422zb0-0049 21-10 14-20 + 0-10 + 0-7 21740 0-458±0-0009 21-10 14-20 + 0-10 + 0-7 23130 0-482±0-0023 20-84 14-03 -0-07 -0-5 25250 0-526±00011 20-84 14-03 -0-07 -0-5 26520 0-551±00010 20-78 13-98 -0-12 -0-9 27240 0-57ldz0-0011 20-9Ö 14-10 0-00 0-0 28680 0-607±0-0020 21-16 14-23 + 0-13 + 0-9 Mean 20-96 14-10 J?, is nearly proportional to H. Plate XI, Fig. 1. The mean valne 20*94 of twelve -#- lO'' from 8220 to 27240 was used to determine the fields '^25020, *25540 and ^26100. In the diagrams of the Plates, the points corresponding to these fields are denoted wit] 1 x. 22 Art. 7.— K. Yamada : The results of former investigators are Investigator Character of Separation H A>, t x''^' Mi >='»" King Kent Van Bilderbeek- van Menrs Triple 16000 28000 32040 0-341 0-501 0-689 21-30 17-89 21-50 14-32 12-03 14-46 The difïerence of my result from that of King is 1*6% (10) ;.:;3S27-, ^x.o. i^x-^ King Kent Van Bilderbeek- van Meurs Triple 16000 28000 32040 0-225 0-346 0-449 14-06 12-36 14-03 9.59 8.43 9.57 (11) ;.: 3825-90. Intensity 8; character h ; King took this for a septuplet (?), but he could not exactly find the separation of parallel com- ponents and only measured the normal component. This resem- bles >i: 441 5* 13. I am of opinion that this ma}^ be further separated in a stronger field or by a spectroscope of higher resolving power. Plate II, Fig. 1; Plate IV, Fig. 4. The results of my observation are: Table XI. Deviation from the Mean AX K^x'»' ^XlO" H ni xio" Deviation Mean Value 8220 0131±0-0008 15-94 10-89 + 0-33 + 31 10880 0-172±0-0010 15-80 10-79 + 0-23 + 2-2 13430 0-206±0-0009 15-33 10-47 -009 -0-9 14500 0-225±0-0008 15-52 10-60 +0-04 + 0-4 16370 0-256±:00007 15-64 10-68 +0-12 + 1-1 17700 0-279±0-0012 15-76 10-77 + 0-21 + 2-0 19300 O-3O7±0-0OC8 15-90 10-86 + 0 30 + 2-8 20000 0-308±0-0020 15-40 10-52 -0-04 -0-4. 23130 0-346dz0-0016 14-96 10-23 -0-33 -3-1 *25020 0-380dzO 0008 15-18 10-37 -0-29 -2-7 25250 0-375±0-0016 14-86 10-15 -0-41 -3-9 *25540 0-401±0-0008 15-69 10-72 + 0-16 + 1-5 *26100 0-403±0-0008 15-44 10-54 -0-02 -0-2 26520 0-396±0-0012 14-93 10-20 -0-36 -3-4 27240 0-425±0-0015 15-61 10-66 + 0-10 +0-9 28680 0-439rfcO-0029 15-31 10-46 -0-10 -0 9 Mean 10-56 ÛI can not be said to vary linearly proportional to H, as the deviations from the mean seem to indicate. Plate XII, Fig. 1. 24 Art. 7.— K. Yamada ; The results of the former investigations are; Investigator Character of Separation H A), t ''''' i^xlO« King Kent Reeso Van Bilderbeek- van Meurs Septuple ? Triple 16000 1 28000 28300 32040 . 0-274W2 0-386 0-452 0-495 17-13 13-78 15-97 15-46 11-70 9-42 10-91 10-54 (12) /l:3820-44. Intensity 10; character b; Plate II, Fig. 1; Plate IV, Fig. 4. King took this for a triplet, hut on my plate this line resem- hles iî: 3825 '90. Although further separation of the inner and outer components could not be detected, this can not be a simple triplet. But here the separations JÀ of two outer components are reported. The determination of the fields is the same as in the case of >î: 3886-29. Table XII. AX H X'^= Ü,:. ««-= Deviation from the Mean H ^xiou Deviation Mean Value 8220 0-147dr0-0ü07 17-88 12-26 +0-77 + 7-0 10880 0-184±0 0007 16-91 11-60 + 0-11 + 1-0 13430 0-222zfcO-0009 16-53 11-33 -0-16 -1-4 14500 0-237±0-0006 16-34 11-21 -0-28 -2-4 16370 0-278±:0-0011 16-99 11-65 + 0-16 + 1-4 17700 0-303±0-0013 17-12 11-74 ' +0-25 + 2-2 19300 0-329dr0-0007 17-04 11-68 + 0-19 + 1-7 20000 0-338±0-0008 16-90 11-59 + 0-10 +0-9 21740 0-370±0-0010 17-02 11-67 + 0-18 + 1-6 23130 O-376±:0.0016 16-27 11-15 -0-34 -3-0 *25020 0-414±0-0007 16-53 11-34 -0-15 -1-3 25250 0-403rt0-0008 16-10 11-03 -0-36 -31 *25540 0-435dz0-0013 17-04 11-68 + 0-19 + 1-7 *26100 0-440 drO-0009 16-87 11-57 + Ü-08 + 0-7 26520 0-428±0-0ül6 16-14 .11-07 -0-42 -3-7 27240 0-453±0-0011 16-64 11-42 - 0-07 -0-6 28680 0-472±00011 16-46 11-28 - Ü-21 -1-8 ^Ican 11-49 _ The increment of Jk is irregular as indicated by the deviations from the mean of -j^^x lO'l No definite conclusion can be drawn from it. Plate XII, Fig. 2. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 25 The results of the former investigations are: Investigator Character of Separation H A>, ^^ X106 ^XIO« King Kent Eeese Van Bilderl>eek- van Meurs Triple 16000 28000 28300 32040 0-282 0-425 0-472 0-536 17-63 15-18 16-67 lG-73 12-07 10-40 11-43 . 11-47 (13) /1: 3815-84. Intensity 10; sharp triplet; character b. Plate II, Fig. 1; Plate IV, Fig. 4. King is also of opinion that this is a triplet. The determination of the fields is the same as in the case of vl: 3886-29. Table XIII. A). ^^ X106 M ^'^, xioi?- Deviation from the Mean H Deviation Mean Value 8220 0-132zfcO-0007 16-06 11-03 + 0-32 + 3-0 10880 0-168±:0-0007 15-45 10-61 -0-10 -0-9 13430 0-204±0-0011 15-18 10-43 -0-28 -2-6 14500 0-227±0-0019 15-65 10-75 + 0-04 + 0-4 16370 0-254±0-0009 15-51 10-65 -0-06 -0-6 17700 0-274±:0-0013 15-48 10-63 -0-08 -0-7 19300 0-305±0-0009 15-80 10-85 + 0-14 + 1-3 20000 0-316drO-0010 15-80 10-85 + 0-14 + 1-3 21740 0-341zh0-0010 15-69 10-78 + 0-07 +0-7 23130 0-361±0-0010 15-61 10-73 + 0-02 + 0-2 *25020 0-3B7drO-0007 15-46 10-62 -0-09 -0-8 25250 0-387±0-0008 15-33 10-53 -0-18 -1-7 *25540 0-399rtO-0010 15-63 10-74 +003 + 0-3 *26100 0-407±0-0007 15-59 10-71 0-00 0-0 26520 0-409±0-0007 15-42 10-60 -0-11 -1-0 27240 0-429±0-0016 15-73 10-81 + 0-10 -0-9 28680 0-451-t0-0013 15-72 10-80 + 0-09 + 0-8 Mean 15-59 10-71 11 ^jj2 is constant within 1 or 2%. results of the former investigations are: Plate XII, Fig. 3. The 26 Art. 7. — K. Yamada Investigator Character of Separation H A). 4-xlcc ,^'- X1013 King Kent Eeese Van Bilclerbcek- van Meurp. Triple 16000 28000 28300 32040 0-264 0-382 0-478 0-505 16-50 13-64 16-90 15-77 11-33 9-37 11-62 10-83 The Dearest value to mine is that of van Bilderbeek-van Meurs. (14) /: 3763-80. Intensity 6; very sharp triplet; character a. Plate II, Fig. 1; Plate IV, Fig. 5. King also took this for a triplet. The determination of the fields is the same as in the case of >^: 3886*29. Table XIV. ^Lxics ^XlOl:^ Deviation from the Mean H A>, H).- Deviation Mean Value 8220 0-112±:0-0011 13-63 9-61 + 0-20 + 2-1 9615 0-134dz0-0008 13-94 9-82 + 0-41 + 4-3 10880 0-145±0-0007 13-33 9-40 -0-01 -0-1 13430 0-l77d=0-0007 13-18 9-30 -0-11 -1-2 16370 0-215dr0-0008 13-13 9-26 -0-15 -1-6 17700 0-231±0-0011 13-04 9-20 -0-21 -2-1 19300 0-263±0-0008 13-62 9-61 + 0-2Ü + 2-1 21740 0-292±:0-0008 13-43 9-47 + 0-06 + 0-6 23130 0-301±:0-002O 13-02 9-18 -0-23 -2-4 *25020 0-329±0-0005 13-14 9-27 -0-14 -1-5 25250 0-335dbO-0008 13-27 9-35 -0-06 -0-6 *26100 0-344±00007 13-lS 9-30 -0-11 -1-2 26520 0-352dr0-0014 13-28 9-36 - 0-05 -0-5 27240 0-367dr0-0026 13-47 9-50 + 0-09 + 1-0 28680 ü-386HzO-0010 13-45 9-49 + 0-08 + 0-9 Mean 13-34 9-41 J/ may he said to increase proportionally to H, though the Plate XIII, Fig. differences from the mean are somewhat great 1. Rla^netic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 27 The results of the previous investigations are: Investi gatoi- Character of Separation H A>, H >^''' King Kent Van Bilderbeek- van Meurs Trij-yle 16000 28000 32040 0-218 0-328 0-431 13-62 11-72 13-45 9-61 8-27 9-48 The results of both King and van Bilderbeek-van Meurs may be said to be in agreement with mine. (15) ;.: 3758-23. Intensit}' ^r ; sharp triplet; character a. Plate IT, Fig. 1. The determination of the fields is the same as in the case of A: 3886-29. Table XV. A> H ^^^^ i:. xio» Deviatioa from the Mean H ■^' X1013 ±l>"-^ Deviation Mean Value 8220 0-140±0-0006 17-03 12-06 + 0-32 + 2-7 9615 0-162±0-0012 16-86 11-93 +0-19 + 1-6 10880 0-l79±00006 16,45 11-65 -0 09 -0.8 13430 0-225±0-0009 16-74 11-86 + 0-12 + 1-0 14500 0-239±:0-000S 16-48 ■ 11-67 -0-07 -0-6 16370 0-267drO-0008 16-32 11-55 -0-19 -1-6 17700 0.291±0-0012 16-44 11-64 -0-10 -0-9 19300 0-321±0-0010 16-63 11-77 + 0-03 +0-3 20000 0-334±0-0016 16-70 11-83 +0-09 + 0-8 21740 0-366±0-0009 16-84 11-93 + 0-19 + i-t5 28130 0-379±:0-0009 16-39 11-62 -0-12 -1-0 *25020 0-412±:0-0006 16-45 11-65 -0-09 -0-8 25250 0-417±0-0006 16-52 11-70 -0-04 -0-3 *25540 0-422±0-0007 16-53 11-71 -003 -0-3 *26100 0-428±0-0006 16-40 11-62 -0-12 -1-0 26520 0-437±0-0009 16-48 11-68 -0-16 -1-4 27240 0-452±00010 16-60 11-75 + 0-01 + 0-1 28680 0-476±0-0005 16-57 11-74 0-00 0-0 Mean 16-57 11-74 JA varies linearl3^ proportional to H. Plate XIII, Fig. 2. ■28 Art. 7. — K. Yauiada : The results of the former investigations are: Investigator Separation H A), iL. 10. •^^' X1013 King Triplet 16000 0-269 16-82 11-91 Kent ,, 28000 0-403 14-42 1019 Eeese Van Bilderljeek- van Älenrs 28300 32040 0-478 0-541 16-89 16-89 11-94 11-94 (16) ;.: 3749-47. Intensity 10; sharp triplet; character a. Plate II, Fig. 1; Plate IV, Fig. 5. The determination of the fields is the same as in tliecase of /:388ß-29. Table XVI. H ^^^^ „^ixio.= Deviation from the Mean H AX ^\ X1013 Deviation Mean Value 8220 0-150±00015 18-25 12-98 + 0-38 + 3-0 10880 0-192±0-0006 17-65 12-56 -0-J4 0-3 13430 0-239±:00013 17-78 12-65 + 0-05 + 0-4 14500 0-255±:0-0004 17-58 12-51 -0-09 -0-7 16370 0-288±:0-0008 17-60 ■ 12-52 -0-08 -0-6 17700 0-311±0-0014 17-57 12-50 -0-10 -0-8 19300 0-342dz0.0005 17-72 12-61 + 0-01 + 0-1 20000 0-353±0 0015 17-65 12-56 -0-04 -0-3 21740 0-388±0-0006 17-86 12-70 + 0-10 + 0-8 23130 0-408±:0-0012 17-65 12-56 - 0-04 -0-3 *25020 0-443d:0-0005 17-71 12-60 0-00 0-0 25250 0-440±0-0005 17-43 12-40 -0-20 -0-6 *25540 0-455±0-0005 17-83 12-68 +0-08 + 0-6 *26100 0-458±0-0009 17-56 12-50 -0-10 -0-8 26520 0'468zt0-0007 17-65 12-56 -0-04 -0-3 27240 0-484±:0-0009 17-76 12-63 + 0-03 + 0-2 28680 0-5122h0-0007 17-85 12-69 + 0-09 + 0-7 Mean 17-71 12-60 AX is linearly proportional to H. Plate XIII, Fig. 3. Magnetic Separations of the Linos of Iron, Nickel and Zinc in Different Fields. 29- The results of the former investigations are: Investigator Separation H A>. M X^°' ^xl„.= King Kent Eeese Van Bilderljeek- Tan Meiers Triijlet 16000 28000 28300 32040 0-289 0-432 0-530 0-580 18-07 15-42 18-74 18-11 12-84 10-97 13-32 12-86 (17) >î: 3737-13. Intensity 6; the separation seems to be a triplet, but each component is diffuse. King took this for a septuplet. Character a. Plate II, Fig. 1 ; Plate IV, Fig. 5. The separations of the outer two components, being consi- dered as a triplet, are as follows: Table XVII. A), ^X10= H-= >"">"• Deviation from the Mean H ,t\ X1013 Deviation Mean Value 8220 0-126±0-0011 15-33 10-97 + 0-17 + 1-0 10880 0-163±:0-0007 14-97 10-72 - 0-08 -0-7 13430 0-204±0-0011 15-18 10-87 + 0-07 + 0-6 14500 0-221rt0'0012 15-26 10-92 + 0-12 + 1-1 17700 0-274±0-0007 15-48 11-08 + 0-28 + 2-6 19300 0-291±0-0009 15-08 10-79 -001 -0-1 21740 0-331±0-0010 15-23 10-91 + 0-11 + 1-0 23130 0-340±0-0014 14-71 10-54 -0-26 -2-4 *23580 0-355±0-0016 15-06 10-78 -0-02 -0-2 *25020 0-367±0-0020 14-68 10-52 -0-28 -2-6 25250 0-363±:0-0009 14-38 10-29 -0-51 -4-7 *25540 0-391±0-0008 15-31 109G + 0-16 + 1-5 *26100 0-406±:0-0007 15-56 11-14 + 0-34 + 3-1 26520 0-392±0.0017 14-78 10-58 -0-22 -2-0 27240 0-415±0-0009 15-24 10-92 + 0-12 + 1-1 28680 0-435±0-0014 15-16 10-85 + 0-05 + 0-5 Mean 10-80 We can not say that JA varies linearly proportional to H, as in all other cases of diffuse triplets. The deviation from the mean can also be seen from Fig. 1 of Plate XIV. 30 Art. 7.— K. Yamacla The results of other investigators are: Investigator Separation H A), ^l X106 ^} X1013 King Kent Ree.se Van Bilclerl)eek- van Meurs Septuple? Triple 16000 28000 28300 32040 0-254 wi 0-371 0-418 0-467 15-88 13-24 14-76 14-58 11-36 9-49 10-57 10-42 (18) /i: 3734-86. Intensity 10; sharp triplet; character a. Plate II, Fig. 1; Plate IV, Fig. 5. The determination of the holds is the same as in the case of ;.: 3886-29. Table XVIII. AX 4-xio« Deviation from the Mean H ^>; X1013 Ma- Deviation Mean Value 8220 0-lo6±0-0009 18-97 13-60 + 0-48 + 3-7 10880 0-198±:0-0007 18-21 13-05 -0-07 -0-5 13430 0-248dr0-0012 18-46 13-24 + 0-12 + 0-9 14500 0-263±0-0009 18-14 13-01 -0-11 -0-8 16370 0-294±0-0013 18-24 1308 -0-Oi -0-3 17700 0-325±0-0014 13-37 13-16 + 0-04 + 0-3 19300 0-354±0-0007 18-34 1315 + 0-03 + 0-2 20000 0-37ldrO-0009 18-55 13-30 + 0-18 + 1-4 21740 0-401±0-0010 18-45 13-23 + 0-11 -0-8 23130 0-421zt0-0008 18-21 13-06 -0-06 -0-5 *23580 0-430drO-0006 18-25 13-08 -0-04 -0-3 *25020 0-455dr00006 18-18 13-04 0-08 -0-6 25250 0-453dr00006 17-97 12-88 -0-24 -1-8 *25540 0-468±0-0007 18-35 13-16 + 0-04 + 0-3 • *26100 0-472dzO-0006 18-10 12-97 -0-15 -1-1 26520 0-480±0-0005 18-13 1299 -0-13 -10 27240 0-498dr0-0009 18-30 13-12 000 0-0 28680 0-524d::0-0009 18-27 13-10 -0-02 ■0-2 Mean 18-30 13-12 Jk may be said to be nearly proportional to H. The greatest deviation is 2%, except at the lowest field that was measured. Plate XIV, Fig. 2. Magnetic Separations of tlie Lines of Iron, Nickel and Zinc in Different Fields. 31 The results of the former investigations are: Investigator Character of Separation H \l ^ X106 ^x.o« King Triplet 16000 0-310 19-37 13-89 Kent ,. 28000 0-453 16-19 11-60 Reese ,, 28300 0-538 19-02 13-63 Van Bilderbeek- van Meurs 32040 0-594 18-55 13-30 (19) ;J:3719-93. Intensity 10; this line does not seem to be a triplet, but rather a sextet, althougli separations of all components can not-be detected; character c. Plate II, Fig. 1; Plate IV, Fig. 5. The determination of the fields is the same as in the case of À: 3886*29. The separations JÀ of the outer two components are: Table XIX. AX 4^ X106 11 MX.^'»" Deviation from the Mean H ^l X1013 Ma- Deviati on Mean Value 8220 0-133±0-0008 16-18 11-69 + 0-19 + 1-7 10880 0-l76±0-0006 16-17 11-68 + 0-18 + 1-6 13430 0-216±00008 16-08 11-62 + 0-12 + 1-0 14500 0-225±0-0005 15-52 11-21 -0-29 -2-5 16370 0-266±0-0013 16-26 11-74 + 0-24 + 2-1 19300 0-311±0-0008 1612 11-64 + 0-14 + 1-2 20000 0313±0-0013 15-65 11-30 -0-20 -1-7 21740 0-353±0-0011 16-24 11-74 + 0-24 + 2-1 23130 0-359±0-0014 15 53 11-22 -0-28 -2-4 *23580 0-382dr0-0010 16-20 11-70 + 0-20 + 1-9 *-25020 0-405±:0-0010 16-18 11-69 + 0-19 + 1-7 25250 0-386±:0-0009 15-29 1104 -0-46 -4-0 *25540 0-4l7±0-0010 16-33 11-80 + 0-30 + 2-6 *26100 0-423±0-0014 16-21 11-71 + 0-21 + 1-8 26520 0-406±0-001I 15-33 11-06 -0-44 -3-8 27240 0-429±0-0008 15-76 11-38 -0-12 -1-0 28680 0-449±00012 15-67 11-32 -0-18 -1-6 Mean 11-50 32 Art. 7. — K. Yamada : We can not say that J?, is proportional to H. The deviations from the mean are also seen from Fig. 1 of Plate XV. The results of the previous investigators are: Investigator Character of Separation H A>. -^j>-^Xl06 ^.v X 1013 King Triplet ? 16000 0-268 16-75 12-10 Kent » 28000 0-393 14-03 10-14 Reese „ 28300 0-448 15-83 11-43 Van Bilderbcek- van Meurs " 32040 0-508 15-81 11-42 (20) /1: 3618-77. Intensity G; diffuse triplet; character b. This line is photo- graphed on the same plate together with the zinc line /:4680"138 as field determination. 4680*138 is of the 3rd order spectrum and 3618'77 of the 4th. A line of other order also appeared close by this zinc line in its red side as seen from the photographs (Plate I, fig. 2 and Plate I, fig. 3). Also Plate III, Fig. 4. In higher fields, as the separations of the zinc line are large, the determination of the field is not affected by the appearance of this line. In lower fields, however, the overlapping of this line witli +ÔÀ component of 4680' 138 was the cause of an error in the measurement of the field, when determined b}' measuring the distance of the outer two components of the zinc line; in such cases, twice the distance between ^^^ and —d?. was taken instead of the separation of the outer two components, as this zinc line has been confirmed to be perfectly symmetrical. J?~ denotes the separation of two outer components as before. Magnetic Separations of the Lines cf Iron, Nickel and Zinc in Different Fields. 33. Table XX. AX tX'"' ^,xio- Deviation from the Mean H Deviation Mean Value 19000 0-197 drO-0023 10-32 7-88 + 0-30 + 4-0 22380 0-221 rtO-0014 9-87 7-54 - 0-04 -0-5 23100 0-228±0-0015 9-87 7-54 -0-04 -0-5 *=23580 0-235dr0-0009 9-98 7-62 + 0-04 + 0-5 26440 0-251±0-0009 9-49 7-25 -0-33 -4-4 27750 0-272HrO-0015 980 7-47 -0-11 -1-5 27920 0-275±0-00]3 9-86 7-53 -0-05 -0-7 28700 0-285±0-0025 9-93 7-58 0-00 0-0 29200 0-288±0-00l7 9-86 7-53 -0-05 -0-7 29400 0-299±0-0025 10-17 7-76 + 0-18 + 2-4 30080 0-30]±0 0011 10-00 7-63 + 0-05 + 0-7 32100 0-315±0-0012 9-81 7-48 -0-10 -1-3 32240 0-323±0-0020 10-02 7-64 + 0-06 + 0-8 33100 0-333drO-0022 10-06 7-68 + 0-10 + 1-3 Mean 7-58 From tJie last column of the table we see that the deviations from the mean are less than 2%, except for three points. Hence we can say that J?^ is nearly proportional to the fields applied. Plate XV, Fig. 2. The results of former investigators are: Investigator Character of Separation H A>, ^ xio« ^,X1013 Kent Van Bilderbeek- van ]\Ienrs Triplet 28000 32040 0-245 0-311 . 8-75 9-70 6-68 7-41 (21) ;:3581-20. Intensity 10; triplet, and each component is intense in com- parison with that of other lines on the same plate; character b. The determination of the fields is the same as in the case of /:3618-77. Plate I, Fig. 3; Plate III, Fig. 4. 34 Art. 7.— K. Yamada: Tap.le XXI. iLxlOe m= X'«" Deviation from the Mean H AX ^_\.X1013 Ha- Deviation Mean Value 143-70 0-207drO-0010 14-41 11-25 + 0-28 + 2-6 18080 0-257±0-0018 14-21 11-09 + 0-12 + 11 19090 O'270±0-0011 14-15 11-04 + 007 +0-6 22380 0-318±0-0016 14-21 11-09 + 0-12 + 1-1 23100 0-329drO-0008 14-24 11-11 + 0-14 + 1-3 *23580 Û-352±0-U009 14-94 11-65 + 0-68 + 6-2 2G440 0-370drO-0009 13-98 10-90 -007 -0-6 27750 0-384d=00006 13-83 10-79 -0-18 -1-6 27920 0-392±0-0007 14-05 10-96 -0-01 -Ol 28700 0-395±0-0011 13-77 10-74 -0-23 -2-1 29200 0-404dr0-0008 13-83 10-79 -0-18 -1-6 29400 0-404±0-0012 13-76 10-73 -0-24 -2-2 30080 0-il8±0-0010 13-91 10-84 -0-13 -1-2 31100 0-441±0-0009 14-18 11-06 + 0-09 + 0-8 31500 0-44üdr0-0013 14-16 1105 + 0-08 + 0-7 32100 0-449±0-0009 13-98 10-90 -0-07 -0-6 32240 * 0-442±:0-0011 13-70 10-68 -0-29 -2-6 33100 0-458±0-0009 13-84 10-80 -0-17 -1-5 Mean 10-97 From the table we see that Ak varies nearly proportionally to the fields apphed, though somewhat great discrepancies are found in thi'ee of these measurements. Plate XV, Fig. o. The I'esults of the previous observers are: Observer Character of ,y Separation AX ^'' XlO« H -^xio« Kent Van Bilderlicek- Titn Meurs Triplet 28000 32040 0-359 0-458 12-83 14-30 10-00 1114 (l>2) /: 3570- 12. Intensity 10; difïuse triplet; cliaracter h. In non-mngnetic field, this line and /',:o58r20 have equal intensity 10; but in magnetic fields, the components of the former are more intense than those of the latter. The determination of the fields is the same as in the case of -^:3G18*77. Plate I, Fig. 3; Plate III, Fig. 4. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 35 Table XXI T. A), ^l X106 H. ^\ xioi« Deviation from the Mean H ^^XlOls Deviation Mean Value 14370 0-195±0-0023 13-58 10-65 + 0-66 + 6-6 18080 0-243dr0-0012 13-43 10-54 + 0-55 + 5-5 19090 0-250±0-0015 1310 10-28 + 0-29 + 2-9 22380 0-288±0-0014 12-87 10-10 +0-11 + 1-1 23100 0-303±00008 13-12 10-30 + 0-31 + 3-1 *23580 0-310±0-0006 13-16 10-33 +0-34 + 3-4 26440 0-329dr0-0019 12-43 9-77 -0-22 2-2 27750 0-347±0-0018 12-50 9-82 -0-17 -1-7 27920 0-352zfcO-0013 12-61 9-90 -009 -0-9 28700 0-359dr0-0029 12-51 9-82 -0-17 -1-7 29200 O-363±0-0013 12-42 9-76 -0-23 -2-3 29400 0-368drO-0015 12-51 9-82 -~0-l7 -1-7 30080 0-375±0-0009 12-46 9-78 -0-21 -2-1 32100 0-395±0-0021 12-31 9-66 -0-33 -3-3 32240 0-399±00016 12-38 9-71 -0-28 -2-8 33100 0-408±0-0019 12-33 9-68 -0-31 -3-1 Mean 9-99 Aa -^ 18 larger in lower ßelds than in tlie higher, as is easily seen in the last column of the table. The deviations from the mean are also seen in Fig. 1 of Plate XVI. The results of the former investigators are : Investigator Character of tt Separation Aa H ^'* y^^^XlOlB Kent Van Biklerbeek- van Menrs Triplet 28000 32010 0-326 0-407 1164 12-71 913 9-96 (23) /i: 3075-725. Intensity 3; sharp triplet; character b. The third order spectrum of this line was photographed together with the second order of /:4(>S0'138 of zinc as field determination. Plate I, Fig. 4; Plate IV, Fig. 3. 36 Art. 7. — K. Yauiada : TvF.Li: XXIII. H A>. ^xio. H^Xl»» 25150 25920 26420 28050 0-330 0-343 0-346 0-372 13-12 13-23 13-08 13-25 13-87 13-98 13-82 14-00 Mean 13-17 13-92 The data are too few to determine the relation between ^^- and H. But in these regions ^ ma^^ be said to be constant. I could not find any published data of the separation of this hue; perhaps it is the first time that its separation has been examined. (24) a: 3050-08. Intensity 3; sharp triplet; character b. Plate I, Fig. 4; Plate IV, Fig. 3. The determination of the fields is the same as in the case of ;.: 3075-725. Table XXIV H A> ^XlOB ^t.xio^^^ 25150 25920 26420 28050 0-330 0-340 0-344 0-365 13-12 13-12 13-02 13-02 1401 14-01 13-91 13-91 Mean 13 07 13-96 In these regions -jj- is constant. Van Bilderbeeh-van fleurs observed this line and reported that H A), ^ XIOC ^xio. 32040 0-437 13-64 14-56 (25) /i:3047-()0. Intensity 3; triplet; character b. The determination of the field is the same as in the case of >i: 3075-725. Plate I, Fig. 4: Plate IV, Fig. 3. Magnetic Separations of the Ijines of Iroii, Nickel ami Zinc in Different Fields. ;-J7 Table XXV. 11 A> -^^ XlOG 11 ^1-xio- ±1A-^ 25150 0-321 12-75 13-73 25920 0-332 12-81 13-79 26420 0-332 12-55 13-51 28050 0-368 13-10 1410 A), Evidently -^ is not constant within these narrow ranges. Jjut the data are too few to decide the feature. For comparison, I give only: Observer H A> ^xio. ^^'°'= Van Biklerbeek-van Meurs 32040 0-432 13-49 14-50 (26) ;.:2756-;:51. Intensity 1; diffuse triplet; character b. Plate I, Fig. 2; Plate III, Fig. 2. The fifth order spectrum of this line is photo- graphed on the plate together with the third order of >?.: 4680" 138 of zinc. Table XXVI. H A>. 4x100 ^.^0.. 26210 0-220dz0-0011 8-42 11-07 27940 0-237±0-00l7 8-48 11-16 29600 0-256±:0-0010 8-67 11-41 30000 0-254±:0-0019 8-47 11-14 30100 0-251±:0-0013 8-34 10-96 Further study is necessary to decide the character, hut in these regions the variation may be considered as regular and approximately IF 15 adopted as the mean value of H>;i~^ 10''. It is to be regretted that I could not find any published data as to the separation of this line. (27) ;:275ry73. 38 Art. 7. — K. Yauiacla Intensit}'- 15; a triplet, whose two outer components are diffuse and wider than the central. Character b. Plate I, Fig. 2; Plate III, Fig. 2. The determination of the fields is the same as in the case of /:27r)G-:31. Tai'.le XXVII. A), -^xios ^' xioi^- Jd),-' Deviaticm from the Mean H m. x^°" Deviation Mean Value 25800 0-226dr0-0005 8-73 11-50 -0-06 -0-5 26210 0-227±0'0007 8-66 11-38 -0-18 -1-6 27940 0-247±0-0007 9-32 12-26 + 0-70 + 6-1 29600 0-258±0-0010 8-73 11-50 -0-06 -0-5 30000 0-260±0-0019 8-66 11-38 -0-18 -1-6 30100 0-261+0-0007 8-67 11-40 -0-16 -1-4 30350 0-263±0-0009 8-67 11-40 -016" - 1-4 31320 0-280±0-0007 8 87 11-68 + 0-12 + 1-0 Mean 11-56 " A> -g- is not straight as in all other cases of diffuse triplets. The following I found reported: Investigator Character of Separation H A>. ^^•XIOC ^XlO« Van Bilderl)eek- van Meurs Triplet 32040 0-272 8-49 11-17 (2^^) yi:274G-Uy. Intensity 8; diffuse triplet; the peculiarity of this line is that the violet side component is weaker in intensity than the red side one, as required by Voigt' s*^ asymmetr}^ theory. Plate I, Fig. 2; Plate III, Fig. 3. Tlie determination of the fields is the same as in the case of ;.:275G'31. 1) ^V. Voigt, Ann. d. rhj-.s., 1 (1900), p. 376. Magnetic Separations of the Lines of Iron, Nickel fncl Zinc in Different Field?. 89 Table XXVIII. 11 H "<"" ^^^^,-XlOi« Deviaticn from the Mt an H ÜA- Deviation Mtan Value 25800 0-238±:0-0023 9-21 12-20 ■ 0-63 -4-9 26210 0-258±0-0009 9-82 1301 + 0-18 + ]-4 27940 0-276±0-0009 9-88 13-10 + 0-27 + 2-1 29600 0-289±:0-0009 9-75 12-92 + 0-09 + 0-7 30000 0 293dzO-0010 9-76 12-93 + 0-10 + 0-8 30100 0-292±:0-00U 9-72 12-87 + 0-04 + 0-3 30350 0-296dr0-0014 9-76 12-93 + 0-10 + 0-7 31320 0-299dr0-0015 9-55 12-65 -0-18 -1-4 Mean 12-83 From the table we see that the variation of JX is not pro- portional to H. Plate XVI, Fig. 3. According to van Bilder- beek-van Meurs: Character oi Separation H A), H X^^^ Triplet 32040 0-305 9-52 12-60 She makes no remark as to the asymmetry of the separation. (29) a: 2746-48. Intensity 10; diffuse triplet, whose outer two components are broader and fainter than the central one. Character b. Plate I, Fig. 2; Plate III, Fig. 3. The determination uf the tields is the same as in the case of X : 2756*31. 40 Art. 7. — K. Yama'la Table XXIX. Deviation from the M^an A) H ^^^^ ,&:x- 11 ^\ xioi« Deviation Mean Vaiue 25800 0-185rtO-0012 7-16 9-50 -0-13 -1-4 26210 o-i87±o-o:o9 7-14 9-48 -0-15 • -1-6 27940 0206±:0-0012 7-38 0-79 + 010 + 1-7 29600 0-216±:0-0007 7-30 9-67 + 0 04 + 0-4 .30000 0-221dr0-0012 7-37 9-77 + 0-14 + 1-5 30100 O-220±0-0O09 7-31 9-69 + 0-06 - 0-6 30350 0-218±:0-0010 717 9-52 0-09 -0-9 31320 0-227±0 0015 7-23 9-60 -003 -0-3 Mean 9-63 Within these regions, -^ may he said to he constant. The result of van l>ilderheek-van iMeurs is Character of Separation H A>. ^'- XlOG «i x^°" Triplet 32040 0-221 6-90 914 Tahle XXX. This hne :^740'4. ^'^ XIO« ±1A- Triplet 32040 0-325 10-13 13-50 (ol) Separations in a single field. Although the magnetic separations of iron lines in a single field have been investigated by many phyicists, it may not be out of place to report here the results obtained in the course of my study of the above mentioned problem. (Plate II, Figs. 2, 3, 4 and 5; Plate V, Figs. 1, 2, 3, 4 and 5; Plate VI, Figs. 1, 2, 3 and 4). 42 Art. 7. — K. Yair.ada: 5h King took this for a sep- tuplet ? ce 'El CO ^ ci a _4; + - '2. s CD -BS 1 3= 'S « a X o g» CO P CD Ö 00 o 1-1 o ÖÖ 1> o CO O ^ Ö L-O O •H Ö L-o o (M o ^ Ö CD O (M O CD O (^a o ^ Ô r-l i> o ^ o I-l IM O Ö ^ r-l o X o CO O O ip O ö Ô 00 o ^ o Ö 6 §8 -1 6 00 T-H t~ o CO O CD O CI O ^ Ô r-l \o o (M O •^ Ö r-l t- o (M O ■^ Ö t~ o rH O -* Ö t-l \a o IM o ■^ ô r-l o X o 1> o 2^ T-l O CQ O I> Ô t~ o -* o ö ô 1-1 CO CO o o 00 o t^ Ö ^ O O o (J5 ô (M 00 O oj O Al Ö (M CD O Oi O Al Ö g8 r-l Ô IM ^8 <î CO (M CD 6 •* o CD O ei o 6 6 CO o (M O '?' o ô Ö 1> o ^8 ô ö CD 00 in 6 iro O ^ o -# o ô ö S8 o o 6 ô o o 1.0 o IQ o ö ö o o Lo o lo o 6 ô -1 ^ Iß O ^ o lo o ô ô CO o -# o Ift o 6 Ö ! H*- = o M o CD o CD lO o CD o C-l o o (M o (M O o o Ol o o C] o 6\ o 0--; O l£0 o IM O il 5=^ O ^ H tH ^ c-l 'S c H In- tensity 1.-3 CC' CD LO o CO r? Tj. ■* ^ % IS CO 00 CO O CD Ö0 CD o 00 lb CD 5^ à) CD Oi CO o CD 10 o 00 ô CO CO Ol t- (M CO IM (M oo o 6 Cl CO 1-1 ô o Cl CO Magnetic Separations of the Lints of Iron, Nickel and Zinc in Different Fields. 43 ac Ö CO 5 c o o 'S* 02 Becquerel and Deslandree took this for an inverse triplet. C<0 r-H O CO o oo ô "be O Ü X ^ o ^ CO o CO o ■^ Ö r-1 03 o - o -I ô (M 00 00 00 CO -# o ■* Ol o (M 0-1 (M 00 o Ci o ô ô I-l CO Cl .-H o o o o Al 6 (M 88 cb ô ift o rf O (M Ô ^ : 05 . ■* • 6 00 o CO O Ks o 6 6 H ^ CO o 6 Ö '!*' 00 Ö ô (MON OtI O CO o o lo ô ô Ö lO o t- o IM o 6 6 00 o CO ô VO O C-J O Wi o 6 6 la o (M o CO o 6 Ö o o o o 00 o ô Ö iM O CO o Ö Ö a o o o O UO o o o o in o IM S IM o 00 CO o 00 CO o t- 00 00 CO IM lO \a CO oc CO CD cp Al '^ 00 CO CO (M CO 00 CO (M 00 C5 ' IM 00 i ^' 1 1 o t> cb O 00 00 LO O 00 00 44 Art. 7. — K. Yainaila : M bD PI CT" -9 OJ 5 ^ •Jl «Ä 3 o o o i) o 'br: 1 :r ^ -^ « o N o o o CO CO O — 1 CO r> o 00 o o o o 'S! ^' ■~> X ^ •* Ô ■* 6 i-i 1 ^ ö C> -^ Ô ■* ö H -* ö -* ô o 1-1 ■* ö ^ •ci - î-l 1 ^^ H "^ <^ ■*v^^^ P ^^ H ^ & , ^ s 2 OC o CO O 00 CO o o o CO o o -H O IM CD o CD ■* o C-J O CO O M CO Ç) 00 CD CO O 00 o o M x> o 1> X 4j< o -* ö ^ ô Ä ■* Ö -* Öi •^ ö ^ ô ô d> •è ö -* ^iq 1-H r-l o o o 53 O CO O O CO O 00 o CO O 00 Ol o lO 00 r-' 00 00 o CD O CO O CO œi o 00 O) CD cp qp t-i o O) CO o CD X Ô 6 S<^ cb ö CO (i) Ö CD CO CD ô CD '^ ö IM CO ô CO 1^^ — 1 00 |_| 00 c-i o f— ( o 2S CO CD O 1> "# CO O 1> lo o 1— 1 O f-H 8 LO ^ LO O IC o ^ o CO r-l O rH CO 1-1 o ^ CO o t, l^' i^ I— 1 •. o ^ Ô ö ô ô ö ô ö 6 ö ô ô 6 6 ö 6 ô '5 *^ '5 pH ö ö H^ H^ H^ -) -1 ^ cu Ä H *5 o o o o o o c- O Ol w (M C] IM C^l IM IM ^ o o o o o o o L'ï lO i.O lO lO o ô lO W CJ IM IM -M IM C>] ?q CM o ö ^ '-' r^ r^ y :;3 ^ ^ o f= y O i) V i i- o o ^ ■^ W O ^ p *, rr' So ^ JC *- u o«^ PI u E3 H^ E-i H o t-i CH o t) Inix'usity lO ■* CO •^ UO 0Î •^ Tji ■c ÖD fl'^ lo o -H 00 o Tjl l.O ^ n:^P LO « ip 00 •H LO M 03 1 ÖJ 00 t~ t- t^ ib do LO >^ 05 o o 00 CO CO Tf< -* 1> Jt> 1>- t^ t^ t^ J:^ t~ rt CO M co CO CO CO CO CO >■ 1 Magnetic Seiarations of the Tiines oï Iron, Nickel and Zinc in Different Fields 45 ■• lii CD ?.2i , ^ br. ic br S"Ï2 ~ :-c E- »-I ;:: 5 J-i ^ ^ o ci tJ 3 t2 O ^ ^ o. Ol c -- > o ^^ O.. _2 P 00 « 1 CO 1 ce 5 t 02 ai S o O (M l-H ^ 02 7? tj » c , ^ _ .^^ • be ^^' ^ ^ lo : O CT) O -^ 00 CO o J> o o SI' O CO .-1 -f< 0 0-. CO o OO 00 o o o i> in OS O CO (M t~ oi> Cl o i> o o CO 00 r-l O CO CO CO o i-H ■ — - ô • Öi -* Ö -# CTl •^ ô -^ t- Ö ^ 60 ^ CT5 -^ 00 ^ Ö ÖO CO -* Ö X H^ H^ '^'^^.^ «^ ^ 1-1 H^ — \ '-^ Ti O t> 00 t- o t- CD a CTi : CT5 t~ CO 1.0 CO o TD o o CO o I—* 00 I-H o o o cc O o LO o 03 - Oi O CO .-1 CO ^H lT, CO o IM O X £- : CT> -^ Ô -^ C5 ^ Ö ^ CO 6 -ä ; ÖD -^ as ^ CT) Ö -j -* ô 4}< 6 ^^ ' s/. ' I-l '^ I-l I-l '"' '"' '"' ^ ^ Ö 00 : O O O O M o o o CO o •e • o o I> »o t> O o 1-1 oî T? O »— < o . t~ T}( p ^ rH -* o 'S* -^ o CO ■ CO o CTH> CTj t- o 1-1 o CO o X ^ : (jq ö Ô Ö M cb Ö '•i' ô 6 6 • cb d> M Ö Cl Ö1 Ö IM Ö Ö5 Ö 1 r-H V^^i^^^^ 0] ■ (M tH rt r— 1 r— 1 î'] r~^ ■^ I-H ^u I> ■ ^ 1-^ o 'H o T-( O --l i> o M • o o CO CO CO CO O -a CO CO o t- o o o kÄ — O o LO 00 o Cj . (M CO o CO o CD O LO 0^' \a o ,*v N : CO r-t o ^ CO --H o — 1 T? o "# : CD Ti< CO ^ 0-5 T? O LO (M ^ O -^ Ô • ô 6 ô ô 6 6 Ö 6 J (M oa (M (M 01 (M =4-< o ri o ^ .s r— • ai j3 ;!_, 13 fS c ^ p, r2 O ^ r^ s r-" _= .8* .E^ ■f. _&. O^ o H & H Intensity O CO o ^Jl 'el ■^ '~* ^ ^ K5 x> C5 CO z> ^ CO -^ kp CO M

b ôo (N t- àl CTl LO Jr~ 1 U 1— 1 'S^ CO (M (M O O CO !>. t^ t~ 1>- t~ 1> t- CO ^ CO 00 CO CO CO CO CO CO ^ r" 46 Art. 7. — K. Yauiada : V 5 Accordino- to Kent, ± & II comps. have the sauio position. But I found that this is a doublet. 'Hi ? a .S V. o 3 3 ^c to Ö . ^ U ifj m CI — ' <^ l) o -5 X -ê o X O O CO O Ô ö CT o 6 Ô + + Öl S8 05 Ö CD ilo H CD O C3 ô i-H 00-2 ■* O 1> o 01 o \o Ö 01 00 t- LO i> Ol CO CO 6 CO CO O o o LO Tp r-l 00 LO 00 o 01 O 00 Ö ffl 00 CO o ^ I> p LO \a LO o X o 01 ö CO CO 'J« CO CO 00 LO 0 o 01 O CO ô 01 00 T-l CO o cb lb 1> 01 o ^ o do 6 01 CD Ol tJI lO Ci p ô CO CO ^ S8 C<5 O Ô ö O CO ö 01 O 0 o 01 O ö ô 01 o 6 H CO ô H 00 01 Ö H o o ■* o lo o ô 6 01 00 00 ^ Ö 6 CO Ö H 00 O CO o CD O Ô Ö LO LO 01 lO CD •<*< !'. - LO 51 o CO 01 o CO CO 01 o CO lO CO 01 o CO LO CO Ol CO Ol o 00 1.0 CO 01 o CO LO CO 01 o CO CO 01 g LO CO 01 o OC LO 00 01 t, o 6'^ o p o o o 3j o ^5' ' Intensity ^ CO CO 01 LO (JO lO lO CO CO PO CD Ol LO CO * CO Ol Ol Iß CO CO 00 J> CT) CO 00 lO ô o CO CD CD Ö 00 Tjl CO * o CO CD CO LO CO Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 47 JA 3 The separation of this line is surely complex ; van Bil derl)eek-van Meurs took it for a quartet. O 'S ••3 1» cq o oo t- 6 r-l -, o ■ O X <^ CC Ci o 00 O CO -H M -^ ô Ö0 '* o Ci o ö ô CO o lO o 00 ô 00 o Ö0 ô 'S! • CO ■ Cl ô X 1* N I> •* -* O

- CO CI .-^ (M O CO O 00 o uo o C3 6 1 Cl o ! CD . cj 6 \ âo ': o o Cl o Ci 6 ^ ±0-647 ±0-399 // 0-181 // 0-090 C3 o CT> o 1-1 o ô 6 r-l O lO O oq o 6 6 •^ o CO o Cl o ô 6 oo • o : (M : 6 ■ i> o —1 o Cl o 6 6 =; CO LO CO 05 o (M CD CJ o Cl CD C3 O 00 LO 00 Cl O 00 lO CO CI o CO ira oo Cl o CO kO 00 Cl ID OO Cl O 00 ira 00 Cl o ira CO Cl Il -J o 1 .1) '■A P o ai 3 l-J ce "S se ce P o EH "6 6* o st: ce r2 Intensity CO oq 1-1 ^ 00 Ift m ^ Wave-length (I. U.) o CO CO o CM * Cl p (M 115 Csl cb * CO cq IM * O Cl CD C3 * 00 CO CD Cl Cl Cl CO CO Cl * 48 Art. 7. — K. Yauiada : ;- r^ ä Zj ." o îd , 1 -4J ,A!ä c3 il ^ ;_, % c3 et o œ O œ o O CD « t- c p 1 1« ^ CS 'r^ > "^ CO fö Ï.J:: - ~^ 04 o ". ^ —' O Oi o ^ i-i CO IM CO i-H ,-< '-< -1 t:: o i> o (M O o Ol o rj3 O IM O |^^ o CD '"' (M o 00 o CO CO O CO O rjU 03 o o !>• X -^ Ô ô ö i> ö 6 Al 6 O ib ô ô tA t-H IM i—i Cl 1—1 '"Ï '-r' <1 ^ 00 C35 03 CO CD O 1-: o O ui O (M O CO ^ 1^ o i^ •^ o o o »o o O o Ô Ô o o £- '-1 N O (M O r~\ (M O o o S-1 Ô Ô 6 6 ö 6 ö 6 6 ô Ö d H- ^ ^ -1 -t *^ H=^ -1^ H o o o o o O 00 00 00 o 00 00 lO l.O LO 1> •o UO p— CO CO CO CO CO Cl IM M S^l IM M 'ti ^ O Ö !^ 2 CV. ^,d , r^ r^ ':s « a ^ « C ^ 0) 5 c2 Intensity 3h o .&• B* K 'C H CO &- O H ^ ci: H -* 00 'S' ■* 00 o ■;r CD ^ ^ CO OJ CO CD M 99 T? 4> t— ' m 9 !■) CO cp 00 o 00 T'-^ ,-H 1— 1 ÖO ö i^ CO t- ^ CO CO Cd IM IM r-H o o > r-i CO CO CO O CD o CO ^ ' Cl OJ (M <^J ÎJ IM (M N ^ ^ ^ r^ Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 49 Of tliese lines, the wave-lengths marked with an asterisk were measured for the first time. Most of these values are in good agreement with King's result. (32) Conclusion, (i) Among the iron hues investigated in different magnetic fields, 4404-75, 4383'55, 4307-92, 3886*29, 3878-78, 3859-90, 3856-38, 3815-84 3758-23 and 3749-47 are all sharp triplets and their separations AX are linearly proportional to the fields applied. Artliur King, using larger grating, also showed that they are all triplets, (ii) 4415-13 and 3825'90 are diffuse triplets, Arthur King took them for septuplets (?). In these lines dX is proportional to H when the fields are compara- tively low, and when the fields become higher, -^ become larger than that in lower fields. 4325*78 and 3820*44 have similar features to these two, although Arthur King thinks they are triplets. 3737*13 is also a line which King took for a septuplet and which I observed as a diffuse triplet, as for example 4415*13 etc. The data are too few in higher fields to compare it with 4415*13 etc., but JX varies so irregularly that it is difficult to consider -^ as constant, (iii) 3734-86 is taken for a septuplet by King, but in my photography it is a sharp triplet and the mean value of -^^ X 10^^= 13' 12 within 2% deviation from the mean, (iv) 3827*83 and 3763-80 are triplets and King is of the same opinion; but —^ is not constant and these two lines have similar features, (v) King reported 3719-93 to be a triplet (?), but in jnj photo- graphs this line seems to be a sextet (?); AX can not be said to vary linearly proportional to H. (vi) The lines from No. (20) to (30) are not reported by King. According to my investigation, AX does not vary linearly proportional to the fields in the case of diffuse triplets and their features are similar to those which are discussed in (ii). (vii) 2746*98 is a line whose violet side com- ponent is fainter in intensity than the red side one. (viii) 3047*60 and 2756-31 are probably lines whose separations were studied for the first time. 50 Art. 7. — K. Yaruada V. The Magnetic Separations of Nickel Lines. The wave-lengths are given in international nnit, aiul the intensities of lines when they are not affected are taken from Kayser's Spectroscopie. (1) /1: 3619-39. Intensity 15; sharp triplet; character b. The fourth order of this line was photographed on the same plate with the third order of a zinc line -^: 4680*138 for field determination. Plate I, Fig. 3; Plate III, Fig. 4. The results obtained are: Table XXXIII. AÀ ^^ X106 Deviaiion from the Mean H HX. x^O" Deviation Mean Value 18080 0-244±0-0019 13-50 10-30 + 0-18 + 1-8 19090 0-257 ±0-00 11 13-45 10-27 + 0-15 + 1-5 22380 0-301±0-0011 13-45 10-27 + 015 + 1-5 23100 0-306±0-0013 13-24 10-11 -0-01 -0-1 26440 0-347db0-0011 1313 10-02 010 -1*0 27750 0-363d=0-0022 13-08 9-98 -0-14 -1-4 27920 0-370dr00008 13-26 10-13 + 0-01 + 01 28700 0-378±0-00l7 1317 10-05 -0-07 -0-7 29200 0-386±0-0009 13-22 10-09 -0-03 -0-3 29400 0-385±0-0026 13-10 1000 -0-12 -1-2 30080 0-398dr0-0013 13-23 1010 -0-02 -0-2 32100 0-426rb0-0010 13-26 1013 + 0-01 + 0-1 32240 0-423dt0-00l7 1314 10-03 -0-09 -0-9 33100 0-439±00014 13-28 10-14 + 0-02 + 0-2 Mean 10-12 AX -g- is constant, as is easily seen from the table (Plate XVII, Fig. 1). Tlie results of the former investigators are: ' Investigator Character of Separaiion H AX ^•-XlO. H^X'»" Graftdijk Reese Kent Triplet 26230 28300 32800 0-393 0-363 0-390 14-98 12-83 11-89 11-44 9-79 9-08 (2) >i: 3566-37. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 5^ Intensity 10; sharp triplet; character b. Plate I, Fig. 3; Plate III. Fig. 4. The determination of the fields is the same as in the case of />>: 3619-39. Table XXXIV. ^^ XlOn i^x^o" Deviation from the Mean H AX ^XIO- Deviation Mean Value 18080 0251d=0-0012 13-98 10-91 + 0-99 + 10-0 19090 0-255dr00011 13-36 10-50 + 0-58 + 5-9 22380 0-282±0-0016 12-60 9-91 -0-01 - 0-1 23100 0-290±0-0032 12-56 9-88 -0-04 - 0-4 26440 0-331±0-0022 12-51 9-84 -0-08 - 0-9 27750 0-339dr0-0014 1222 9-60 -0-32 - 3-2 27920 0-350-b0-0009 12-54 9-87 -0-05 - 0-5 28700 0-347 ±0-0030 12-10 9-52 -0-40 - 4-0 29200 0-364±0 0012 12-34 9-70 -0-22 - 2-2 29400 0-362±:0-0019 12-31 9-68 -0-24 - 2-4 30080 0-374dr0-0029 12-43 9-78 -0-14 _ 1-4 32100 0-402±0'0022 12-53 9-86 -0-06 - 0-6 32240 0-408±0-0026 12-67 9-96 + 0-04 + 0-4 33100 0-413rfc0-0024 12-48 9-81 -0-11 - 1-1 Mean 9-92 AX H may be supposed to be constant when H is greater than 22380; but when it is smaller than this, -^ is larger than that in stronger fields. Plate XVII, Fig. 2. The results of the former investigations are: Investigator Character of Separation H AX ^^ X106 id ^XIOIB Graftdijk Reese Kent Triplet 26230 28300 32800 0-356 0-338 0-318 13-57 11-95 9-54 10-67 9-40 7-50 (3) /1: 3524-53. Intensity 15; somewhat diffuse triplet; character b. Plate I, Fig. 2 and Plate I, Fig. 3; Plate III, Fig. 2; Plate IV, Fig. 1. The determination of the fields is the same as in the case of >'^: 3619-39. 52 Art. 7. — K. Yauiada : Tarlj: XXXV. Deviation from the Mean M >J.X106 ^XIO« H to X'»" Deviation * Mean Value 18080 0-262±0-0009 14-49 11-65 + 0-62 + 5-6 19090 0-269±0-0010 14-09 11-32 + 0-29 + 2-6 20340 0-285dr0-0009 14-02 11-27 + 0-24 + 2.2 20600 0-289dr0-0011 14-03 11-28 + 0-25 + 2-3 22380 0-311±:0-0011 13-88 11-17 + 0-14 + 1-3 23100 0-321 ±0-0006 13-90 11-17 + 0 14 + 1-3 *23580 0-328±0-0006 13-90 11-17 + 0-14 + 1-3 25800 0-346±0-0010 13-40 10-77 -0-26 -2-4 26210 0-355drO-0008 13-55 10-90 -0-13 -1-2 26440 0-363d=0-0011 13-73 11-05 + 002 + 0-2 27750 0-377±0-0014 13-59 10-94 -009 -0-8 27920 0-3S4±0-0010 13-76 10-06 + 0-03 + 0-3 27940 0-383±0-0008 13-71 11-03 0-00 0-0 28700 0-386±0-0008 13-45 10-82 -0-21 -1-9 29200 0-393±;0-0009 13-45 10-82 -0-21 -1-9 29400 0-398dr0-0010 13-47 1083 -0-20 -1-8 29600 0-398±0-0010 13-44 10-83 0-21 -1-9 30000 0-409±0-0013 13-63 10-96 -0-07 -0-6 30080 0-411±0-0006 13-67 10-99 -0-04 -0-4 30100 0-410±0-0010 13-63 10-96 -0-07 -0-6 30350 0-413±0-0016 13-63 10-96 -0-07 -0-6 31100 0-424±0-0013 13-65 10-98 -0-05 -0-5 31320 0-430 ±0-0008 13-73 11-01 -0 02 -0-2 31720 0-433±0-0012 13-66 1099 -0-04 -0-4 32100 0-441±00011 13-75 11-05 + 0-02 + 0-2 32240 0-437±0-0016 13-56 10-91 -0-12 -l-l 33100 0-449±0-0015 13-58 10-93 -0-10 -0-9 Ml* an 11-03 In weaker fields the variation of J}, is more or less irregular, but in strong fields it is regular and JX is proportional to H. Plate XVII, Fig. •^. The results of formier investigations at single fields are: Investigator Character of Separation Triplet H M -^Vxl06 -,t,^,X10l3 Graftdijk Keese Kent 26230 28300 32800 0-409 0-391 0-410 15-60 12-55 13-82 11-12 12-.50 10-05 (4) ;(: 3515-06. Intensity 10; very sharp triplet; character b. Plate I, Fig. 2 Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 53 and Plate I, Fig. 3. The determination of the fields is the same as in the case of X: 3524*53. Tajîle XXXVI. ^ X106 A^ X1013 HX2 ■ Deviation from the Mean H AX H^^XIO» Deviation Mean Value 18080 0-226+00002 12-50 10-12 + 0-45 + 4-7 19090 0-234±0-00l7 12-26 9-91 + 0-24 + 2-5 20340 0-245±00011 12-05 9-75 + 008 + 0-8 20600 0-245±00008 11-89 9-62 -0-05 -0-5 22380 0-268±00005 11-98 9-70 + 0-03 + 0-3 23100 0-279±0-0007 12-08 9-78 + 0-12 + 1-2 *23580 0'276± 0-0004 11-71 9-48 -0-19 -2-0 25800 0-305dzO-0006 11-82 9-57 -0-10 -10 26210 0-311±00008 11-86 9-60 -0-07 -0-7 27750 0-327±0-0012 11-78 9-54 -0-13 -1-3 27920 0-337dzO-0010 1208 9-77 + 0-10 + 1-0 27940 0-333 ±0-0009 11-92 9-65 -0-02 -0-2 28700 0-337+0-0015 11-74 9-50 -0-17 -1-8 29200 0-348dt00010 11-92 9-65 ^0-02 -0-2 29400 0-353±0-0020 12-01 9-72 + 005 + 0-5 29600 0-356±00008 12-02 9-73 + 0-06 + 0-6 30000 0-359±00014 11-97 9-69 + 0-02 + 0-2 30080 0-358rt00010 11-90 9-63 -0-04 -0-4 30100 0-357d=0-0009 11-86 9-60 -0-07 -0-7 31320 0-373±00021 11-91 9-64 - 0-03 -0-3 31720 0-380±00012 11-99 9-70 + 0-03 + 0-3 32100 0-379±0-0015 11-80 9-55 - 0-12 -1-2 32240 0-385±00021 11-93 9-66 -001 -0-1 33100 0-390±0-0023 11-78 9-54 - 0-13 -1-3 Mean 9-67 AX H XVIII. are: is nearly constant as in all cases of sharp triplets. Plate Fig. 1. The results of already published investigations Investigator Character of Separation H AX ^ X106 JlI ^xioia Graftdijk Eeese Kent Triplet 26230 28300 32800 0-346 0-324 0-339 13-19 11-45 10-33 10-68 9-26 8-36 (5) /1: 3492-96. 54 Art. 7. — K. Yauiada : Intensity 10; diffuse triplet; character b. Plate I, Fig. 2 and Plate I, Fig. 3; Plate III, Fig. 2; Plate IV, Fig. 1. The determination of the fields is the same as in the case of '^^:3524'53. Takle XXXVII. Deviation from the Mean AX ^^ XIOH 11 ai >"»'" H ^'^ xioi^t Deviation Mean Value 18080 0-218dr00021 12-05 9-88 + 0-29 + 30 20340 0-233±0-0010 11-47 9-40 -0-19 -2-0 20600 0-230±:0-0015 1116 915 -0-44 -4-6 22380 0-263dr0-00l7 11-75 9-63 + 004 + 0-4 23100 ■ 0-276db00014 11-95 9-80 + 0-21 + 2-2 *23580 0-270±0-0016 11-45 9-39 -0-20 -2-1 25800 0-297±00007 11-51 9-43 -0-16 -1-7 26210 0-301±00015 11-48 9-41 -0-18 -1-9 26440 0-312±00023 11-79 9-67 + 008 + 0-8 27750 0-32BdrO-0049 11-82 9-69 + 0-10 + 1-0 27920 0-334rfc0-0018 11-97 9-82 + 0-23 + 2-4 27940 0-328dr0-0015 11-74 9-63 + 0-04 + 0-4 28700 0-339dr0.0026 11-81 9-68 + 0-09 +0-9 29200 0-345dr0-0014 11-82 9-69 + 010 + 1-0 29400 0-344d=0-0019 11-70 959 0-00 0-0 29600 0-339dt00011 11-45 9-39 -0-20 -2-1 30000 0-358dr00016 11-93 9-79 + 0-20 + 2-1 30100 0-347±0-0012 11-53 9-45 -0-14 -1-5 31320 0-369d=0-0012 11-78 9-66 + 007 + 0-7 31720 0-372rbO-0026 11-73 9-62 + 003 + 0-3 32100 0-380dr00023 11-84 9-71 + 012 + 1-3 32240 0-380±0-0028 11-78 9-66 + 0-07 + 0-7 33100 0-382 ±0-0021 11-54 9-46 - 013 -1-4 Mean 9-59 AX can not be said to be proportional to H. Plate XV III, Fig. 2. The results of former investigations are: Investigator Character of Separation. H AX H X^^" ni xioi'^ Reese Kent Triplet 28300 32800 0-301 0-311 10-64 9-47 8-72 7-77 (G) /l:34(')l(;)5. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Diffei-ent Fields. 55 Intensity 20: very sharp triplet; character b. Plate I, Fig. 2 and Plate I, Fig. :J; Plate III, Fig. 2; Plate IV, Fig. 1. The determination of the fields is the same as in tlie case of l\ 3619*39. Taj'.le XXXVIII. Deviation from the Mean AX 4^X10. ^xio. H ^^" xiois Deviation H>.-' ^ " Mean Value 18080 0-257±000l7 14-21 11-85 + 0-50 + 4-4 19090 0-262±0-0014 13-72 11-45 + 0-10 + 0-9 20340 0-277rfc0-0009 13-62 11-36 + 0-01 + 0-1 20600 0-281±0-0008 13-64 11-38 + 0-03 + 0-3 22380 0-304±0-0009 13-58 11-33 -0-02 -0-2 23100 0-318±0-0015 13-77 11-48 + 013 + 1-1 25800 0-349±0-0009 13-53 11-28 -0-07 -0-6 26210 0-355±0-0007 13-54 11-29 -0-06 -0-5 26440 0-351± 0-0011 13-28 11-08 -0-27 -2-4 27750 0-37ldrO-0019 13-37 11-15 -0-20 -1-8 27920 0-381rt0-0008 13-65 11-38 + 0-03 + 0-3 27940 0-382±0-0007 13-68 11-42 + 0-07 + 0-6 28700 0-385drO-0016 13-42 11-19 -0-16 -1-4 >9200 0-397±0-0014 13-60 11-34 -0-01 -0-1 :9400 0-392d=0-0014 13-34 11-12 -0-23 -2-0 ^9600 0-404±0-0008 13-67 11-40 + 0-05 + 0-4 30000 0-406±0-0015 13-53 11-28 -0-07 -0-6 30080 0-410±0-0011 13-63 11-37 + 0-02 + 0-2 30100 0-411±0-0011 13-65 11-38 + 0-03 + 0-3 30350 0-417±0'0019 13-57 11-32 -0-03 -0-3 31320 0-424d=0-0012 13-53 11-28 -0-07 -06 32100 0-44G±0-0016 13-71 11-44 + 0-09 + 0-8 32240 0-441d=0-00l7 13-68 11-42 + 0-07 + 0-6 33100 0-453±:0-0019 13-70 11-44 + 0-09 + 0-8 Mean 11-35 Al is linearly proportional to the fields applied as in all other cases of sharp triplets. Plate XVIII, Fig. 3. The results of former investigators are: Investigator Character of Separation H A>. •^ X106 ^XlO« Uraftdijk Reese Kent Triplet 26230 28300 32800 0-386 0-377 0-397 14-71 13-32 12-10 12-28 11-11 10-10 (7) /Î: 3458-45. 56 Art. 7. — K. Yainada; Intensity 10; diffuse triplet; character h. Plate I, Fig. 2; Plate III, Fig. 2. The determination of the fields is the same as in the case of >^: 3619-39. Tatilf. XXXIX. A). T«»' ^.XIOIB Deviation freui the Mean H Deviatifin Mean Vaiue . 20340 0-194dr0-0013 9-54 7-98 -0-74 -8-5 22380 0-235±0-0028 10-48 8-77 + 0-05 +0-6 25800 0-27l±0-00]3 10-48 8-77 + 0-05 + 0-6 26210 0-27'4±0-0009 10-45 8-74 + 0-02 + 0-2 27940 0-29.5±0-0013 10-65 8-91 + 0-19 + 2-2 28700 0-289±0-0020 10-16 8-43 -0-29 -3-3 29200 0-311±0-0015 10-63 8-89 + 0-17 + 1-9 29600 0'313±0-0014 10-58 8-86 + 0-14 + 1-6 30000 . 0-318=t0-00I3 10-60 8-86 + 0-14 + 1-6 30080 0-322drO-0027 10-45 8-74 + 0-02 + 0-2 30100 0-318±0-0014 10-58 8-86 + 0-14 + 1-6 33100 0-350±0-0025 10-56 8-84 + 0-12 + 1-4 Mean 8-72 AX Deviations from the mean are too large to take -g^ for a constant; hut similar features are seen in the cases of diffuse triplets (Plate XIX, Fig. 1). The results of former investigators are: Investigator Character of Separaction H AX A) ^xio.. Graftdijk Reese Triplet 26230 28300 0-30 0-292 11-44 10-32 9-6 8-63 (8) /: 3446-27. Intensity 9; chffuse triplet, especialty the central component is hroad; character h. Plate I, Fig. 2; Plate III, Fig. 2. The determination of tlie fields is the same as in tlie case of -^.:3524'53. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 57 Tai'.le XL. A). T^'O- l^x-" Deviation from the Meau H ^^ vlQlS Deviation • HX2 '^^" Mean Value 20340 0-247±0'0012 12-15 10-24 -0-09 -0-9 20600 0-247 ±0-0011 11-97 10-09 -0-24 -2-3 *24700 0-309dr0-0015 12 52 10-55 + 0-22 + 2-1 25800 0-322dz0-0013 12-48 10-52 + 0-19 + 1-8 26210 0-321dr0-0010 12-25 10-33 000 0-0 27940 0-347±0-0006 12-41 10-45 + 0-12 + 1-3 29600 0-358±0-0007 12-11 10-20 -0-13 -1-3 30000 0-367±0-0012 12-22 10-30 -0-03 -0-3 30100 0-369±0-0012 12-23 10-31 -0-02 -0-2 Mean 10-33 JÀ in nearl}^ proportional to H. Plate XIX, Fig. 3. The results of forDier investi2;ations are: Investigator Character of Separation H Aa H XIO» ^x'°" ^ Graftdijk Eeese Kent Triplet 26230 28300 32800 0-350 0-350 0-350 13-34 12-37 10-67 11-24 10-43 8-99 (t:)) /: 3414-82. Intensity 10; sharp triplet; character b. Plate I, Fig. 2; Plate III, Fig. 3. The determination of the fields is the same as in the case of ^:3524"53. 58 Art. 7. — K. Yamada : 'Iaiîlk XLL AX -^A-xio6 H ^'^ xiois Deviation from the Mean H «i xio.» D(»viation Meaft. Vahie 20340 0-272±0-0010 13-38 11-48 + 0-10 + 09 20600 0-273drO-0007 13-26 11-38 0-00 0-0 *24700 0-331±0-0014 13-40 11-50 + 0-12 + 1-1 25800 0-339rb0-0005 1313 11-27 -0-11 -1-0 26210 0-344dr0-0004 13-18 11-30 -0-08 -0-7 27940 0-372drO-0005 13-32 11-43 + 005 + 0-4 29600 0-391±0-0005 13-21 11-34 -0-04 -0-4 30000 0-397±0-0010 13-23 11-36 -0-02 -0-2 30100 0-399dr0-0004 13-26 11-38 0 00 0-0 30350 0-402±0-0007 13-23 11-36 -0-02 -0-2 31320 0-414±:00006 13-22 11-35 —0-03 -0-3 Mean 11-38 The variation of JA is proportional to tlie fields applied as in all other cases of sharp triplets. Plate XIX, Fig. 2. The results of former investigators are: Investigator Character of Separation H AX ^; X106 Hi ><'»" Graftdijk Triplet 26230 0-39 14-88 12-76 Keese „ 28300 0-374 13-22 11-34 Kent " 32800 0-406 12-37 10-61 (10) /i::]31)2-')7. Intensity 7; triplet, Imt each component is somewhat wide; character^. Plate I, Fig. 2; Plate III, Fig. 3. The determina- tion of the fields is the same as in the case of À: 3619'39. Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 59 Tahle XLII. A), ^XlOB ^xio^. Deviation from the Mean H m» ><">" Deviation Mean N'alue 20340 0-295±0-0020 14-53 12-62 + 0-46 + 3-8 20600 0-294±0-0009 14-27 12-40 +0-24 + 2-0 25800 0-358dr0-0012 13-88 1205 -0-11 -0-9 26210 0-365dr0-0008 13-92 1210 -006 -0-5 27940 0-387zt0-0009 13-85 12-03 -013 -1-1 29600 0-410dz0-0012 13-85 12-03 0-13 -1-1 30000 0-418±00016 13-93 1211 -0-05 -0-4 30100 0-418±0-0008 13-88 12-05 -Oil -0-9 30350 0-426rt0-0023 14-03 12-19 + 003 + 0-2 31320 0-432dr0-0011 13-80 11-99 -0-17 -1-4 Mean 12-16 In weak fields, AX is larger than tliat in the stroni^er. Results of former investigators are Investigator Character of Separation H AX ^^^ X106 ni x'°" Graftdijk Kent Triplet 26230 32800 0-399 0-399 15-21 j 13-21 12-16 i 10-56 (11) /: 3380-58. Intensity 6; very sharp triplet, this is the sharpest line among the nickel lines which I have observed; character b. Plate I, Fig. 2; Plate III, Fig. 3. The determination of the fields is the same as in the case of ^^: 3619*39. '60 Art. 7. — K. Yauiada ; Table XLIII. Deviation fr im the Mean A), f-XlOO à >="'" H ^^ XlOi:' Deviation Mean Value 20600 0-223dr00009 10-82 9-47 -018 -1-9 25800 0-286±00010 1108 9-69 + 0-04 + 0-4 26210 0-288±0-0008 10-99 9-61 -004 -0-4 27940 0-311±00013 1113 973 + 008 + 0-8 29600 0-328±00012 11-08 9-69 +004 + 0-4 30100 0334±0-0009 1109 9-70 + 005 + 0-5 31320 0-345±00015 1102 9-63 -0-02 -0-2 Mean 1102 9-65 JA is linearl}^ proportional to the field applied, as in all other cases of sharp triplets. Plate XX, Fig. 1. Graftdijk is the only one wlio has studied this line; she obtained: Character of Separation H AX ^l X106 11 m. «»" Triplet 26230 0-315 12-00 10-49 (12) '<:3ü5U-«2. Intensity 6; diffuse triplet; character b. Plate I, Fig. 4. The third order spectrum of this line is photographed on the same plate together with the second order of '^:4680"138 of zinc. Table XLT\^ H AX ^XIO. 25150 25920 26420 28050 0-233 0-242 0-244 0-260 9-28 9-36 9-24 9-27 9-97 10-05 9-93 9-95 Mean 9-29 9-93 Miii^-netic Separations <>f the Lines of Iron, Nickel and Zinc in Different Fields. {j\ .» The data are too few for a discussion of tlie character. Ac- cording to (Ti'aftdijk. Character of Separation H A>, '^'- X106 M ai xi°" Triplet 26230 0-272 10-37 11-14 (l."l) Separations at single fields. Table XLV. Wave- j jr lenüfth ê Character of Separation H A>, - XIC m^ >"'''' â-x-" (Graft- dijk) 3858-29 8 Triple 25020 ±0-401 // 0-000 1603 000 10-77 000 11-29 000 3807-14 8 Triple 25020 ±0-209 // 0000 11-96 000 8-26 000 16-07 0 00 3510-33 10 Triple 23580 ±0-143 // 0-000 608 000 4-61 0-00 6-5 0-0 3483-78 4 Triple 23580 ±0-217 /■ 0000 918 0-00 7-57 000 7-60 0-00 3472-53 5 Triple 24700 ±0-222 // 0-000 8-97 0-00 7 46 0-00 1413 000 3452-88 r, Triple 24700 ±0-357 // 0000 14-46 0-00 12-14 0-00 12-50 0-00 3437-29 5 Triple 24700 ±0-331 /^ 0-000 13-40 0 00 11-35 000 12-68 000 3433-58 6 Quadruple ? 24700 ±0-235 // ... 9 52 8-07 12-49 4-98 3423-71 5 Unaffected ' 24700 + 6-44 + 0-00 339104 4 Triple 24700 ±0-354 // 0-000 14-31 000 12-69 0-00 12-60 0-00 3380-58 6 Triple 24700 ±0-296 // 0-000 11-57 0-00 10-12 0-00 10.51 0-00 3369-57 4 Triple 24700 ±0-310 // 0-000 12-55 000 1104 0-00 11-91 0-00 •62 Art. 7. — K. Yamacla (14) (i) Among all nickel lines which I have studied in différent magnetic fields, U:3619'39, 3515'UG, 3461-65, 3414-82 and 3380-58 are sharp triplets and their separations are pro- portional to the fields applied, as in the case of all sharp triplets of iron lines, (ii) Ikit the separations of the sharp triplet 3566-37 are linearly proportional to the fields applied at stronger fields, while -g- is larger in lower fields than in the higher, (iii) -^-^i 3524-53, 3492-1)6, 3458'45 and 33U2-97 are diffuse triplets and their separa- tions are not proportional to the fields, as we have already seen in the iron difïuse triplets, (iv) But 3446-27 is the only diffuse triplet whose separations are nearly proportional to the fields. VI. The magnetic Separation of a Zinc Line. /: 3345-13. >^: 3345*13 of zinc is a line of the first subordinate series of zinc, and its magnetic separation, has already been studied by ]\liller'^ wlio obtained the result AX X 10'= 11 -27, A>. xlO^'=10-ll jj " xv^ XX _. , JJJ2 In the course of my investigation of iron and nickel lines in different magnetic fields, this line is also photographed on the plates, and I think it may be of interest to report the results of measurement. The fourth order of this line was photographed and the fields were measured with the aid of >^: 4680-138 (3rd order) on the same plate. Plate I, Fig. 2. Taj'.li: XLVI. H AX -M X'°' i^-'»'' 20340 0-251±00016 12-34 11-04 20600 0-255dr00011 12-37 11-06 25800 0-323rt00004 12-52 1119 26210 0-329±0-0019 12-55 11-22 27940 0-351dr00023 12-56 11-23 29600 0-372±:00015 12-57 11-23 .30100 0-385rb00012 12-79 11-43 30356 0-384rb00022 12-65 1131 31320 0-394±0 0021 12-58 11-24 Mean 12-54 11-22 1) Miller, Ann. d. Phys. (4), 24 (1907), pp. 105—136. Magnetic Separations of tlie Lines of Iron, Nickel and Zinc in Different Fields. 63 The discrepanc}^ of the value obtained by Miller from that here given is about 10%, but the simultaneous and accurate determination of magnetic fields leaves no room for doul^t as to the value here given. (Plate XX, Fig. 2). Summary. The object of the experiment was to investigate whether the separations of iron and nickel lines are proportional to the fields applied. For measuring the strength of the fields, the zinc line -i: 4680*138 was adopted, as the separation of the line has been accurately determined by man}^ observers. In groups "g", " h " , ' i " and " j " , this line ivas simultaneously photographed on the same plate with iron and nichel lines. The simultaneous photography, however, limited the number of lines to be studied, consequently I was obliged to photograph this line l^efore and after the photography of iron lines (groups " e " and " f "). Concave grating was used for obtaining spectra of the lines from the second to the fifth order. The sharp triplets show separations of their outer two com- ponents proportional to the fields applied ; but in the case of most of the diffuse triplets, these separations are not proportional to the fields; in some diffuse lines, -^ is constant in weak fields but becomes larger in the stronger; in other diffuse lines, the devia- tions from the mean are too large to assume -g- as constant. Arthur King observed these lines with larger grating and took these diffuse triplets for more complicated separated lines. In the course of investigation, the range of spectra photo- graphed was taken as wide as possible at single fields. The results thus obtained are compared with the published data. -^:3345'13 of zinc is also measured in different fields. The separation of this line is linearly proportional to the fields. (34 Art. 7.— K. Yaniatla : In conclusion 1 express my sincere thanks to Professor Nagaoka, at whose suggestion I undertook this investigation and by whose kind guidance I have been able to carry out the experiment. Published Februaiv 28th, 1921 J Jour. sa. Coll.. Vol. XLI., Art. 1 , PI. I. Fig. 1. (Group g) H =34120 Fig. 2. (Group h) H = 30100 ^ ^ d. ? _• q &ib U^ t Fi^'. 3. (Gl'oup i) H -23100 0) (-'• î^ f& 1-7 CO ts -t< ut 32 M Fl g. 4. (Group j) H -=36420 C*, et -T O O OO o K. Ya.ma(.la. Magnetic Separations of cho Lines of Iron, Nickel and Zinc in Differrent Fields. 1 Jour, Sei. Coll,, Vol. KU., Art, 7, PI, II, Pig. 1. (Group f) H = 19300 »» o w b- :û ift X X X i^ t, t- QÇ X x X X X 5^ X OS (C -^ Ôi i ôi — i- X X X CD CI CI X XX X XXX (m2) H = 24560 O ■* tN ^ Fig. 3. (m 6) H = 25Ö40 Fig. 4. (ni 7) H = 23580 CC -r- ^ CC t- M — • X CO tC iQ OS fi*1 X W I - o o !zi g. 2 'Z X ec cc" ■XI cc ÇÏ p X CC X »a t- ic Tt4 w ID m o t« c^ Ô fC c^ 1 S la CO '(f* 1 CO X CO cc — cc X X Ï I- eft t}< c: -* - wr cc c^ ic 'Jr <: tc la- -^ -j- -f • Fig. 5. IUI 8) H = 24700 cccccocc CCÇC cceccccccc -c o ê t* cc t^ ^ :: 4; Ç ^ ^ ■c fc ec CO cc ec « -r t~ (M — 0 C-. •M — ■= OÎ a X ;r .i* rj^ ^ ÇC cc cc cc 1 ■ 1 1 1 1 1 1 1 1 — — ' O OS C4 c« e^ 34 S. Tamada. Magnetic Separations of the Lines of Iron. Nickel and Zinc in Ditîerrent Fields. Jour. Sei. Coll.. Vol. XLi.. Art. 7 PI. III. Flg. 1. (Enlargonipiit of Pl;ite 1. F..,'. t 1 Fig. 2. (En1iu-|,'emint of I'liUo I. Pig. 2.) 1^ Fig. 3, (Enliirgeni. nt of Plate I, Fj. II Fis;. +. (Euliirg.-im.nt ..f Plato I. Pic. 3.) pr- »V I II K. Yaijiaclii. M.iuneti»: Separations of the- Lines of Iron, Nickel and Ziue iu Diffirrent Fields, Fig. 1. (EnlargeuHut i.f PlaU- I. Fi^. 3 ) Jour. Soi. Coll., Vol. XLI.. Art. 7, PI. IV. I Fig. 2. (Enlargement of Plate I. Fig. i.) II f f Fig. 3. I Enlavgemeut of Plate I, Fig. -1.) Fig. i. (Enlargement of Plate II, Fig. 1.) I Fig. 5. (Eulargeui.nt of Plate II, Fig. l.| K. YiiuiaUa. Magui-tic Separations of the Lines of Iron. Niekel and Zinc in Ditterrent Fields. Fig. 1. (Enlargonient m1 FhUo II, Fig. 2.) ^01//-. Sei. Coll., Vol. XU., Art. 7, PI. V. rr7 f'n'twctrs:^: Fig. 2. (Enlargement of PliUe II, Fig. 2.) 1 ! f 1. Fig. 3. (EnUi-gement of Plate II, Fig. I» ' I :i n Fig. 4, (Unlargement ..f Plate II. Fig. 3.1 f ' f I Fig. 5. (Enlarg.iiient of Plate II. Fi. rjM i^imm 'I i. i 8 i 1 Î - ïi 1 i . i l K. YamaJa. Magnetic S. parafions of tte Lin. s of Iron, Sie2il anil Zmo in Difterront Fiel.ls, J'ii,'. 1. ;EnIar,r,,„i.-.it .,f I'liik- II. Fi>;. -l ) Jour. Sei. Coll.. Vol. KLI.. Art. 7, PI. VI. Fij,'. 2. iKulu-gcui.'lU uf riati- II, FiK'. 4.) F,j;. 3. {Eulnrg.'iii.'Ut ..f I'liitc II, ¥, Fis,'. 4. lEnlarf,'oui.-iit uf PUto II, Fig, 5.) I IC. Yii.im.la. Masjnctic Separations of tlic I/incs of Iron, Xi.-kcl iin.l Zinc In DiftV,rr„nt Fields. Jour. Sei. Coll., Vol. XLI., Art. 7, PI. VII. X : 4415-13 (Fe) 0-000 -»H X : 4404-75 (Fe) 0-800 0-700 0-600 0-500 0-400 0-300 0-200 0-100 0000 1 1 \ 1 1 \ 1 r — I r- — I 1 — 10000 20000 30000 -H Fig. 2. K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei. Coll., Vol. XU., Art. 7, PI. VIII. 0-800 0-700 ' 0-600 0-500 0-400 0-300 0-200 0-100 0-000 k : 4383-55 (Fe) \ \ \ \ \ \ \ \ \ 10000 20000 30000 -H Fig. 1. À : 4325-78 0-600 0-500 0-400 0-300 0-200 0-100 0-000 10000 20000 30000 Fig. 2. K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei. Coll., Vol. KU,, Art. 7, PI. IX. 0-700 0-600 0-500 0-400 0-300 0-200 0-100 0-000 À : 4307-92 (Fe) 10000 20000 Fig. 1. 30000 ->H ; : 3886-29 (Fe) AX 0-600 0-500 0-400 0-300 0-200 0-100 0-000 10000 20000 30000^ H Fig. 2. K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei. Coll., Vol. XU., Art. 7, PI. X. ; : 3878-78 (Fe) 0000 30000 -> H X : 3859-90 (Fe) 0-600 0-500 0-400 0-300 0-200 0-100 0-000 ^--^ 10000 20000 30000-^ H Fig. 2. K. Yamacla : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei. Coll., Vol. XLL, Art. 7, PI. XI. À : 3856-38 (Fe) AX I 0-600 0-500 0-400 0-300 0-200 0-100 0000 30000-^ H 0-500 0-400 0-300 0-200 0-100 0-000 X : 3827-83 (Fe) 10000 20000 Fig. 2. 30000^ H K. Yamacla -. Magnetic- Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei. Coll., Vol. KU., Art. 7, PI. XII. 0-500 0400 0-300 0-200 0.100 0000 0-500 0-400 0-300 0-200 0-100 0-000 0-500 0-400 0-300 0-200 0-100 0-000 X : 3825-90 (Fe) 10000 20000 Fig. 1. X : 3820-44 (Fe) 10000 20000 Fig. 2. k : 3815-84 (Fe) 10000 20000 Fig. 3. 30000 -> li. ,-V^ ^ -^ ^ J- ^ ^ ^- -^ ^ -^ 30000^ H 30000-^ H K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Différent Fields AX 0 ^ -^ ^ >-' ^--^ r^ ^ --- ^ 30000 -> H K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei., Coll. Vol. XU., Art. 7, PI. XIV. 0-500 0-400 0-300 0-200 0-100 0-000 X : 3737-13 (Fe) 30000-^ H 0-600 0-500 0-400 0-300 0-200 0-100 0-000 k : 3734-86 (Fe) it ^_^.^ I 1 1 I I I 1 i I l i I 10000 20000 Fis. 2. 30000 -> H K. Yamacla : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. 0-500 0-400 0-300 0-200 0-100 0-000 0-500 0-400 0-300 0-200 0-100 0-000 "^ Jour. Sei. Coll., Vol. XLI., Art. 7, PI. XV. /.: 3719-03 (Fe) 10000 20000 Fig. 1. /: 3618-77 (Fe) lUUUD 20000 Fis. 2. / : 3581-20 (Fe) 10000 20000 Fig. 3. 30000^ H 1 I 30000 -> H ...... — ^ 1 ! r^ ^ Ci — ^ 1 ^ --- -^ ^-^^ ^■-^ ^^ -^ "^ _-^ 30000 ->H K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Jour. Sei. Coll., Vol. XLI., Art. 7, PI. XVI. 0-500 0-400 0-300 0-200 0-100 0-000 0-500 0-400 0-3(-iCi 0-200 0-100 0-000 ; : 3570-12 (Fe) ^^^. ,^» 10000 Fis. 1. / : HH')-l.\n (Fe) luOOO 20000 300CI0 -> H 1 1 1 1 1 '■ 30000 -> H Fig. 2. 0-100 0000 ; :2/4C-WS (Fe) 30000 ->H K. Yiuuada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in DiÊferent FieJdß. Jour. Sei. Coll.. Vol. XL I., Art. 7, PI. XVII. A) 0-500 o-4o;t 0 300 0-203 0-100 0000 0-500 0-400 0-300 0-200 0-100 0-000 G-000 ;. :8ßl9-8!) (Ki) I 1 ^p--^" I 10000 20000 Fitj. ^. / : 856Ü 87 (Ni; 1 1 iij( lO 20i )( lO Fier. 2. ;. : 3524-58 ÇKi) 1000(1 20000 Flg. 8. 30000 ->H 30000 ]\. Yamuda : Magix-tic Separation of tl)e Lines of Iron, Nickel and Zinc in Difi'erent Fields. Jour. Sei. Coll., Vol. XLI., Art. 7, PI. XVIII. 04Ck:i OoÜO 0-200 0100 0000 / : 3515-06 (Ni) ^a-'î'^ ^^^ ^ __^ _^ ^ '— IdOOO 20000 Fig. 1. ?,U000 ■H 0-400 0-300 0 200 0-100 0-000 or.oo 0 400 0-300 0-200 0100 0-000 ?. : 3492-96 (Ni) 10000 20000 Flg. 2. k : 3461-65 (Ni) 10000 20000 Flg. 3. 30000 ^H ^ _^ -^ "^ ^ ^ l^ 30000 H K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. Al t Û-4O0 0-300 0-LMO 0-100 0000 Jour Sci. Coll., Vol. XLI., Art. 7. PI. XlX. À:S-l^S-li) ilsï} f ^ i 1 \ 1 I . ^h-'--^ lOUOO ' lOUO Fia. l 30000 0000 /:8-4:l4-.S-2 (Ni) -^0000 *-H 0-400 0-300 0-200 0-100 0-000 / : 344Ö-27 (Ni) J ' -^ ..— » ' \r- 1 ^^ ^ " ^^ ^^ -^ "^ 1 loOOO 20000 Flg. 8. 30 JOO -> H K. Yamacla : Magnetic Separations of the Lines of [ron, Nickel and Zinc in Different Fields. Aï. 0-4C0 Ü-3C0 0-200 0-100 0-000 Jour. sa. Coll., Vol. XLL, Art. 7, PI. XX. À : SaSO-ôS (Ni) 10000 20000 Fis. 1. 30000 H À : 3345-13 (Zn) Ü5C0 0-400 0-300 0-200 0-100 0-000 10000 20000 Flg. 2. 30000 K. Yamada : Magnetic Separations of the Lines of Iron, Nickel and Zinc in Different Fields. .TOriiNAL OF ■I'llK COLLllGK OV S IKNCE. ■L'uKVO IMPERIAL rMVKHSlTY. VOL. XLI., ARTICLE 8. Magnetic Separations of Iron Lines in Different Fields. Ey Yutaka TAKAHASHI, Uiyakushi. With 13 Plates. Introduction. As a recent development of the atomic tîieoiy, various hypotheses have been proposed to illustrate tlie atomic structure from the point of view as revealed by radiation. In spite of the divergence of the various theories, they all agree in attributing the origin of the spectrum lines to the moving electrons in the atom. Prof. Nagaoka has suggested that, if such an atom is placed in a magnetic field, the mutual influence between the electrons may give rise to the separation of some spectrum line, which is not proportional to the magnetic field. In some series lines with complex structure many in- vestigators^^ have observed anomalous Zeeman separations, and it is known as a general ruie that each component, showing its own Zeeman separation in weak fields, either disappears or unites with others when the field is increased, and as a whole tends- to the normal triplet in a sufficiently strong field ; consecjuently the separation is not generally proportional to the field. But to 1) Pa!scli< n u. Back, Ann. d. Physik, 39 ;i912) p. 897 : 40 (1913) p. 960. Fortrat, C. R., 156 (1913; p. 1607 ; 157 (1913) p. 636. Na;aokaand Takamine, Proc. Tokyo Math.-Phys. Soc, [2], 7 (1913) p. 188; (1914) p. 331 ; Phil. Ma?., 27 (1914 p. 333 ; 29 (1915) p. 241. Wood and Kimura, Astrophys. Jour. 46 (1917) p. 197. 2 Art.S.— Y. Taktihashi : make exact measurements of such lines seems to be very difficult. As for tlie single lines, lleese'' and Kent^^ found that the separations of // 4680, 4722 and 4810 of zinc are not proportional to the ^ekh hut Cotton and Weiss'^* showed that })ropo]-tionality holds good for these lines. The deviations from the proportion- ality given by Reese and Kent for other lines are much smaller than those for the above lines. Several investigators showed that magnetic separations of single lines are in general nearly proportional to the field. 8o far. however, as the present writer is aware, those lines for wiiich the separations are accurately measured over a wide range of magnetic fields are very few, so that it requires more extended experiments covering a wide range of magnetic fields as well as numerous spectrum lines to ascertain whether a linear relation between the sej^aration and the magnetic field exists or not, wliicli seems to be very interesting inasmuch as it may throw some light on the structure of the atom. The following investigation was undertaken with this object in view. For this purpose, elements rich in lines are convenient, as the chance of coupling among the electrons may be large in such an atom, for it is not probable that these lines are emitted from separate atoms eacli l\y itself; indeed they give man,y Zeeman triplets having divergent values of separations,"^ which might be the result of coupling. Iron, manganese, calcium and titanium were examined, but the experiments with the latter three elements being yet incomplete, only tlie results obtained with the first are given in the present report. Besides the fact that iron is rich in lines which give divergent values oï magnetic separations, its ferromagnetic property — though we do not know the property at the temperature of the spark — may show some special character in its magnetic spectrum, 1) Reese, Astrophys. Jour., 12 (1900) p. 120. 2) Kent, Astrophys. Jour., 13 (1901) p. 289. 3) Cotton et Weiss, Jour, de Phys., [4], 6 (1907) p. 429. 4) Zeeman, Magnetooptics, (1913) p. 157. ]\I:i.gut;tic Stparptions of Irou Lines in DiflVreut F:t.Lls. 3 and il coinparisoii with other para-ur diaiiiagnetic elements, when sufficient data are obtained, will he interesting. Source of Light. The soui-ce of hght used was the spark discharge between 25 percent nickel-steel terminals excited b}- an induction coil, the primary circuit being fed with G amperes of alternating cun-ent of 110 volts, 50 cycles, obtained from the city main. 4 Leyden jars were connected parallel to the spark gap directly between the poles of the induction coil to obtain a condensed discharge. An adjustable inductance and auxiliary spark gap with small capacity were inserted in the spark circuit in series; the increase of the inductance, giving the spark the character of an arc, made the spectrum lines sharper and put out the air lines, with, however, a considerable sacrifice in the intensity of the iron lines also; the auxiliary spark gap served to control the spark under examination in a favoural^le state for various magnetic fields. It is desirable to ifive the spark sonorous character with steady greenish ap- pearance. When the magnetic field was not strong the spark continued in a favourable state for a long time, but in a strong field tlie terminals soon became dark red and the spark formed an arc of violet tint, making the iron lines weaker and increasing the luminosity of the continuous spectrum in the back ground, — this was especially remarkable in tlie green region — , so that it was necessar3" to polish the tips of the terminals every 5 or 10 minutes. The terminals, on being polished, I'ecovered their ferromagnetic property, and it was very troublesome to place them in pi'oper position between the poles of the strong electromagnet, which was, however, overcome l^y heating the polished terminals Avith a Bunsen burner. Small cylinders about 1 mm. thick attached to brass reds were used as terminals for most uf the experiments, and wedge shaped ones were used with the strongest fields to prevent the current from pa-sing through the poles of the magnet. 4 Art. 8.— Y. Takahaslii : During the whole course of the experiment, it was desirable to regulate the length of the spark gap, which was adjustetl by clamping the terminals to the poles provided with an arrangement like a spark micrometer, b}^ which the distance between the sparking terminals was changed so as to obtain a spark best suited for the exferiment. Electromagnet. The electromagnet used was so constructed that the cores with the coils can be displaced along their common axis ami rotated as a whole about the vertical axis, which enabled us to bring the middle of the magnetic field on the line of collimation of the optical system, keeping this line and the magnetic lines of force at right angle with each other. A current of I'o to 23 amperes from the secondary battery was used according to the magnetic field desired, and its constancy was carefully observed by means of a small adjustable resistance and an ammeter by Siemens & Halske. Conical pole pieces ending in circular sections were used during the whole course of the experiments, the diameter of the faces and the air gap between them were 2 cm. and 1 cm. respectively for weak fields, and 0*3 cm. and 0'12 cm. for the strongest ones, the diameter and the gap being changed between these limits. For the purpose of examining the character of the magnetic separation, it was desirable to extend the range of the magnetic field as wide as possible, but it was difiicult to apply a field stronger than 37230 gauss with tlie electromagnet in our laboratory for the spark gap giving a source of light of consideralde intensity, the diameter of the faces of the pole pieces and the air gap between them being limited as m-entioned above, for further approach of the pole faces caused short-circuit of the sparking current through them, and further diminution of the diameter ZVIa^netic Separntious of Ii\)ii Lines in Different Fields. 5 miglit liave (iisturbed tlio uniformity of the magnetic field in the s]-)ark. An electromagnet witli water-coohng arrangement and ft-rro-eohalt pole pieces would have been more suitable for experiments in stronger fields. Echelon Grating. The echelon grating by Hilger, frequently used by Prof. Nagaoka" in tJie study of tlie structure of mercury lines and other works, was used m the present researcli also. Tiie con-tant-; of tliis instrument are as follows : — Thickness of plate 9"350 mm., Number of plates 35, Steps I'O mm.^ Length 32-73 cm., A=T5ô5055, B = 5-9595x]0% C = l-95l4xlOi-, where. A, B, C are the constants in Cauchy's formula fen- the index of refraction <« = -^ + ^ir- + -^7-' o / being expressed in A.U. The wave number intervals corresponding to the distance of two ^uccessi^'e orders were calculated from the usual formula okmar 1 o for the wave lengths of constant intervals of 50 A.U., and their values are 2;iven in Table I. 1) Nac^aoka and Takamine, Memoir of the Iaip3rial Acxiemy, S->c. II, Vol. I., Xo. 1, 1913. Alt. S.— Y. Takaliashj Tahli- t. xxio^ X'-' 1-4739 >,X10^ 4550 X- 1-6103 XX 10^ 5X,i„ X 3800 5300 1-G948 3850 1--4853 4600 1-6172 5350 1-6992 3900 1-1963 4650 1-6239 5400 1-7035 3950 1-5070 4700 1-6304 5450 1-7077 4000 1-5172 4750 1-6367 5500 1-7 U7 4050 1-5272 4800 1-6428 5550 1-7157 4100 1-5368 4850 1-6487 5600 1-7195 4150 . 1-5460 4900 1-6545 5650 1-7233 4200 1-5551 4950 1-6600 5700 1-7269 4250 1-5637 5000 1-6655 5750 1-7305 4300 1-5722 5050 1-6707 5800 1-7339 4350 1-5803 5100 1-6758 5850 1-7373 4400 1-5882 5150 1-6808 5900 1-7406 4450 1-5958 5200 1-6856 5950 1-7438 4500 1-6031 5250 1-6002 6000 1-7469 Actual v^alues of oh for iron lines were obtained by interpo- lation, and, if desired we can get the wave lengtli intervals by multiplying hy )^\ tbis is unnecessary, as it is more rational to express the Zeeman separation in cbange of frequenc^T^ tban in wave length. Tbe cbief advantage of using tiie echelon spcctri_>scope consists in its higb resolving power, tbe great intensity of bglit and the comparatively large vahie of -^ ^^ '' . Tbe first allows us to observe fine separations in weak fields, tbe second is convenient especially in investigations witb strong fields, and tbe last is important in tbe investigation of spark spectra, for tbe lines in such spectra bave considerable widtlis. l''a])ry-Perot"s mterfero meter and hummer Gebrcke's ])!ate satisfy tbe first and the second, but not tbe tbird condition. Kven witb tbe ecbelon gratmg - — ^^ — is not sufiiciently large to enable us to measure the separation in some fields wben one component falls npon or close by the otber. Complex separations giving moie tban foui" Magnotic Separations of Irou Lines iu Different Fields. 7 component-; willi tlie ^^anio polarization were not measnred in different tields. Tliungli the appearance of the components belonging to the next order is troul)lesome in the measurement of complex separations, it lias some advantage. Irregular contraction of the gelatine film of a photograpliic plate may cause some displacement of the silver deposit, and tiie error coming from tliis may bo proportional to the separation, giving a large errur for a strong tield. With echelon spectra we need not mea-ure a separation larger than — ~—' but may simply add ^he multiple of this quantity to the measured fraction, and can tlius get an accurate vahie for the separation, provided that Jy^^ is calculated witli sufficient accuracy. ^ Apparatus and Method of the Experinnent. The spai'k placed between the magnet poles was focused l)y a lens on the horizontal slit s, of the echelon spectroscope, and the image formed by the echelon was projected on the vertical slit So of a Hilger constant deviation spectroscope Ijy means of a Eudolph's planar of 5 cm. focus. The telescope of this spectroscope Avas removed and its place was taken l:)y a }>hotographic camera with an objective of 05 cm. focu-:. The rougli sketch is given in Fig. 1. Fio-. 1. C Oe I induction coil C I^eyJen jars L adjustable inductance S spark under examination S' anxjliary spark gap yi electromao-net Si horizontal slit E echelon irratin»- A Kudclph's planar W Wollaston prism ?2 vertical slit K constant deviation prism P photographic camera Art. 8.— Y. Takabashi : When the horizontal sht was opened wide only the prism spectrum was observed at tlie camera, and then the sht was slowly closed until the vertical lines contracted to dots of echelon spectra. Thus many lines were photographed at a juxtaposition, admitting exact comparison of the magnetic separations among these lines. Proper adjustment of the inclination and position of the photographic plate and the position of the planar Itehind the echelon spectro- scope made it possible to photograph green lines in one and the same exposure with violet lines, if we allowed the disturbing of the sharpness of the prism specti'um, but the difference in the intensity and the photographic sensitivity made it convenient to separate the exposure into several steps, each lasting 10 to 3()0 minutes according to the magnetic held and the lines to be photo- graphed. In some |)lates I have photographed the echelon spectra of the whole region in one focus with disturbed prism spectrum, and in other plates with sliarp prism spectrum, changing the posi- tion of the planar from exposure to exposure. For the larger part of the experiment a WoUaston prism AV Avas inserted between the planar and the vertical slit, by which it was possible to photograph both the parallel and perpendicular Zeeinan components at once. For some exposures the Wollaston prism was removed and a nicol was placed in fr(.»nt of the horizontal slit. Some photos were taken without separating the polarized components to see if any dissym- metry of the resolution exists, l)ut without any conclusive result. The others were photographed without the planar to shorten the time of exposure at the cost of small magnihcation. In order to make .the apparatus free from mechanical and thermal disturbances, the whole arrangement was placed in a cellar, and the l»ox containing the echelon grating was protected by cork plates. On tine days, dry air was allowed to enter the room through tlie windows; then shutting tlie windows and the door tightly, the temperature in the room was kej)t as constant as possible, and the image of the echelon spi'etva cHd not suffer any sensible disturbance even if tlie exposure was eontinued during the whole day. In order to eliminate errors arising from irregular contraction Magnetic Separations of Inm Lines iu Different Fields. 9 of the gelatine film of the photographic plate and other sources, photos were taken repeatedly, increasing and decreasing the magnetic field. The photographic plates mostly used were the panchromatic and process plates hy Wratten cfe Wainwriglit. Wratten douhle instantaneous and Ilford procès plates were used occasionally for the stronger and the weaker fields respectively. Result. In the earlier course of the experiment it was my chief object to study the behavior of the nine strong lines in the violet region //I 4415-13, 4404-75, 4383-55, 4325-78, 4307-92, 427175, 407r75, 4063*6], 4045"82'^ over a wide range of magnetic fields. Three of these lines are arranged in a particular spacing, thus. 1 1 |, the distance between the second and the third lines being twice as large as that between the first and the second. Another aim of the investigation was to observe the se]:)arations of the principal lines in the green region, as former results in this region showed some discrepancies, which may have been due partly to experi- mental errors and partly to the difference of the magnetic fields. Tn the course of the experiment it seemed to me important to study tlie separations of weak lines, for a weaker line may l»e afïected more than, a stronger b}^ mutual action, though we do not know the mechanism of radiation in the atom. Thus the resolu- tions of the weak lines, so far as observed with the present instru- ment, are also here given. Many weak lines lost their intensity with the increase of the magnetic field and became very diffuse, some showing complicated resolutions. Especially in the green region the increase of the intensity of the back ground disturbed the echelon spectra so much that they appeared to melt into it, — this phenomenon is somewhat due to the character of these lines, whose components are diffuse or complex and one falling close l)y 1) The wave lengths of these lines and the others criven in the present report were first identified in the map of the iron spectrum l:>y Buisson and Fabry in " Recueil de Constantes Physiques," and then the exact numbers were taken from the international in Kayser's „Handbuch der Spectroscopic." 10 • -^it- 8.— Y. Takahashi : tlie otliei" in tlio increased field — , and the separations of tliese lines were not measnred for strong fields. As 4404"75 came out most sharply in many plates, its separa- tion was taken as the standard, and tlie separations of tlie other lines were compared with it, without any anticipation of the absolute value of the magnetic field. The magnetic fields given in the following were calculated from the separation of the above line assuming the specific separation -'^' = 1-0G4X10-*. /-H which was borrowed from the result of Mr. YannuhV^ who found that the separation is proportional to the magnetic field by com- paring the separation of this line with that of tlie zinc line 4ß''^0. As my observation extends both in weak and strong fields beyond the limits in his experiment, it may be objected that the calcula- tion of the field for these portions is an extrapolation, but the linear relation between the magnetic field and tlie separations of many sharp lines shows that the separation of 4404 "75 is propor- tional to the maiiiietic fieLl. /. Nine Strong Linc^ in the Violet R(yio/i. ;> 44I5'!3, at the end of the first three lines towaid the red, appears as a triplet in weak fields. The intensity of this fine is comparable with that of 44(-)4'75 in the spark spectrum witbout or with a weak magnetic fichb but, when the field is increased, all the three components become diffuse and weak, so tliat it is difiicult to stud}' the behavior fully in strong fields with tlie eclielon spectroscope. On AVratten double instantaneous ])Iates pho- tographed with fields of .'îoBôU and o72oO gauss 1 liave ol »served its fine resolution, giving at least 3/)-an!, tliongh* van Bilderbeek-van Meurs'^ and Graftdijk'^ found this to bo triple even in a field as strong as :j2040 ganss. Tajjli: II/' / = 44l.5"18, 8^>-and 4??-comps. ov more. H ^: X10-. wx- H ^.? ..0. ex- f4730 (outer) ^3780 (inner) 1-270 = 5x0-254 7690 754 0-980 1-015 = 4x0-254 37230 2) 980 0-263 7580 807 1-064 n (3660 (inner) 1-027=4x0-257 6880 725 1053 35650 ji I 928 0-261 6850 764 1-115 20100 2280 1-134 9 6820 716 1 -050 18^90 2104 1-126 >, 715 1049 18070 2072 1147 6770 693 1-023 10980 1251 1-140 6750 700 1037 10510 1155 1-099 6670 715 1071 10200 1039 1-019 6500 649 ö-:)9S 9130 979 1-071 6450 678 1051 9060 964 1-064 6800 664 1-054 8830 884 1001 5910 666 1-12S ■ 8800 878 0-998 5670 592 1044 8580 940 1-095 5590 572 1023 8410 935 1-111 5260 552 1-050 8350 870 1-042 5000 556 1-112 8230 873 1-061 4730 1 484 1-022 8180 855 1-045 4650 502 1-080 7810 810 1-038 1350 474 1-090 7740 792 1-023 1 4110 447 1-015 1) Van Biklerl5eek-van Meuss. Arch. Néerl., (2), 15 (1911) p. 353. 2) Oraftdijk, Arch. Néerl., (3), 2 (1912) p. 192. 3) In this and the following- tables the separation is for the pair of components .sym- metrically situated about the initial line. In the case of an ordinary triplet no remark with regard to polarization is given. 12 Art. 8.— Y. Takahashi : Tap.lk III.'> Comparison with former results. H ^'- XIO' X-'H Takaliashi H^ Cl 1000 1-05G van Bilclerbeek-van Meurs 32040 1-06 Graftdijk .. 1095 King (7 comps. ? 1 16000 1-084 Hartijumn 1-048 111 Table II. and Fig. 2 it can be seen that the specific separation measured as a triplet increases with the magnetic field, which ma}'' be attributed to the complicated separation. / 4404*75, the middle hue of the first group, is resolved into a sharp triplet. The separation of this line, being measured most accurately, is taken as the standard of the separation, so that neither table nor diagram of the separation is given for this line. But the linear relation Ijetween the separations of the principal lines in the following and the magnetic field — separation of 4404*75 — shows that the separation of this line is exactly proportional to the magnetic field. According to Mr. "^^iinada, the separation of this line is given l)y J/ ni = 1-064x10-*. As the absolute magnitude of the specific separation is not the chief object of the present research, I have calculated tlie magnetic field Irom the separation of this line by means of tlie above value for the specific separation, williout any absolute determination of the magnetic field. 1) The magnetic fields applied by former investigators are not clear, as the separations of sv)Uie lines, being measured at different fields, ai-e reduced to correspond to their standard fields under the assumption tliat the separation is proportional to the magnetic field. But, as the difference l^etween the stiintlard and the observed fields may probably not be large, the standard fields are here quoted. As Hai-tmanns original paper is inaccessible, his result is taken from Lütig's paper in Ann. (1. Physik, 38 (1912) p. 43; the ma<4netie field is therefore unknown. The luean specific separation is here olitained fr».)U> AX \^ AX /v Matj^netio Soparations of Iron Liues in Different Fields. 13 >> 4383*55, at the ond of the tir^t group toward .the violet, is resolved into a sharp triplet. The result is given in Table I\'. and Fig. 3. Table IV. ;.= 4383-55, triple. H 37230 36330 35650 34830 3t490 259G0 25350 25140 24870 24350 24040 23840 23600 23160 22660 22090 21940 21360 20880 20760 20550 19860 19510 18860 18800 18690 18540 18070 11050 10640 10600 10580 10330 10180 9730 9rt20 9590 9420 9130 AX X2 xio; 3982 3870 3830 3814 3696 3669 2781 2745 2688 2692 2639 2580 2560 2541 2498 2451 2355 2366 2303 2246 2229 2208 2147 2134 2032 2019 2030 1990 1960 1170 1143 1143 1127 1130 1108 1046 1005 1011 992 978 A>.-xioi \m H 1070 9010 1-068 8590 1-074 8580 1-070 8520 1-061 8410 1064 8350 1-071 8270 1-083 8260 1-069 8250 1-082 8180 1-084 8040 1-073 7830 1-074 7810 1-077 7740 1-079 7690 1-082 7580 1-066 7170 1-079 7150 1-078 6850 1-075 6820 1-074 6770 1-074 6700 1-081 >> 1-094 6670 1-077 6560 1-074 6470 1-08Ö 6300 1-073 5910 1-085 5890 1-059 5670 1-075 5330 1-079 5250 1-0Ö5 5000 1-093 4730 1-080 4650 1-075 4590 1-045 4520 1-055 . 4350 1-053 4020 1-071 ?9X) M XIO: ■km XJO' 965 1-071 919 1-069 913 1-064 900 1-055 910 1-081 895 1071 880 1064 884 1070 888 1-076 875 1-069 887 1-079 827 1-056 838 1-073 815 . 1-052 830 1079 849 1120 795 1-109 774 1-081 730 1-066 729 1-069 754 1-114 710 1060 704 1051 716 1-074 703 1-071 675 1-043 674 1-070 623 1-054 637 1-081 601 1-060 575 1-078 562 1-070 562 1-124 480 1-014 515 1-108 493 1074 492 1088 507 1-165 467 1161 463 1-187 u Art. 8.— Y. Tiiktihasbi : Tahlk V. Comparison with former results. H 'J'akaliashi mean van Bll« Kini;- 16000 'H;i,rtnianii >,-'H XlO' 1-075 1-110 1-095 1-078 0-998 It can be observed that the specific separation is a httle hirger at about 250U0 gauss and a little smaller in the stroiigest fields tlian tlie mean value, but the discrepancy is within the limits of experimental errors, and we may consider the separation of this line to be exactly proportional to tlie magnetic field within the range between 3900 and :)72:)0 gauss. / 4325*78, at the end of the second group toward the red, appears as a sharp triplet in weak fields and becomes diffuse witli the increase of field. Though tlie broadening of the ??-components is not so remarkable, the jp-component becomes so broad in strong fiekls that we may expect some fine resolution of this component. This resolution was not actually observed, but the distance between the two successive apparent maxima, wlien the echelon grating was adjusted in its double order position for this com- ponent, was much shortened and tlie decrement in the field of 36.330 gauss was nearly IdXmax corresponding to the separation of o 0*008 A.U. ; moreover tliis component was almost resolved m a field or 37230 gauss. Judged from appearance, this line seems to be si)lit into a quintuplet (or sej^tuplet) with 3/^-and 2 (or 4) 7i- components. The result obtained with this line is given in Table VI. and Fig. 4. Magnetic Separations of Iran Lines in Dift'orcnt Fields. Tahli: vi. / = 4825'78, op-, 2n-comps. ? H — -XlO: X2 ^Vxio* >.'-'fr H --^^XIO^ >i XIO' 37230 3500 0-940 8750 806 0-921 36330 3384 0-931 8580 746 0-870 34830 3230 0-927 8410 706 0-840 3-4490 3279 0-951 8350 749 0-896 33440 3142 0 940 8270 708 0-856 25960 2330 0898 8260 706 0-855 24870 2232 0-897 8250 690 0-836 24040 2130 ; 0-886 8180 684 0-836 11120 990 0-890 8040 720 0895 11050 1 970 0-877 7810 675 0-865 10980 962 0 876 7740 707 0-914 10660 944 0-885 7710 651 0845 10640 944 0-887 7170 594 0-828 10630 i 942 i 0-886 6850 616 0 899 10600 956 0 902 6810 578 0-849 10580 924 0 873 6300 573 0-842 10510 899 : 0-855 6770 616 0-910 10330 922 0892 6750 583 0-864 10200 : 836 0 820 6700 580 0-866 10180 876 0860 .. 571 0-852 9900 881 0-890 6300 542 0-860 9760 863 0-884 5910 530 0-897 9620 i 812 ; 0-844 5890 486 0.825 9610 812 0-845 5670 488 0-861 9590 828 0-863 5260 457 0-869 9420 754 0-800 5230 445 0-850 9280 765 0824 4650 398 0-855 9250 781 0-844 4610 378 0820 9130 : 810 0-887 4590 389 0-846 9010 746 0-828 4520 349 0-772 8830 787 0-891 4020 386 0-960 8800 \ 778 [ 0-884 3900 304 0-779 Art. 8.— Y. Taka.hashl : Table VII. Comparison with former results. Takahashi van Bildeibcfk-ran Meurs <îraïtxiijk King HartUiaun H>-11120 32040 16000 K-H ;10' (I 865 0-873 0-874 0-817 0-862 As may be seen in the above table and in Fig. 4, tiie separation is proportional to the magnetic field below 11000 gauss, showing good agreement with Hartmann' s result. Though King gives a considerably smaller value of the specific separation, the result of van Bilderbeek-van Meurs, of Graftdijk and of the writer show that the specific separation is larger in the stronger field. The separation of the //-components in the strongest field obtained l)y the writer is equal to that of the normal triplet, -though the measurement is not accurate. The character of this line somewhat resembles that of 4415*13. >^' 4307-92, middle line of the second group, is resolved into a sharp triplet. Table VIIL /Î =4307-92, triple. 11 ^^XICS 3958 -^^ X104 X2H .'/ >^' X1Û3 . ^Vxio« . -X2H 37230 1-063 25350 2667 1-052 36330 3869 1065 25140 2599 1034 35650 3744 1050 24870 2630 1-057 » 3729 1-046 24350 2558 1-051 34830 3640 1-045 24040 2536 1-055 34490 3605 1-045 23840 2510 1-053 26230 2742 1-045 23600 2467 1-045 25060 2690 1-036 23160 2143 1-055 4 Miiffnetio Separations of Iron Tjini-s in Diffei-ent Piolds. 17 H "^1 X 103 2410 - ^>' XlO. a2H 1064 H 8250 ■^} xipg X2 ^^-XIO* 22660 859 1-041 22220 2340 1-053 8180 867 1-060 22090 2308 V045 " 866 1-059 21960 2304 1-049 8040 S68 1-079 21940 2308 1-052 7830 844 1-077 21600 2261 1-047 7810 818 1-048 20880 2206 1-057 7740 818 1-056 20760 2173 1-047 7710 800 1-038 20550 2132 1-038 7690 839 1090 19860 2109 1-062 7580 813 1-072 19640 2060 1-049 7300 760 1-041 19510 2061 1-056 7170 758 1-057 18860 2020 1-071 7150 774 1-082 18800 2017 1073 6850 706 1030 18690 1983 1-061 6820 697 1-022 18540 1981 1-069 6800 695 1022 11120 1176 1-058 6770 732 1-081 11050 1161 1-051 6750 723 1-071 10640 1137 1-069 6700 709 1-058 10600 1094 1-032 .. 706 1-054 10580 1130 1-068 6670 700 1-050 10510 1114 1-060 6650 728 1-094 10330 1111 1-075 6560 670 1-021 10200 1088 1-066 6300 667 1059 10180 1083 1-064 " 655 1-040 9730 1019 1-047 5910 622 1-052 9620 1001 1-042 5890 620 1-052 9590 988 1-030 5670 592 1-044 9420 982 1-042 5590 584 1-044 9130 956 1-047 5260 565 1075 9010 965 1-070 5230 558 1-067 8830 907 1-027 5000 561 1-122 8800 924 1-050 4790 536 1-119 8590 897 1044 4730 477 1-007 8580 876 1-021 4650 492 1-058 8490 912 1-074 4610 469 1-018 8480 871 1-027 4590 485 1-055 8410 876 1-042 4520 470 1-040 8350 896 j 1-073 4350 447 1028 8270 866 1 1047 4020 442 1-099 8260 880 1-065 3900 452 1-159 18 Art. 8.— Y. Takahashi : Tatîlk IX. Comparison with former results II ^'' xio* X2H Takahashi mean 1-054. van Bilderljeek-van Meurs 32040 1-080 GrafUlijk .. 1-081 King 16000 1-078 Hartuianu 1-012 Table VIII. and Fig. 5 show that the separation is proportional to the magnetic field. ^ 4271*75, at the end of the second group toAvard the violet, is resolved into a sharp triplet, the specific separation being the largest of the nine lines. Tajjlk X. / = 4271-75, triple. H .~-:^Xl04 X2H H ^J- X103 ^^''- XlOt X2ff 35650 4173 1-171 19640 2268 1-155 34830 4065 1-167 19510 2252 1-154 34490 4030 1-168 18860 2188 1-160 34110 3973 1-165 18800 2160 1-149 33440 3910 1-169 18660 2192 1175 32800 3838 1-170 18540 2200 1-137 22660 2668 1-177 18070 2080 1151 22090 2593 1-174 18000 2076 1-153 21940 2602 1-186 17420 2049 1-176 21360 2480 1-161 17340 1994 1-150 20880 2418 1-158 16910 1980 1-171 20760 2396 1154 10640 1239 1-164 20550 2316 1-127 10630 1233 1-160 19860 2315 1-166 lOüOO 1249 1-178 Maii'iu'tic Separations ^ 4071*75, end hne of the third group toward the red, is resolved into a sharp triplet, the specific separation being the smallest of the nine lines. Table XII. / = 4071 -75, triple. H ^-XIO^ X2 , --^'-XlO: X2« H X3 ■^"^ XlOi 37230 2288 0-615 10580 666 0-629 34890 2156 0-618 10330 648 C-o27 34110 2120 0-622 9420 587 G-823 18860 1185 0-628 » 584 0-620 18000 1142 0-634 8260 521 0-631 17420 1072 0-615 804O 500 0-621 17340 1086 0-626 7160 468 0-654 16910 1061 0-627 6820 430 0-631 15120 935 0-618 .. 424 0-622 14770 909 0-615 6700 430 0-642 Taiîle XIII. Comparison with former results. H >^ XIO* X'-î / Takahashi mean 0-623 van Bildei-l>cek-vau Meurs 32040 0-645 King 16000 0-641 In Table XII. and Fig. 7 some systematic deviation of the spécifie separation may be observed, but it lies within the errors of experiment, and the separation may be considered proportional to the magnetic field. >^>4063 6I, middle line of the third group, is resolved into a sharp triplet, and the separation is proportional to the magnetic field as may be seen in Table XIV. and Fig 8. Magnetic Separations of Iron Lines in Different Fields 21 Taijlk XIV. / =4063-61. triple. H -■^-^X10 = -^'' -xio-t H ^^-XIO: X2 ^-^XIO* X2H 37230 3796 1-020 10640 1118 1051 34890 3538 1-014 10630 1090 1-025 26230 2659 1-014 10600 1098 1-035 25960 2588 0-997 10330 1046 1013 25350 2540 1-002 9420 945 1003 25140 2504 0-996 » 944 1002 24870 2520 1-013 8590 907 1-056 24350 2467 1-013 8260 849 1-028 24040 2480 1-032 .> 840 1-017 23840 240<3 1-Ü07 8250 825 1-000 23600 2380 1-008 8180 845 1-032 23160 2340 1-010 8040 821 1021 22660 2340 1-033 7300 763 1-045 22220 2286 1029 7170 762 1-063 22090 2256 1-021 6820 685 1-004 21960 2276 1-036 „ 678 0-994 21940 2244 1-023 6700 699 1-043 21600 2152 0-996 .. 698 1-042 21360 2148 1-006 6300 640 1-015 20880 2101 1-006 5890 589 1-00(3 20550 2084 1014 5670 545 0-961 20100 2048 1019 5590 576 1-030 19860 1990 1-002 5230 535 1-022 19660 1986 I'OIO 4590 436 0-949 18800 1950 1-037 4520 423 0-936 18540 1916 1033 3900 437 1-120 Taklk XV. Comparison with former results i ^' ^^ xio* X2ff Takahashi j mean 1-016 van Biklerbeek-vau Meurs 32040 1-037 Kiut^- 16000 1-017 22 Art. 8.— Y. Takahashi ^4045*82, end line of tlie third group toward tlie violet, is resolved into a sharp triplet, the separation being proportional to the magnetic field as may be seen in Table XVI. and Fig. '.). Table XVI. ;=4045-82, triple. H ^>- xio^ '^'^ xio* H ■^' X10Î A>, XIO* A-i- 34890 4015 1-151 8590 968 127 34110 3912 1-147 8-260 947 117 33440 3900 1-166 .. 929 124 32800 3770 1-149 82r,0 985 193 22090 2580 1-168 8180 029 135 21360 2500 1-170 8040 950 181 20880 2415 1-157 7300 838 148 20550 2339 1-138 7170 891 242 20100 2316 1-152 6820 767 125 19860 ' 2359 1-188 '. 766 124 19640 2299 1-171 6700 771 151 18800 2174 1-156 6300 719 141 18660 2179 1-168 5890 675 145 18540 2138 1-153 5670 636 1-22 18000 2052 1-140 5590 654 170 17420 2022 1-161 5230 612 170 17340 2008 • 1-158 4590 492 071 16910 1975 1-168 4520 481 064 10640 1228 ' 1-154 3900 486 246 9420 1088 1-154 Tahli: X\MI. Comparison with fonuer results AX xio* moan »'A Takahashi 1-156 van Bilderbuuk-van Meurs .■^2040 1-196 Kinc; 16000 1-138 Mas;uetic Soparatiüns of Iron Lines in Différent Fields. 23 //. OtJur less Stro)uj Lines. / 5615*661 is separated into a sharp triplet. ÏA1ÎLE XVIII. ;.==5()l5-6(il, triple. H ^■' xio- 944 ^^ xioi X2H 1-130 H AX ^^Q, X2 '^^ xio* X2H 8350 6300 1 721 1-144 8230 904 1-098 4730 534 1-128 7560 819 1-123 4240 478 1-127 6880 753 1-094 i Table XIX. Comparison with tbriiier results. H mean Takabashi Graftdijk ' 32040 King 16000 Hartmann A>. X104 1-130 1-170 1-161 1-052 King's value of the specific separation is a little larger than tile writer's, Graftdijk' s being still larger: the discrepancy maj^ be due to the difference of the magnetic i\Q\à applied. If so, the curve in Fig. lU must turn upwards, but, so far as my experiment goes, the separation is proportional to the magnetic field as shown in Table XVIII and Fig. 10. The a1)0ve discrepancy is probably due to expei'i mental errors. / 5586*772 appears as a diffuse triplet. 24 Art. 8.— Y. Takaliashi : Tajîle XX. « /=5586-77'2, triple. H -^' XIO* 8230 862 1-047 6300 677 1-074 4240 465 1-096 Tabek XXI. Comparison with former results. J^ ■^' X 101 >.-'H Takahashi mean 1-068 Graftdijk 32040 1-19 King (7 comics. ?) 16000 1-021 Hai-tuiann 0-901 As ma}' be seen in Table XX., XXI. and Fig. 11. my points lie witli King's on a straight line, which intersects the line of separation shghtly above tlie origin, but tlie deviation is toof small to assume that this is not due to experimental errors. Graftdijk gives a much larger specific separation. / 5572*86 appears as a diffuse trijjlet. Taiuj: XXII. / = 557-2-86, triple? y/ ^XIO' 8350 588 0-704 7560 541 0-716 4730 :;sl' i")-S07 Mui;u<.'tie Separations of Ivou Lines in DilYevent Fields. Taj5li: XX II I. Comparisoii with former results. 25 xioi Tibkahashi Kin'^- (7 couips. r ) mean 16000 0-732 0-945 The points by the writer lie on a straight hne as may be seeii in Fig. 12. King's value of the separation being innch larger. ?^ 5455*614 is separated into 2p- and o??-coniponents. TAI5LE XXIV. '^ / = 5455"()14, 'Ip-, 8?i-cümps. H AX X- XlO' AX X2H xio* n p n P 9690 1608 1-659 8350 1148 1145 1-374 1-371 8230 1160 1170 1-409 1-421 6880 910 932 1-322 1-355 6300 884 853 1-402 1-353 4730 724 638 1-.530 1-350 445 t 712 619 1-600 1-391 424 1 534 1-377 Table XXV. Comparison with former results H -^-'' xio* X2if 11 P Takahashi King- mean 16000 1-469 1-452 1-376 1-429 1) The .separation of the unter pair is i^-iven for the ?; -components. 26 Ari. 8. -Y. Takahaslü : 111 Tal)lL' XXI\\ and Fig. 13 it may 1)0 observed that the separation of tlie ^^-coniponents is proportional to the magnetic field, tliough that l)y King is a little larger. One half of the separation of tlie outer //-components — the separation l^etween the undisturbed and one of the displaced com])onents — is given in Fig. 14 to prevent confusion, which sliows that the separation curve is concave upward between 50()() and OlJOO gauss, but, if we take King's point into consideration, it must go down again, showing a wavy form. / 5446*92 appears as a quadi-uplet in weak fields. TJie n- ccmponents become very difïuse when the magnetic field is increased, which may be considered as the result of complex separation as observed by King. King giA'es 4 ^/-components in his table, but only two sharp p -components are observed on n\y plates, the specific separation being a little smaller between 5500 and 8000 gauss and a little larger above lOOOO gauss than King's for the outer pair, as may be seen in Table XXVI. and Fig. 14. Taiîle :$xvi. >?=.5446'92, 1p-, 271-comps. H ^'- xio: ^'"^ xio* H y? X2H p P p 545 P 10760 1109 1-030 6400 0-851 10510 1072 1-020 .. 531 0-830 9880 953 0-965 6300 561 0-890 >» 028 0-939 5520 510 0-924 9690 941 0-971 .. ' 1 484 0-877 9390 878 0-935 4940 446 0-901 » 870 0-926 4730 428 0-905 8860 778 0-878 4480 412 0-919 8350 773 0-926 4450 383 0-861 8230 775 0-941 » n 567 n 1-274 7560 682 0-903 4240 391 0-921 6880 561 0-815 4110 374 0-910 6690 573 0-856 " 361 0-879 MiiLiUctic Separatious ot Iron Lines iu Diti'ort'Ut FieWs. Taj'.li- XXVTI. Comparison with former results. 'Il Takahashi KlUs;- (8h-. 4;;)-3rtUips.) 0-942 0-476 / 5429*70 appears as a quadruplet in weak fields. ÏAJ1LE XXVIII. >?. = 5429-70, 2p-, 271-comps. H AX ^X2" XIO' AX xio* im // p n P 11630 10760 10510 9690 850 803 789 727 0-731 0-746 0-750 0750 9390 591 « 0-629 8860 558 . 0-630 8350 508 0-608 8230 520 0-631 7560 501 0-663 6880 774 429 1-125 0-624 6670 449 0-674 6400 425 0-664 6300 414 0-657 4940 520 1-052 4730 515 ' 1-089 4480 560 301 1-249 0-671 28 Art. 8.— Y. Takaliashi : Tabli: XXIX. Comparison witb former results XlO' Takahashi H<:9500 H>.9500 Graftdijk 32040 King (6 or 8 )i-, 4 j)-c-omps.? ) 16000 Uartniaun '/ßH ■^• P 0-744 1126 0-645 1-54 0-0 1-286 0-636 0'981 GO The separation of tlie //-components was not measured accurately owing to tlie broadening, hut the discrepancy among the above four results in Table XXIX seems to be too large, giving greater specific separation for the stronger field, to attribute it to an error of measurement. This may be due to the comphcated resolution, though Graftdijk has not observed any further resolution with a field of 32040 gauss. The separation of tlie j)-components increases suddenl}^ at about 9500 gauss, for each side of this field the observed points in Fig. 15 lie on a separate line which does not pass tlirough the origin. ?^ 5397*12 appears as a quadruplet. Tai'.li: XXX. /=:5397T2, 2p-, 2n-comp8. A>. xio- A>. XIO* H X2H n p ;/ P 11630 675 0-580 10760 588 0-546 10510 ' .621 0-590 9880 473 0-484 8230 1120 451 1-360 0-548 6880 888 1-210 4730 575 1-215 4450 601 1-361 Magnetic Separations of Iron Lines in Diffei-ent Fields. 29 Table XXXI. Comparison with former results. H AX xio* im n p Takaliushi mean 1-288 0-551 (■Jraitdijk 32040 • 1-46 00 Kin^■ (6 conips..'^) 16000 ] -SS'Z 0-476 The measurement is not quite accurate, but the separation of the M-components agrees with that of King. The p-components indicate a wavy fluctuation of the specific separation, though the data are too scanty for minute discussion. ^- 5371*495 is separated into a sharp triplet. Table ;> = 5371 XXXII. 495, triple. H ^'' XIO^ ).2 -A^ xioi V-'H H ^^ XlO^ X'-i ^^ XIO* 9880 898 0-909 6500 555 0-854 9690 885 0-913 6400 581 0-908 9610 903 0-940 G300 575 0-912 9390 787 0-838 5520 501 0 908 9350 825 0-891 5490 496 0-903 8860 723 0-816 5080 435 0-856 8410 66i 0-790 4940 396 0-801 8350 72* 0-866 4730 392 0-829 8230 708 0-860 4650 350 0-753 7830 649 0-829 4480 372 0-830 " 7690 649 0-844 4450 426 0-957 7560 607 0-803 4240 362 0-853 6880 576 0-838 4110 365 0-889 6690 553 0-826 3580 282 0-788 6670 517 [ 0-775 2480 222 0-895 6560 543 0-828 30 Art. 8.— Y. Takahashi : Tahlk XXXIIL Conjparison with former results. Takahashi CTvaftdijk (p-eouip. prob, decomi^osedj King i9 couips. ?) H in eau 32040 16000 ^' XIO* 0-855 0-80 0-890 Fig. 17 shows that the specific separation suddenly falls at a])Oiit 7000 gauss. / 5328*06 appears as a sharp triplet. Table XXXIV. x = 5328-06, triple. H ^^XIO'. ).2 "^ xio* xm H -"^/ XIO'^ '^"' xio* X^H 9880 ♦ 1093 1-106 6670 645 0-967 9830 1060 1-079 6650 700 1-052 9690 1009 1-040 6560 687 1-048 9610 994 1-034 6500 659 1013 9390 990 1-054 6450 Ö46 1-001 9280 1022 1-101 6400 698 1091 9250 989 1-069 6300 735 . 1-166 9060 876 0-968 5520 576 1043 8860 904 1021 5490 580 1-056 8790 994 1-130 5060 496 0-980 8480 936 1-103 4940 490 0-991 8410 869 • 1033 4790 511 1-065 8350 910 1090 4730 521 1-101 8230 86» 1-048 4480 496 1-105 7830 820 1-046 4i50 513 1-153 7690 811 1055 4240 i 428 1-010 7670 840 1-095 4110 460 1-120 7660 805 1-065 3920 415 1-059 7150 752 1-051 3580 352 0-983 6880 677 0-984 2950 280 0-949 6690 722 1-079 2480 282 1-137 Mag-îietic Séparations ot Iron Liueô in Doft'erent Fields. 31 Tahli: XXXV. Comparison with forim r i-e!>ults. Takahashi Graftdijk (j(>eouip. prob, decomposed) Kint;- i9 comps. ?) H mean 32040 16000 À-'H XIO* 1-057 1-03 1-03 i The separation is proportional to the magnetic field as may be seen in Fig. 18. / 5269*53 is separated into a sharp triplet. Table XXXVI. ;. = .52(59-53, triple. H '^'' X103 ),2 - ^^ xio* X2H H J^Lxio^ X2 _^'^ xio* X2ff 9880 1130 1-144 6670 677 1-015 ,. 1125 1-139 6650 698 -1050 9830 1104 1123 6540 ■ 781 1-194 9690 1093 1-128 6500 655 1-007 9610 1101 1-147 6450 706 1095 „ 1040 1-082 6400 700 1-093 9390 1057 1-126 6300 726 1-152 9280 1060 1-142 5680 676 1-190 9250 1033 1-117 5520 584 1-058 8860 980 1-106 5490 615 1-119 8790 991 1-128 5060 535 1-058 8490 9]L5 1-077 4960 545 1-099 8480 906 1-069 4790 517 1079 8410 916 1-089 4730 544 1-150 ,. 915 1-088 4650 447 0-961 8350 939 1-124 4480 505 1-127 8230 910 1-105 4450 514 1-155 7830 829 1-058 4240 458 1-080 7690 837 1-088 4110 475 1156 7670 826 1-077 3920 453 1155 7560 786 1-040 3580 388 1-084 7150 746 1-043 2950 342 1-159 6880 708 1-029 2480 316 1-274 6690 700 1-045 32 Art. 8. — Y. Takahîtshi : Tablk XXXVII. Comparison with former results. H ^ xio* Takahashi mean 1101 <>raftdijk (f-comp. prob, decoiuposed) 32040 1-166 King 16000 1-128 It may be observed in Table XXX VI. and Fig. 19 that the specific separation suddenly falls at about GoOO gauss, which niay be the result of the superposition of the famt component of /I 5270'oö on the measured one. This hne appears in the natural state as if it w^ere the satellite oi the line in question, and, in a magnetic held, one of the >i-components i^ masked by a con^- ponent of tlie line in question and the other y^-coraponent, which is expected to appear, is not found, thougli the p-component is clearty observed. These /i-components seem to be very faint in a weak field, but one of them can be observed in my photograms taken ^vith sufficiently long exposure applying a field near 10000 gauss. If we assume the result of King, the separation of this line is nearly one half that of 5269-53, and the red component of the former just overlaps with the violet component of the latter at 4800 gauss, so that the separation of the latter must appear too large below the field and too small above it. The first appearance of the fall at 6500 gauss seems to be too late to be considered merely as the effect of the superposed line, and it comes too suddenly compared with the slow recoveiy in the stronger field, the reverse being expected in the above case. It seems nece.ssary to study with an instrument of different constants to decide the l)ehavior of the line 5269*53. / 5232*957 appears as a triplet. Magnetic Separations of Iron Lines in Different Fields. 33 Table XXXVIII. ; = 5232-957, triple. H ^^ XIO' X2 >^^ -XIO* \m H A^- XIO« X^ >^ XIO* •km 8350 1046 1-252 6300 758 1-20» 8230 985 1-196 4730 528 iiiff^ 7560 859 ■ •1-136 4450 539 1-211' Table XXXIX. Comparison with former results. H ^'^ XIO* xm Takatashi mean 1-190 Graftdijk 32040 1-24 King (7 coinps. ?) 16000 1-158 Hartmann 1-106 Though the measurement is not accurate, it may be seen that the separation is approximately proportional to the field. ^ 5227*20 appears as a triplet. Table XL. ;. = 5227-20, triple. E A' XIO'- /2 ^^ XIO* xm H ^^ xio = X^ ^^ XIO* xm 10760 1117 1-038 6880 591 0-859 10510 1149 1-092 6690 595 0-889 9880 962 0-974 6400 576 0-900 „ 945 0-956 6300 621 0-986 9690 959 0-990 5520 562 1-019 9390 868 0-924 4940 465 0-941 9250 826 0-893 4730 470 0-993 9060 765 0-845 4480 394 0-879 8860 772 0-871 4450 425 0-955 8350 765 0-916 4240 368 0-867 8230 761 0-925 4110 398 0-969 7560 752 0-995 34 Art. 8,— Y. Takahashi; Tauli: XLI. Comparison with former result^ H xio* Takahaslii mean 0-947 Graftdijk (^;-coinp. decouipostnl) 32040 1-01 King 16000 0-946 Hartmann 1-202 As may be seen in Fig. 21, the separation curve is concave upward between 8000 and 10000 gauss. Thougli King's result coincides with the mean value of tlie writer, (Iraftdijk and Hartmann give larger values. / 5167*492 appears as a triplet. Table XLII. / = 51G7-4<)'2, triple. H ^''- XIO^ >2 ^' XlO^ )2H H ^"^ XIO' X2 Z^^' XlOt x2ir 10760 1119 1-040 6650 725 1-090 9880 1042 1-055 6450 725 1-124 » 997 1-009 6400 ! 675 1-054 9610 1006 1-047 6300 701 1-112 9390 940 ]-00Û 4940 504 1020 9060 906 1-000 4730 509 1-076 8350 849 1016 4480 480 1-070 8230 869 1-055 4450 531 1-192 7560 767 1-015 4240 457 1-077 6880 689 1-001 4110 457 1-112 Table XLIll. Comparison with former results. H -■^Vxio* X^H Takahashi mean 1050 Graftdijk (j,-couii). pruli. (=4957-62, triple. H ^^ XIO' X2 ■km H ^^ xio X2 ^^ XIO* ■km 9880 1071 1-085 6690 773 1155 9610 1027 1-069 6670 734 1100 9390 1022 1089 6650 714 1073 9280 975 1050 6500 677 1041 9250 999 1-080 6450 660 1-037 9060 930 1027 6400 727 1-135 8860 924 1-042 6300 716 1137 8410 893 1061 5520 645 1-169 8350 924 1-106 5490 610 1110 8230 943 1-145 5060 514 1-015 7830 878 1121 4730 526 1-111 7670 862 1-124 4550 456 1-001 7560 827 1-094 4480 490 1-091 6880 765 1-111 4450 506 1138 6750 757 1-121 4240 481 1-133 Table XLVII. Comparison with former results. H ^^ X104 \m Takahashi mean 1091 Grat'tdijk 32040 1-144 King 16000 1-441 Hartuiann 1-112 'J l.c ji] puent K] iuaiK 1 u v} It ;f'u'Ucl ly ll;e nti^l.l t i.rujg line 41^J57-31, lut tie lemli^ t,l Guiiulijk aid tlniin {iiu] fi^rec witli tl:eii.({iii value Icie (liaiiitcl, ileugb Kiug gives a niueli larger seiîaialion. MagTaetic Separations of Iron Lines in Different Fields. ol / 4920*52 appears as a triplet. Taiîle XLVIII. / =4920-52, triple. H AX X2 xio^ 9880 1148 9390 1019 9060 910 8860 913 8410 882 8350 889 8230 930 7690 787 7560 749 6750 67-2 A^ xio* 1-000 1038 1-049 1-070 1-032 1-058 0-94-2 1-041 1-030 Table XLIX. Comparison with former results. H - AV^XlO* Takabashi mean 1-047 «jiraftdijk 32040 1065 King (prob, complex) 16000 1154 The apparent separation may be affected by /4919'007, though the mean value here obtained agrees with Graftdijk's result. If we accept King's result, tlie specific separation in a field stronger than 9000 gauss is greater than that in a weaker field. À 4891*51 appears as a triplet. 38 Art. 8.— Y. 'J akahashi Table L. /=4891-51 triple. H .^^ Xl(ß j ^^ xioi a2H H ^; X10-. "^^ xio* 9880 877 0-888 6690 58-2 0-870 9390 892 ' 0-950 6670 618 0-926 9280 821 0-884 6400 618 0-966 8860 808 0-911 6300 568 0-902 8410 780 0-927 5520 540 0-978 8350 783 0-937 4240 392 0-924 8230 771 0-936 4110 392 0-954 7600 740 0-962 Tablk LI. Comparison with former results. H Takahashi moau 0-925 . «îrafMijk 32040 1-06 Kin<4- 16000 1-012 The apparent separation may ^e affected l)y the neighbouring line 4891-78. / 4583*83 is separated into a triplet. Tahlj: LI I. / =4588-88, triple. II ^^ xio^ 6450 >- 716 • 8230 954 1-159 1-110 6880 751 1-091 4450 523 1-175 6820 725 1-063 Magnetic Separations of Iron Linos in Différent Fields. 39 Taf.li: lui. Comparison witn former results. H ^^ xio* im Takahashi mean 1118 Graftdijk 32040 1-168 King- 1 6(300 1-118 The separation is approximately proportional to the magnetic field. / 4528*622 appears as a triplet. Table LIV. ;. = 4528-622, triple. H , ^^-X103 X2 - ^'^^xio* im H -"^^ X 10^ X2 ^^'^-X10< X2ff 9280 1 994 1-070 6540 742 1135 9060 , 924 1-020 6450 733 1-136 8860 ' 974 1-099 6400 747 1167 8410 ; 915 1-087 6300 761 1-208 8350 902 1-080 5590 683 1-221 8230 916 1-113 5520 643 , 1164 8180 , 880 1-075 4730 568 1-200 7690 857 1-115 4650 580 1-247 6880 765 1111 4480 510 1-138 Ü820 773 1134 4450 533 1-198 6750 813 1-204 4240 499 1-176 6670 715 1-072 4110 458 1-115 Table L\\ Comparison with former results. n ^ ^^ X104 im Takahashi mean 1-127 van Bilderbeek-van Meurs 32040 116 Graftdijk .. 1-213 Kin;^ (7 conips. ?) 16000 1-090 40 Art. 8.— Y. Takahashi : Fig. 28 sliovvs that the separation curve is convex upward between 4000 and 8000 gauss. If we accept the results of van Bilderbeek-van Meurs and Graftdijk, it must turn upwards at a still stronger field showing a wavy form, X 4494*572 appears as a triplet. Table LVI. /I = 4494-572, triple. H ^'l X103 >>■ XlO* y?H // >>• . X 10- ; X2 — xio* 9880 980 0-991 6450 663 1-028 9060 816 0-901 6300 725 1-150 8230 805 0-978 4730 545 1-151 8180 811 0-991 4650 546 1-175 7690 785 1020 4450 466 1-047 6880 682 0-991 '].240 466 1099 6750 695 1-030 4110 430 1-047 6540 714 1091 ! 1 Tap.lk LVI I. Comparison with f(;i-nier results. Takahashi Graftdijk Kin^ (7 couips. ?) in tan 3204<) 1 600) ^'' xio* 1032 1-28 0-935 The separation curve of this Hue (.]uite resemhlrs that of À 4528-fKÎ. À 4476'03 appeal's as a triplet. Miii^-netic Separations of Iron Lines in I>itYeront Fields. 41 Table LVIII. /= 4476-03, triple. H ^^xio-- >2 -^ XIO* im H ^XIOS a2 4^- XIO* X2fl 9880 891 • ' 0-902 6540 640 0-979 8230 818 0-994 64.50 661 1-025 8180 798 0-976 4450 435 0-973 6880 6i8 0-942 4250 375 0-882 Table LIX. Comparison with former results Takahashi Graftdijk Kino- (7 comps. ?) XlO* It may be seen in Fig. oO that the separation curve is sliglitly convex upward between 4000 and 10000 gauss, but the deviation is too small to warrant the drawing of any definite conclusion, and the separation is nearly proportional to the magnetic field / 4466'556 appears as a triplet. Table LX. /î=4466 5 56, triple. H ■ XIO'"- ^ - X 10* im n X2 -XlO» ^A xio* 9060 916 1-011 6450 748 1-160 8230 925 1-123 5490 593 1-080 8180 858 1-049 4650 497 1-069 7690 838 1-090 4450 536 1-204 6880 758 1-101 4240 516 1-216 6820 717 . 1051 4110 503 1-2-24 6750 734 1-087 42 Art. 8.— Y. Takahashi Table LXI. Comparison with former results. Takahashi Graftdijk King (6 couips. ?) H mean 32040 16000 ^'' XIO' A'-iH 1-101 1-189 l-OT-i As the points in Fig. ol are scattered, we can sa}^ nothing about this Hne except that the separation is approximately proportional to the magnetic field. ;. 4447*72 seems to l^e separated into a sextiiplet, but, the yi-compononts being too diffuse to be separated sufficiently, this line was measured as a quadruplet. Table LXII. /l=4447'72, 2^-, 4w-comps., measured as a quadruplet. H ^'- xio:= X2 ^^- xio* n P ■ n P 9690 993 1-024 8180 1890 798 2-308 0-976 7530 1584 2-102 6880 1205 637 1-751 , 0-926 6020 1108 1-840 4730 743 502 1-570 1-060 4-i.'')0 677 413 l-r)21 0-92i1 Taiîle LXIII. Comparison with former results. H Takahashi King (Q comps.) mean 16000 A-'H XIO* 1-907 2-280 1-419 0-985 0-971 Magnetic Separations of Iron Lines in Different FieMs. 43 Fig 32 shows that tlie separation of the j[?-components is approximately proportional to the niagnetic field, but that of the /i-components is concave upward. Judged from the broadening, further resolution of these ?z-componénts is probable, and, if we accept King's result, the measured mean position moves from tlie inner component to the outer with the increased field. Though the curvature is clear, it can l)e accounted for, if we assume that the relative intensity of the components changes with the magnetic field. / 4442*34 appears as a quadruplet. Taiîle LXIV. / = 4442-34, 1p-, 2n-comps. AX X103 AX XIO* H X2H n P n 1 p 11630 827 [ i 0-711 10760 808 0-750 10510 745 0-708 9690 681 0-703 8230 597 0-726 8180 1325 1-620 1 7530 538 0-715 4730 811 401 1-713 0-847 4450 812 1-824 4240 721 1-700 Table LXV. Comparison with former results. H AX xio* y?H n P Takahashi King (671-, 42J-comps. ?) mean 16000 1-699 1-533 0-729. 0-583 44 Art. 8.— Y. Takal.ashi III Fig. oo it ma}' be observed that the separation of tlie ^-components is proportional to the magnetic field, but, if we take King's result into consideration, the specific separation must be smaller for the stronger field. The straight line connecting King's point for the n-component with those of the writer does not pass through the origin. The apparent separation may be affected by the neighbouring line 4443*19. ?. 4427*314 appears as a triplet. Table LXVI. >î = 4427-314, triple. u ^"'- X103 X2 - ^^A^xio' X2ff H ^' X103 ' X2 1- - - -- - 1 ■ ^ X104 X2H 8230 1190 1-445 G020 879 1-459 7530 1155 1-533 4730 (385 1-447 13880 910 1-322 4650 661 1-421 (5750 930 1-377 4450 661 ' 1-485 Table LXVIl. Comparison with former results. Takahash King mean 16000 \iH XlO* 1-436 1-371 It may be seen that the separation is approximately proportional to the magnetic field. X 4375*934 is resolved into a triplet. Table I.XVIII. / = 4375-934, triple H ^'' xlO» X2 ^^ XIO* X2H H 6020 •^'^ XIO'^ X^ 825 ^'^ xio* X^H 8230 1165 1-415 1-370 7530 1068 1-416 4650 684 1-470 Ö820 1008 1-477 I Magnetic Separations of Iron Lines in Different Fields, Table LXIX. Comparison with former results. 45 H ^^ XIO» Vi Î Takahashi King mean 16000 1-4.29 1-384 The separation is proportional to the magnetic field. ^ 4337*04 is resolved into a quadruplet. Table LXX. /=4337-04, 2j9-, 2w-comps. u ^^ XlOS X2 ^^ XIO* n P n P 10510 984 0-935 8230 825 1-002 6820 668 388 0-980 0-569 4650 413 0-88S Table LXXI. Comparison with former results. H ^^ xio* X'-2H n P Takahashi King mean 16000 0-957 0-878 0-569 0-512 Though the number of the observed points is not sufficient to infer anything from them, the separation curve of the n- components seems to turn downwnids at about 8U()0 gauss to reach the peint given by King, and tlie line connecting the lower 3 points does not pass through the origin, as shown in Fig. 36. ^ 4315*089 is resolved into a quadruplet. 40 Art. 8.— Y. Takahaslii Tahle LXXII. /=z 48 151)89, 2p-, 'iîi-comjjs. ).2 XlOS AX XIO' 21960 20880 7530 6850 6820 6020 2> 557 p 540 1397 1112 1162 1074 /) 0-254 ]j 0-258 1-852 1-623 1-704 1-783 7/ XIO'J X-H XlOt 5590 964 4650 ; 835 „ 805 4460 776 4350 760 3580 626 n 1-722 1-795 1-730 1-740 1-747 1-750 Table LXXllI. Comparison with former results. H mean AX XlO* AîH n P Takahashi 1-745 0-256 van BiMerbtek- van Meurs 32040 1-77 00 King 16000 1-735 0-302 The separation of the vi-components is proportional to the magnetic field. The straight line connecting the points for the j9-coinponents does not pass through the origin, but we cannot say much about this. / 4299*26 is resolved into a triplet. Table LXXIV. >^ =4299-26, triple. u ^^ X103 X2 ^^ xio* )2ir H ^^ xio» X2 -^'-XIO* X2H 11120 1525 1-371 6300 835 1-325 8360 1064 1-274 6020 844 1-401 6880 945 1372 5910 785 1-329 6860 909 1-327 5590 757 1-352 6820 949 1-390 5490 692 1-259 „ 932 1-367 4730 621 1-312 6750 966 1429 4650 687 1-478 6540 848 1-296 4360 600 1-377 Magnetic Separations of Iron Lines in Different Fields. 47 Table LXXV. Comparison with former results H xio^ Takaliashi Van Bilderbeek-van Meurs King mean 32040 16000 1-353 1-37 1-372 The separation is proportional to the magnetic field. / 4294*13 appears as a quadruplet. Table LXXVI. : 4294- 13, '2^j-, 2n-comps. H ^^ XlO;-! X2 -^^ X104 X2H H ^^- xio-i X2 » n P P 9150 1058 1-156 21960 1083 0-493 8580 907 1-056 20880 979 0-46S 8350 999 1196 18G90 850 0-155 8230 872 1-060 18660 953 0-511 7690 826. 1-074 18070 785 0-434 6880 707 1-027 11120 533 0-479 6850 754 1-100 10980 561 0-511 6820 708 1-039 10660 498 0-467 6750 753 1-115 10200 495 0-485 6670 777 1-165 9730 490 0-503. 6540 761 1163 9150 439 0-480 6300 723 1-147 8800 408 0-464 5910 669 1-132 8580 387 0-451 5590 608 1-087 8350 369 0-442 5490 655 1-192 4730 536 1-133 4350 543 1-249 48 Art. 8.— Y. Takahashi : Table LXXVII. Comparison with former results. H A). X2If XlO* n P Takabashi mean 1119 0-475 van Bilderbeek-\an Meurs 32040 1-15 decomposed King (6 comps. ?) 16000 1-081 0-468 The ?i-compoDents become diffuse when the magnetic field is increased, indicating further resolution; the separation of tlie p-components shows slightly wavy fluctuation. }. 4282*408 appeal's as a triplet. Table LXXVIII. /}= 4 282-408, triple. H A^ X103 X2 ^^ X104 H ^^ X103 X2 ^^ xio* X2iT 7690 • 855 1-111 5910 740 1-252 6880 820 1-191 5680 701 1-234 6850 817 1-192 5590 664 1-187 6820 788 1-155 4730 559 1-181 » 783 1-149 4650 61(i 1-325 6750 749 1-110 4460 5D9 1-141 6540 774 1-182 4350 524 1-204 Table LXXIX. Comparison with former results. H ^'- X104 Takahashi mean 1-182 •van Bilderbc(k-van Meurs 320'10 1173 King ^7 comps. ?) 16000 1-058 MagTietic Separations of Iron Lines in Different Fields. 49 Fig. 40 sliows that tjie separation curve if^ slightly convex upward hetween 4000 and 8000 gauss. À 4260*48 is resolved into a triplet. Table LXXX. / = 4260-48, triple. H ^^X103 t 1316 ^A xio. H ^XIO» 1 X2 ^Axio* X-'H 8580 1-532 5590 815 1-456 8180 1208 1-475 5230 752 1-437 6850 1000 1-460 5060 745 1-472 6820 1000 1-166 4730 686 1-450 6750 1002 1-482 4650 689 1-481 6670 1020 1-529 4520 614 1-359 6300 965 1-5.31 4350 605 1-390 " 935 1-483 4020 598 1 1 1-488 Table LXXXI. Comparison with former results. H vm Takaliaslii mean 1-475 van Bilderbeek-van M(Mn-s 32040 1-526 King 16000 1458 Hartmann 1-436 The separation is proportional to the magnetic held. X 4250*79 appears as a quadruplet. Table LXXXII. / =4250*79, 2p-, 2w-comps. H ^'^ X103 X2 ^'^ XIO* rm H ^^XlOB X-' ^'^ X104 X-fl r V v' V 10980 879 0800 6850 616 0-899 10660 816 0-765 6300 532 0-845 9130 735 0 805 5910 n 690 n 1-168 8350 646 0-774 50 Art. S.— Y. Takahashi Table LXXXIIl. Coinparisoii with former results. H Aa -XIO* ).2H n p Takahashi mean 1168 0-808 Tan Bilderboek-van Meurs 32040 0-97 0-713 Kinji- (12 comps. ?) 16000 0-851 0-730 The Ti-compoiients are too diffuse to be measured accurately. The points for the 7)-components He on a straight hue, wliicli does not pass througli the origin. The apparent separation may be affected by the neighbouring hue ?. 4250" 15. /■ 4235*94 is resolved into a triplet. Table T.XXXIV. / = 4235-94, triple. H -^X103 )2 --— -XIO* >2H H ^>- xio^ )2 — ^ X 10* X2H 6300 5590 965 853 1-531 1-525 4730 4650 693 696 1-464 1-496 Table LXXXN'. Con:iparison with former results. H -^A X104 Takahasb i van Bilderbeek-van Meurs Kins mean 32040 16000 1-508 1-62 1-572 The straight hue connecting the observed points with King's does not pass through the origin, but we may consider the separation to be proportional to the magnetic field, as the deviation is very small. I\luii,-nt.'tie Separations of Iron Lines in Different Fields / 4219*36 is resolved into a triplet. Table LXXXVI. /- 4219-36, triple. H ^' X103 )^2 "^^ XIO* H ^'- XIO« >2 "^-^ XlQi - . 21960 10660 8180 2150 1049 790 0-979, 0-984 0-966 6820 5590 715 503 1048 0.900 Table LXXXVII. Comparison \^ath former results. U ^'- xio* Takahaslii van Bilderbeek-van Meurs King- mean -32040 16000 0-987 0-959 0-996 The separation is proportional to the magnetic tleld. / 4202*04 appears as a quadruplet. Table LXXXVIII. / = 4202-04, 2p-, 2n-comps. H ^^- X103 Vi -^A. xiot H ^''- X103 ^i XlOi n n p p 8350 941 1-126 20880 964 0-461 8040 918 1-141 20100 916 0-456 6820 767 1-124 10660 488 0-458 » 760 1-114 10330 4G4 0-449 6700 751 1-121 8350 410 0-491 6300 » 731 703 1-161 1-116 8040 377 0-469 5590 617 1-103 5230 564 1-078 4730 507 1-071 4520 440 0-973 4020 413 1-028 52 Art. 8. — Y. Takahashi Tablk LXXXIX. Comparison with former results. H ^^ -XlOi n P Takahashi mean 1105 0-462 van Bilderbeek-van Meurs 32040 1-098 0-383 King (10 comps. ?) 16000 1-142 0-5-2 The separation of the ?i-components and that of the j>- components are both proportional to the magnetic field abovt- 6000 gauss, while the diagram for the former components curves downwards at the weaker field, as can be seen in Fig. 45. À 4199*09 is resolved into a triplet. Table XC. ;. =4199-09, triple. H ^^ X103 ^^ xio* H ^^ xio:^ ^^ xio* 21960 2168 0-988 6300 599 0-950 10660 1050 0-985 '. 590 0-936 8350 808 0-967 5590 558 0-999 8180 770 0-941 5230 497 0-950 8040 799 0-991. 4730 419 0-886 6820 662 0-971 4520 443 0-980 6700 675 1-008 Table XCI. Comparison with former results. H -^^ ,X10> X-'H Takahashi mean 0-971 van Bilderbfek-van Meurs 32040 1010 Kin» IGOOO 0-978 Mag-netic Scparatious oï Iron Lines in Different Fields. The separation is proportional to the magnetic field. / 4181*76 is resolved into a triplet. Table XCII. À = 4181-76, triple. 53 H ■^'^ XlOa X2 '^'-XlOt H ^'' X10> X3 ^-XlO* X2fi 21960 20880 2Ü100 2612 2-i70 2348 l-18t» 1-182 1168 8180 6820 55iK) 982 807 680 1-200 1-183 1-215 Tabli: XC'III. Comparison with former results. H ■^'' XlO* Takahashi van Bilderbeek-van Meurs Kiui;- mean 32040 16000 1-185 1-225 1-210 The separation is proportional to the magnetic field. / 4143*88 has at least 3p-and 4/('-components. Table XCIV. /.=r J:]4rV88, at least 'dp-, 4«-comps. H •^'^ XIO« X2 -^-^■--XIO' W ^"' XIO^' ^À X104 21960 f 3820 (2380) \3103 p 692 f 1-739 = 5x0-348 ^1-413 = 4x0-353 p 0-315 5590 5230 768 673 1-373 1-287 6810 915 1-343 4730 662 1-400 6700 1004 1-499 4520 588 1-301 6300 905 856 1-436 1-360 4020 542 1-349 54 Art. 8.— Y. Takahashi : Table XCV. Comparison with former results. H -^'^ xio* Takahashi H<7000 1-377 van Bildcrliix'k-van Meurs 32040 1-435 King (7 comps. ?) 16000 1-43 Appearing as a triplet in a weak field the separation is approximately proportional to the magnetic ßeld. When the magnetic Held is increased, both j9-and ;^-components become diffuse, indicating further resolution. On a few plates photo- graphed with the field above 22()0ü gauss, 3^-and 4>i-components Avere observed, though it was difficult to measure them accurately, as they were faint and diffuse. ^^ 4132*08 has at least 5/?-and 47i-eon)punents. Table XCVI.'^ ^ = 4132*08, at least bp-, 4n-comps. H _^X10^' - -^Lxioi H ■^' XIO^ À- ■^' XIO» 21960 20880 20100 f4000 " \3085 p 390 f 3620 "X3000 p 387 " \2735 p 390 fl-821 = 5X0-364 "\l-404=4x0-352 p 0-178= èxO-3.56 fl-732 = 5x0-346 "Xl'43ti = 4x0-360 p 018r)-=ix0-370 r 1-716 = 5x0-343 "\l-360 = 4x0-340 p 0 194-^x0-388 6820 5590 5230 4020 1000 824 713 610 1-466 1-473 1-363 1-518 1) Tl\c interval bjtwot>n two adj-7000 (1-400 = 4x0-351 P 0-186= èxO-371 van Bilderbeek-Tun Isleuts 32040 1-62 King (13 comps. P) 16000 1-860 (outer) Though this hne appears as a triplet in a weak field, both jo-and /i-componeiits become diffuse when the magnetic field is increased, and finally the}" are resolved into inan}^ fine com- ponents. I have measured 5 ji9-components in equal spacing and 4 (or more?) /^components. / 4118*552 is resolved into a triplet. Takle XCVIII. /z=4118'55'2, triple. H ^''- xiœ ^'' xioi H ■^ÀXIOS A- ^^ xio« X-'H . 21960 20880 10660 2174 2056 984 0-989 0-984 0922 8180 6820 5590 793 634 533 0-969 0-930 0-953 Table XCIX. Comparison with former result;- H ^^ XIO* X2H Takahashi mean 0-968 van Bilderboek-van Meurs 32040 1-024 King- 16000 0-998 56 Art. 8. — Y. Takahashi : The separation is proportional to the magnetic field. >i 4005-26. Taf.lk C.'> / = 4005 •26. H ■^'"^ XlO'î ^' XIO* X2H H ^■' XIO« X2 "^^ xiot 20250 6820 p 303 n 1000 0150 1-467 5G70 5590 n 77-2 11 771 l-3tî2 1-379 Table CI. Comparison with former results. H ^''- xio* Takahashi uiean n 1-407 P 0-150 van Bilderbeek-van Meiirn 3-2040 1-55 King 16000 1-797 Though this line appear.s as a triplet in a weak field, the components become diffuse when tlie magnetic field is increased until they are resolved into many components. 5 yr components in equal spacing were measured, tliough the correspondmg nr components were not measured. The resolution of tliis line quite resembles that of À 4]3:^'0- ;;-eouipouoiits is <^iven iu the taI)1o. à Magnetic Separations ot Iron Lines in Different Fields. 57 TAr.LE CIL ^\ xio* A Takahashi King- Graftdiik van Bilderbeek-van Meurs 5434-527 unaffected unaffected 5341-03 n diffuse 2^^ 0-86 /( 1-703 p 0-939 5192-363 1-57 n 1-738 p 0-494 5169-03 1-44 1-317 4736-786 , 1-15 1-185 4647-439 1-09 1-139 4549-47 1-01 0-964 0-974 1482-27 4 J 1"^6 ^ "\ 0-49 2 p 0-97 n 0-433 p 0-712 4i69-39 1-01 1-37 4461-65 1-31 1-366 4^1-59-12 2 p 0-44 /( 1-410 p 0-397 4423-57 2 n 1-40 2 p 0-86 I 1-380 I 0-492 p 0-895 4369-77 0-93 0-923 ■4352-741 1--58 n 1-371 p 0-466 4233-615 [ 3-04 6 ni 2-05 I 0-94 3 p 1-04 ( 2-720 n\ 1-921 I 0-923 p 0-974 4227-44 roi 1-080 1-120 4210-36 2-81 2-841 4191-442 * »{ JS 3 p 1-04 "\ 0-940 /J 0-989 4175-64 1-04 1-06 4172-13 1-10 1101 4156-81 1-14 « 1-328 p 0-437 4134-685 1-08 1-109 4021-872 1-05 1-051 4014-53 0-98 0-970 3997-41 104 1-04 3969-26 1-37 1-405 1-472 3956-67 1-08 1-154 3930-30 1-41 1-424 1-429 3927-94 1-40 1-428 1-427 3922-92 1-39 1-428 1-412 3920-26 1-34 1-420 1-444 58 Art. 8.— Y. Takaliashi : ///. Nickel Lines. As tlie source of light was the spark between nickel-steel terminals, some nickel lines were photographed on the same plates with iron lines. / 5477*12 is resolved into a triplet. ÏA1ÎLE cm. / = 5477-l-2, triple. H ^^- X103 >,2 ^' XIO» >2H H ^'- X103 X- ^^- X104 10760 1181 1-097 6400 607 0-940 10510 1141 1-086 6300 600 0-952 9880 1036 • 1049 5520 521 0-944 9690 985 1-016 4940 421 0-852 9390 862 0-918 4730 432 0-913 8860 798 0-901 4450 444 0-998 8350 806 0-965 4240 409 0-965 8230 820 690 0-996 0-913 4110 410 0-998 7560 mean 0-976 6880 599 0-871 H'-component of 4325'78 indicates some fine resolution which, I think, consists of three componunts, while 4071'75, the first line of the third group, appears as a sharp triplet of the separation proi)ortional to the magnetic field. These lines seem to have 7, 5 (or 7) and 3 components respectively. The rest of these nine lines give sharp triplets, the separations being exactly proportional to the magnetic field. ( )f tlie nine strong lines, seven, which were measured most accurately', give the separations proportional to the magnetic field as if no mutual influence existed between the radiating electrons, yet the divergent values of the specific separations show that Runge' s rule does not hold among these fines. The ratios of these separations to the normal value are given in the I'ollowing table. 1| Fui- till- liuo 4415'1.3, the specific sciHiratuni of thf ;i-cotiiponents in a weak field is takon . Magnetic Separations of Iron Lines in Different Fields. 61 Table CVI. AX 1) IL oth =- — - ~' lines. Though we find I I considerable deviations for ^^""^"""^^ ' the separations of the 1 !, I p-components, i. e. the -''» -^ ^'' ^' *^" distance between the outer p-components of the former line is 0*315 x 10"*, and those between the outer and inner pairs of the latter are 2x0-371x10"*, and 0-371 xiO"*, respectively, the discrepancy can be accounted for by the errors of measurements, as these separations are too small to be measured accuratel3^ Hence we may take 0*35l x 10"* as tlie unit of the separations of these two lines. But we can find no relation ])etween a and /j. As for the nickel lines, the curvature in the separation curve of 5477' 12 is clearly marked, Ijut we must study the separation with the spark Ijetween pure nickel terminals or with concave grating to be able to fix it definitely. 4714-68 shows a separation curve similar to the iron line 537r405, but the measurement is not so accurate. 440 1*77 shows a smooth curvature in the separation curve, Ijut the observed points are not numerous enough m Art. 8.— Y. Takahashi and the deviations are too small to speak of its existence; in fact, we can connect tliem l)y a straight line which does not pass through the origin. (Jf the separations which are not proportional to the magnetic field, some may he caused hy true coupling l)etween radiating or non-radiating electrons, and others b^^ experimental errors. Among the conditions giving rise to the appearance of false coupling, we may count the followii^ : — 1. p]rrors in the determination of the magnetic field. 2. Apparent displacement of the mean position of a l)road line in the double order position. 3. Errors in measuring the broad line. 4. Disturbance by other coincident or neighbouring lines known or unknown. 5. Complicated separation whose components are not thoroughly resolved. ^6. Irregular contraction of the photographic film. Those lines photographed on one plate with the same exposure as the standard line ai"e free from the first error, as the standard line was measured most accurately'. For those which were not photograj^hed with the same exposure as the standard line, we can determine the field ])y measuring the photos of the standard line taken before and after the exposure for those in question without breaking the current through the electromagnet and carefully observing the constancy of the current. The separation of the standard line measured on the former agi'eed well Avith that on the latter in ahnost all cases, so tliat we may rely on the determination of the relative field. The second is important for broad lines. This can be eliminated by taking the photos in different positions of the echelon spectra, bringing the undisturbed line in single order for some exposures and in double order for others. The third is unavoidable, but the use of different types of cross wire somewhat improved the result. The cross wire of the micrometer used by the writer was constructed as showu in Fig. 55. a was mostly used for sharp lines to bisect the lino in the Magnetic Stiaiatious vî Iren Lines in Different FieLl.s. 67 Fig. 55. mean position, // was used for In-uad lines bringing the point of intersection of the wire on tlie mean position, and c was used for those which were photographed with the narrow second sht s^ bringing the spot in the middle of the two adjacent wires. c ■■{■ «c Tlie fourth is unavoidal)le, unless we use other instruments with different constants, or by crossing them. Plane grating crossed with echelon or other interferometer seems to be most convenient for such an element rich in lines as iron, though the experinent failed in the present work owing to insufHcient intensit}'. While 4957*62, 4020'52 and 489 To 1, which jnay be disturbed more or less by neighbouring lines, show separations approximately proportional to the field, 5477-12 (Ni), 526953, 4442':;4 and 4250-79 give separations which are not proportional to the field with some indication of the disturbance caused bv the neigh1)0uring line. In order to overcome the fifth difficulty, an instrument of high resolving power with considerably large value of o?.ma.v i^ necessary, for we have to resolve the components in différent positions. If dÀmax is small, many components fall close to the others, and it is difficult to study their l-ehavior with a sufficient number of fields. It is also necessary to apply a strong magnetic field. In the present experiment 4415*13, giving a separation which is not proportional to the magnetic field in weak fields, is resolved into seven components in fields above 35000 gauss. The separation of the n-components of 4447-72, given in Fig. 32, also showa false curvature, the mean position of the two componefits moving from the inner to the outer. The separations of other diffuse components may be subject to the same error, if they are formed of assemblages of several lines, and the measured positions correspond to the means. To overcome the last difficulty it is necessary to measure the separation on different plates and increase the number of the observed points. In the present investigation I took great care to avoid these errors. 68 Art. 8.-Y. Takiiliaslü : In conclusion the writer wishes to tender his coordial thanks to Prof. Nagaoka, at whose suggestion the present experiment was undertaken. Summary. 1. The hght emitted Ijy the spark between non-magnetic nickel-steel terminals has been examined in different magnetic fields. 2. ]Man3^ lines photographed at a juxtaposition with a prism spectrograph placed Ijehind an echelon spectroscope have enabled us to compare their relative separations exactly. 3. The separations of the lines pliotographed were compared with that of the line 44(.)4'75, and, assuming the latter to be proportional to the magnetic field, the behavior of the otliers was studied. The assumption is verified by the fact tliat the separations of many sharp lines are proportional to the separation of the standard line. 4. The magnetic fields are calculated assuming the specific separation of the standard line 4404*75 given by Mr. Yamada. 5. Nine strong lines in the violet region being studied carefully, the separations of seven of them were found to he exactly proportional to the magnetic field, wliile the others gave larger specific separations for stronger fields, the components appearing more and more difîïise witli the increase of the field, and the line 441. ")• 13 was rosolverl into seven components in strong fields. G. Other less strong fines were also studied. Some of them give different types of separations, which are not proportional to the magnetic field, and the others show that their separations are proportional to the field. Specific separations are given for those measured with less accuracy or with only one or tAVO fields. 7. Nickel lines photographed on tlie same plate as the iron lines are also given. 8. jMany sharp lines give separations proportional to the inagnetic field as if there was no mutual influence between the Magut'tic Separations of Iron Lines in Different FicMs. (j9 mdiating electrons, but tlie (li\ergeiit values of the specific separations seem to indicate the existence of coupling. Moreover the separations of a number of lines which are not proportional to the magnetic held, seem to indicate the existence of some mutual influence, though the measurements are not sufhciently accurate. U. Many iron lines apparently show simple resolutions into diffuse components, which are likely to be resolved again into hue components, if they are examined in stronger magnetic fields with an instrument of higher resolving power. Published December 30Lh, 1920. Jour. Sei. Coll., Vol. XLI.. Art. 8, PI. I. A), 4000 3000 1000 Fio-. 2. ). = 4415-13, Zp-, 4îi-coinps. y :- -■ 1 n y / / . ■' -^ • /' ( ■ ^, ^ y -- •^ ^ " 1/ ' ^ -- ' , " -'^z ^ ^ -- ^ > / • ' y /- ^ /- / ■^ r' ^ ^ ' ^^ ■y' ;. " / " y ^ '' -'' r ^ "^ ^^ -^ ^■^ --^ [^ _ . -- - " ^ -' __ - - ■ - " ^ — - - ~ ^ 10000 20000 40000 AX 4000 3000 2000 1000 Fig. 3. ) -4383-55, tripl. , ^A- J J -^ ^y^K 10000 20000 30000 40000 (J) denotes an asseuihlage of two ov uiore.'iioints. B, G and K denott^ the points taken from the results of van Bilderljeek-van Meurs, Graftdijk and King respectively. Y, Takahashi. Magnetic Separations of Iron Lines in Different Fields. Jour. Sei. Coll., Vol. XLI., Art. 8, PI. II. ^xio« 4000 3000 2000 1000 F if.-. 4. >.-4325-78, (luintiiple? 1 -^ ^-; --'' ,^:"'^ / ^-B- ,- '' ' -> r' ^' ^- ^ - ,--- •" ^ '^ -- — ^ -> '^" — - - , "^^ .^ - '^- -^"i 'k ^ ^ ^ ''^-.'' %-' - — - — ^^ ^■ ^* ,^'. ^' L/ 10000 20000 30000 40000 A), XlO^ 4000 3000 2000 1000 X = 4307-92, triple. ^ ^ .^ 3 ^ ^ /-' 1 ^ ^-'• *' ^ ^ ^ ^ ^ ^■ ^ ^ /^ iT / -^ r^ r^ , • ^ ^ *^' ^ ;^' ^^ -^ 10000 20000 30000 40000 Y. Takahashi, jMagnetic Separations of Iron Lines in Ditferent Fields. Jour. Sel. Coll., Vol. XLI., Art. 8, PI. III. M X 10" Fio-. G. ),- 127 1-75, triplo. 1 1 ^ ' 4000 "1 i fi ' ' / ^ /■ ' ^ -^ 1 /- 3000 ^ / ^ ^ V ^ '<'' L_ ,-^ /'' — 2000 iC ./ '' ^^ .^ isi- ^^ — — — ^ — — 1000 ^ ' ,- S^ 3f «■•- - y -- '- 7^ — _ ... _ _ __ ._ __ __ — — — — 10000 20000 30000 40000 -r-XlO" 2000 1000 Fio-. 7. >, = 4071-7n, trii.le. r- - -^ B ^ ^' ' ^ -^ -- ,^ '^'' ^ -^ 1 ^ -^' _^ ^ '^ ^^ ,^ ^ .^ ^ ^^ ^ -^ __ __ _ L _ 10000 20000 30000 40000 Y. Takahashi. Magnetic Separations of Iron Lines in Different Fields. Jour. Sei. Coll., Vol. XLL, Art. 8, PI. IV. M xiüs 400Q 300Q 2000 1000 Fi--. 8. >,-=4.Ü6;:i-Gl, triple. 1 [■ -^ B /^ <^ ^ r ."- "^ ^-^ .— >' ^ ■^ ^•■ » ^'- ' ,^ -'' ,.■' ''^ Jr- - - — »^' x^ ^^ ^ '-'■' ( ,.. k" .- H 10000 20000 300D0 40000 XlO^ Ti'io-. 9. ^4045-82, tripl. 4000 30Q0 2000 1000 ' " • / _| ~ ' P ^ ^ / /-• — / /' /^ // / / / ^ > ^ - r^ /^ ^ 7^ K ^ ^ /^ [I ~~ ~" •/ ®' ~" r* Z^ y^ ~~^ ^ ^ -:^ ZI H 10000 20000 30000 40000 Y. Takahashi. Magiutic Separations of Iron Lines in Different Fields. ^xio^' F.-. 10. X = 5615-661, IripL ' K, ^ •^ ^ ^ /■ y^ ^ ^ ^ ^ ^ L^ ^ \y L _ 10000 Jour. Sel. Coll., Vol. XU., Art. 8, PI. V. F.-. 13. ),= 5455-614, 'Jj;-, 37i-r(jmps. ^^xio^ .K^ 2000 />-> ■-' — 1 1 ^. / yV'' ^^ >^ / 1000 _,/ X ^ '^■n K-^ TL '^ y / /"' ^ ^^ „S^- y / p^ .^ y , _ _ H 10000 -^XlOS Fl-.; 11. A = 5586-772, triple'? Fio-. 1,4. --1 [K / -^ --■ " y X /^ y ^ ^ A), "^ y lO-'' '/.= 5446-92, 2p-, ;2ïj-couiiis. 2000 1000 r.^ / ^ "^ ^ > ■^ K >- ^ >^ ^. / y ^ ^ -^sr'• y y^ »■' - ^ ^ H 10000 10000 A>, XlO' Fig-. 12. X = 5572-36, triple K. Aa Fiu-. 15. ),-5429'7ü, 2y>, 2;j-,'o;iips. — XlO'- 2000 •i j^ - \j "l ' ,; >^ , ^ ^ "1:: -^ -£- f^'""" H 10000 10000 Y. Takahashi. Magnetic Séparations of Iron Lines in Different Fields. Fio-. 16. ^X10^> I ^•- y ^ 1 1 y y / jy y y 1Q00 •y y- y y' ^ ^ -^ .-- ■Kf> y y- ^ "^ y ^ -^ ^ ^ _ H A), ~rj X 10! ./Oi//'. Sc/. Coll., yo\. XL I., Art. 8, PI. VI. Fio-. 19. ),-52G0-53, triple. 2000 1000 ^ -'' ^'' X ' y^' .'^ i^^- ^"^ ' ^-^ ^^ 10000 II 10000 A), XiO' 1000 Fis. 17. X = 5371-495, ti-ipk'. ~ • K — — r^ — - J ^ ^^ ~- ZP - .^ ■* • Fio-. 20. Xio-5 ). = 5232-957, triple. 2000 1000 10000 ir_ I :^ H 10000 Fio-. 18. — XlO» ),-5328 06, triple. 1000 ■ [y'^'^ \.y ■ J ^r^ >r ^^ IT ^^ ^'^ H AX )2 Fitr. 21. X103 X = 5227-20, triple. 1000 1 ''' ^'/^ ' /\ 1 -/ 1 1 ■^ ^ i y -^ ^ zy' y U H 10000 y. Takahashi. Magnetic Separations of Iron Lines in Difïerent Fields. Jour. sa. Coll., Vol. XLI., Art. 8, PI. VII. AX X103 100Q I Fio-. 22. X--51G7-492, triple. n ,^/ ^ ^ ^ ^ ^ K<'' >^- — ^ ^ ^ ^ " 1 L. 1000Q Aà Fio-. 25. -XlOT >, = 4920-52, triple. 2000 — — y — — 1 ^ -^"^ -^ /'' - / ^ -- - ^: '-' >^ - / - /]• y ^ 1 1 1 1 10000 H Fig. 23. \= 5018-45, tripL / A, X ^XIO:' / 2000 Â-a / / / • / - 1000 ^ / ^ / L^i (/ / K ' /\ / ! / 1 1 10000 10003 Fio'. ^4. ), -4957-02, IripL ^"^X103 K 2000 '- / 1000 ^ /^ -f^ ^^ Ur vr' /-- ^ /• ^ ).-' xio^ Fip;. 27. X= 4583-83, triple. 2000 K 1 y ^ " ^ - L ■ ^ y 1300 y" ^ y' yf y /- /- /^ 1 _ II 10000 10000 Y. Tak.\hashi. r\r;if,nietic Separations of Iron Lines in J)itïereijt Fields. -fxio. Fig. 28. ), =4528-622, tripK -^ '■' ^ - ^ .^ " ^ ^ / / y 1\ xiO'- X103 IDQOO Fio-. 29. X~=44.04-572, triple. -^r^'- ^ I I I I I I I I I 1D000 Fig. 30. V76 03, trill — — — — — -^^^ — Jour. Sei. Coll., Vol. XLI., Art. 8, PI. VIII. Fig-. 31. — X103 X = 4466-556, triple. 2000 1000 _ ^ 3z:tt:zzzzzzzzzz"z H 10000 Fig. 32. ) — 1447 72, 'ly-, 4/i-eouips., Hiiasurt;«! as a quadruplet. 400Q 3000 2000 1000 /k'' / / / / / / '■ / ^ ■Un n /' _ — - /7 ! ' - / / n. / ■^^ •H / / / ^ ^ 1 / / / y ^ / / y ' ■^ j / / ./ ^..--^ — / / ■^ ^'\ 1 6^ c H 1 1 ^ ^ : \ i H 10000 10000 Y. Takahasht. iNIagiiotic Separations of Iron Lines in Different Fields. Jout, Set, Coll., \/ol. XLl., Art. 8, PI. IX. Fi-. 33. X =3 4442-34. 2f-, 2?i-comps. 10000 FiK^ 34.. X-4427-314, triple. AX ^^XiU' 2000 i\ ~ / ' /- / y ,/ /- 1000 y /< y / y ^ . Fig. 3G. -TT-XlO'i > =4337-04, 2/;-, 2n-couips. 1000 ~ , r — ^ ^==■^1^ ^ -^ n^ -^ ^ ^ ^ ^ ■^ ■^ ^ ^ _^ -^ ^^ ■^ p ~ ^ =1 ^ I_ _ __ ^ L 10000 10000 Fi.r. 37. X-4315-089, 2p-, 2«-coHips. Il r 1 ' ' /!«' ti xio. y TV / ?nnn / , / / / • ^ / ~ 1000 k- -ji /• -^ ^ ~ ^ _. ji, ^ - _ xt = ^j i ^ — — — -1 ^ — - — " 1 -1 0 inr inn onn nn H yic^. 35. ), = 4375-934, triple. AX -T^XIO'! / / 2000 / y / /^ X — / /" lono _j X z' ^ / / ^ P ^-- Fig. 38. X=4299'26, triple. 10000 ".: X1G3 ^ 2000 ^K ^^ -" // y ^ y^ p— 1000 y \^ ^ / ■ ^ .-_ 1 H 10000 Y. Takahashi. Magnetic Separations of Iron Lines in Different Fields. Jour. Sei. Coll., Vol. XL I., Art. 8, PI. K. Fig. 39. M -X10-- )>=4294-13, 2p-, 2n-comps. 1000 10000 M Fig. 42. ^Qs X = 4250-79, 2^>, 27i-comps. /• / ' /- — ^ ^ -' -^ •y VÏ h — ' % -^ ^ ^ - ^ ^ r^ "' , \l 1000 20000 __^ ■ ^n __^ --' '^ 1^ -^ > " ' ^ > "-' "^ y ^ ^ .^ y 10000 H )2 xio:- 1000 Fijy. 40. ).- 4282-408, tripL^. , ^ ^ '' ^ . '' -' — - ' X — -- /' ^ 10000 Fig. 43. ), = 4235-94, triple. r^ r^ ^!ïxio^- / Â2 / 2UÜU ,/ y / / ,/ — 1000 r' A > /^^ /■ '"■ / 10000 2000 1000 Fig. 41. ),-42G0-48, triple. ■^>- ,^. TvX10> K A / '' y ^ y ~ / y" A y y ^ \A AX H F) g. 44. — ^XlO"' ). = 4219-36, triple. ^ ^ ^ v^ ^ ^ ^ ^ K ^ 1000 ^ ^ • ^ ^ -^ ^ ■^ ^ ^ T[ 10000 10000 20000 Y. Takaiiashi. Magnetic Separations of Iron T.iiios in l^ifferent Fields. Jour. Sei. Coll., Vol. XU., Art. 8, PI. XI. M XIO' Fio-. 45. X = 420'2-04, 2/,-, 2;i-coiiips. 2000 1000 jrSn / ^ /- ^ TV, ^ ^A ' ^ K.f. '" -' 1 r ,»' > ^ -— ^'■ -^ -^ / ^ -—^ __<: ' """^ H 10000 Fig-. 48. X = 4143'88, at Itast 3^--, 2«-comps. The sej^aration at 2196 g-auss is not given in the tiijuro. 20000 ^X10:= K. ;!; A-i /;/ 2000 [/ y^ ^ '^ n ^ M -^ 1000 ,^ y 7^ ^" J ^ 10000 20000 ^XIO:) 2000 1000 Fig. 4G. X = 4199-09, triple. ."■ ^ - ,-/ ■^ ^ r ^ ''\ ^ -^ ^ -^ ,r* -■< ^ "• ^ ■^ ^ M 10000 20000 AX Fig. 47. ).-=4181-76, triple. - r" y ■' ' 2000 / /" y ^ X // ' ,.- 1000 y /^ X / y ^ / ^ -^ II 10000 20000 Y. Takahashi. Magiiotic Se]mi'ations of Iron Lines in Different Fields Jour. Sei. Coll., Vol. XLL, Art. 8, PI. XII. AX Fig. 49. X = 4132'08, at least 5}}-, 4ra-couips. ^:rXlO-« 40QQ K^ n. / / 7^ / / / / — 3000 ^ / n '/ / ^ — / -^ y _ / / y / / ^/ 2000 / y / / y / y y ^^ /v ■^ y '/'' 1 -^ 1000 y ^ ^ ,. •'■' ^^ ^' y ^ ■-- _— t^ '-^ r - ^ -^ . — ■ — / / ^ -— ' À ^ p_ J H 10000 20000 ^XlOB Fig. 50. X = 4118-552, triple. 2000 ^ ' /^ '' K^ >-- ^ — 1000 J .^ .^ ■^ -^ -^ /- ^ ^ f ^ y H A\ Fig. 02. X = 4714-GS (Ni), triple. 1000 10000 H 10000 20000 Fig. 51. AX — -xio^ ' ^' ' ■ 2000 /■2 /' ''^\ •' / ' / / / 1000 / ^^ / ^ ^ ^ ^. •^ H 10000 A> Fig. 53. XiQs X = 440] -77 (Xi), triple. 10Q0 , / / y y y y y / y ,^ -^ 10000 Y. Takahashi. Magnetic Separations of Iron Lines in Different Fields. (b) lü il '4 Jour. sa. Coll., Vol. XLI., Art. 8, PI. XIII. ■ « ■ :« i: r ■ '■. (h) I I I I !> I ' . . Id» |SS 1^ ci r-'in m CM M 05 N- ¥ si e se e>j h« tn es e>j 05 CO mrrin co*5? M m M - m en 00 h. ION- 7- in Ol il? m «M 00 •- r>- m « • (a), H=0. (b), (c), H = 7l70, each line is resolved into a sharp triplet. (d), (e), H = 37230, both p-and n-comps. of 4415'13 are resolved into 3 and 4 lines respectively, p-comp. of 4325*78 is frin^ (f), H = 24040, p-comp. of 4325-78 is diffuse. (g), H = 35650, both p-and n-comps. of 4415*13 are resolved. [D-comp. of 4325*78 is diiîuse. (h), H = 6820, both jj-and n-comps. of 4415*13 are sharp. (i), H=20880, „ „ „ diffuse. l||t 'K^i P i \ [ < I ' (»I R^ • tii«*i« (i) (k) 0) I » ^^ ^ cn 01AC*9 bS 03 ^ «O C^ CO to CM r*. N" r— cj ey CM cn eo ^ m ^mnm mm m •— CM CM m m ep ^ ô h« 03 ea m • • : • CT» CO en « tninm h. in (j), H= 21960, p-and n-comps. of 4143-88 and 4132-08 are resolved into many lines, fk), H = 8230. (1), H = 6400. Y. Takahashi. Magnetic Separations of Iron Lines in Different Fields. .TOl'KNAL Ol" TirF. Cur.f.KGE OF SCrENCE, TOKYO IMPERIAL UNIVERSITY. VOL. XLI., ARTICLE 9. Ueber eine Theorie des relativ Abel'schen Zahlkörpers. Von Teiji TAKAGI, Uigakuhakiishi, Professor der Mathematik an der Kaiserl. Universität zu Tokyo. Der vorliegende Aufsatz ist die ausführliche Darlegung einer Theorie des relativ Abel'schen Zahlkörpers, deren Umriss vor einigen Jahren in den Proceedings der hiesigen Mathematisch - Physikalischen Gesellschaft selir knapp und mangelhaft skizzirt worden ist. Diese Theorie stützt sich auf den verallgemeinerten Begriff der Idealclassen, welcher sich in der modernen Theorie der algebraischen Zahlen allmählich entwickelt, und durch Heinrich AVeber eine explicite Formulirung in der sehr allgemeinen Form gefunden hat. Es werden danach zwei Ideale eines algebraischen Körpers nur dann als aequivalent betrachtet und in dieselbe Idealclasse gerechnet, wenn ihr Quotient durch eine Zald dargestellt werden kann, welche gewisser Congruenzbedingung nach einem A'orgeschriebenen Idealmodul des Körpers genügt. Es existirt alsdann zu einem beliebigen algebraischen Zahlkörper ein bestimmter relativ Abel' scher Oberkörper von der folgenden Beschaffenheit: 1) Die lielativdiscriminante des Oberkörpers enthält die und nur die Primideale als Factor, welche in den Idealmodul des Grundkörpers aufgehen, der der Ciasseneinteilung in demselben zu Grunde gelegt wird. 2) Die Galois' sehe Gruppe des Oberkörpers in Bezug auf den Grundkörper ist holoedrisch isomorph mit der Classengruppe (im verallgemeinerten Sinne) des Grundkörpers. 2 Art. 9.— T. Takagi : 3) Diejenigen Primideale des Grundkörpers, welche der Hauptclasse (im verallgemeinerten Sinne) angehören und nur diese erfahren im Oherkörper eine Zerlegung in die Primfactoren der ersten Relativgrade; allgemeiner hängt die weitere Zerlegung der Primideale des Grundkörpers in dem Oberkörper nur von der Classe ab, der die Primideale im Grundkörper angehören. Es ist dies eine naturgemässe Yerahgemeinernng der Grundeigenschaften des Classenlcörpers, welcher zuerst von D. Hilbert eingeführt wurde und die Theorie desselben von Ph. Furtwängler weiter fortgeführt worden ist. Jener Oberkörper sei daher als der allgemeine Classenkörper für die zugehörigen Idealengruppe des Grundkörpers bezeichnet, welche Gruppe die Hauptclasse (im verallgemeinerte! i Sinne) des Grundkörpers bildet. Eine wichtige Tatsache in der Theorie des relativ Abel' sehen Zahlkörpers ist nun die, dass umgekehrt zu jedem relativ Abel' sehen (Jberkörper eine bestimmte Classengruppe nach einem geeignet gewählten Idealmodul in dem Grundkörper existirt, welcher jener Oberkörper als Classenkörper zugeordnet ist, so dass die relativ Abel' sehen Oberkörper einerseits und die Idealen- gruppen in dem Grundkörper anderseits einander characterisirend in wechselseitig eindeutiger Beziehung stehen. Ich habe so w^eit als möglich diese Theorie ohne die üljlichen Voraussetzung entwickelt, dass der Grandkörper die Einheitswur- zeln enthalte; liierbei haben sich die von Hilbert eingeführten, einem i*riniideal in relativ normalen Körper zugehörigen Körper, Avelche die weitere Zerlegung des Primideals des Grundkörpers behersschen, als ein sehr nützliches Hülfsmittel erwiesen. Unter den Anwendungen dieser Theorie sei der Existenz- beweis für die unendliclivielen J^rimideale ersten Grades in jeder Classe (im verallgemeinerten Sinne) eines beliebigen algebraischen Zahlkörpers hervorgehoben; es ist dies eine schöne Verallge- meinerung des classischen Dirichlet' sehen Satzes über die Prim- zahlen in einer arithmetischen Reihe. Als ein Beispiel und eine naheliegende Anwendung der allgemeinen Tlieorie habe ich die der relativ Abel' sehen Körper Ueljoi- eine Thoorio des relativ Aborschon Zahlkövpers. 3 in Bezug auf einen imaginären quadratischen Körper in einem besonderen Capitel behandelt. Es gelang die Bestätigung der berühmten Kronecker' sehen Vermutung über die aus der Theorie der complexen Multiplication der elliptischen Functionen entsprin- genden Körper vollständig durchzuführen, was durch H. Weber und R. Fueter (in der unten citirten Al)handlung) nur zum Tuil geschehen ist. In Verzicht auf die vollständige Litteraturangabe seien die folgenden Werke angeführt, die, sei es als Grundlage, sei es als Anregung, für diese Untersuchung von Wichtigkeit gewesen sind: H. Weber, Ueber Zahlengruppen in algebraischen Körpern. Math. Ann. 48, 49, 50. (1897-1anit}ieit aller Ideale, welche einem gegebenen aequivalent sind, fassen Avir in eine Idealclasse zusammen. Dann ist die- Anzabl h der Idealclassen im Körper k endlich. Diese Classen lassen sich durch Multiplication zusammensetzen: sind nämlich A, IÎ irgend zwei Classen, n, h beliebige Ideale dieser Classen, dann gehört das I*roduct n h einer durcli die Classen a, r. eindeutig, bestimmte, von der Wahl der Representanten a, ,(i unabhängige Ciasse AB. Die h Classen bilden in der Tat eine Abel' sehe Gruppe, in Avelcher die Multiplication als die Regel der Zusam- mensetzung gilt, und die Hauptclasse die Stelle des Hauptelementes einnimmt. Man kann auch die Gesamtheit der ganzen und gebrochenen Ideale des Körpers k als eine (unendliche) Abel' sehe Gruppe g auffassen, indem wir die Ideale durch Multiplication zusammen- setzen. Dann bilden eben die Gesamtheit der ganzen oder gebrochenen Hauptideale eine Untergi'uppe o vom Index k] sind ûi, a2,...(ih ein System der Representanten der A Classen, dann ist in einer, in der Gruppentheorie üblichen, Bezeichnungsweise: G = oai + ono-l f-oa/.. (1) Eine engere Fassung des Classenbegriffs hat sich bei den verschiedenen Problemen als von Nutzen erwiesen. Es werden die Ideale n, b nur dann als aequivalent aufgefasst und in eine- und dieselbe Classe gerechnet, wenn ihr Quotient einem Haupt - ideale {>c) gleich ist, wo ?' gewisser Bedingungen betreffs des Vorzeichens unterworfen ist. Es ist zum Beispiel verlangt, dass ?f positive Norm habe", oder dass n total positiv sei,''' d.li. die mit H conjugirten Zahlen in den sämtlichen mit k conjugirten reellen Körpern kj, ks.-.k, jwsitiv seien. Solche Vorzeichenbedin- gungen lassen sich in allgemeinster Weise wie folgt auffassen: Das System der Vorzeichen, welche die mit « conjugirten Zahlen in kl ks.-.k;. aufweisen, sei mit l)ezeichnet, wo e=±l ist; wir wollen es kurz die Vorzeiduitccm- 1) Vgl. Hil}>ert, éericlit, § 24. 2) Hilljort, Eclativ Abel. Zalilkch-per, § 5. Uobei- eiuo Theorie «los i-clativ Abol'scheu Zalilkorpers. 5 ■ hhuhtloit dyr Zalil n ncimeii. J.)aiiii l)ildeii die '1'' möglichen Vor- zeic'ieiicoinbiuationen eine (Irappe nach ^[ultipUcation, welche mit der Gruppe der entsprechenden Zahlen homomorph ist, d.h., 'ist (s/,^/, s/) die Vorzeichencomhination von ;"■', ') enthält, welche durch die Zahlen n mit den Vorzeichencombinationen von it erzeugt werden. An Stelle von (1) hat man nunmehr die neue Ciasseneinteilung: G = o'ai + o'no+ ()'a;.'> wo h' die Classenzahl von k im neuen, engeren Sinne ist, und es .zerfällt jede Classe oa im alten, weiteren Sinne in eine dieselbe /// / . Anzahl ^— von den Classen o a im engeren Sinne, wo die Zahl li' . . —j— offenbar ein Teiler von dem Index der Gruppe h, d.h. von 2'"^'"<^ ist, we on 2''> die Ordnung der Gruppe h ist. Eine andere Ju'weiterung des Classenbegriffs erblicken wir in •die sogenannten Ringclassen. " Es sei r ein Zahlring im Körper k, f der Führer desselben. Zwei zum Führer f relativ prime Ringideale a^ und 6^, werden dann aequivalent genannt, unraisclion Kürperu, Math. Ann. 43-50. Lehrbuch, der Algebra, III., § 161. Uebor eine Tlicoric der relativ A))cr sehen Zahlkorpers. 7 Die Hixuptidealc, welche in ii enthalten sind, bilden für ^ich eine Gruppe Hq, offenbar vom endlichen Index. Definiren wir dann die Classen nach iio, so sind die Classen nach n nichts anders als die Zusammenfassung einer gleichen Anzahl der Classen nach ii,,; mit anderen Worten, die Classengrui)pe nach h ist die comple- mentäre Gruppe o/ir, wenn die Classen nach itq zu Grunde gelegt werden. Jedem Hauptideal (a) von Ho entspricht nun ein System von associrten Zahlen e«, wo e Einheiten von k bedeutet. Betrachten wir nun diese Zahlen einzeln für sich, dann bilden sie in ihrer Gesamtheit eine unendliche Abel' sehe Gruppe, deren Elemente einzelne Zahlen sind, und in welcher die Multiplication die Compositionsregel abgibt. Daher kann man mit Weber zur Definition des Classenbegriffs eine Zahlenp'uppe zu Grunde legen. Die Gesamtheit^ / der ganzen und gebrochenen, zu dem gegebenen Ideal in primen Zahlen des Körpers k ist eine Gruppe; es sei o eine Untergruppe derselben, von welcher der Index (z: o) endlich ist. Jede Zahl von o definirt ein zu m primes Hauptideal, die Gesamtheit desselben ist dann eine Idealen- gruppe, die wir vorübergehend mit ö bezeichnen wollen. Dann bilden nach Weber die Ideale eines Complexes öa eine Idealclasse nach o, also speciell die Ideale von n die Hauptclasse. So werden die sämtliclien zu m primen Idealen von k in Classen verteilt. Die Beschränkung, dass nur die zu m primen Ideale in Betracht gezogen werden, ist für die Classeneinteilung ohne Belang, denn jede Idealclasse im absoluten Sinne enthält die zu m primen Ideale. Erst durch die Einführung der Zahlen- gruppe o Avird jede absolute Idealclasse in eine dieselbe Anzahl d von den Classen nach o zerlegt. Diese Anzahl d bestimmt sich nach Weber durch die Formek-* ,7- (z:o) (E : Eo) wenn i: die Gruppe der sämtlichen Einheiten in k, Eo diejenige der Einheiten in o, und allgemein (a:t^) den Grupi^enindex bedeutet. 1) H. Weber, Math. Ann. Bd. 48, S. 443. Lehrbuch, III, S. 598. 8 Art. 9.— T. Takagi : §. 2. Congruenz-classengruppen. Von einer besonderen Wichtigkeit ist nun der Fall, wo die Zalilengriippe o die folgende Bedingung erfüllt: ^^ Es sei a ein beliebiges ganzes Ideal in g, und T{t) die Anzalil der in ô enthaltenen durch a teilbaren ganzen Hauptideale, deren Norm nicht grösser als die positive Grösse t ist. Dann soll N(aj und folglich Lim ^=-J!— t=x t N(a) sein, worin g eine endliche von Null verschiedene positive Grösse ist, (h'e nur von den Gruppen q und o, aber nicht von / und von der Wahl des Ideals a abhängt, wälu'endJ/ eine Function von / ist, welche mit unendlich wachsendem t nicht unendlich wird, und o endlicli eine nur von dem Körper k abhängende positive Grösse bedeutet, die kleiner als 1 ist. Unter dieser Voraussetzung folgt, wenn für ein variables s>\ N(ir gesetzt wird, worin j die sämtlichen ganzen Ideale einer Classe a nach (» durcli läuft, .s— 1 wo G{s) eine inunction ist, welche für .s=l in einen endlichen Grenzwert üljei-geht."-' Hieraus folgt zunächst, dass die Classenzahl nacli <> endlich ist.^'^ 1) H. Webex-, Ueber die Zahlengruppen usw., Math. Ann. 49., S. 84. 2) Do. S. 85. 3) Die Vorans-setzimg 2. bei Weber, a. a. O. ist in der Voraussetzung 3. enthalten. Uülier eine Theorie ilös relativ Atersclicn Zahlkörpors. 9 Es sei nun ii eine Untergruppe der Classeiigruppe nach o vom Index h. Dann gibt es bekanntlich A S3'"steme der Gruppen- charactere Zi' /-'>••■ /,„ welche für die ('lassen in h den Wert 1 liaben. Dementsjirecbend •definiren wir nach Weber die h Functionen Ç,(.s) durch die unend- lichen Eeihen: Qis)='ixix)Ä{s)=LML, (f=i, 2, h) wo sich die erste Summe auf die h Classen a, die zweite auf die sämtlichen ganzen Ideale von g erstreckt. Diese Reihen con- vergiren absolut wenn .5>1. Ist 2, der Hauptcharacter, dann geht für .5=1 in den endlichen von Null verschiedenen Grenzwert ^A über: für die A— 1 anderen Charactere gehen die Functionen Qi{s) (i=2,8, h) gleichfalls für 5=1 in die endliche Grenzwerte über, die jedoch auch verschwinden können. Die Functionen Qls) lassen sich, so lange .s>l, in unendliche Producte entwickeln: 1 _ /.(y) ' •wo p die sämtlichen Primideale von g durchläuft. Definiren wir demnach die Function log Q.,{s) durch die eben- falls für s^>\ unbedingt convergente Reihe: loga(.)=-Vlog(l-^) 2Q Art. 9. -T. Takagi : SO erhalten wir, indem wir nach 7 sumrniren wo links unter log. der reelle Wert des Logaritlinius zu verstehen ist, und wo die erste Summe rechts sich auf die sämtlichen in h enthaltenen Primideale \\ erstreckt, während sich die zweite Summe auf alle Primideale \\, erstreckt, von welchen erst die zweite Potenz in n enthalten sind, usw. Da nun (s — 1) H QX."^) lür s=\ endlich ist, so erhalten wir die für .s>l geltende fundamentale Beziehung ^' ^ ^ log -_L_+ /•(.-, (1) N(p)^ h 5-1 wo sich die unendliche Sunuue auf die sämtlichen in ii enthaltene Primideale p erstreckt, und wo /(.s) eine Function von s ist, Avelche für .5=1 nicht positiv unendlich wird.'^ Die oben fin- die Zahlengruppe o gestellte Forderung wird erfüllt, wenn o die Gruppe der zu m primen Zahlclassen nach dem Modul ii; ist, mit oder ohne Vorzeichenbedingung von der in § 1 erwähnten Art, und dementsprechend g die (lesamsheit der zu lit primen Ideale des Körpers ist. In dem Falle, wo o die Gruppe der sämtlichen Zahlen o. ist, welche die Congruenz a = '[, (m) befriedigen, also aus einer einzigen Zahlclasse mod. m besteht, dem Falle, worauf es im Wesentlichen ankommt, bestätigt man durch die bL'kannte Methode der Volumenbestimnnmg,"'' dass wo n den (Irad iXar^ Ivörpers k, ^ die Anzahl der Paare con- jugirt imaginären unter den mit k conjugirten Körpern, d die 1) Diese Sclilüsse bleibt offenbar gültig, wenn nur die Priuiideale ersten Grades in die Suuiuie anfgenouiuicn -werden. 2) Vgl. H. Weber, Lelirbucli der Algdna, II. 20 und 21 Absei-., auch ZahUngruppcn,. Math. Ann .49 S . 90-94. Uelior oliiL' Tlu'dvie iks relativ Alierschen Zahlkorper?. 2.1 Discriminante des Körpers k. N(m) die Norm des Ideals m im Körper k, w die An/ai d der Einlieitswenzeln in o, L den Regulator'^ des Systems der Fundamentaleinheiten in o Ijedeuten; es ist vorausgesetzt, dass für die Zahlen in <> alle Vorzeichencom- binationen zugelassen werden. Eine Idealclasse nach <>, d.h. die Gesamtheit der Ideale o. j, wo i 'ein gegebenes zu m primes Ideal, « eine ganze oder gebrochene zu in prime Kurperzàhl ist, derart, dass (A = \, (in) neanen wir eine Congruenzclasse nach dem ÙModul m. ein System solcher Classen, welche sich durch Multiplication und Division reproduciren eine Comjruenzclassenl '' ^ ^ log A +fis), N(p)^- h s-l 1) Diriclilet-Dedekind, Vorlesungen über Zahlentheorie, 4. Aufl. S. 597. 2) Gemeint ist der Index von h in Bezug auf die Oruppe der sämtlichen Classen von k, eine abkürzende Bezeiclmung, die in den folgenden durchgehend teiljehalten wird. 12 Art. 9.— T. Taka.^i : (vo fiß) eine Funcüou der reellen Ver'aaderüchen s ist, welche nicht positiv unendlich wird, wenn sich s abnehmend der Grenze 1 nähert. Ist nun a ein zu m relativ primes Ideal, dann gibt es in der -Zahlengruppe o eine durch a teilbare ganze Zahl a. von der Art, dass a-.ix relativ prim zu einem l)eliebig vorgeschriebenen Ideal c ausfällt. Denn sind q, q',... die von einander verschiedenen Primfactoren xow c, welche nicht in m aufgehen, dann gibt es bekanntlich eine durch a teilbare ganze Zahl a^ derart dass «o^a durch keines der Ideale q, q', .. teilba]* sind. Bestimmt man ■dann o. aus den Congru en zen (Hl). wo [j eine in o enthaltene, folglich zu in prime Zahl bedeutet, dann befriedigt a die gestellten Forderungen. Aus dieser Tatsache folgt unmittelbar, dass jedes zu m prime Ideal a als den grössten gemeinsamen Divisor zweier in o enthal- tenen ganzen Zahlen ;r, p dargestellt werden kann. Ist nämlich j! eine durch a teilbare Zahl in o, p ebenfalls eine solche Zahl, dass jedoch p : a prim zu ;<■ : a ausfällt, dann ist in der Tat Ferner folgern wir noch die folgende wichtige Tatsache: Satz 2. In Jeder Classe a nach o gibt es Ideale, die zu einem beliebig gegebenen Ideale c relativ prim sind. Be\veis. Sei a ein beliebiges Ideal in der zu a reciproke Classe A~', a eine durch a teilbare Zahl in o: a = a b, derart, dass h prim zu c ausfällt. Da dann b der Classe a ange- hört, so ist der Satz bewiesen. Wenn daher von den Idealen jeder Classe einer Classen- gruppe ir nach dem Modul m, nur die beibehalten werden, welche relativ prim zu einem beliebigen Ideal c sind, dann bleiben die Classenzahl ungeändert. Eine solche Classengruppc kann aber auch aufgefasst werden, als eine Classengruppc nach dem Modul lieber eine Theorie dos relativ AlicrHchen ZahlkcJrpers. ] 3. m', wo in' (las durch in teilliaro Ideal bedeutet, welches dadurcli ans m entsteht, wenn demselben alle in c enthaltenen Primideale als Factoren hinzugefügt werden, die nicht in m enthalten waren. In diesem Sinne ist eine Classengruppe nach dem Älodul in zugleich eine Classengruppe nacli jedem durci i m teilbaren Modul m'; nur spielen dabei einige Factoren von in' die Rolle der zur Ciasseneinteilung unwesentlichen Excludenten. Ist allgemein it eine Classengruppe sowohl nach dem Modul lUi als nach iiio, und ist m der grösste gemeinsame Divisor von nii und in2. dann ist h eine Classengruppe nach m. Denn sei «o eine zu nil und nto prime Zahl, die der Congruenz: '^0=1, (in) (2) genügt, also «0 = 1 + /^, wo /^ durch \\\ teilbar, folglich in der Form darstellbar ist: /^ = "'^l + ''Î2. wenn mit '-^i und ^2 bez. durch nii und wu teilbare Zahlen be- zeichnet werden. Setzt man daher dann bestehen die Congruenzen « = «0, (nil) ; a = l, (nu) ; folglich ist a prim zu nti und zu nu-. Nach der zweiten Congruenz ist das Ideal («) gewiss in h enthalten, und weil n auch eine Classengruppe nach dem Modul nti ist, so folgt aus der ersten Congruenz, dass («0) in 11 enthalten sein muss. Da aber «0 eine beliebige der Congruenz (2) genügende Zahl ist, so ist unsere Behauptung nachgewiesen. Demnach gibt es unter allen Moduln nt, die dieselbe Classen- gruppe H definiren, einen bestimmten von kleinster Norm. Der- selbe nennen wir der Führer der Classengruppe 11. ] 4 Art. 9.— T. Takagi : Ein Fundamenlalsatz uber die relativ normalen Körper. Satz 3. \\\:n)i K ein relativ normaler Körjier vom Relativgrade n l/i Bezug auf dem' Körper k ist, und wenn p, alle Frimideale vom Grundk'Jrper ]c durchläuft, ivelche in ]v /y/ die von einander verschiedenen Primideale des ersten Tielatirr/rades zerfallen., dann ist für s>-] ^^'1 1 1 2' = loo- + Fis), um F{s) eine Function des reellen Veränderlichen s ist, die endlich bleibt, icenn sich s abnehmend der Grenze 1 näherte Beweis. Das für s:>l absolut convergente, auf alle Primideale ^4> von K mit Ausschluss von den endlichvielen, in die Relativdif- ferente von K/k aufgehenden, zu erstreckende, unendliche Product % 1 11 1-N^e-P)- wo Nk die Norm im Körper K bezeichnet, lässt sich wie folgt umformen : wo sich das erste Product rechts auf alle Pi'imideale p, von k, das Product Jl auf alle Primideale P/ von k, welche in K in e von einander verschiedene Primideale des/ten Pelativgrades zerfallen, Av<> /' = >1, endlich das Product ^ sicli auf alle von 1 ver.-chie- denen Teile]' /' von y?, erstreckt. Geht man in die Logarithmus über, so erliäk man 1) Für den al:)sulut normalen Körper, vgl. Hilbert, Bericht, S. 265 (Satz 84). Dieser Satz bleibtauch gültig, -wenn nur die Priuiidealo pi vom ersten (absoluten) Grade in die Summe aufgenommen werden, worauf es im wesentlichen ankommt ; vgl. die Fussnote ') aif S. 10. wo Ueber eine Theorie dos relativ Aliel'schen Zahlkörpers. 15 S-rA 1 V 1 1 1 v__L_ + + 1^-- i + 1 V JL + ] ' 1 i. 1 \ N(i)--'^- +- N(iy^^ + j Î 1 il = " ^' =^ <2^^2' N(i/[N(J>>-1] ^ "- NO)- ' wenn 2' eine über alle von dem Einheitsideal verschiedenen ganzen Ideale von k zu erstreckende Summe l)edeutet. S ist also eine für -s> j— absolut convergente Dirichlet'sche Reihe, und gelit für s=l in einen endlichen Grenzwert über. Da bekanntlich Lim endlich ist, so ist unser Satz bewiesen. Von diesem Satz machen wir eine Anwendung auf einen Specialfall, um eine Tatsache herzuleiten, die wir später einmal benutzen werden. Sei K relativ Abel'sch über k vom Relativgrade /'. Avelcher aus t von einander unabhängigen relativ cyclischen Korpern vom Primzahlgrade l zusammengesetzt ist. Sehen wir von den in einer endlichen Anzahl vorhandenen, in die Relativdiscriminante aufgehenden Primidealen ab. dann zerfällt ein Primideal von k in K entweder in /' von einander verschiedenen Primideale vom ersten Relativgrade oder in /'"' vom / ten Relativgrade; dieses letztere zerfällt dann in einem Unterkörper K' vom Relativgrade /'"' in die Primideale vom ersten Relativgrade; es ist nämlich K' der Zerlegungskörper für jedes der Z'~^ relativconjugirten Primideale von K (K muss relativ cyclisch in Bezug auf K', also hier vom Relativgrade / sein). Bezeichnen wir die Primideale der ersten Art durchweg mit Pi, die der zweiten Art, welche einem bestimmten Köii:)er K' 16 Art. 9.— T. Takagi : entsprechen, mit p.,, dann folgt ans Satz 3, angewandt auf K und K', dass Vi ] 1 1 N(ftr / ^ s-i ' n\ i V2 1 . 1 1 ( ^ h 1 1 — log- folglich auch ^{ViT N(p,)^ / V-' ^ s-1 ' ^ 1 I-l . 1 endlich bleiben, wenn sich der reelle A^eränderliche s abnehmend der Grenze 1 nähert. Die Primideale \h sowie \h sind daher in unbegrenzter Anzahl vorhanden. Enthält k die primitive f° Einheitswurzel, dann lässt sich dieses Ergebnis wie folgt ausdrücken: Es seien «i, «o, «^ ganze Zahlen des Körpers k, welche die primitive V Einheitswurzel enthält, wo / eine natürliche Primzahl ist, von der Art, dass keine der /'— 1 Producte mi die man erhält, wenn man jeden der Exponenten die Werte 0, 1, 2, /— 1 durchlaufen lässt, mit Ausschluss eines Wertsystems 0711=7712=... =7711=0, dieZ*" Potenz einer Zahl in k wird, ^^ind dann Çi, 1^2, ^i beliebig vorgeschriebene/''' Einheitswurzeln, dann gibt es in k stets unendlich viele Primideale p vom ersten Grade, für welche ilH'irH (-?-)=^" wo (~„ ) den /'"' Potenzcharacter und e eine gewisse von p abhän- gige nicht durcli / teilbare ganze rationale Zahl ist. '^ In der Tat, wenn zunächst ç„ çg ^/ sämtlich gleich 1 sind, werden durch die gestellte Forderung diejenige Primidealo von k characterisirt, die im relativ Abel' sehen Oberkörper K=k (^^, -i/a., v^«t) vom lielativgrade l' in die Primideale vom ersten 1) Vgl. Hubert, Bericht, Satz 152. Ueber tine Theorie des relativ Abel'schen Zahlkürpers. ]^7 Relativgrade zerfallen. Ist dagegen etwa Çi =1= 1, dann Ijestimme man ^—1 ganze rationale Zahlen 7u Ut so, dass ^1 s.i — 1, Si Çt — -•-> nnd setze dementsprechend ■II ■> Q lit O Dann lässt sich die gestellte Forderung umformen in: Sie werden durch diejenige Primideale p von k erfüllt, Avelche in dem relativ Abel' sehen Körper K'=k (-v//\ ^ßt) vom Relativ- grade V~\ nicht aber in K, in die Primideale vom ersten Re- lativgrade zerfallen. Die über diese Primideale erstreckte Summe wird daher nach Satz 3, für s=l unendlich wie - N(p} V/'-i V- I ^ s-l womit unsere Behauptung bestätigt wird. §• 4. Der Classenkörper. Es sei K ein relativ normaler Oberkörper von k vom Relativgrade ?^; die Idealclassen in k seien nach dem Modul m definirt. Die Gesamtheit derjenigen Classen von .k, welche Relativnormen der zu m primen Ideale des 01)erkörpers K enthalten, bildet dann eine Classengruppe, die wir mit ii bezeichnen, und es sei h der Index von ii in Bezug auf die vollständige Classengruppe von k. Der Körper K und die Classen- gruppe H bezeichnen wir als einander zugeordnet. Die zu m primen Primideale von k, welche in K in die Primideale des ersten Relativgrades zerfallen, sind demnach sämtlich in den Classen von h enthalten, womit nicht gesagt wird, dass umgekehrt jedes in einer Classe von h enthaltene Primideal 18 Art. 9.— T. Takagi : Von k ill die Priinideale des ersten Relativgrades in K zerfällt. Wenn der Relativgrad des relativ normalen Körpers K und der Index der zugeordneten Classengruppe u von k einander gleich sind, dann soll K der Classenkörper für die Classengruppe H genainit werden. jMit Hülfe der Sätze 1 und 3 folgt aus der obigen Definition der folgende Satz, welcher in der Folge von einer fundamentaler Bedeutung ist. Satz 4. Der Belatlvgrad des relativ normalen Körpers ist niemals klenier als der Index der zugeordneten Classengruppe des Gruîididhjyers. Beweis. Nach Satz 1 ist, Avenn p die sämtlichen in der Classengruppe n enthaltenen Primideale von k durchläuft, wo h der Index der Classengruppe h ist, und/(s) eine Function der reellen Veränderlichen s, welche für s=l unter einer endlichen positiven Schranke bleibt. Die sämtlichen zu m primen Primideale von k, welche in K in die von einander verschiedenen Primideale vom ersten Relativgrade zerfallen, die wir durchweg mit Pi bezeichnen, sind in h enthalten; wir bezeichnen die übrigen in ir enthaltenen Primideale durchweg mit p'. Dann ist V 1 V] 1 v[ 1 ~ N(p)^ =" ^ N(pO- + ~N(p')^' und nach Satz 3 . Vi 1 w log — + F(.s), (.s>l) Ni^Pi)* n s—1 wo n der Relativgrad von K/k ist und F(s) eine Function von s, die für s=] endlich bleibt. Demnacli liat man für .s>l. Da/(s)— 7^(.s) nicht positiv unendlich wird, wenn sich .s' abnehmend der Grenze 1 nähert, so folgt hieraus Ue':H^i" eine Theorie des relativ Aliel'sclieu Zahlkörpers. ][9 1 1 oder k n n ^ h, womit der Satz bewiesen ist. Dieser Schluss bleibt, wie man sofort erkennt, auch dann gültig, wenn nur vorausgesetzt wird, dass die in ii enthaltenen Primideale vom (absolut) ersten Grade in die Primideale vom ersten Grade in K zerfallen, sogar mit einer endliche^ Anzahl Ausnahme, oder unendlichvielen, wenn nur die über diese Ausnahme-ideale erstreckte Summe --^rr-y-für .s=l endlich bleibt. Eine wichtige Folgerung des obigen Beweises ist die, dass, wenn n=h, also wenn K Classenkörper für die Glassengruppe it ist, die Function /(.S-) notwendig für s=l endlicli bleibt. Dann sind die Grenzwerte für s=l von den Reihen Q,{s) (. = 2, 3, h) (§ '2, ^. 0) von Null verschieden, und liieraus folgt, die folgende wichtige Tatsache": Satz 5. I/i einem beliebigen algebraischen Körper eMstirt in jeder Classe nach dem Jlodid m eine unbegrenzte^ asymptotisch gleiche j'^ Anzahl von Primidecden ersten Grades; speciell existiren, wenn tx eine beliebige, a eine zu ij. prime, ganze Zahl des Körpers ist, unendlichviele ganze Zahlen, öj in dem K>r]jer, die der (Jongruenz x:j =.a, (fi) genügen , und von der Art sind, dass {vs) unendlichinele Primideale des ersten Grades darstellen; (dies unter der vorläufigen Annahme, dass es für jede Glassengruppe II eines beliebigen Körpers einen entsprechenden Classenkörper gebe, was tatsächlich der Fall ist, wie in der Folge bewiesen werden wird). Wir fügen liier noch einen Hülfssatz hinzu, d^n w^ir später nicht wohl entbehren können. 1) H. Weber, Zahlengruppen, Math. Ann. 49, S. 89. 2) E. Landau.Ueber die Verteilnno- der Primicleale in dm Idealklassen eines algebraischen : Zahlkörpers, Math. Ann. 6?.. S. 196-197. 20 Art. 9— T. Takagi: Hülf.ssatz. Sei K/k ein relativ normaler Körper vom Relativgrade >^ ir eine Classeiigruppe in k vom Index /?, welche nicht dem Körper K zugeordnet zu sein braucht. Dann gil)t es in k unendlichviele Primideale (ersten Grades), die nicht einer Classe vom h angehören, und auch nicht in K in die Primideale vom ersten Relativgrade zerfallen.'^ Beweis. AVir beweisen diesen Satz nur in dem Falle, wo h:::^'!. weil wir ihn spater nur für eine Classengruppe eines ungeraden Primzahhndex anwenden werden. Nach Satz 1 gilt für die ül.>er alle nicht in h enthaltene Primideale erstrecke Summe wo •^(n) l'ür s=l endlich oder positiv unendlich wird. Anderseits ist -log^- 4-FGs), N(p,)* n ° s-l wo 7''Çs) für .$=1 in einen endlichen Grenzwert übergeht, wenn die Summe auf alle Primideale \\ erstreckt wird, die in K in die Prim ideale des ersten Relativgrades zerfallen. Wenn nun A>'2, dann ist jedenfalls h-l 1 woraus der Satz fokt. h Eindeutigkeit des Classenkörpers. Satz 6. fSeien ir, ji' Cldssengrvppen in k; K, K' Icz. die Klassenkwj)er für dieselben. Ist dann ii' Unter gru'p'pe von ir, dann ist K' Oberkörper von K. Fiir eine Classengrnpp)^ kann es daher nicht mehr als einen Classenkörper geben. Beweis. Seien K/k, K'/k bez. vom Relativgrade 7i, n'; der 1) Vo-l. Ph. Furtwiingicr, Math. Annalen 63, S. 23. Ueber cino Thooric ilfs relativ Alicrsf.lieu Z:ihl!i<)rpers. 21 aus K und K' zusammengesetzte Körper K* ist dann wieder relativ normal, er sei vom Relativgrade n.''' Seien ferner aS'i, ä, Sx die auf die Primideale p von k erstreckten : Summen .• 1 und zwai- erstrecke sich A', auf die sämtlichen Primideale, die sowohl in K als auch in K', folglich in K* in die Primideale des ersten Relativgrades, S^ auf die, welche in K aber nicht in K', S^ auf die, welche in K' aber nicht in K, in die Primideale des ersten Jielativgrades zerfallen. Dann ist nach Satz 3 'S\= \ log-I— + F,(4 .S', + ,S',= ^ log— 1— + F,Ù), Il s — i S, + S,=^ \ log^_+K,(,), n s — i ^.Avo die Functionen 7^(.s^) für .s=] endlich bleiben. Hieraus erhält ,man ,S, + S,+ß, = (^^ + -l^ \A^og-J— + G{s), (1) \ 11 II. If / S — L WO auch G(s) für s=] endlich ist. Anderseits ist, nach Annahme, die Classengi'uppe ir vom Index n; ferner soll ir alle oben in die Summen S\, 1S2, /S3 r aufgenommenen Primideale, und möglicherweise noch die anderen, ^ enthalten, von welchen letzteren auf einer ähnlichen AVeise die Summe iS' gebildet sein möge. Alsdann ist nach Satz 1. :S, + S,^S,+ S'= ^ log-J_ +/(.), (2) n s — i ■.wo /(.5) eine Function von s ist, welche unterhalb einer endlichen positiven Schranke bleibt, wenn .l woraus zu schliesseu ist, class oder n nr Da aber n"^ n, so erliiilt man Also fällt der Körper K'^'' mit K' zusammen, d. h. K ist in IC enthalten. Wenn nun K' aueh der Classenkörper für h ist, dann nuiss- nach dem eben bewiesenen K' in K enthalten sein. Daher fällt K' mit K zusammen: es kann daher nicht mehr als einen Classenkörper für h geben. Wir bemerken noch, dass die obigen l>chlüsse gültig bleuten, Avenn nur vorausgesetzt wird, dass die Primideale von k, welche bez. in den relativ normalen Körpern K und K' in die Primideale vom ersten Relativgrade zerfallen mit endlicher Anzahl Ausnahme bez. in h und h' enthalten sind. Dasselle gilt auch dann noch, wenn nur die Primideale ersten Grades von k in Betraclit gezogen, werden. CAPITEL II. Die Geschlechter im relativ cyclischen Körper vom Primzahlgrade. § G. Einige allgemeine Sätze über die relativ Abel'schen Zahlkörper. 1ji diesem Artikel fassen wir einige Krätze ül>er die relativ Abel' sehen Körper zusammen, die wir in der Folge wiederholt Ui'liei- oiue Theorie dis relativ Aliel'selien Zalilkörpers. 23 anzuwenden haben. Es sind die Sätze, welche die Zerlegungs- Trägheits- und Verzweigungs-körper eines Primideals betreffen, die zuerst von D. HilberL'^ für die absolut normalen (Galois' sehen) Körper aufgestellt, und von IL Weber^^ für die relativ normalen Körper verallgemeinert worden sind, und die wir hier für die relativ xVbel' sehen Körper specializiren werden. Sei K/k relativ Abel' seh vom Kelativgrade n. Ein Primideal p vom Grundkörper k wird in K auf einer folgenden Weise in die Primfactoren zerlegt: v={%%. %y, wo n-egf, und /" der Relativgrad"'^ von jedem der relativ conjugirten Ideale "^v '^^2, -4?« von K in Bezug auf k ist. Die Zerlegungskörper von diesen relativ conjugirten Prim- idealen in Bezug auf k sind, wenn K relativ Abel' seh ist, ein und derselbe 0]:)erkörper von k, so dass wir berechtigt sind, ihn als der Zerlegungskörper für das Primideal p im Oberkörper K zu bezeich- nen. Gleiches gilt für den Trägheits-, und Verzweigungs-körper. Der Zerlegungskörper K- für p ist vom Relativgrade e in Bezug auf k, er ist der grösste in K enthaltene Oberkörper von k, in welchem p in die von einander verschiedenen Prim ideale des ersten Relativgrades zerfallt. Der Trägheitskörper K^ für p ist vom Relativgrade ef in Bezug auf k, und relativ cyclisch vom Grade /' in Bezug auf den Zerlegungskörper K,. Er ist der grösste in K enthaltene Ober- körper von k, dessen Relativdiscriminante prim zu p ausfällt. Der Verzweigungskörper K„ für p ist relativ cyclisch in Bezug auf den Trägheitskörper K,, dessen Relativgrad ein Teiler von^-^^'-I ist, wo j9^ die Norm von p in k, alsop'-^' die Norm von '-P in K ist; dieser Relativgrad ist als der grösste Teiler von y bestimmt, welcher 1) D. Hilbert,Grunclzüge einer Theorie des Galois'schen Zahlkörpers, Göttinger Nachrich- ten, 1894; vgl. auch Bericht, §§ 39^7. 2) H. Weber, Lehrbuch der Algebra, 11. (2 Aufl.) 19. Abschnitt. 3) H. Weber, 1. c. S. 645. 24 Art. 9.— T. T.ikai?i prim zu p ist. Wenn (/ durch p teilbar ist, dann sind zwischen K, und K die Verzweigungskörper höheren Grades K',, K^, ein- zuschahen ; die Kelativkörper K',/K,,, K ['I K ', sind relativ Aber seh und aus nicht mehr alsjgf" von einander unabhängigen relativ cyclischen Körpern jö'"" Grades zusammengesetzt. Es ist ^-l^'' ein Primideal in K,, dasselbe wird in K/K< in die g te Potenz eines Primideals ^^5 zerlegt, welches vom ersten Uelativgrade in Bezug auf Kl ist. Wir heben speciell die folgenden Sätzen hervor. Satz 7. Ist K/k relativ ci/cUsch vom Primzahlpotenzgrade V, und gellt ein zu l primes Prlmkleal \> von. k hi die Rehtivdiscrimi- nante des in Iv enthaltenen relativ cyclischen Oberkörper von k vo7n Relativgrade l auf, dann ist die Relativdiscriuiinante von K/k genau durch die V — 1" Potenz von p teilbar ; ferner ist N (p)=i, in, wo N die in k genommene Norm bedeutet. Satz 8. Es sei K/k relativ cgcUsch vom Primzahlgrade l, ferner sei i ein in l aufgehendes Primideal von k. Wenn dann die Relativdis- criminante von K/k durch i teilbar, dann ist sie genau durch die {v+ 1) (Z— 1) te Potenz von i teilbar, wo v :> 0. Die Zahl v ist dadurch characterisirt, dass für jede ganze Zahl A von K die Congruenz besteht: lüo s eine erzeugende Substitution der Galois' selten Gruppe des Relativ- körpers K/k, s.l die relativ conjugirte Zahl von A, und i' das in i aufgehendes Primideal von K bedeutet. /Speciell ist, wenn A genau durch die erste Potenz von S teilbar ist, hA-A genau durch die ?•+!''' Potenz von. Ü feilbar. '-* Für die Zahl v gilt die Beziehung wenn s der Plvponent der höchsten in. l aufjeheiulen Potenz von i id. Ferner ist v nur dan?i durch l teilbar, wenn 1) Hubert, Bericht, § 44, 47 ; es ist v + 1 tier dort mit L })t'zeiclmete Expanent. Ueber eino Theorie des relativ Abel'scheu Zahlkörpers. 25 ^,_ si l-\' (al^;o wenn •>> duivh /— 1 teilbar ist). Beweis. Es genügt, den zweiten Teil des Satzes zu beweisen. Sei A eine genau durch die erste Potenz von .^ teilbare Zahl von K. Ist dann A genau durch 2' teilbar, dann kann man eine zu 2 prime Zahl B so Ijestimmen, dass ä=Bä% (fi"), (1) won ein Ijeliebig grosser Exponent sein kann. Ist nun ^'^(». (/), dann ist sj'— j' genau durch S''+'' teilbar, daher auch sA-A=B(sA'-Ä'') + (isB-B)sA", (S") genau durch S''+' teilbar, weil das zweite Glied rechts wenigstens durch i:î''+'+^ teilbar und nach Annahme u>v + e ist. Ist aber .6=0, (/), dann kann man in (1) a' durch eine Zahl À von k erset- zen, welche genau durch die eiT" Potenz von l teilbar ist; man erhalt dann sA-A = (^B-By, (S"), folglich ist s^I— ^1 gewiss durch eine höliere als die v+e^"" Potenz von 2 teilbar. Bildet man daher aus der Zahl A:=hA — ä wieder die Zahl ^Tl<;=s.ii— .II, und sofort, bis man erhält ^„=syi„_,—^4„.i, welche letztere Zahl A„ symbolisch mit bezeichnet sein möge, dann ist dieselbe genau durch die e + iiv^ Potenz von ,ß teilbar, wenn keine der n Zahlen e, e+v, e+'2v, ■e+(n—l)v durch / teilbar ist, andernfalls aber gCAviss durch eine höhere als die e + nv^" Potenz von S teilbar. Vermöge der Identität +i(x-iy--+(x-iy-' schreiben wir nun die Relativs^^ur von A in der Form: 26 Art. 9.— T. Taka^-i : Das ers^e Glied auf der recliten Seite ist genau durch die sl+V^, alle folgenden Glieder bis auf das letzte durch höhere Potenzen von i^ teilbar; das letzte Glied aber niöge genau durch S" teilbar sein. Nach dem vorhin Bemerkten ist dann a.>l + v{I — i), ausser wenn ^' = 0 oder =1 (/). Da der l^xponent der höchsten in S(/l) aufgehenden Potenz von v; durch / teilbar sein muss, so ist jedenfalls sl + l^a. (2) Hieraus folgt für ^'^O, ^1, (/), s/>v(/-l). Dassell)e muss aber auch für r=], (/) gelten, weil dann durch / teilbar, folglich das Gleichheitszeichen in (2) ausge- schlossen ist. Wenn endlich ^ = 0, (/), so ist a=l+v(l—l) nicht durch / teilbar, daher muss in (2) notwendig das Gleichheitszeichen gelten, also womit der Satz bewiesen ist. Wenn k die primitive r Einheitswurzel C enthält, und wenn ein Primideal ( genau zui- ^**" JV)tenz in (1— C) aufgeht, dann ist s=(7 (/—]). Ist dann /^- eine Zahl von k die genau durch eine Potenz von i teilbar ist, deren Exponent zu / prim ist, dann geht I in die Kelativdiscriminante (\e>^ relativ cyclischen Körpers K = k (^/i) auf, und die entsprechende Zald v nimmt den grösstmög- licheii Wert al an. Wenn dagegen /< nicht durch / teilbar ist und m der höchste Exponent bedeutet, fin- den es eine Zahl a in k gibt, so dass // = «/(r"), dann ist die Kelativdiscriminante von K = k(^//) nur dann durch l teilbar, wenn j/Knl. Jn diesem Uel)cr c'ino Theurie des relativ Alielsclun ZalilkiJqX'rs. 27 Falle ist cihov m notwendig prim zu /. Für die entsprechende Zahl V erhält man den ^\'ert r=al—m,. Denn die Zahl A==a — ^t von K ist genau durch l^", und ■: Endlich sei noch das folgende bemerkt: Ist K/k relativ cyclisch vom (Irade /'' , und wird mit K^"^ der in K enthaltene relativ cyclische Oberkörjier von k vom Relativgrade /" (v=l, 2,--- •••//) bezeichnet, und geht { in die Kelativdiscriminante von K^^Vl"^ auf, dann zerfällt l in K in die Z'"" Potenz eines Frimideals; die Kelativdiscriminante von K/k enthält dann l genau zu der Potenz mit dem Exponenten: /"-\'^ + l)(/-l) + /"-\r,+ l)(/-l)+ +K_i+1)(7-1) = /"-l + (/-l)[r/"-i + r/-^+ +n.i], WO ^'l, v.y, dieselbe Bedeutung für K '" '/K* ". \\}''^ jl\-^''% haben, wie v für K*^'7k, und es ist 1 ^ Vs die Relativnorm in Bezug auf k bezeichnet wird. 1) Vgl. Hubert, Bericht, Satz 148, wo die hier angedeuteten Tatsachen für den Kreiskörper k Iiewiesen wird ; dieser Beweis ist leicht auf den allgemeinen Körper k zu übertrasren. 0,S Art. 9.— T. Talcagi : lieber die Normenreste nacli Primidealpotenzen in k gilt der folgende fundamentale Satz. Satz. 9. Ks sei Kik relativ cycüsch vom Prlmzcthlgrade l. (I) Wenn dann \> ein Primideal in k ist, welches nicht in die Relatlv- (liscrimi'ncinte von K/k aufgeht^ dann ist jede zu p prime Zahl in k Normenred des Korpers K nach jeder Potenz von p. (II) Wenn dagegen p in die Relativdiscriminante aufgeht, jedoch p jyrini zu l ist, dann ist, von allen zu p primen und einander nach p incongruenten Zahlen in k gemiu der V" Teil Normenreste nach p , liier bedeutet e eine beliebige naiùrliche Zahl. (III) Dasselbe gilt auch für die Potenz {"eines in l aufgehendes Priniideals i von k, falls i zur Potenz i^-'+ix^-i) in die Relativdiscriminante aufgeht, und eJ^-v ist. Dagegen ist jede zu i 2Jrime Zahl in k Normenrest nach V, ivenn e^v ist. Hier hat die Zahl V die in Satz 8 angegebene Bedeutung. ^^ Beweis (I). Wir unterscheiden vier Fälle, jenachdem p in / aufgeht oder nicht, und p in K zerfällt oder nicht. Zunächst sei p prim zu /, und es zerfalle p in K in / von ein- ander verschiedene Primideale: Sei /der Grad des Primideals p in k, also auch der Primideale ^;p, ^:p', in K, und es sei /> eine Primitivzahl nach p. Jede Zahl u in k, die zu p prim ist, genügt dann offenbar einer Con- .ü;ruenz der Form •ö wo n- eine Zahl aus der Reihe 0, 1, li, ;>--, und a^ eine ganze Zahl in k ist, welche die Congruenz ao=\, (p) beh'iedigt. Demnach ist «o ein /ter Potenzrest nach p' : 0.0 = /, (p"). Ferner sei eine Zahl P in K so bestimmt, dass 1) Vgl. Hubert, Bericht, §130, wo der Satz für dou Kreiskörper der Zt,«u Einheitswurzeln ■auff^estcllt ist, allerdinj^-s ohne i>-enaue Angabe des critisdien Wertes des Exponenten ein (III.) Ue1)er eine Theorie des relativ Abel'sehen Zahlkürpers. 29* P = ^>. 0^5"). =1, OrT"^ ); dann ist demnacli womit der Satz im vorliegenden Falle bewiesen ist. Zweitens sei p prim zu /, und es bleibe p=^^ prim in K. Ist dann P eine Primitivzahl nach "^ in K, dann ist offenbar eine Trimitivzald nach p in k. Da jede zu p prime Zahl a in k einer Congruenz der Form genügt, wo «0=1 (p) ^iiiti folglich «o=/''. (v"), in k? !=^'> ist auch in diesem Falle a = ^(rP'% (pO. Drittens, sei p=i ein in / aufgehehendes Primideal von k, welches in K in / von einander verschiedene Primideale zerfällt, l — QQ/Qrf «(^-1). Da jede zu i prime Zahl in k offenbar P''' Potenzrest von t ist, so ist unser Satz richtig für die erste Potenz von i. Angenommen nun, es sei eine zu ( prime Zahl a Normen- rest nach t. Wir setzen wo ?^ eine genau durch die erste Potenz von [ teilbare Zahl in k ist.- Bestimmt man dann eine Zahl ß in K gemäss den Congruenzen: 8=1, (ß), =0, (2'2'' ). so dass für die Pelativspur von 6 gilt: S(Ö) = 1, (i), dann ist •N(l + |6';0 = l + .-^% (V''), 30 Art. 9.— T. Takag-i : wenn ç eine beliebige Zahl in k ist. Demnach hat man Da man nnn ^ gemäss der Bedingmig bestimmen kann, so ist erwiesen, dass « Normenrest nach der höheren Potenz ï'+^ von t ist, nnd hiermit ist der ^^atz bewiesen. Zuletzt, sei I ein Primfactor von i in k, und f=V prim in K. Der Beweis verlauft genau wie im vorhergehenden Falle; nur muss die Existenz einer Zahl 0 in K, deren Relativspur prim zu I ausfällt, besonders bewiesen werden. Sei also P eine Primitiv- zahl nach Ü und P^ + «iF-i+ + «,=0 die Gleichung l^ Grades in k, welche durch P befriedigt wird. Wäre nun S (P") für «=1, 2, ^-1 durch I teilbar, dann musste, nach der Newton' sehen Formel für die Potenzsummen, die Coeffi- cienten «i, «o» 'h-\ durch t teilbar sein, also P' = N(P), (I). Alsdann Aväre WO /der Grad von I in k, also // der Grad von V in K ist, folglich z=i+z/+F+ +/"-^n (./'-IX was offenbai- immöglich ist. Daher gibt es in der Tat eine Zahl 6 in K derart, dass S(0) = 1, (l). Hiermit ist dei- Teil (I) unseres Satzes vollständig l)ewiesen. Beweis (IL) Sei v ein zu / primer Primfactor der Ilelativ- discriminante. iJann ist Uebcr eine Tlioorie «les I'elativ Alierschon Zahlkoi'pevs. Q]^ ■\vo ''l^ ein PriDiideal in K, nnd , (^:)3'=) =^(p«) =p^<^-'^f{2/- 1), wenn *I» bez.

(p)7 ^^iid 7«, eine Zahl aus der Reihe 0, 1, 2, p^— 2 ist. Es ist nun «o offenbar ein T"" Rest von p". Da nach Satz 7. 2?''— 1=0, (/), so ist /)" dann und nur dann ein T^" Rest nach p", wenn n durch / teilbar ist. Hiermit ist der Teil (II) unseres Satzes bewiesen. Beweis (III). Sei t ein Primfactor A'on / in k, welcher zur (v + l)(^— l)ten Potenz in die Relativdiscriminante von K/k aufgeht, ferner sei I=£^ ,wo S Pi-imideal in K ist. Wir bezeichnen in den Folgenden durchweg mit /, und j,, eine genau durch die e te Potenz bez. von l und ß teilbare Zahl von k und K. Für die Relativspur von Ae erhält man dann, wie beim Beweise des Satzes 8 ^Ae)=LL+[l){^-^)X-\-[l){^-irA,.+ + (s-iy-u,. Das erste Glied rechts ist nun genau durch die .s/+e** Potenz von S, alle folgende Glieder bis auf das letzte durch die höheren Potenzen, das letzte Glied aber wenigstens durch die e + v{l—lY^ Potenz von ( teilljar. Daher erhält man, wenn man die Relation: sl^v{l-l) 32 Art. 9.— T. Takagi. berücksichtigt (Satz 8), wenn er : N(l + Jj = l, (("+1). (4) Dies geschieht am einfachsten dadurcli, dass man mit Hülfe der Newton' sehen Formel über die Potenzsummen die Teilbarkeit der elementarsymmetrischen Functionen von Ac^ M^, «''M^ durch die entsprechenden Potenzen von l nach (1) bestätigt. Nach (2) und (3) folgt nun, dass N(i+/i„)=i, (r), dann und nur dann, wenn e ^ V, woraus weiter, dass für zwei zu ki prime Zahlen A, B dann und nur dann, wenn A~B, («'■). Berücksichtigt man daher die Relation dann ersieht man, dass jede zu l prime Zahl in k, Normenrest nach V und folglich nach jeder niederen Potenz von i ist. J)a, nach (4), auch für den Modul T^^, aus der Congrue nz A~B, (.Ü^+^), die andere: N(^)=N(£), (l"-*^) Uelier eine Theorie des relativ Abel'schen Zahlkörpers. 33 ZU folgern ist, so wird unser Satz für l'^^ bewiesen sein, wenn nachgewie^^en wird, dnss die Bedingung N(^) = l, (D+i) (5) durch genau / einander nach t'^^ incongi'uenten Zahlen befriedigt wird. Nach (2) kommt hierzu nur die Zahlen von der Form 1+A (6) in Frage. Es gibt nun in der Tat eine Zahl von dieser Gestalt, welche der Congruenz (5) genügt. Es ist nämlich yli— syli genau durch -''^^ teilbar. Bringt man daher den Bruch ^Ai'-Ai indie Gestalt s yJi _ Aq Al « wo a und .^0 ^'ti S prime ganze Zahlen bez. in k und K sind, und worin «=1 nach einer beliebig hohen Potenz von I angenommen werden kann, dann ist N(/lo) = a' = l, (V^'l Anderseits folgt aus aB Ai = AoAi, oder Ai (Ao — a) = a (s Ai—Ai), dass Ao—a genau dnrcli ^^" teilbar ist, demnach nach Annahme über « Nach (3) genügt diese besondere Zabi Af^ der Congruenz ^(Al°^j + ^(A'n=0. (r+1). (7) Für jede Zahl A von der Form (6) gilt nun A~l+pA?\ (V'^^), (8) also nach (4) und (.")) N(^) = 1 + ^. S( A^ + p' ^i Ai''), (1"+^). 34 -^rt. 9.— T, Takagi: ]L)alier ist dann und nur dann, wenn oS(.if:^>)+/>'N(,i^))=o, (r^), oder nach (7), wenn oder was dann und nur dann der Fall ist, wenn /> einer rationalen Zahl nach I congruent ist. Die Congruenz (5) wird daher genau durch l nach i"^^ incongruente Zahlen befriedigt, die man erhält, wenn in (8) ,0=0, 1,'2, /— 1 gesetzt wird, wie zu beweisen war. Ferner ist, wenn t eine positive ganze rationale Zahl, p eine zu I prime Zahl in k ist, N(i+M^iJ=i+M '^(AX (i"+'n also, da nacli (7), S(.lij genau durch î" teilbar ist, N(l+^o/,./lJ = l+/<+t. (9) Ist also a Normenrest nach t''^\ und zwar wo ,2^ zu (prim, und für /'.„^ dieselbe Zahl wie in (9) angenonnnen wird, dann ist Da man p aus ap + ß~0, (Q bestimmen kann, so ist a Normenrest nach I"^*^^ . Jeder Normen- rest nacli I'"''' ist daher Normenrest .noch jeder höheren Potenz \on I, und weil jeder Normennichtrest nach I"^^ umsomehr Normen- nichtrest nach jeder höheren Potenz von I ist, so ist liiermit unser Satz in allen seinen Teilen vollständig bewiesen. Ue1)ür eine Theorie des rekitiv Al^eFschou Zahlkörpers. 35 In der Folge benutzen wir den Satz 9 in der folgenden vemljgemeinerten Form: Satz 10. Sei K/k relativ cijdhck vom Primzahlgrade l, die Belativdiscriminante b von K/k enthalte d von einander verschiedene Primideale von k als Factor, derart, dass h = f-\ f = //p. //['^\ wo die Producie ^1 W ^^t"'^ bez. auf die zu l primen und in l wo Ç eine primitive T'" P]inlieitswarzel bedentet, letzteres nur dann, wenn K=k([c']), und folglich C=[^']'^'' in o enthalten ist. Indem wir auf diese Weise fortfahren, erhalten wir 7?=iT^(^) Hy^^H^' '^'-'\ßl (5) WO und die Coefficienten «i, ?// sämtlich Zahlen aus der Reihe: 0, 1, 2, l-l sind. Wir untersuchen nun die Annahme: es sei i=£r^(^) ifr^'^H^-'^'-\^l (6) Aus der Bedeutung des Einheitensystems //i, i/2, //« folgt zunächst Ui = ih= =7^„=0, so dass, für /=2, schon Fi(s) = 0, F,.(s)=0, und für ein ungerades /, i=(iî?'(^) iî?"(«^iz-(i-^)'-^'y-^[,-], (7) wo Eine Relation von der Gestalt wo H eine Einheit in 0 bedeutet, ist aber offenbar nur dann möglich, wenn N([^])=CT=], so dass [^] eine Z*' Einheitswurzel ist. Ist [^] = 1, dann ist // selbst, ist aber [^]=C eine primitive l^ Einheitswurzel, // eine Einheit in o; jedenfalls ist // selbst 3S Art. 9.— T. Takagi : einu Einheit, die wir mit [^] bezeichnen können. Demnach kann man statt (7) einfacli setzen: Die Einheit // anf der rechten Seite bringen wir wieder auf die Form so dass wir erhalten .l=iî^'(^) H^^'^'W-'^'-\$l wo F/Cs)-^^i' + ^^Al-s)+ + ut'Kl-^y-'' + v,(l-s/-', alniliclie Bedeutung wie Ei(8) haben. Daher folgt weiter v{=iiJ^ uj=0. So fortfahrend sieht man ein, dass, auch für ungerades /, aus ((3) notwendig folgt : Fi(«)=0, F,(s)=0. _ Daher sind die vi(/— 1) Einheiten H„ uJ-^ H^-^'-'^ iï„, Hi'-\ Hü-^y" unal)]i;ingig in Bezug auf die Gruppe der Einlieiten: Diese Gruppe ist aber identiscli mit der Gruppe der Einheiten: weil (i-s)'-^=i+s+ +s'-^(/), und anderseits (l-s)'-V(s) + (l + s+ +s'-iX^(s) = /, wu v'(s), v''(^) ganzzahlige ganze rationale Functionen von s sind.'^ 1) Für 'li(s) kann, man die / — 1 ersten Glieder der formalen Entwickelnng von //(1 + S + +s'-l)nach steigenden Potenzen von 1— snelimen. Uelior cine Theorie tks relativ Aliel'schen Zahlkörpers. g9 Daher lässt sich jede Einheit K von O in der Gestalt E=hI-^'^ Hf-^^fî'[çJ darstellen, wo Fi(s), E,.(^) niit E eindeutig bestimmt sind. Da die sämtlichen Einheiten in O und die Emheiten [?] //' bez r^^ und l''^\ oder /''" und /'' Einheitenverbände '^ in O aus- machen, jenachdem eine Einlieitswurzel, deren Ordnung eine Potenz von Hst, in O vorkommt oder nicht, so ergibt sich B-r=n{l-\). Wenn man hierin den Wert von R—r nach (2) oder (3) einträgt, so erhält man den im Satz angegebenen Wert von 71. Satz 12. 3Iachen die B elativnormen sämtlicher Einheiten in O l''" Einheitenverbände in o aus, dann gibt es in O f> Einheiten Ei, E2J Ep mit der Relativnorm 1, von der Beschaffenheit, dass jede Einheit in O mit der Relativnorm 1 in der Eorm : K'E'!r E''/H^-' (8) darstellbar ist, wo Ui, v->,\ n^ Zahlen aus der Reihe 0, 1, 2, l—l sind, und H eine Einheit in O bedeutet; diese Einheiten E^ E.y, Ep sind in dem Simie von einander unabhängig, dass eine Einheit der Form (8) nur dann gleich 1 sein kann, wenn Ui = n-i= Die Zald i> hat den Wert : f> = r + 1 + o — Vq, 'irenn l ungerade ist y ^/ = y + 1 + 1> — V — Vq, loenn I = 'l, wo 0=1 oder 0=0 zu setzen ist, jenachdem. die primitive l'' Ei nheifs- wurzel hl o vorkommt oder nicht. Beweis. Hier wiederum handelt es sich nur um die Be- stätigung des für /' angegebenen Wertes, da die Existenz des Einheitensystems E^, E., E^ mit der im Satz angegebenen 1) Unter einem Einheiten verband in O verstehen wir ein System der Einheiten in C' von der Form EW, wo E eine gegebene Einheit in O ist, und H alle Einheiten von (!• durchläuft. Vgl. Hubert, Math. Ann. 51, S. 21. 40 Art. 0.— T. Takagi : Eigenschaften ohne weiteres klar ist. Wir unterscheiden nun drei Fälle: Erstens sei vorausgesetzt: die primitive V^ Einheitswurzel C kommt niclit in o vor. Dann kann die Einheit [^J in (3) nur die Einheiten in o bedeuten, und weil es keine Einheit in o gibt, ausser der Einheit 1, mit der Relativnorm 1, so kann man E,, E^, Ep für i> der Einheiten H^, IL,, H„ in (8) nehmen, es seien diese Jf„.f,^i, If,^, sodass jede Einheit in O in der Form: E=H'^^ H'^iyE';' E'/H'-'l^] (0-Cw,^ (o^iL^i) und somit woraus nach Hinsetzen des im Satz 1 1 angegebenen Wertes von 7i und Berücksichtigung von «=0 der gesuchte Wert von p sich ergibt. Zweitens sei vorausgesetzt: es konnne C in o vor, jedoch sei K nicht durch die /** Wurzel einer Einheit in o erzeugt. Hier ist wieder die Einheit [?J in (3) die Einheit in o, und es ist C in dem System der Einheiten [?], nicht aber in //'"' enthahen. Wir setzen demnach sodass jede Einheit E in ( ) sich in der Form E=H^' H'J^1-/X E{' E^ IP-' [,-] (0 < M, y < /) +l, woraus mit d=l der gesuchte Wert von /> sich ergibt. §• ^). Formulirung eines Fundamenlalsatzes. Nachdem in den vorhergehenden die vorbereitenden Sätze erledigt worden, sind Avir nun im Stande, einen Fundamentalsatz zu formuliren, dessen Beweis das Hauptzweck der nachfolgenden Paragraphen dieses Capitels sein soll. 1) Wenn 1=2, ist -/jo durch — vio zu. ersetzen. 42 Art. 9.— T. Takagi: Satz 13. Die Relatlvdiscrhiünante des relativ cycUschcn Körpers K/lv vom ungeraden PrimzaJdgrade l ^ei b = f"\ ico \=IIX^. Il['^\ ICO p ein zu l primes^ mid i ein in l anfgeliende.A Primideal von k hedeidct. Die Idealclassen von k seien nach einer Zalilengrwppe o défini rt, welche aus den Zahlen a hestehi, die der Congrnenz: « = 1, (m) genügen, wo der Modul m ein beliebiges durch f teilbares Ideal von k id. Dann sind die Relativnormen aller zu \\\ primen Ideale von K in einer Classengruppe vom Inder l in k entltalten. Dasselbe gilt auch, für den relativ quadratischen Körper K = kCV/^). wenn an Stelle von o eine ZaJdengruppe ü mit gewisser Vorzeichenbedingung angenommen tcird. Es soll ndnilicJi nur die- jenigen Zahlen von o in n aufgenommen werden, ivelclie wenigstens in allen denjenigen mit k conjugirten reellen Körpern, worin g. negativ ausfällt, positiv sind.^^ Mit andern Worten : Jeder relativ AbeVsche Körper vom Prinizahlgrade l mit der Jïelativdiscriminante f'^ ist der Classenlcörper für eine Classengruppe nach dem Modul f.^-* §. 10. Die Anzahl der ambigen Classen im relativ cyclischen Körper eines ungeraden Primzahlgrades. Es sei K/k ein relativ eyclischer Körper von einem nngeraden Prinizahlgrade /, nnd es sei s eine erzeugende Substitution der Galois' sehen Gruppe des ]xelativkürpers K/k. Eine Idealclasse C des Körpers K heisst ambig, Avenn sie mit der i-ekativ conjugirten Classe sC identisch ist; im Zeichen: 0^-^=1. 1) Wenn ki ein mit k conjugirter reolk-i- Körper ist, dann soll eine Zahl a von k abkürzend als „ ijositiv oder negativ in ki " bezeichnet wei-dcn, wenn die mit a conjugirte Zahl in kl Ijositiv bez. negativ ausfüllt, ungeachtet des Vorzeichens von ot selbst oder auch wenn a selbst imaginär ist ; diese Abkürzung wird in den folgenden durchgehend beibelialten werden. 2) Vgl. § 4. Uelier eine Theorio dos relativ AI)crselu'Ti Zalilkiiri^ers. 43' Eine Classe ist ami »ig, wenn sie ein Ideal des Griindkürpers k, oder ein anibiges Ideal des Relativkörpers K/k, oder aber ein Product eines anibigeii Ideals und eines Ideals in k enthalt, nicht aber umgekehrt. Ueber die Anzahl der ambigen Classen im KTa-per K gibt dec folgende Satz Aufschluss, Satz 14 Wenn h die Classcnz((]il des Ki'iiyers k, r die Jinzahl der (TTundeinheiien. in k, 0 die Zahl 1 oder 0, jcnaehdein \^ die primitive V'' Eiiiheitsiüiirzel '^ enthält oder nichts d die Anzahl der von einander rerschiede/icn amhigen Primideale des ICörj)ers K/k, 1 die Anzähl der Einheitenrerlànde in k, die dirreh die Belativnormen 'von Einheiten und von gehroehenen Zahlen des K'nrpers K gebildet sind, a die Anzahl der ambigen Classen des Körpers K ist, dann wird In diesem Satze solle'n die Idealclassen der Körper K und k im absohden Sinne genom-men werden. Beweis. Wir zählen zunächst diejenigen ambigen Classen. des Körjoers K ab, welche durch die ambigen Ideale von K/k und die Ideale von k erzeugt werden. Die Ideale wo S ein ambiges Ideal von K/k (oder das Ideal 1) und j ein Ideal in k bedeutet, bilden, weil X^ ein Ideal in k ist, in ihrer Gesamtheit eine Gruppe der Ordnung l'h, worin der Inbegriff der ganzen und gebrochenen monomischen (Haupt-) Ideale von k das Hauptelement der Gruppe ist. Diese Gruppe sei mit D Ijezeichnet. Diejenigen der Elemente dieser Gruppe, welche in K in die Hauptclasse übergehen, bilden dann eine Untergruppe Do von D. Dann ist offenbar die Anzahl (^o der aus S und j entspringenden ambigen Classen von K gleich dem Gruppenindex (D : Do). Es seien nun, wie in Satz 12, wo jetzt O und o sämtliche- Zahlen des Körpers K bez. k umfassen sollen, E^, E.2, E^ 44 Art, 9.— T. Takagi : die Einheiten des Körpers K mit der Relativnorni 1, von der folgenden Beschaffenheit: 1°. Jede Einlieit E von K mit der Relativnorm 1 ist in der Form darstellbar: E = E,^h E.j"-2 Ep^'fH, wo Ui, U2, W/, Zahlen aus der Reihe: 0, 1, 2, /— 1 sind, und II eine Einheit von K bedeutet. 2°. Diese /> Einheiten sind in dem Sinne von einander unabhängig, dass niemals eine Beziehung von der Form l=Eni E.^h E,^>p h'' (0^?i) und zwar ist nach 2'' ^4^ nicht eine Einheit in K. Das Hauptideal (J,) ist daher von der Form Sj und es ist (Si)'=N(yl,) ein Hauptideal in k. Da eine Beziehung von der Form : ^j"i Ä._^2 ^/'/. =Ha, (0^?6) so erzeugen diese Ideale genau /'' Elemente der Gru})pe Dq. Ist aber umgekehrt ein Ifauptideal in K, so ist l-s Ä = E, 1) Vgl. Hubert, Bericht, Satz 90. Uel)er cine Theorie des relativ Aliel'sohcn Ziihlkörpers. 45" WO E eine Einlieit in K ist, für welche N(^)=l ausfällt. Daher ist nach 1° E=E,»i Ej<^ E/'r H, ^" (0^2f<0 wo 7/ eine Einheit in K ist, oder l-s l-s ^ =(^i«. ^2"- --l/'' ^)' folglich A=A i«i ^2"- -4 /'f Ha, wo ß eine Zahl in k bedeutet. Das Ideal Sj ist daher unter den oben erwähnten Z' Elementen der Gruppe D,, enthalten. Hiermit ist nachgewiesen, dass die Gruppe Do von der Ordnung /'' ist; für den Gruppenindex «.o=(D: Do) ergibt sich daher ao=hIf-'' (1) Wenn mit Vq die Anzahl der Einheitenverbände in k, die aus- den sämtlichen Relativnormen der Einheiten von K bestehen^ bezeichnet wird, dann gibt es nach Annahme noch v—Vq unabhän- gigen Einheiten in k, welche Relativnormen der gebrochenen Zahlen von K sind: von der Art, dass jede Einheit e von k, welche Relativnorm einer gebrochenen Zahl von K ist, in der Form darstellbar ist: £ = Si»! £."2 N(iï), (0<«,"i(^j'-. H), l-s 6 = ßui (-}U2 jji^ WO Ä eine Zahl von K ist, 3l=(2ti«i2l3 A)," 3l=9V'i3to"2 .1S|. Demnach ist also nach (1) Da nach Satz 12 so ist wie zu beweisen war, ft V i 11. Die Anzahl der ambigen Classen im relativ quadratischen Körper. Satz 15. Wenn K = k (a//-?) relativ quadratisch In Bezug aufl^ ist, und wenn v die Aîizakl derjenigen mit k conjugirten reellen Körper ist, worin die Conjugirten von fj. negativ ausfallen, dann ist, -unter Beibehaltung der 'uhrigen Bezeichnungsweise von Satz 14. Die Classen in K wie in k sollen luiederum im absoluten Sinne genommen werden. Der Beweis verläuft genau wie bei Satz 14; nur soll am ^-Schlüsse für die Zahl /> der im gegenwärtigen Falle gültige Wert: 48 Art. 9.— T. Takagi : eingesetzt werden (vgl. Satz 12). Es sei noch bemerkt, class im Falle, wo die mit k coiijugirteii Körper sämtlich imaginär sind, dieser Satz genau mit Satz 14 zusammenfällt, weil dann v=0 und die Zahl d in Satz 14 gleich 1 zQ setzen ist, da die Einheitswurzel —1 in k vorkommt. §• l:^. Die Geschlechter im relativ cyclischen Körper eines ungeraden Primzahlgrades. Es sei K/k relativ cyclisch vom ungeraden Primzahlgrade /, 'i) = f-'^ die Relativdiscriminante desselben, o die Zahlengruppe in k, die aus der Gesamtheit der zu f primen Normeiireste des Körpers K/k nach f besteht. Die Idealclassen in k seien nach o definirt, so dass zwei Ideale ji und jo in k dann und i^ir dann aequivalent sind, wenn die Idealgleichheit besteht: ii=j,« und « = N(zl\ (f), wo a und A zu f prime ganze oder gebrochene Zahlen von k bez. K sind. Wenn dann zwei Ideale Si und % \on K im absoluten Sinne aequivalent sind, und einer Classe (im absoluten Sinne) C von K angehören, dann fallen die Relativnormen dieser Ideale in eine und dieselbe Classe c nach o hinein; diese Classe c heisse die Relativnorm der Classe C; im Zeichen c=N(C). Da o eine Congruenzgruppe nach dem Modul f ist, so ist Satz A anwendbar, demzufolge die Classengruppe von k, welche sämtliche Relativnormen der Classen von K enthält, von einem Index / sein muss, welcher den Relativgrad / des Kelativkörpers K/k nicht übertreffen kann : i^l (1) Seid die Gruppe der sämtlichen C'lassen von K, H die Ijitcrgruppe von G, welche aus der Gesamtheit derjenigen Classen UoKt eine Theorie ilos relativ Alx-rschm Zahlkörpers. 49 von K besteht, deren Kelativnornien die Hauptclasse nach o sind. Dann ist der Gruppenindex (G: H) offenbar gleich der Ordnung derjenigen Chissengruppe von k nach o, welche aus der sämtlichen lîelativnormon der Classen von K l)esteht. Daher folgt aus (1) (G:H)=4-^^^--, (2) i l wenn // die Classenzahl von k nach o bedeutet. Ferner sei Ho die Gruppe der Classen von K, welche syiubolische 1—s*'' Potenzen der Classen von K sind, so dass der Gruppenindex der Anzahl der anibigen Classen von K ist. Da offenbar Ho eine Untergruppe von H ist, so folgt nach (2) « = (G : Ho) .^ (G : H) .^. -^, (3) Nach Satz 14 ist nun'' «=:ÄZ'' + '-('+l+*) (4) wenn h die Classenzahl von k im absoluten Sinne bedeutet. Anderseits ist, wenn o' die Gruppe der sämtlichen zu f primen Zahlen in k bedeutet, nach Satz 10 (o':o) = /', - (5) wo d die Anzahl der von einander verschiedenen in f aufgehenden Primideale in k ist, also dieselbe P)edeutung hat, wie in (4). Ist ferner e' die Gruppe der sämtlichen Einheiten in k. und ]■: die der Einheiten in o, dann ist offenbar (e':e) = /'-"'-", (6) wenn /• und ^ dieselbe T3edeutung haben, wie in (4). und /" die Anzahl der Einheitenverbände in o ist. Deîunach ist"^ nach (5) und (G) 1) Die Besohränkung-, dass wir hier nur die zu f primen Ideale von K in Betracht ziehen, hat keinen Einfluss auf die Anzahl a geMisser Classen von K, die ja im absoluten Sinne genommen wird, vgl. Satz. 2. 2) Vgl. § 1, S. 7. 50 Art. 9.— T. Tabagi : Aus (:0: (4), und (7) folgt ä + v-{r-\-\^- n)-^d-\-n-{r + «)- 1, oder J)a often ]-)ar m— y=0, so erhält man • n = v. ... (8) Dies 'hat zur Folge, dass in (3) und somit auch in (2) und (1) notwendig das Gleichheitszeichen gelten muss. Demnach ergibt sich a= Y, (9) H=Ho. (10) i=L (11) Hiermit ist der Fundamentalsatz 13 für einen relativ cyclischen Körper vom ungeraden Primzahlgrade bewiesen, denn wenn die Classen von k nach einem beliebigen durch f teilbaren Ideale tu definirt werden, so mag sich jede Classe nach o in gleichviele Classen nach m auflösen, jedocli ohne dass der Index einer Classengruppe verändert wird. Aus dem vorhergehenden Beweis von Satz 13 ziehen wir noch einige wichtige Schlüsse: Alle diejenige Classe von K, deren lielativnorm eine und dieselbe Classe von k nach der Gruppe o der Normenreste nach f ist, fassen wir in ein Geschlecht zusammen, und definiren speciell das Ilavptgeschlecht als den Inbegriff derjenigen Classen von K, deren Relativnormen die Hauptclasse von k nach o sind. Das Ilauptgeschlecht ist also die Classengrupj)e H, und das Geschlecht, welchem eine Ciasso C angehört der Classencomplex HC. Also folgt aus (9) und (10): Satz 16. Die Anzahl der G eieJd&'Mer in K ist (jleicJt dem V'" Teil der Classenzahl von k nach o. Ueliei' cine Tlion'ic /,«) i'^k^-tiv quadratisch in Bezug auf k ist, und v/enn V die Anzahl derjenigen mit k conjugirten reellen Körper ist, worin die Conjugirten von /y. negativ ausfallen, dann rechnen wir nur diejenigen Normenreste nach f, welche in diesen '-^ Körpern positiv ausfallen, in die Zahlengruppe o^, und legen dieselbe der Ciasseneinteilung in k zu Grunde. Da die Relativnormen der zu f primen Zahlen \"oii K offenbar < der Zahlengruppe o"^ angehören, so fallen die Relativiiormen aller 52 Art. 9.— T. Takayi: Ideale einer Oia^r-e (im absoluten ^inne) C ^•ou K in eine und dieselbe Classe c von k nach o''; dieselbe nennen wir demnach -C'ii)^ Die Bedingung a-^ — ■— 2 ergibt, wenn man darin fin- a den in Satz lö augegebenon A\'erl : einsetzt. ^ + ?• + ^' - (/■ + •:!^)^d + ^ + v/ - (;?■ + 1 ) - 1 , oder Avoraus wie vorhin Ueltev oinc Tlie.irio «Its relativ Aliol'^x-heii ZahlkOrpers. 53 M=r, 'Hid folglich 2 ; — •> Die letzte Gleich) leit beweist Satz l-"] für einen relativ quadra- tischen Körper. Wenn die Gesamtheit derjenigen Classen von K, deren Ivelativnornien eine und dieselbe Classe von k nach o"^ sind, in ein •Oeschlecht, diejenigen, deren Relativnormen die Hauptclasse von k nach o' sind, in das Ilauptgeschlecht gerechnet werden, dann gelten die Sätze: Satz 19. Die An-a/d der Geschlechter la elneiii relativ quadra- tischen Körper ist yield t der Hälfte der Classenzaht von k nach o"^. Satz 20. .Tede Classe des Ilairptyeschlechts In elnersi relativ quad- .rati sehen ICôrj^er 1st die st/mbollsche 1 — s'' Potenz rnwr Classe von K. Ferner gilt. Satz 21. Wenn eine Einhell von k oder eine Zahl von k, welche Idealquadrat In k ist, poslflr In den mit k coiijwjirtéiti reellen Kö^yern worin die Zahl ft negatlr ausfälW und N^ormenrest des relativquad- ratischen Körpers K = k(v^//.) nach dem Ideal f (^b') ist, dann ist sie wir /dich Relcitlvnorm einer Zahl von K. Eine Verallgemeinerung des Geschlechterbegriffs. Es sei f'"' die Relativdiscriminante des relativ cyclisclien Kör- pers K/k \'om Primzahlgrade /, m ein l:)eliebigos durch f teilbares Ideal in k, o die Zahlengruppe in k, welche aus der Gesamtheit derjenigen Zahlen co in k besteht, die der Congruenz: <-/; = !, (m) 1) Vgl. Fussnote 1), S. 42. 54 Art. 9.— T. Takag-i : genügen, und im Falle: / = 2, überdies total positiv sind. Wir legen diese Zahlengruppe o der C^lasseneinteilung im Grundkörper k zu Grunde, und verallgemeinern den Begriff der Geschlechter in K dahin, dass die Ideale in K nur dann in ein Geschlecht gerechnet werden, wenn ihre Ixelativnornien i]i eine und dieselbe Classe nach o hineinfallen. Insbesondere ist dem- nach das Hauptgeschlecht die Gesamtheit der Ideale S in K. deren Relativnormen in der Hauptclasse nach o liegen, d. h. N(3) = (o;), wo oJ~^, (m), und, wenn /==2, überdies noch w total positiv ist. Dass die Anzahl der Geschlechter gleich dem /'"" Teil der Classenzahl nach o ist, dass also die Sätze 16 und 11) auch fiu- die Geschlechter im verallgemeinerten Sinne gehen, ist einleuchtend, nach einer Bemerkung in § 12 (S. 50). Zweck dieses Artikels ist es nun, nachzuweisen, dass es möglich ist, eine geeignete Zahlen- gruppe O in K so zu bestimmen dass, wenn die Classen in K nach - derselben definirt werden, jede Classe des Hauptgeschlechtes in K die symbolische (l-s)''~ I*otenz einer Classe von K wird, dass also auch die Sätze 17 und 20 ihre Gültigkeit beibehalten werden... AVir müssen uns aber zunächst mit einigen Hulfssätzen beschäfti- gen. Hülfssatz 1. Ist q ein Frimideal in k, 'welches nicht in die lielativdiscriminunte des relativ cifclischen Körpers K/k vom Prim-^ zaldgrade l (iitfgeht, S eine ZaJd in, K, welche der Bedingung ^{(i)~\, Cq') - (1) genügt, ico e ein beliebiger positiver Kvponenf ist, dann gilt ex in T\ eine zit il prime Zahl A, derart, dass O-A,'-^ (cf). l>eweis. Wir bedienen uns auch hier mit Vorteil des Gruppen- begriffs. Sei G die Gruppe der sämtlichen zu 0 primcn Zahlclassen von K nach dem INIodul (f, IT diejenige der Zahlclassen, deren Ueber eine Theorie des relativ Abelsclien Zahlkörpers. 55 Zahlen die Bedingung (1) befriedigen, endlich Ho die der Zahl- classen, welche durch die Zahlen A^~^ re]>resentirt werden. Es ist dann zu beweisen, dass Da offenbar Ho eine Untergruppe von H ist, so gilt für die Gruppenindices (G : H) ^ (G : Ho). Berücksichtigt man nun, dass, wenn 6=1, (if), offenbar N((9)=l, (q') ist, so sieht man ein, dass (G:H) gleich der Anzahl der Normen- restclassen in k nach ([% also nach Satz 9 (G:H)=Kcr). Anderseits ist (G:Ho) offenbar gleich der Anzahl der Zahlclassen, deren Zahlen der Bedingung ^^-^-1, (^r) (2) genügt. Unser Satz wird daher Ijewiesen sein, wenn gezeigt wird. dass jede Zahl Ä, welche der Congruenz (2) genügt, notwendig congruent einer Zahl in k nacli dem Mochd iV ausfallen nuiss. Dies ist einleuchtend, Avenn ^] prim zu / ist, denn aus ('2) folgt Ä = A^~A^'---~A,^''' (cf), daher lÄ = HiA), (cf), wo S(^) die Relativspur von A, also eine Zahl in k ist. Da / prim zu (\ ist, so folgt hieraus das Gesagte. Wenn q=t ein in i aufgehendes Primideal ist, unterscheiden wir zwei Fälle, jenachdem t in K in / von einander verschiedene Primideale zerfällt, oder prim bleibt. Im ersten Falle, sei i=S(s2) (s-iO 56 Art. 9 —T. Tcikaori : Da dann «P(i.'0=*v«ö')= =^(f ) ^o ist für jede Zald A in K A = a, (2'),=a', (s.Ö^) ^vo ^>!, «', Zahlen in k sind. Ist also A = Ä\ (I), dann muss, da A'=a, (sS*), «=«', (^2"), also « = «', (Î'O; ebenso «'=«", (f), usw. Folglich ist A~a, (['■). Zweitens sei I=S prim in K. Ist dann l Primideal/**'' Grades in k, also If ten Orades in K, dann ist bekanntlicli für jede zahl Ä in K Ist daher A — A'' {2), dann ist A^a!' (V). also, wenn A prim zu i ist, Dies ist aber das Kriterium dafür, dass A einer Zahl a in k nach k* congruent sein soll. Sei ferner A = A' («'), dann kann man setzen A^a + W, {\i'), wo a eine Zal)! in k, À eine durch die erste Potenz von t teilbare Zahl in k, und B eine Zahl in K ist. Dann folgt B^B,' (2). also nach dem vorhergehenden B:^ß, (2), wo ,9 eine Zalil in k ist. Es ist also audi A=a', (v2), Ueber eine Theorie des relativ Abel'scheu Zahlkörpers. 57 "WO o'. eine Zahl in k ist. So fortfahrend l)eweist man den Satz für jede beliebige Potenz von l als Modul. Es sei noch bemerkt, dass dieser I>e\veis für das Priniideal i auch für die zu / primen Priniideale p seine ( lültigkeit Iteihcbält. Hülfssatz 2. Es sei \-> ein zu l pri/tw-^ Primideal in k. loelchoi in die Relativdiscriminante des relativ cijclixchen Körpers K/k vom Primzahlgrade l aufgeht, so dass p die V Potenz eines Priviideah ^^s in K ist. ferner sei 0 eine Zahl in K, 'welche der Bedingung N(6»)=l, (pO genügt, u:o e ein beliebiger positiver Exponent ist. Dann gibt e>< eine Zahl A in K. derart, dass wenn für A auch eine durch '4> teilbare Za.hJ zugelassen wird. Dasselbe g ill auch dann, wenn p = l in l aufgeht, vorausgesetzt, dass fur den Modid der ersten Congruenz V'''\ für den der zweiten ^^''"^"' ange- nommen wi7'd, wo n eine beliebige positive g(fnze rationale Zahl ist. und ■V die bisherige Bedeutung Jür das Primideal {=\t' hp l)ez. S primen Zahlclassen nach dem :\rodnl ^^s^'-'^'+' bez. V^''^ deren Zahlen J der Congrue nz bez. • Jez.4% (Ö'+"0 (3) genügen, genau ^(p') 1)ez. ^(l' '") betrügt. Für das zu / primes ^^p ist dies einleuchtend, wie beim Beweis des vorhergehenden Ifülfssatzes. Um den Satz für das Primideal 2 zu beweisen, sei At t-ii^e genau durcli die t^''' Potenz von Ü' teilbare Zahl. Setzt man eine Zahl A in der Form an: A=a-\-At, wo a eine Zahl in k ist, und t für gegebenes A den möglichst grossen Wert haben soll, so dass t nicht durch l teilbar ist, dann genügt ^1 dann und nur dann der C-ongruenz (3), wenn Diese Zalden A werden also durcli gegeben, wenn für a„ die ^(1"+^) eintinder nach f+Mncongruenten zu t prime Zahlen in k, für ß^, ß^.^ je ein S^^stem der f einander nach l incongruenten Zahlen in k gesetzt werden. Es ergibt sich. also fiii' die Anzald in I'^rage der Wert wie nachzuweisen war. '^ 1) Ohne Siitz 9 zu benutzen, zeigt man leicht, wie aus der vorhergehenden BeAveise ein- zusehen ist, dass der Normenrest nach ^'^ ''cz. ï'"i" /c't'/i-s^cHS den Zten Teil der silmtlichen Zahl- classen nach pe bez. l^+n ausmachen kann. Mit dieser Oljcrgrcnze für die Anzahl der Normen- reste kommt man alwr beim Beweis des Satzes 13 in §12 aus. Denn alsdann ist auf der rechten Seite von (,") (S. 40) d+x statt d zu setzen, wo .(-rsO. Dann erhält man zunächst 0=71 -r + ,r, woraus notwendig îi— 1' = 0 und .r=0 folgt. So wäre der Satz 10 auf diesem Umwege vrin noiiem bewiesen sein. Diese Bemerkung füge ich zu, als eine Verificiruug des- Satzes 10. lieber eiuo Theoi-if , (8) ^^'0 Demnach folgtnach (G) n'nd (8) woraus, in der Tat, Avenn 3'=3i 33 3 gesetzt wird. Wir haben oben die Vorzeichenbedingungen ausser Acht gelassen. Ist nun K=k(V/^. ) relativ quadratisch, dann ist in (5) total positiv, also e positiv in jedem mit k conjugirten reellen Körpern, worin r- negativ ausfällt. Dalier gilt nach Satz 21 die Gleichheit (6) auch in die-em Falle. 1 )a ferner nach (7), die Zahl -p- dieselbe Vorzeichen in jedem Paare zu K conjugirten Ober- körpern von k' hat, wo k' ein beliebiger zu k eonjugirter reeller Körper ist, in welcher /^ positiv ausfällt, und weil Ä in (8), nach Hülfssatz o, beliebig vorgeschriebene Vorzeichencombination haben kann, so kann man A so wählen, dass ä^~^ dieselbe Vor- zeichencombination wie -jY l>êkommt, so dass die Zahl iJ in (8) total positiv in Bezug auf K wird. Hiermit ist unser Satz in allen seinen Teilen vollständig bewiesen. CAPITEL III. Existenzbeweis für den allgemeinen Classenkörper. • §. 1"). Formulirung des Exislenzsatzes. Satz 23. Tu einem (dijebraischen Körpei' k ,''el cuw Clissea- gruppc H nach dem Modul m mit oder ohne Vor zeiche nhedingung vorgelegt. Dann existirt stets ein Classenh'ùrj)er K für diese Glassen- ^ nippe n, welcher die folgenden Eigenschaften besitzt: Ueber eine Theorie des relativ Abel'soheu Zahlkörpcrs. ß3 1) K Ist relativ Abel sch in Bezug (U(f\<. 2) J)ie Galois' sehe Gruppe des Relativk'ôrpers K/k ist holo- edrisch isomorph mit der coiuplementären Griqope (;/n, ico g die Gruppe der sämtlieheu Classen von k bedeutet. o) Die Relatirdiscriminante von K/k enthält kein Fj- im ideal als Factor, ivelches nicht in den Modul m aufgellt. Dieser Satz ist die naturgemässe Verallgemeinerung des zuerst von 1). Hilbert '^ für den Fall: m=l, also für den Classenkörper im absoluten Sinne ausgesprochenen Satzes, welcher von ihm in den einfachsten Specialfällen, dann später von Ph. Furtwängier -^ für beliebige Grundkörper k bewiesen worden ist. Der Beweis des oben aufgestellten Existenzsatzes für den allgemeinen Classen- körper gelingt durch die gehörige Erweiterung der Hubert' sehen Methode; eine grosse Erleichterung erzielen wir aber durch Zahülfenahme des Fundamentalsatzes 13. §. 16. Rang der Gruppe der Zahlclassen. Es sei l eine gerade oder ungerade natürliche Primzahl, l ein Primideal des Körpers k, welches zur sten Potenz in / aufgeht, und vom/""" Grade ist. Es existirt alsdann in k ein System von /> Zahlen ^1, y., T/., Avelche sämtlichst, (l), und so beschaffen sind, dass für jede zu I prime Zahl y von k eine Relation von der Form besteht, wo g eine beliebige natürliche Zahl ist, und die Expo- nenten Ui, Ui, u^ fiir gegebenes y eindeutig bestinnnte Zahlen ans der Reihe : 0, 1, 2 l—\ sind. Die Zahl .> ist der Rang von der Abel' sehen Gruppe, der Ordnung V^-'^^^', deren Elemente diejenigen Zahlclassen nach dem 1) D. Hubert, Uebor die Theorie der relativ Abel'schen Zahlkörper, Göbtinger Nach- richten, 1898. 2) Ph. Fnrtwängler. Allgemeiner Esistenzbeweis für den Klassenkörper usw. Math. Ann. 63. 64 -A^rt. 9.— T. Takaj^i : Modul f sind, die aus den Zahlen = 1. (() bestehen. Daher bestimmt sich (> daraus, dass V die Anzahl der einander nach f incongruenten Lösungen der Congi'uenz: f' = i. (ro (1) ist. Hülf Es ist (' = \jl - -^ J f, u-enn ~-^ ^ f/ > 0 ; ' ■ ^ l-\ (spccifU ;i=0, ivcnn g='[). wo €=\, oder—O, jenachdcin die Congruenz /;/ k lösbar ist, oder nicht ; das Zeichen \_x~\ lud die geivühnliche Bedeidinig der grössten ganzen rationalen ZaJiL die x nicht übertrifft. Der F(dl e=l ist nur dann möglich, trenn s = n{l-\) durch l—] teilbar ist. Speciell ist e=\, werui der Körper k die primitice V' EinJteitsumrzel C enthält, also stets, wenn 1=2. Beweis. Bezeichnet man allgemein mit /„ eine genau durch die //'° Potenz von ( teilbare Zahl von k, dann ist, wie leicht nnch/uweisen ist, (1 + /;/ = ! + /... (2) (1 + À.V-1+/, ., (8) (i+/,y=i + //,+;!. rr'+M. (4) wenn s won 11 ^'-,-1 und wenn — ^ — =n t-l Ist also sl ::::g>0, 1) V-al. T. Takenouchi, diese .Tournai, vol. 36, Art. 1. üeber eine Tl.eorie des relativ Abel'scht u Zahlkörper?. (55 dann ist, nach (2) dann und nur dann, wenn nJ^fj, oder 71 ^g^,, wo (/o die kleinste natürliche Zahl ist, die iiocli ^ -~- ist. Die Lösungen der Congruenz (1) sind dalier die Zahlen: s' = l, (P), welclie nach dem ^Modul V genau ['^''■''■'^ incongruenteii Zahlen abgeben. Daher ist in diesem Falle r=(fj-9o)f= \j--j~]^^- Ist zweitens _ sl aber s nicht durch /— 1 teilbar, dann sind nach (3) die Lösungen der Congruenz (1) die Zahlen: ç = l, ([-). (5) Daher ist in diesem Falle Wenn aber s durch /— 1 teilbar, also gxyJ, dann kommen nach (4) ausserdem noch die Zahlen von der Fornn 1 +/a in Betracht, wenn für dieselben //. + /i = 0, (1"'^^), oder au-fällt. Ist nun fin' eine dieser speciellen /, t36 Art. 9.— T. Takagi : SO kann man, wie leiclit ersiclitlic]), in f) so bestimmen, dass «"=1(1"^-) wird. So fortfahrend erhält man eine Zahl welclie für beliebig grosses ?„ kann aber in der Form dargestellt werden : Soli diese Zahl die Congruenz (1) befriedigen, so muss jedenfalls weil aber auch so ist notwendig folglich r=c, (I), wo c eine zu / prime ganze rationale Zahl ist. Denniach ist also wo n > a. 1 )amit diese Zahl ß der Congruenz (1) genüge, ist aber nacli (3) notwendig und him-eichend, dass ?^^^— .s. Man sieht, dass im gegenwärtigen Falle, alle Lösungen von (1) durch die Producte der Zahlen (5) mit einer der / Zahlen 1, /i, /e ß\-^ gegeben wcrdrn. Es ergibt sich also /' = s-./'+l. Uclier eine Thoorio dos rdativ Aljclscheu ZaVilkiirpers. Q.'J Wenn die primitive /*" Einlieitswui'zel C in k vorkounnt, dann Avird die Congruenz dnrcli r = l— r (wenn /=2. dureli ç = 2) befriedigt, weil (i+r)---(i+c+--'+c'--) ~ ^-1 =-1, (mod. 1-^) In diesem Falle ist daher stets c=l. i. 17. Rang der Classengruppe. Wenn die Idealclassen des Körpers k nach der Zahlengru^Dpe o der Zahlen =1 (m) definirt werden, nnd ist die Ordnung der Gruppe (I der sämtlichen Classen von k, d.h. die Classenzahl von k nach o genau durch die h^ Potenz einer geraden oder ungeraden Primzahl / teilbar, dann bezeicJmen wir mit g« die Untergruppe von G von der Ordnung l\ und mit j) den Inbegriff aller Classen, deren ( )rdnungen prim zu / sind, so dass G=GaD das di]'ecte Product der beiden (Tru})pen Gq und n ist. Im folgen- den spielt der Rang dieser Gruppe Gq eine fundamentale Rolle. Satz 24. /Sei t die Anzdhl der in Gq enthaUencn unabhängigen Idealclassen im absoluten Sinne ; Xi, x-p v^ ein Sgstem der Represen- tanten dieser Classen^ die prim zu m sind ; pi, p.,, pt die niedrigsten Potenzen dieser Ideale, welche monomisch sind; s^, eo,......e,_|„- ein System der Grundeinheiten von k, zu welchem wir eine derjenigen Einheits- wurzeln mitrechnen, deren Ordnung eine Potenz von l, und zwar die höchste in ^^ ist, so dass d=l oder o=0, jenachde^n die primitive V^ Einheitswurzel in k vorhamlen ist oder nicht, und, es sei V die Anzahl ■:dôr V' Potenz re.te nach m, welche in dem System von l "'*' Zahlen : (jg Art. 9.— T. Takaoi: er e„;"^vr rP (1) enthalten siml. Dann ist der lUing der Classing nippe Oo T=d-vni{,j) + n-{r^o), (2) wo d die Anzahl der in m auf (j eilenden^ von einander verschied enetr zu l primen Primideale v^ für welche 9{v) durch l teilbar ist, Fi{g) der im Hïdfssaiz des §16 angegebene Bang der Zahlengriqjpe nach dem 3Iodid (" ist, und die Bummedion über alle in m aufgellenden Potenzen V erstrecht icerden soll. Beweis. Da nach Voraussetzung N=d+lR{fi) ' (3) unabhängige P Niclitreste iiaeli m gibt und ]^' = r + d + t-n (4) von denselben durch die Zahlen des Systems (1) gegeben werden,, so lässt sich ein System von i\^ Zahlen aufstellen, von denen die J\" letzten aus dem System (1) entnom- men werden sollen, derart, dass sich jede zu m prime Zald ;' von k. in der Form darstellen lässt: .. '1 - . ■•''.V- .V y- .'/l -^ .'/A- ^J (,.,.) / —11 /.v-.v VI V-v '' V"'.' oder y— /i /.v-.v ^1 Vv '^■^, vv ^vo die Exi)onentcn x, g l'iir jedes gegebene /' eindeutig l>esliinmte- Zahlen aus der Iveihe: <>, ], 2, l—l sind und « eine '/^\\\] in o bedeutet; « = 1, (m). Ist (hdier r ein beliebiges zu m primes Ideal y<^\\ k, dann. besteht eiiie Idealgleichheit von der Form r = r/'^ r/";-/'^ YsJ^^-'af,. (ß^ Uebcr cine TIkomo dew relativ A)jcrs.'hi'ii Zalilklirpev?. ß9 "WO \ ein zu m primes ganzes oder gebrochenes Ideal von k be- deutet. Kin Ideal von der Form (b) ist aber nur dann gleich 1, wenn die Exponenten «i, (/j sämtlicli verschwinden, also eine Zahlen- gleicliheit von der Form bestellt: ■ oder 1 = ;-/'^ yyJ"-' [^,/d^'. (m), wo mit [-'/'] eine Zabi des Sv-stems (1) bezeichnet wird. J)a nun /'i, Ts. /'.V-.V sowobl von einander als von [-, ,"] unabhängige Nichtreste sind, so bedingt diese Congruenz, dass auch die Exponenten ö Z'v-v sämtlich verschwinden. Hiermit ist gezeigt, dass für jedes gegel)ene Ideal r, die Exponenten «, b auf der recbten ^eite von (6) eindeutig bestimmt sind, dass dalier der gesuclite Kang der sodann, wenn m der Grad des Körpers k ist, w = 2(r+l), ferner ist für jedes Primideal (, e = \, ' und s=a[J-l) •^> durch 1— 1 teilbar. Der Modul m enthalte d von einander verschiedene zu /prime Primideale: p, p', p^''"^^ als Factoren, für jedes derselben f(p) durch / teilbar ist.^^ Von den in / aufgehenden Primidealen seien diejenigen, die in m aufgehen, deren Anzahl d' (mit Einschluss des Wertes: d'={)) sei, durclnveg mit I) die übrigen mit V be- zeichnet. Einfachheitshalber wollen wir zunächst annehmen, dass jedes Primideal I, wenn überhaupt, wenigstens zur aZ + P'" Potenz in m aufgehe, so dass in der Formel (2), § 1 7 für den Rang der Classen- gruppe (^o ly Vgl. Formel (2) §17. 2) Vgl. Hülfssatz, §16. 3) Dies folgt aus der Tatsache, dass die Norm jedes Primideals in dem dnreli l erzengteiL Kreiskörper congruent 1 nach l ist ; vgl. Hilhert, Bericht, Satz 119. 72 Art. 9.— T. Taka>,n : yAi setzen ist. Dieselbe Formel lautet »laber im gegenwärtigen Falle: r= d + d' + Isf-V n -(>• + ! ), (1) wo die Summation auf alle in m aufgellenden Primideale ( zu erstrecken i-t. ] )ie /" in dem System e,'^ e,,,^'+Vi'^ Pt'' ( zu ersetzen, so dass t unverändert bleibt. Dies hat zur Folge, dass jede der ^* — 1 : /— I Classengruppen vom Index / nach dem Modul m auch Classengruppen nach mist, wobei die Primideale q, (\' q^""^^ als unwesentlicher Excludenten auftreten.'-* Nu m mehr sei Vri"uv^j iV't i'=(öö), qri'-ir./2 Vt'ti"^ = {),), 1) Vgl. § 2. S. 13. UüV>-'r oino Tiie^riij ^i zu einer Potenz mit einem durch l teilbaren Exponenten als Factor enthalten sollen, so dass die Ex- ponenten X, y, u, t\ w den t linearen Kongruenzen: (Aiu -\ h /->! y + + Clic + =0, (0 <(tii + hiv + + Ct>c-\- =0. y.w genügen haben. Wir verlangen sodann, dass die Zahlen (o) noch pt-imär in Bezug auf jedes l' d.li. /'° Rede nach der a'l^^ Fotenz von. V sein sollen. Bezeichnen wir für einen Augenblick mit i^=s'/ den Rang der Zahlengruppe, mit yi, y-,, j',ein S3'stem der unabhängigen /'^'' Xichti-este nach dem Modul V". und ist demnach _ .1 e-, I \ fh^Yi^ '■:\i,- v^-yr-- /^r/^- .^^r.^^- ••■;•/■■'=', (I-) 74 Art. 9.— T. Takagi : SO ist eine Zahl (o) dann und nur dann ein Z**"' Rest nach i",. Avenn die Exponenten x, y, u, v, w, dem Sj^stem von v hnearen Congru en zcn: ei-TiH — nyi-i Vpyii +/ii'-i \-l\w-\- = 0, A ^^^1+ f,v/i+ pji-\- /,v+ lu'^ü-\- =0, ; genügen. Unter den Zahlen (o) gebe es nun /' Zahlen, Avelche diese- t+-s'f Bedingungen genügen, die wir dann in der Form l^''!h\ K" {O^e^l) (4) darstellen können, wobei die Zalden //.i, /a, pt in dem Sinne von einander unabhängig sind, dass eine Zahl (4) nur dann /''' Potenz einer Zahl in k sein kann, wenn die sämtlichen Exponenten (\, ('i, (',, verschwinden; und es i^t t'^r + 1 + Z^ + fZ + tf +;«-(/ + Is'f), (5) woraus folgt t'^. (0) In der Tat: nach (1) und (5) e -T>::2 r + 1 ) - ( -!>/■+ ^s'f) = 2 ;r + 1) - m = 0. AorsclK'H Zalilkiiriiers. 75^ aufgeht. Aveil jedes der Ideale Vi, r«. r« genau zu einer Potenz in n aufgflit, deren Exponent Vielfaches von l ist, und überdies //. /^' Polenzrest nach jedem {"'■ ist. Da ferner fin- jedes wirklich in die Pclativdiscriminante aufgehende Priniideal ( die ent- sprechende Zahl v^tJ (Satz 8), und anderseits nach Vorausset- zung der Modul m dasselbe Ideal i wenigstens zur aZ + l''" Potenz als Factor enthält, so ist Satz 13 auf den Körper K anwendbar, demzufolge K Classenkörper für eine der Classengruppen h sein muss. Da es genau /— 1: /— 1 Classengruppen it, und nach (G) wenigstens ebensoviele Kdrper K gibt, da ferner nach Satz G für jede Classengruppe nicht mehr als ein Classenkcirper existiren kann, so folgt, dass jeder Classengruppe h ein Classenkörper K zugeordnet sein muss. Es bleibt noch übrig, nachzuweisen, dass die Kelativdiscrimi- nante des Körpers K durch keines der Primideale q, q', q^""^^ teilbar ist. Wäre aber der Gegenteil der Fall, so wähle man ein zweites, vollständig vom ersten verschiedenes System der n Primideale q.q,' q^'''^^, welche den Bedingungen (2) genügen, und bilde darauf die entsprechenden Körper K, deren Discriminanten dann sicher niclit durch q, q', q^""^^ teilbar sind, und folglich notwendig von K verschieden sein mussten. Da auch diese Körper K Classenkörper für je eine der nämlichen Gruppen h sein müssen, so führt die Annahme zu einem Widerspruch gegen. Satz G. Hiermit ist im gegenwärtigen Falle unser Existenzsatz bewiesen. Wir haben zu Beginn dieses Beweises angenommen, dass jedes in / aufgehende Primideal Ï entweder gar nicht oder wenigs- tens zur ffZ+l*"' Potenz in m als Factor enthalten sein soll. Es ist nun leicht, diese Beschränkung aufzuheben. Es sei nämlich iit^ ein Teiler vom m derart, dass \\\^ genau durch I' teilbar ist. wo ij'^enutzten Methode hinreiclien würde. Ausser den d' Primideal potenzen [', für welche f/xrl, mögen noch gewisse andere, sie seien durchweg mit (f' bezeichnet, wo gi^Tj, in m aufgellen; die übrigbleibenden in / aufgehenden Primideale seien, wie vorhin, durchweg mit V bezeichnet. Indem wir die sonstingenBezeiclmungsweisc des vorhergehenden Beweises beibehalten, io ist nucli in diesem Falle iiucli Kiitluilt nun die Ivelativdiscriininante eines dieser Körper K den Primfactur (i dann ist wegen (7) die entsprecliende Zahl n^9i—^' r)aher ist Satz lo noch anwendbar, nnd es folgt, genau wie vorhin, die Existenz der ^ unabhängigen Classenkörper für die Gruppen h, welche alle Forderungen des Existenzsatzes befriedigen. Das Ergebnis dieser Betrachtungen sprechen wir in den folgenden Satz aus: Satz 25. (reht ein in I aufgeh ndcs Frimuhal l zw (/"'' Potenz in Î11 avj\ ICO g^al, und ist die lielatbxlkcriminante eines dassenhor'pers für eine Classenyruppe vont Index l nach dem. 3Iodid m durch dieses Primideal Ï teilbar, dann ist die entsjyrechende Zahl v kleiner als g y Dasselbe gilt offenbar auch, wenn g—aj-\-\. Ferner ist, wenn g=^, die Relativdiscriminante des Classenkörpers prim zu I. Ein einfacher Factor l von m macht keinen Beitrag zu der Rangzahl von G« Ferner gilt'^ Satz 26. Jfat der relativ cyclische Körper K/k vom Primzahl- grade l die Relativdiscriminante b = f"\ dann, ist f der Führer'^^ der zugeordneten Classengrupj^e vom Index l im Grund kihper k. Das soll heissen: Um die Relativnormen aller zu b primen Ideale von K in eine Classengruppe vom Index / in k einzu- schliessen, genügt es nach Satz 13, die Classen von k nach einem durch f teil1)aren Modul nt zu definiren. In Satz 2G wird nun umgekehrt behauptet, dass es auch notwendig ist, dass jii alle Primfactoren von f, speciell jeden Factor l wenigsten zur v+V"'' Potenz, als Factor enthalte. Beweis. Betreffs eines zu l primen Primfactor p von f ist dies evident; denn wäre u eine Classengruppe \'om Index /nach einem zu p primen Modul iii. dann nuisste nach dem vorhergehen- 1) Dies zunäcbsfc unter der Auuuhme, dass l ungerade ist, und k die /^ primitive Ein-- heitswurzel enthält ; diese Beschri'inkiiui;- w rd später aufgehoben werden. Vgl. § 19. 2) Vgl. § 2, S 13. 78 Art. 9. T. Takagi : den Beweis ein Classenkörper K' für ji existiren, dessen Kulativ- discriminante zu p prim ist, der folglich gewiss von K verschieden ist. Enthalte aber m einen Primfactor I /ai einer Potenz, deren Exponent kleiner als ■y-fl ist, dann musste nacli Satz 2-") ein •Classenkörper K' für h existiren, für welchen die entsprecliende Zahl v'. §. 10. Forlsetzung des vorhergehenden Artikels. In dieser Fortsetzung des vorhergehenden Artikels behandle ich denjenigen Fall des Existenzsatzes, wo der Index der Classengruppe ii eine ungerade Primzahl / ist, aber der ( îrund- körper nicht die primitive Z*^ Einlieitswurzel T enthält. Für den Fall, wo 111 = 1, also für den absoluten Classenkörper hat Herr Ph. Furtwängler'^ den Existenzbeweis dadurch geführt, dass er zunächst dem Körper k die V^ Einheitswurzel C adjnngirte, dann einen geeigneten Oberkörper zu den so erweitei'ten Grundkörper k' •construirte; sodann zeigte er, dass dieser Oberkörper den gesuch- ten Körper als Unterkörper entlialten muss. ])iese Beweis- metliode bewährt sich auf in un-erem Falle. Indem irli hier dieselbe Methode anwende, schicke ich einen Hülfssatz voran, welcher eine gewisse Vereinfachung des Beweises bewirken wird. Hülfssatz. JiJs sei k' rclatlr cyclhch vom lichit'iryradr n In BtziKj auf k, Jv relativ cydisch vom Grade l in. Tuziiij oi/fk\ und relativ normal aber n'icht relativ Abel' seh in, Beziuj auf k: niid r.9 sei l eilte Primzahl, die njelit in n s nach Voraussetzung ^ cyclisch von der Ordnung /, und die comple- mentäre Gruppe ©/§ ebenfalls c^^clisch von der Ordnung n ist. Dalier gibt es in @ eine Substitution T, von der Art, dass die T^erlegung gilt: Die Ordnung der Substitution T, welche durch n teilbar und in nl îiufgelît, muss notwendig gleich n sein, weil die Gruppe @ nicht •cycliscli sein soll. Ist aber H eine beliebige Substitution von .§, dann ist HT auch von der Ordnung }i., weil HT in der oljen angegebenen Zerlegung von © an Stelle von T treten kann. Ebenso folgert man, dass die Ordnung jeder nicht in ^ enthaltenen Substitution ein Teiler von n sein muss. Sei nun p ein Primideal von k, welches die Voraussetzung ■des Satzes genügt, und es gelte in K die Zerlegung V = '^.%s '^.^ so dass wenn / der Relativgrad der Primideale ^^i, ^2. i'i Bezug auf k ist. Nach Voraussetzung ist also/:>l. Die Zerlegungsgruppe des Primideals ^^1, welche von der Ordnung / ist, muss hier eine €_7clische Gruppe sein, weil die Trägheitsgruppe die identische ist. Xacli dem vorhin bewiesenen, muss daher / ein Teiler von n^ oder gleich / sein. Die letzte Eventualität ist aber ausgeschlossen, weil alsdann ^ die Zerlegungsgruppe ist und folglich p in 11 Factoren in k' zerlegt werden muss. Da also / ein Teiler von >^ ist, so muss V durch / teilbar sein, womit der Satz bewiesen ist. Wir gehen nunmelir zum Beweis des Existenzsatzes über, unter der Voraussetzung, dass der Grundkörper k nicht die primitive P" Einheits\vurzel enthält. Durch Adjunction derselben erweitern wir k zum Köri)er k'. welcher relativ cyclisch über k von einer Ordnung n ist, wo n ein Teiler von /— 1, folglich prim zi\ l ist. Die Idealclassen von k seien nacli der Zahlengruppe o 80 Art. 9.— T. Takao-i : der Zahlen = 1 (in) definirt, wo der Modul lu ein jedes in l aufgehendes Primideal I mindestens zur ersten Potenz als Factor enthalten soll, eine Annahme, die ohne Schaden der Allgemein- heit geschieht, weil die Hinzunahme eines einfacJien Factors t zu in, falls m nicht durch t teilbar sein sollte, offenbar den Rang t der vollständigen Classengruppe g„ von k nicht beeinflussen wird.'^ Legt man dann der Ciasseneinteilung in k' die Zahlengrujjpe o' der Zahlen = 1 (nt) zu Grunde, dann fallen die Kelativnormen der Ideale einer Classe nach o' in eine und dieselbe Classe nach o in k hinein, so dass wir berechtigt sind, von den Eelativnormen der Classen von k' zu sprechen. Dasselbe gilt offenl>ar aucli fin* jedem in k' enthaltenen (3berkürper von k. Sei nun in leicht verständlicher Bezeichnungsweise G=[Ci, Cs- Ci;D] (1) die vollständige Classengru})pe von k, wo n wie in § 17 die (Trui)pt' der Classen, deren Ordnung zu / jn-im sind, und c,, ('•_-. ein System der P)asisclassen der Gruppe o,. bedeuten, die so gewählt sind, dass eine gegebene Untergruppe n von <; vom Index /in der Form dargestellt werden kann: H=[ci,Co, Ci\V>]. ('2) Anderseits bezeichnen wir mit j\, diejenige l^ntergruppe von I), welche aus allen Pelativnormen der Classen von k' besteht. Obgleich es sich später herausstellen wird, dass der ({ruppenindex (d: j'„) gleicli // ist, sind wir in dem gegenwäitigen Stadium niclit berechtfei'tigt, (Hes vorauszusetzen, weil Satz l.'> nur für einen Oberkörper vom Piimzahlgi'ade bewiesen worden ist. W'w wissen aber, dass gewiss (n: r\,)>], also f\, nicht mit d zusanmienfälll. eben zufolgt! jenes Satzes, weil derselbe auf jeden rnterkor])er k,' von k' angewandt werden kann, welcher von einem Prim- zahlgrado im iîezug :iuf k ist, da in jedes in / aufgehendes l*rimi(leal als k'act( i- enlhält. 1) V-I. H ill f. -salz in § 16 uiul Satz 24. Ucl'or eine Theorie des relativ Abel'sclien Zahlköipers. 31 Bezeichnen wir ferner mit d' die Gruppe derjenigen Classen von k' , deren lielalivnormen in Po hineinfallen, dann lässt sich die vollständige Classengriippe g' von k' in der Form darstellen : g'=[ci,c.,, D']. (3) Denn, ist c' eine beliebige Classe in k' und n(c')=cl''c^^' [D„l, Avo n die im Relativkörper kVk genommene Eelativnorm be- zeichnet, und [i\,] eine Classe in der Classengrnppe d,, bedeutet. Setzt man dann c'=cf c-p u, (4) so dass n(c')=cr'cr- ii(u)- Bestimmt man dann .t,, .To, so, dass was ja möglich ist, weil n prim zu / ist, dann folgt I3(u) = [Do]. also, dass in (4) u der Gruppe d' angehört. Zugleich sieht man ein, dass die Classen c,, c.,, in k'^ unabhängig in Bezug auf der Gruppe d' sind. Denn die Annahme cf^cp [D'] = l, wo mit [n'] eine Classe der Classengruppe 1/ bezeichnet wird, bedingt, dass C^Cr^ [l>o] = l, was nur dann der Fall ist, wenn Cr^ = l, cr^=], ; [Dj = l in k; also, weil n prim zu / ist, wenn cf'=l, cp=l, 82 Art. 9.— T. Takagi : Die Classen Ci, Ca, erleiden also in k' weder die Verlust der Unabhängigkeit noch die Erniederung der Ordnungen. Demnach wird durch H'=[cLco, d'} (5) eine Classengruppe vom Index l in k' definirt. Für diese Classengruppe n' existirt nun nach dem vorherge- henden Artikel ein zugeordneter Classenkörper K vom Relativ- grade / in Bezug auf k', weil k' die primitive t" Einheitswurzel enthält. Weil aber die Classengruppe ii' gegenüber der Substitutionen des Relativkörpers kVk invariant ist, so fallen die in Bezug auf k mit K relativ conjugirten Körper" als Classenkörper von h' nach Satz G mit K zusammen; also ist K relativ normal in Bezug auf k. Aus der Tatsache, dass die Relativnormen der Idealen von K in Bezug auf k in die Classengruppe [cLco, ; D„] (6) hineinfallen, ist aber zu schliessen, dass K relativ Abel' seh in Bezug auf k sein muss. In der Tat, sei p ein Primideal von k, welches nicht in die Primideale des ersten Relativgrades in k' zerfällt, und zugleich in einer Classencomplex ct'c./ D mit ri^O, (7) (7) enthalten ist; die Existenz solcher Primideale folgt aus dem Hülfssatze des § 4. Wäre nun K nicht relativ Al)erscli in Bezug auf k, dann musste jedes in p aufgehende Primideal p' von k', nach dem vorhin bewiesenen HüHssatze in / von einander ver- schiedene Primfactoren in K zerfallen. Folglich musste p' einer Classe der Gruppe (5) angehören, und infolgedessen n(p')=:v' in eine Classe der Gruppe (0) in k liineinfallen. Da al)er p der Classe (7) angehört, und da / als Teiler von ii prim zu / ist, so ist dies unmöglich. Üobor eine Tliooric dis relativ Aliel'schou Zalilkörpori?. 83 Da also K relativ Abel' seh vudi (irade id in Bezug auf k ist, so enthält K einen Unterkörper Ko, welcher relativ C3^clisch vom Grade l in Bezug auf k ist. Dieser Körper Ko muss nach Satz 13 (vgl. weiter unten) einer Classengruppe in k vom Index l als .Classenkörper zugeordnet sein. Diese Classengruppe muss aber, da Ko in K enthalten ist, offenbar die Classengruppe (6) enthalten, kann also keine andere sein als die vorgelegte (rruppe h. Die Ivelativdiscriminante des Körpers Ko/k enthält offenbar kein Primideal als Factor, welches nicht in m aufgellt. Geht insbesondere ein Primideal ( genau zur (f""^ Potenz in m auf, wo li''^ (h' (0^«.i^<'2) 1) Vgl. Formel (8) in § 17, wo jetzt an Stelle von S/?(f;) und \\ -ro bez. i:(s/+l) + S/i'(.(/;) IUI 1.1 V gesetzt werden müssen. Ueber oiue Theorie des relativ Abel'scbeu Ziibikörpers. 35 «enthalten sind, und in k,, ks, ...k^ positiv ausfallen. Ferner sollen die Zalilen des Systems (3) in § 18, ausser den dort erklärten t+ l's'f + l'B{2.si—gi+l) Bedingungen (vgl. S. 76), noch )\—> weiteren unterworfen sein, in den von ki, ks, ...k^; ver- schiedenen ri— i- reellen mit k conjugirten Körpern positiv zu ^sein. Da diese letzteren Bedingungen j-i— v lineare Congruenzen mod. 2 involviren, welche die Exponenten x, y, it, i\ /r, ... der Zahlen (3) in § 18 zu befriedigen liaben, so liaben wir jetzt an :Stelle von (5) in § 18, t'^r + l + t + d + d' + n-[t+Isy' + lR('2s,-g,+ l) + r,-^^]. (b) Man erhält aus (a) und (b) t'-t^'2(r + l)-)\-m, woraus, weil bekanntlich aioch immer *Da die Zahl /x (vgl. (4), § 18) nun höchstens in den y Körpern ki, k.,, ...k^ negativ ausfallen kann, so ist die Anwend- barkeit von Satz 13 gesichert, und man überzeugt sich wie in § 18 von der Richtigkeit des zu beweisenden Satzes. Der obige Beweis bleibt gültig, wenn v— 0, was den ersten 'Teil unseres Satzes bestätigt. j\Ian erhält alle für ein gegebenes m überliaupt möglichen Tclativ quadratischen Classenkörper, wenn man in u^ nur die total positiven Zahlen von o zulässt, und für jede Classengruppe ■vom Index 2 nach o^ den entsprechenden Classenkörper con- struirt. § -^1- Relativ cyclische Classenkörper vom Primzahlpotenzgrade. Wir wollen nunmehr den Existenzsatz in dem Falle beweisen, •wo H eine Classengruppe nach dem Modul m von einem geraden 86 Art. 0.— T. Takag-i : oder UDgeraden rrimzahli)otGiizindex /'' und die compleinentäre^ Gruppe (r/n cyclisch ist. Es sei also c eine solche Classe in k, dass erst die f te Potenz von (■ in n enthalten ist; ferner sei <;„ die Classengruppe vom Index /. welche h in sich entludt, so da- der Körper K = l\, K0...K,. in der Tat der Classenkörper für die Classengruppe 11 ist. Ueber eine Theorie îles relativ Abel'schen Zahlkörpers. g9 • Hieraus folgt aber weiter, dass K voim Relativgrade n ist und Classengruppen nach dem Modul iiio aufzufassen. Unter der Voraussetzung, dass K, K' keinen gemeinsamen Unterkörper über k enthalten, folgt, dass die Gruppe [ir, ii' ] mit der vollständigen Ciassengrappe g von k zusammenfällt, weil sonst der zu der Classengruppe (ii, ii' } geluirige Classenkörper nach Satz (') notwendig sowohl in K als auch in K' enthalten sein musste. Da nun ii, ii' bez. vom Index n, n' sind, und {ii, n']=(r, so muss die grösste gemeinsame Untergruppe iin von II und \\ notwendig vom Index nn' sein. Weil aber die lielativuormen der Ideale von dem zusammengesetzten Körper KK' VDiii Belativgrade nn' sämtlich in Uq enthalten sind, so folgt, dass KK' der Classenkörper für die Classengruppe iio ist. Da dasselbe auch von mehreren relativ Abel' sehen Kör])ern K, K,' K"... gilt, so folgt das Gesagte. 1) D. Hilpert, ü1>er die 'J'hoorie der relativ Aberschen Körper. Göttiiiger Nachr. 1898. Uel>er eine Theorie des relativ Aberschen Zahlkörpers. 91 Da Satz 2S sdioii fin- die relativ cycliscliGii Kürper vom Primzalilgrade in Satz lo Ijewiesen worden ist, so handelt es sich jetzt darum, den letzteren auf die relativ cychschen Körper vom Primzahlpotenzgrade zu verallgemeinern. §. 24. Lieber die Geschlechter im relativ cyclischen Körper eines Primzahlpotenzgrades. Um am Ende des vorigen Artikels angezeigten Beweis durchzuführen, stellen wir den folgenden Satz auf. Satz 29. Ks sei K/k ein relativ cycUscher Körper vom Prim- zahlipotenzgreide /''. Dann yiht es stets ein Ideal m in k, welches jedes in die Relativdiscriminante von K auf V, (2) .und setzen t7=(t7'-ü)/+r, (3) wo V die mehrmals erklärte Bedeutung in Bezug auf das Primideal 2 und den Relativkörper K/K' besitzt: es ist die Relativdifferente von K/K' genau durch die (v-[-l){l-\y^ Potenz von 2 teilbar. (Ist also U'=v + n, dann ist U=v + )il, wo n>0). Wir setzen diesen Wert von U in den Ausdruck von m in (1) ein, und definiren die Classen von K nach diesem Modul m. Dann kommt Satz 22 in Anwendung, demzufolge die Relativnormen 1) Hiermit ist nicht gesagt, class jedes \) und jedes l schon in die Eelativdiscriuiinante von K' aufgeht. Auch sollen, wenn mehrere von einander verschiedene ^' in ein p aufgehen, ■das Product n%' und JI-1> in (1) auf alle diese Primfactoren von p erstreckt werden ; gleiches ^gilt für die Pri in ideale (. Uel)t'r eine Theorie des relativ Aliel'schen Zalilkörpers. 93 (1er Classen von K in Bezug auf K' eine Classengruppe H' vom Index / nach d)l' ausmachen, und speciell die Classen von K, deren Relativnormen die Hauptclasse nach W sind, symbolische 1— s' te Potenzen der Classen von K sein müssen. Da nun die Classengruppe H' ihrer Bedeutung nach offenbar gegenüber s invariant ist, so ist zu schliessen, dass die (1— s) te Potenz jeder Classe von K' notwendig in li' enthalten sein muss. In der Tat: sei C eine nicht in H' enthaltene Classe von K', so dass auch C" nicht in H', folglich in einem Classe ncomplex H'C" enthalten sein muss, wo'« eine Zahl aus der Reihe: 1, 2, .../— 1 f» 71 bedeutet. Da dann C" in H'C" enthalten ist, so folgt, wenn man n — f'^ macht, dass die (1— a")*^ Potenz von C in H' enthalten ist, d. h. es ist «''"' = 1, (/), woraus folgt, dass « = 1, (/), also a=\ sein muss. Es ist daher C^~' in H' enthalten, wie behauptet wird. Demnach folgt, nach Annahme, dass alle Classen von K', deren Relativnormen in Bezug auf k die Hauptclasse nach r.i in k sind, in H' enthalten, folglich, da H' nur den /*"' Teil der sämtlichen Classen von K' ausmacht, dass die Relativnormen aller Classen von Iv in Bezug auf k eine Classengruppe h vom Index Z'' in k ausmachen, welche in der Classengruppe n' enthalten ist. Nunmehr ist noch zu zeigen, dass die Classen von K, deren Relativnormen in Bezug auf k die Hauptclasse in k sind, notwendig die symbolischen 1 — s'^" Potenzen in K sein müssen. Da, wie vorhin bemerkt, die complementäre Gruppe g/h cyclisch ist, so kann man in k eine Basisclasse c angeben, deren Ordnung eine Potenz von /, und von der erst die V' te Potenz in ii enthalten ist. Demnach hat man, in einer leicht verständlichen Bezeichnungs- weise n A-i G = [c, d], h=[c',d], h'=[g' ,d]; dementsprechend lässt sich die vollständige Classengruppe von K' in der Form darstellen: [c, D'}, (4) ■91 Art. 9.— T. Takagi : WO D' (leu Inbegriff der Classen vun K' Ijedeutet, deren Relativnurmen in k in d hineinfallen; so dass jede Classe in D' Kelativnorni einer Classe von K in Bezug auf K' ist. Sei nun G eine Classe von K, deren llelativnorm die llauptclasse in k ist. Dann ist, nach Annahme 3t(C) = C'r (5) wo l)i die in K genommene Kelativnorm in Bezug auf K', und C •eine Classe von K' bedeutet. Da aber nacli (4) C'=c"[D'] (6) wo mit [])'] eine Classe von K' bezeichnet wird, Avelche der Gruppe D' angehört, folglich Relativnorm einer Classe D von K ist: D' = 9Î(D}. . (7) Aus (5), (G), (7) folgt Setzt man daher di(C)=mçD'-'). C=T>'-'A, (8) so ist A eine solche Classe von K, dass dl (A) die Hauptclasse von K' ist. Folglich ist A eine 1— s' te Potenz, also auch eine 1— s*" Potenz einer Classe von K. Dasselbe gilt daher nach (8) auch von C selbst, wie zu beweisen war. Um eine untere Grenze für den Exponenten u zu bestimmen, sei angenommen, dass das Primideal .!!j von K genau zur l'-'ten Potenz in ( aufgeht, sodass die Verzweigung von i erst in dem Unterkörper von K vom Relativgrade /'"'"^ über k beginnt. Indem wir allgemein mit Ky den in K enthaltenen relativ cyclischen Oberkörper vom Grade T über k bezeichnen, seien ?',, r2,...v .die Zahlen, die mehrmals erklärte Bedeutung'^ in Bezug auf die Relativkörper K/._,+i/K,,_^, K/,.,^.jIK;,.j.i, K,,/!"^;,-! l^^T^^en, so dass Ijekanntlich l^Vi<'y2< v,. (10) In die Rekitivdiscriminante von K/k geht (genau zur o(/—]) ten Potenz auf, wo'-* r;=[(ü,+ l) + (l^,_l+l)/+ (^,^ + 1)^1]/"-. -{^.+vi^+ +t\p-'+^^Y-\ Nach (10) kann man daher einen Wert von a finden, derart, dass ausser wenn h=g = lf wo notwendig d=u=i\ + l. Ohne nähere Kenntnis über die Zahlen v-,, r., r.,. kann man eine untere (Irenze für u angeben, Avelche sich für alle Fälle bewähren wird: nämlich u>gs-h-^, (11) wo s der Exponent der liöclisten in / aufgehenden Potenz von I bedeutet. Denn es ist nach Satz 8 -V,, -^v.„ T^V„, i-i— ' z-1— - i-r so dass aus (11) folgt uP:>'gsP + -^^(l-l)[vg + v^,.il-ï + i\I'-^] + i\j, wodurch (10) befriedigt wird. Wir haben bisher den Fall ausser Betracht gelassen, wo I='2 und unter den mit k conjugirten Körpern reelle vorhanden smd, 1) Vgl. Hill:)ert, Bericht. Satz 79. 96 Art. 9.— T. Takagi : wu also unter Umständen eine Vorzeichenbedingung für die Ciasseneinteilung in k unentbehrlich werden kann. Gebe es nun in diesem Falle einen mit k conjugirten reellen Körper k'V für welchen der entsprechende mit K conjugirte Körper K"' imaginär ausfällt, dann ist notwendig der in K* enthaltene mit K' conjugirte Körper K"" vom Relativgrade 2"'^ reell. Es ist daher leicht, in Bezugnahme auf Satz 22 einzusehen, dass unser Beweis seine Gültigkeit beibehält, wenn in der Zahlengruppe, welche der Classeneinteilung in k zu Grunde gelegt wird, nur diejenigen Zahlen, die in allen vorhandenen Körpern k"'" positiv ausfallen, umsomehr also, wenn nur die total positiven Zahlen zugelassen werden. Durch das Vorhergende ist, nach der Bemerkung am Ende des § 23, Satz 28 allgemein bewiesen worden. Es ist jeder relativ Abel' sehe Körper K/k Classenkörper für eine Classengruppe h in k, deren Führer jedenfalls ein Teiler der Eelativdiscriminante von K/k ist, wie man sich auf Grund des vorhergehenden Beweises leicht überzeugt. Nach Satz 23 ist die Galois' sehe Gruppe des Eelativkörpers K/k holoedrisch isomorph mit der complementären Gruppe G /ii. Allgemeiner ist jec^f^r Unterhörper K'/k von K/k als Classenhörper einer Classengrupjje h' zugeordnet, welehe i\[ in sich enthalt und ungekehrt; es ist dabei die Oalols sehe Gruppe des relativ ÄbeVschen Köipers K/K' holoedrisch i.^omo/ph mit der complementären Gruppe h'/h. § 25. Der Zerlegungssatz. Wenn K der Classenkörper für die Classengruppe ii des Grundkörpers k ist, dann ist jedes zum Führer der Classengruppe relativ prime Primideal von k, welches in K in die Primideale des ersten Relativgrades zerfällt, in einer Classe von ir enthalten. Umgekehrt gilt der folgende sehr wichtige Satz. Satz 30. {Der Zerlegungssafz). Jedes in einer Classengrvpjye eines heUehigen Körpers enthaltene Primideal zerfallt in die von I'olior oino Thoovic des relativ Aliel'sclicn Zalilkörpers. Çf^ c'inandei' vergeh icä im » Primîdcalr drs crsfcn lU'hii'wcjiadc^^ hi (hm Chi'<>^enk'ôr2ier für diese C/a^isengnippe. Be^yeis. Es genügt, diesen Satz für den Fall zu beweisen, wo der Oberkörper relativ cyclisch von einem Primzahlpotenz- grade ist. Denn die dem Oberkörper K zugehörige Classen- gruppe II ist die grösste gemeinsame Untergruppe der Classen- gruppen, weleheden relativ cyclischen Körpern von den Primzahl- potenzgraden zugeordnet sind, aus Avelchen K zusammengesetzt wird. Zerfällt anderseits ein Primideal des Grundkörpers in allen jenen Körpern in die von einander verschiedenen Primideale des ersten Relativgrades, so muss dasselbe auch in dem zusammenge- setzten Körper K gelton, wie leicht einzusehen ist. Sei also K relativ cyclisch vom Kelativgrade l" in Bezug auf k, II die zugehörige und o die vollständige Classengruppe von k, so dass die complementäre (îruppe aju cyclisch von der Ordnung /" ist. Wir setzen a G = 2'ha% {0^a nicht in die Primideale vom ersten Relativgrade in K, zerfallen, was ein Widerspruch ist. Es ist daher unsere Annahme zu verwerfen: p muss notwendig in /" von einander verschiedene Primideale in K zerfallen. Somit ist der Satz im gegenwärtigen Falle bewiesen. Wir gehen nun zu dem Falle über, wo c /'^ Potenz einer Classe in k ist. Dann muss es eine Zahl w in der Zahlengruppe o geben, die der Ciasseneinteilung in k zu Grunde gelegt ist, von der Art, dass Avo i ein gewisses Ideal von k bedeutet. Zum Modul m der Zahlengruppe o sei alsdann ein l'rimfactor q hinzugefügt, von der Beschaffenheit, dass jede Einheit e und jede Zahl o, welche /''' Potenz eines Ideals von k ist V^' Potenzrest nach q, dagegen die Zahl vj ein /'"' Nichtrest nach q ist. Definirt man dann die Classen von k nach dem Modul m=inq, dann wird das Primideal p gewiss in einer Classe (enthalten sein, welche nicht die /^° ]*otenz einc^r Classe ist, und wir können den Beweis des Satzes genau wie oben durchführen. iOs kommt also darauf an, die Existenz des Primideals q nach- zuweisen. Ijithält k die jH-imitive V^ Einheitswurzel, dann ist dies evident, weil eine Gleichung von der ({estait ca=t"--.r---^- (0^v,vt. '^ Enthält aber k nicht die primitive /"' Einheitswurzel, dann adjungire man dieselbe dem Körper k, und ei'weitre ihn zu k'. Da der Relativgrad von kVk prim zu / ist, so kann eine Relation von der Eorm (1) auch nicht in k' bestehen. Daher gibt es in k' ein Primideal ersten (jfrades q', für welches (y>'' U)=^' (-r)=>' Ist dann q das durch q' teilbare Primideal von k, dann ist offen) lar q ein Primideal von der geforderten Beschaffenheit. Nur scheinbar allgemeiner als der vorhergehende ist Satz 31. Ist K der Classe7ih'ôrper für die Classengruppe ii vo)i k, dann iverden die Primideale von k, welche einem und demselben C lasse ncomjjlex hc angeJiören, in K auf derselben Weise zerlegt, d. 1i. sie erfahren in K eine Zerlegung in dieselbe Anzahl von Primidealen derselben llelativgrade. Beweis. Ist p ein Primideal, welches der CMasse c oder einer Classe des Complexes hc angehört, dann ist der Zerlegungs- körper für p in K der umfassendste in K enthaltene Oberkörper von k, in welchem p in die Primideale des ersten Relativgrades zerfällt. Dieser Körper ist daher der kleinsten Classengruppe in k zugeordnet, welche ir und v enthält, d. h. der Classen- gruppe [h, c]. Ist daher n der Relativgrad des Körpers K/k, also der Index der Classengruppe h, und ist / der kleinste positive Exponent, für welchen c^ in n enthalten ist, dann ist der Index der Classengruppe [n, c], und demnach auch der Relativ- grad des Zerlegungskörpers für p gleich e=—r\ und das Prinndeal p zerfällt in K in e von einander verschiedene Primidcalt' vom /"^ Relativgrade. AVir erläutern noch kurz das Zerlegungsgesetz fiir da- in die Relativdiscriminante aufgehende Primideal. Das t die naturgemässe N'erallgemeinerung des (lesetzes, welches die Zerlegung der natürlichen Primzahlen in dem Kreisteilungskörper regeln. Der durch die primitiven m^""' l^inheitswurzeln definirte Kreisteilungski'irper (p {mj"'^ Grades ist der vollständige Classen- korper K(//^), werm der (irrundkörper k der natürliche ist. Ist|7 •eine nicht in m aufgehende rationale Primzahl, dann ist die in Satz 31 mit/ bezeichnete Zahl der kleinste positive Exponent, für welchen p^=l, {m) ausfällt; p zerfällt daher in K(??i) in e=(p()/f):f von einander verschiedene Primideale. Ist ferner l eine genau zur /t""' Potenz in m=l"m., aufgehende natürliche Primzahl, dann ist der Trägheitskörper für / in K{m) der Körper K(//?„), d. h. der ■durch die mT primitiven Einheitswurzeln definirte Körper. In K(7/i) zerfällt / in ein Product von ç(/") ten Potenzen der e von einander verschiedenen Primideale, wo r genau wie oben zu bestimmen ist, indem man nio an Stelle von m setzt. '^ Ij Vo'l. Hubert, Bericht, Satz 125. 102 Art. 9.— T. Talcaj,ä : Als ein weiteres Beispiel sei der Teiluiigskörper der leni- ni>katisclieD Functio]i sii (?/.; i) aiigefülirt. Der Grnndkürper k ist der Gauss' sclie; sei [=(!+/), und iii = (/':) ein ungerades Prim- ideal in k. Der Teilungskörper zum Divisor m'^ ist dann der Classenkörper K(r'iu) voin Eelativgrade f(iii) = N(iii)— L I»er Trägheitskörper fin- t ist K(iii) vom liekativgrade ç-(m):4. Ist al,-o /der kleinste positive Exponent, für Avelehen (1 + //=/,« (in) ausfällt, wo r die Eiidieiten von k bedeutet, und setzt man dann zerfallt t in K(Pi)i) in ein Product von 4'"'" Potenzen dur e von einander verschiedenen Primideale. "^ § 26. Ein Crilerium für den relativ Abel'schen Zahlkörper. H. Weber'^ hat den Classenkörper durch die folgende l)efini- tion eingeführt, Avelche, offenbar auf der Analogie mit gewissen in der Theorie der complexen Multiplication der elliptischen P'unctionen vorkommenden Körpern beruhend, von der unsrigen griindlich verschieden ist. Es sei im Grundkr>rper k eine Zaldengruppe n nacli dem ]Modul m \orgelegt, welche eine Idealengruppe vom Index h erzeugen möge; ferner sei k ein ()l»erkörper von k vom Ivelativgrade n, welcher aber nicht als relativ normal vorausgesetzt wird. Dann heisst 51 nach Weber Classenkörper für die Zahlengrup]ie ii. wenn die folgenden Bedingungen erfüllt sind: 1) Vgl. weitor xmten, § 32. 2) Dieses Ero-e).uJs isb durch direkte IJeehuimg hergeleitet iu der Alihiiudlung : T. Takagi, Über die iui Bereiche der rationalen complexen Zahlen Ahel'schen Zahlkörper, diese Journal, vol. 19. (1903) Vgl. dasolUst S. 25, wo jedoch ein Fehler zu corrigiren ist : es soll statt (1 + ?■)/= 1, die richtige : (1 + ?y= 2" zu setzen. 3) Lehrbuch der Algebra, III. S. 607 : Vgl. iiuch Über Zahlengruppen usw. Math. Ann. Bd. 49, S. 87. Ueber eine Theorie des rtlutiv Al^tTschou Zahlkorpers. 103 1) Älh Primideale ersten flradcs von k, die in it enthcdten Aind, zerfallen in 9i iii ein Prodidcf roii Imdei' Primidealen ersten Grades. 2) J^ein Primideal ersten Grades von 51 geht in ein Primideal von k auf. welclies nicht in ir enthalten ist. In den beiden Forderungen 1) und 2) wird eine endliche Anzahl Ausnahme zugelassen, die dann als Factor in den Modul m hingenommen Averden, weil von den in den Modul aufgehenden Primidealen von k überhaupt abgesehen werden. Auf dieser Definition gestützt, beweist Weber'^ die folgenden Tatsachen: 3) E^htn^h. 4) Für ein gegebenes h, kann es nicht mehr als einen Classenkörper 51 geben. 5) ^ ist relativ normal in Bezug auf k. Ferner spricht er die Vermutung aus: 6) Für jeden Classenkörper 51 ist n=h.~^ Es sei nun K der Classenkörper (in unserem Sinne) für die Classengruppe ir. I)a die Forderung 2) in unserer Definition des Classenkörpers enthalten ist, und da nach Satz 30 auch 1) erfüllt ist, so ist die Existenzfrage"'-' für den Körper 5Î nach Satz 23 gelöst, und zwar wie aus 4) folgt, mit der Eindeutigkeit der Lösung. Ferner ist die Vermutung 6) bestätigt, und das Priidicat in 5) zu ,, relativ Alwlsch'' precisirt. Nachträglich folgt noch aus Satz 30, dass für alle nicht in dem Führer der Classengruppe ii aufgehenden Primideale von k ohne Ausnahme die Bedingung 1) erfüllt sind. In der Weber' sehen Definition des Classenkörpers 5£ ist die Forderung versteckt, dass 51 relativ normal in Bezug auf k ist, eine Forderung, die von der Classeneinteilung in k unabhängig ist. In der Tat, besagen 1) und 2), dass überhaupt jedes Prim- ideal ersten Grades von k, welclies bei der Zerlegung in 51 ein 1) Lehrbuch, TU. S. 607-611. 2) In der S. 102 citirten Abhandlung-, nimmt Wel>er diese Beziehuno- als eine Forderuno- in der Definition des Classenkörpers auf. 3) Eine Frage, in die Weber nicht eingeht, indem er sich nur mit den von der Theorie der elliptischtn Fuuctiomn gellefeitjn aktuell vorhandenen Köriiern beschäftigt. ] 04 -"^i"*^« 9.— T. Tcikagi : I'rimideal ersten Grades von Äl unter den Fiictoren aufweist, notwendig in lauter Primideale ersten Grades von ^ zerfallen muss; dies ist aber ein Critérium dafür, dass St relativ normal in I^ezug auf k ist. Denn aus dieser Voraussetzung folgt IL ( ^^^ 1 \" Avo das Product links auf alle Primideale ersten Grades von 51. und das Product redits auf alle Primideale ersten Grades von k. welche in n Pj-imideale ersten Grades von AÎ zerfallen, erstreckt wird, und wo N das Zeichen für die in dem bezüglichen Körper genommene absolute Norm ist. Daher ist, wenn ro 1 Po{s)=ll^ ^ i-Nug-^ gesetzt wird, Lim (s-lj» PJ^s)=a (l) endHcli nnd von Null verschieden, wenn sich der reelle Veränder- liclie s al^nehmend der Grenze 1 zustrebt. Ist nun 9i' ein beliebiger mit .vt relativ conjugirter Körper, dann zerfallen alle Primideale Po ii"! lauter Primideale ersten Grades in Ä', welche letztere mit einer endlichen Anzahl Aus- nahmen alle Primideale ersten Grades von ^' erschöpfen. Gleiches gilt daher von dem zusammengesetzten Körper 5Î5Î', für welchen also die entsprechende Relation (1) bestehen nuiss, demzufolge der Relativgrad von ^R' notwendig gleich n ist. Daher fällt Si mit S\ znsanunen, ist folglich relativ normal in Bezug auf k. l'nter Hervorhebung dieser Forderung kommt die Weber sehe Delinition des Körpers ^ auf das folgende hinaus: Ein relativ nor- maler Körper ^Xjk. soll in dem zu Beginn des ^ 4. erläuterten Sinne der ( 'lassengruppe ii zugeordnet sein (Weber' sehe Bedingung 2); in Bezug auf diesen Körper Sl und diese Classengruppe u soll das in Satz 30 ausgesprochene Zerlegungsgesetz gelten (\Veber'sche Bedingung 1). Wie oben bemerkt, folgt ans diesen Bedingungen die Uebereinstimmung des Körpers iî mit unseren Classenkörper Uober eine Tiieorie des relutiv Al)erschen Zahlkjrpers. 105 für H. Wir sind a1)er umgekolirt au- dem Zusammenfallen des Körpergrades und des ( îruppenindex als Definition des Classen- körpers ausgegangen und durch eine Reihe von Schlüssen an den Zerlegungssatz gelangt. Für den Existenzbeweis hat dieser Weg als eine grosse Erleichterung erwiesen. Immerhin gil)t die Weber' sehe Definition ein Critérium für den i-elativ xVbel' sehen Körper, welches sich fiir die Anwendung auf die 'i'lieorie der •complexen ^.lultiplication besonders eignet. W^ir wollen dieses Critérium noch als einen Satz aussprechen. Satz 32. Wr//// K i-chd'ir normal in Bizwj nuf k /.sV, und ir. iiii ■alle in einer Classeur/ nipe ii ro// k enthaltenen Primideale erden (nunles, und nvr diese, wieder in die Primideale ersten Gradex in. K zerfallen, dann ist K rehdiv AheV seh in Bezug auf k, u/nl der Relatirgrad von K stimmt mit dem Index der Glassengruppe n üher- •ein. Zum Schluss sei noch das folgende bemerkt. Hei immer K/k relativ normal vom llelativgrade //, und die Classen xon k nach •einem Modul m de fini rt. Der Inl)egriff aller Classe von k. die ein Primideal enthalt, welches in die Primideale ersten ( Irades in K zerfällt, bildet eine Classengrii])pe ii. Denn sind c und c' zwei beliebige dieser Classen, dann entlialt die Ciasse ('c' gewiss ein Ideal i, welches Kelativnorm eines Ideals 3 von Iv ist. Denkt jiian sich nun die Classen von K auch nach dem Modul m definirt, so enthält die Classe von K, welche eben das Ideal 3 enthält, nach Satz 5, ein Primideal ersten (îrades ^I?, derart, dass ^:p=/i3, wo A~l, (m). Hieraus folgt: j = 9î(^^I^) = ^>!p, wo a = dt(A)=l, (m), ;)=9î(^:p), wenn 5i die Relativnorm in Bezug auf k Itedeutet. Daher •enthält die Classe ce' auch ein Primideal ersten Grades von k. Der Index // dieser Classengruppe n hängt von der A\'ahl des Moduls lu; nur bleibt stets h^n (Satz 1). Erreicht nun h für •einen gewissen Modul m die obere ( îrenze n, dann ist K relativ Abel' seh, er ist der Cla-senkörper für n. Wird dagegen die obere •Grenze n nie erreicht, dann sei ir diejenige offeidjar eindeutig bestimmte Classengruppe, l^ei der der Index h den möglichst grossen Wert hat. Dann ist der C lasse nk(>rper für 11 der grösste relativ Abel' sehe Körper, w^ elcher in K enthalten ist; K selbst ist 103 Art. 9.— T. Takagi : folglich nicht relativ Ahehech. In diesem Falle müssen also in ir iinendlichviele Primideale enthalten sein, welche in K nicht in die Primideale ersten Grades zerfallen. Mit andern Worten, die Primideale von k, irelclir in einem relativ normalen aber nicht relativ AbeV mlien Oherh'n-per in die Primideale des ersten Grades zerfallen, lassen sieh nieht darch eine Cong men zhedingung characte- risiren, trie sie in unseren hisJierigen Betrachtungen zu Grunde gelegt worden ist. CAPITEL V. Anwendung auf die Theorie der complexen Multipli- cation der elliptischen Functionen. Absolut Abel'scher Zahlkörper. Wenn der Grmidkörper k der natürliche ist, dann ist der vollständige Classenkörper l\{m) derjenige Zahlengruppe o{m) zugeordnet, welche aus den positiven Zahlen a hesteht, die der Congruenz «=1, (^m) genügen. Ki- ist also von der Ordnung f(w.). Der Führer für den: Köj-per K(///) ist m, ausgenommen der Fall, wo m = 2m', und?// ungerade ist, wo \\{ni)=l\{m'), und der Führer für denselheii gleich m' ist. Der Körper \\.{in) ist der Kreisteilungskörper, welcher durch die primitive m'"" Einheitswurzel erzeugt wird. Denn sei r eine solche, und '^ ein Primideal erstens Grades des Kreisteilungskör- pers, welches in die rationale Primzahl p aufgehen mag. Dann ist notwendig also, von einer endlichen Anzahl der in die Zahlen der Form. 1-"" aufgehenden ^>p abgesehen, j9 = 1 , {m). Uel>cv eine Theorie des relativ Alierschcn /alillc'irpevF. 107 Der Kreisteilung.skorper ist daliev der durch die Zahlengruppe oÇ-iit) definirten Idealengruppe von k zugeordnet. Berücksichtigt man daher nur die Tatsache, dass der Kreisteilungskörper liGchstens von der (Ordnung ) aus K(rt) und K(/>) zusammengesetzt: K(«6)=K(a).K(/j), (1) weil die Gruppe >0. Daher ist \\{t)i) imaginär, enthält aber einen reellen Körper vom halben Grade, welcher durch cos -^ erzeugt wird. Bezeichnen wir denselben mit K„(w), dann gelten für diesen das Compositionsgesetz (1) nicht mehr. Denn der zusammengesetzte Körper Y^ia).\\Sj>) ist der Zahlengruppe zugeordnet, deren Zahlen den Congruenzen genügen: x~\, («.), =±1, W, oder a-^-l, (a), =±1, (6). Diese Gruppe ist daher als eine l^ntergruppe vom Index 2 in der Zahlengruppe für K„(«/>) enthalten, welcher folglich relativ quad- 1) Vgl. H. Weber, Lehrbuch, IT. (2. Aufl.) S. 728. 1Q3 Art. 9.— T. Takagi : mtisch ill Bezug iiuf K„OO.Ko(A) ist.'^ Sind aber a, h ungerade -und relativ prim, dann ist, wie man leiclit einsielit, Alle reelle Aber.sclie Körper lassen sich daher auf die Körper zuriickftiliren. Diese sind cyclisch vom ( Irade 'l''"\ ç{p''), und bez. durcli sin-T^Jr^ ^^^d sin- ',^ erzeugt. Ich habe diese an sich triviale Tatsache erwähnt, weil sie ein gewisses Analogon in der Theorie der complexen Multiplica- tion der elliptischen Function hat, welches dort eine bedeutende Kolle spielen wird. §. 28. Relativ Abel'sche Oberkörper eines imaginären quadratischen Körpers. Nebst dem Körper der rationalen Zahlen zeichnen sich die imaginären quadratischen dadurch aus, dass sich die relativ Aber sehen Oberkörper derselben auf gewisse von den Primideal- potenzen im Grundkörper abhängende elementare Körper zurück- führen lassen. Es sei k ein imaginärer quadratischer Ivörper von der Discri- minante J, m ein beliebiges Ideal in k, dann ist der vollständige Classenkörper K(m) zum Modul m vom Ilelativgrade w ■wo Ji die Classenzahl von k im al)Soluten Sinne, 'l> die Euler' sehe Function in k, mid w die in § 2'> mit (k: i:„) bezeichnete Zahl, hier also die Anzahl der nach m incongruenten Einheiten von k bedeutet; es ist demnach, 1) Ausgenommen der Fall, wo a cAev b=2 ist. Uebor cniie Thouvic ilos relativ Aliorsclun Zalilköiliers. lOD^ wenn J-< — 4, Avciin J = — 4, wenn J=~3, H-=2, im allgemeinen, ic = l, wenn m in 2 aufgellt ; 'lü = 4:, im allgemeinen, =2, wenn in = (2), =1, ween m = (l + /) oder i w=G, im allgemeinen, =s, wenn m = (2), =2, wenn ]n = {^ — 3), =1, wenn m = (l). Das Ideal m ist nicht notwendigerweise der Führer für die Classengruppe, welche dem Körper K(iii) zugeordnet ist. Ist aber f der Führer, so muss, weil f Teiler von m ist, K(f) in K(m) enthalten sein, und da K(f) der umfassendste Classenkörper für den AEodul: f ist, so ist notwendig K(f)=K(m); also a>(in) _ 2cm •I'(f) «-î wo îi'ni die oben angegebene Bedeutung hat. Im folgenden geben, wir die Tabehe für sämtliche Fälle, avo m nicht mit f zusammen- fällt. Darin w^erden mit p, p' und q, q' die Primideale ersten Grades von k bezeichnet, welche bez. in 2 und 3 aufgehen, so dass (2) = p- oder = pp'; (3) = q- oder==qq'. J=0, (4): (2) = p^ K(iii) = K(pm), wenn m ungerade ist. k=:K(l) = K(p) = Ivq)=K(pq) K(2)=K(2p)=K(2q). //=!, (8): (2)=pp'. K(m) = K(pm), =K(p'm), = K(2ni), wo 111 prim bez. zu p, p', 2 ist. k=K(l)=K(Pj-K(q)=K(pq)=K(p'q) = K(2q), wo P die 7 eigentlichen Teiler von 4 bedeutet.. ^]0 Art. 9.— T. Takagi : J=D, (8): k=lHl)=K(ii). K(2)=K(2q). J=—i: P = (l + 0 K(ni)=K(pm), iii, ungerade k=K(l)=K(p")=K(l±2 V\ O^n^ 3. J=-3: ci = (v'"=8) k=K(l)=K(q) = K(cf)=K(2: = K(2if=Iv(2±v/-3) Sind nun a, b relativ prim, dann ist der aus K(a) und K(b) zusammengesetzte Körper der Classengruppe in k zugeordnet, welche aus den monomischen Idealen («) bestellt, wo fiir a eine Zahl gesetzt werden kann, die den Bedingungen } oder > ^S (b). J =^, (b) ) genügen, wenn mit e, ei, e. I)elie1)ige Einheiten von k ]>rzi'ichnet werden. Für den Körper K(ab) dagegen müssen ^i—^ oder l = e sein. Es gilt demnach Satz 33. Wenn a, h relativ p/'mc Ideale in einem imajinären rjuadralischen Körper siml, dann ist, abgesehen von gewissen trivialen speciellen Fällen, der ans K(a) und K(b) zusammengesetzte Körper als echter Unterk'örper in K(ab) ( iithaltcn. Der JUiativgrad ron K(Qb) in Bezug auf K(a)K(b) ist - ' — '— ^ wo Wm. die in S 100 cr/auterte Bedeutung hat. (Wenn von den speciellen Fällen: J=— 4 und .J= — 3 abgesehen wird, ist dieser Relativgrad gleich '1, ausser Avenn a oder h in 2 aufgeht). Dagegen ist, wenn a, b, c relaliv prim sind und a in'clit in 2 aufgeht (für J=-3, auch noch nicht gleich (\/ — 3) ist) K(ii(\')=K(ali) K(nc). Denn der zusammengesetzte Körper K(ab). l\(ac) ist dci' Z;ililen- gruppe zugeordnet, welche durch das Congruenzensystem «ES 1, (ab), =E, (nc) Ueber eine Theorie des relativ Abel'schcn Zahlkörpers. Xll definirt wird, wo e eine Einheit von k bedeutet. Es muss dalier und wegen der dem Ideale n auferlegten Beschränkung folglich a = 1, (nk\ Um mich bestimmt anszudriickeii und in Hinsicht auf die Beziehung auf die Theorie der complexen Multiplication der elliptischen Functionen, setze ici! « = (, wo ( ein in 2 aufgehendes Primideal von k l)edeutet, und (? = 3 oder 2, jenachdem J^O oder 1, (4), so dass r nicht in 2 aufgeht. Dann ist, wenn i, m, iiii, wu, zu je zweien relativ prim sind K(iMiiit2--), c die Euler'scben er fine Thoorio des I'elativ Aljel'schen Zahlkorpers. 113 //•^:z=l, im allgemeiei], = 2, wenn J= — 4, = 3, wenn J=-S. i> Dlt Führer ??i des Ringes ist begrifflich N'erscliieden von dem Führer der Classengruppe, welcher der Körper K(m) zugeordnet ist. wie wir ihn in § 2 definirt haben. Diesen letzteren be- zeichnen wir mit f. Es ist wiclitig, denselben für M('m) zu l)estimmen. Da M()if^ jedenfalls Classenkörper nach dem Modul in ist, so ist f ein Teiler von w. und wie aus der Natur der zugehörigen Classengruppe ersichtlich, ein invariantes Ideal von k. Wir setzen vi = \a=fa, (1) wo/ die kleinste durch f teilbare natürliche Zahl bedeutet. Dann niuss durch jede Zahl y von k, die der Congruenz /=1, (f) (2j genügt, auch die andere: y = re, {m) (3) befriedigt werden, wenn r eine rationale Zahl und e eine Einheit von k ist. Daim allgemeinen £=±1, so ersetzen wir (3) durch Vergleicht man die Anzahlen der nach vi incongruenten Lösungen von (2) und (4) mit einander, so erhält man N(a)=c«. Da aber nach (1) n durch a teilbar ist, so folgt hieraus a = 1. alsr> ist im allgemeinen ^=»1. In dem speciellen Falle: J=— 4, sind noch in (3) die AVerte e=±i zu berücksichtigen; weil aber nach (2), (3) l = r£ (f) so kommen nur die Möglichkeiten: \=il) und f=(l + 0 in Betracht. 1) H. Weber, Lehrlmch, IIT. S. 366. Für m = l ist der E' lativo;rad immer gleich /(, also ist «0=1 zu setzen. ]]^4 ^^^- 9-— T. Takagi: Da K(1 + /)=K(1), so kann (l + i) überhaupt nicht als ein P'ührer «1er Ciassengrappe auftreten. Daher bleibt nur noch ein Fall: f = (l) zu luitersuchen übrig. In diesem Falle, niu-s offenbar M(//0 = K(l)=k, also woraus als der einzig mögliche Fall, m=2 sich ergibt. In dem zweiten speciellen Falle: J=— 3, erhält man durch genau dieselbe l-berlegung die Bedingung: f=l. M{m)=k. woraus so dass man erhält: m=2 oder m=o. Daher haben wir nach § 24 Satz 35. In die Relativdisc7-lmi7iante von M(//i) tjcJieti alle und nur die Primideale von k avf, welche in m aufgehen ; nusgenominen sind nur die drei Fälle, wo M(m) mit dem (rruiidki}rper k zusammen- Jällt : J=z-4, m = '2,; J= — 3, m=2 oder 3. Als ein Beispiel für die am Ende des § 2(*> gemachten Bemerkung behandeln wir noch kurz eine von H. Weber gelöste Aufgabe: Alle in M(/) enthaltenen absolid AheV selten Körper zu finden. Es handelt sich darum, den grössten Abel' scheu Körper zu bestimmen, Avelcher in dem (absolut) normalen Körper M(/) enthalten ist, der daher nach § 20 Classonkörper für die dort mit ir bezeichnete Gruppe in dem absoluten Bationalitätsbereich ist. Diese Classengruppe ii ist aber offenbar durch die rationalen Zahlen a definirt, welche Normen der Zahlen « von k sind, die nach/ mit rationalen Zahlen r congruent ausfallen: also a>0, a^r\ if) a:^Norinenrest nach A. ^^ \) V-l. § 7. UebfV oint; Thforic dos relativ Al>orKclieu Zahlkörpers. X15 Ist dalier /o tUis kleinste gemeinsame Vielfaclie von/ und J, dann soll a zAiiuiclist quadratisclier Rest nach jeder in /, aufgehen- den ungeraden Primzahl sein, und ausserdem noch in Bezug auf die in/, aufgehende Potenz von 2 die folgenden Bedingungen • befriedrigen: 1) wem., / = 4, (8), ctTil, (4); •2) wenn, /„=0, (8j, aber /^O, (4), folglich JsO, (8), ft = Normenrest nach 8, -±1, (8), wenn -^ = 2, (8), 4 = 1,3, (8), wenn -A^ -2, (8^ ; 4 3) weini /o = 0, (8), und / wenigstens durch 4 teilbar,^' a=l, (8). 4) wenn /o nur durch 2 teilbar ist, st ist« nur der irrelevanten Beschränkung unterworfen, ungerade zu sein. Der gesuclite Abel' sehe Körper ist denuiach zusammengesetzt aus den unabhängigen Cjuadratischen Körpern, die durch die folgenden Zahlen erzeugt werden können:'"^ /(— 1) '-^ 2h wo j) die in/, aufgehenden ungeraden Primzahlen sind ; und 1) V — 1 ' ^^^^'^'^ /o^^. (öj; 2) ^±2 , wenn /^O, (4) und J = (), [S), jenachdem ^=±2,(8); 4 >3) v'^^undV2, wenn /=0, (8), oder f=é, {S) und J~0, (8). 1) Wenn /" nur durch 4 teilliar ist, , d.h. <:'j genüge einer primitiven quadratischen Gleichung mit ganzen rationalen Coefficienten wo I)=B--^AC=f-J, wenn J die Discriminante des Körpers k bedeutet. (Demnach ist w ein (Quotient zweier Zahlen des Ringes mit dem Führer /, speciell ist Aw eine Zahl, die mit 1 eine Basis des Einges bildet). Wir wollen die Wurzel der Gleichung (3) mit dem positiven. imaginären Teil mit (o=[A,B,C] bezeichnen; dann ist \iA.-lB,c], {iÄ,B,^], oder {^, -f , -f-} also der Discriminante, 4/>, 1), odei- -^ zugehörig, jenaehdem (7=1, (2), C='2, (4), oder /)-0, 6'=0, (4). Ist dann n[co) der Modul der Jacobi'schen Function, und adjungirt man dem Körper k jr{M) oder 7.'(w), so ist nach Weber'^ \[n\o>y\=u(:if), (G) ferner ist k[>r(r.>)] = M('2/j (xler M(4/-) (7) jenaehdem (7geiade oder ungerach' ist. Wendet man dieses Resultat auf a-(-^], dann folgt mit Hülfe der Formol (der (.îauss'sclien Ti'ansformation) Ty l+;r( + n{o)) 1) H. Weher. T.ehrbuch, III. S. 50.j 507. Uc'ber ciue Tiuvirie 'les relativ Abi'l'sjlitii Zahlkurpors-. WJ k[v'H^)] = Mi2/), M('if), M(8/), (8) Jena :-! Klein C = 0, (4), (7 = 2,(4), C=l,(2). Nun sind, wenn D=~). (8), .4, (/ notwendig ungerade, in anderen Fällen kann man stets ein co so bestimmen, das A ungerade und 0 gerade und zwar C = 0, (4) wird, ausgenommen der Fall: J=0, (4) und/=l, (2), wo notwendig (7=2, (4) ausfällt. Unter dieser Voraussetzung folgt aus ((>), (7), (8): weiii) f=\, (2), J=0, (4), k[;,\..)] = k[;<..)]=M(2/) ; k[V ;7J=.M(4/) ; (9) wenn /=!, (2), J = 5, (8), k[;rJ = M(2/), kM = M(4/j; k[V';r] = M(8/; ; (10) wenn f=0, (2\ J = 0, (4) oder Jeeö, (8), oder wenn J^l, (8), fiir beliebiges/, k[;rJ = kM=k[v'T] = M(2/). (11) § ••')<». Gleichzeitige Adjunction der singulären Moduln und der Einheitswurzeln. Wenn der ( )rdnungskürper ^l(in) durch die Adjunction der primitiven 7?^'"" Einheitswurzeln erweitert ^vird, so entsteht ein relativ Abel' scher Körper üljcr k, den wir mit bezeichnen wollen. Da M(W) in M(///) enthalten ist, wenn ni in m aufgeht, und ähnliches für die Kreisteilungskörper gilt, so ist das Gleichsetzen von dein Führer des Ordnungskörpers und dem 'Grad der zu adjungirenden Elinlieitswurzel offenbar keine wesent- liche Beschränkung. Der Körper M{/)i) ist der Classenkörper für die Idealengruppe, •welche durch die Zalilen a detinirt wird, die der Congruenz a^r^, (m) (1) 213 Alt. 0.— T. Takao-i : geniiiien, wo i:, eine rationale Zahl liedentet, derart, dass ^0^ = 1, (m). (2) Wenn vun den in Satz o-") angegebenen drei trivialen Fällen abgesehen wird, ist in der Führer für den Classenkürper M (w).'^ Der Relativgrad von ^{m) ist. in der T)ezeichnungsweise des § 29, wo '1^ die Anzahl der nach m ineongrnenten Lösungen der Con- gruenz (2) bedeutet- Wenn m—])' eine ungerade Frimzahlpotenz ist. dann ist in (1) >'o =±1 zu setzen, so dass M(^/')=K(^y'). (4) Ebenso ist M(4) = K(4); (5) dagegen ist, wenn u^è M(2")<:K(2")-e root of unity, dioso .Tournai, vol. 37. Art. 5 (S. 70), 1916. lieber eine Theorie des relativ Abcrschiu Zahlköipeis. 11<) M(4/?) ist allgemein der CIasi?eDgriippe /Aigeordnet, die durch die Zahlen a definirt ist, welche der Coiigruenz a = l, 1 + 2;?, (4j?>) genügen. Da anderseits M(v/?,) nur dann Ap /.um Führer hat, wenn j)i=4p, so ist K(4^) niemals in einem Körper N[(m) enthalten. Man sieht hieraus, dass, von den in Satz 34 angegebenen elementaren Körpern, die Ijeiden ersten Typen K(2") und K(p"),. nicht aber der letzte K(4j>) durch die singulären Moduln und die KinheitsAvurzeln zu erzeugen sind, dass um K(4p) zu erhalten, weitere Au>ziehung einer Quadratwurzel unumwendbar notwendig ist.'> Allgemeiner ist, wenn m (>-2) eine ganze rationale Zahl ist, K(7>^) Oberkörper von M(w?) vom Relativgrade 2'"^ welche aus p—l unabhängigen relativ quadratischen Körpern über M(m) zusammen- gesetzt werden kann; hierbei hat die Zahl p dieselbe Bedeutung wie oben in (.")). Das lùgebnis dieser Betrachlungen formuliren wir als Satz 36. Jeder in Bezug auf einen imaginäre n quadratisehen relativ AheV sehe Zahlk'ôrper rom ungeraden Relaiirgrade lässt sich durch Einhcitsivurzeln und singidüre Werte der 3IoduIfnnction J(~) erzeugen, (rlciches gilt auch im Falle eines geraden Belatirgrades, wenn die Belativdiscrrininaide Iceine anderen Printfactoren enthält, als ■solche^ die in eine nnd dieselbe natürliche Primzahl aufgehen; im gegenteiligen Falle alnr kann noch die Adjunction gewisser Quadrat- wwzeln notwendig werden, deren Anzahl im änssersten Falle Iris zu der Anzahl der von einander verschiedenen, durch die Primjactoren der Pelativdiscriminante teilbaren, rationalen Primzahlen ansteigt. Wie in den folgenden Paragraphen nachgewiesen -werden soll, können alle relativ Abel'sche Oberkörper erzeugt werden, wenn man noch die Teilwerte der Perioden der Jacobi'schen Function ^n{u) zu Hülfe nimmt. .1) Eine z lerst von E. Futter entdeckte Tatsache ; vgl. Math. Ann. 75. 120 ^ft- 9-— T. Tafcagi: § 31. Ueber die complexe Multiplication der Jacobi'schen Function. Um die zuletzt erwiihnte Frage zu erledigen, betrachten wir die Teilungsgleichung der Jacobi'schen Function sn(ic) mit einem singulären Modul }({(o) durch ein ungerades Ideal. Da es aber nicht in unserer Absicht liegt, die Theorie des Teihmgskörpers für sich ausführlich zu entwickeln, so begnügen wir uns damit, nach- zuweisen, dass der Elementarkörper K(4p) oder Iv(hu) (vgl. §2.S) durch die Teilwerte von nnCti) erzeugt wird, indem wir das hierzu nötige Material aus dem Weber sehen lîuclie'^ entnehmen. Sei ojz=[^, B, C] (1) tîine zur Stanimdisci'iminante J gehörige Irrationalzahl von k. so dass J=B'-4AC, und [1, Äco] eine l)a>is des Körpeivs k ))i!det. Für die Function '%(v\oj) und einen ungeraden complexen Multiplicator ri, welcher dem Ringe mit dem Führer 2 angehört, also (,. = ,, + Im, (2) wo a eine ungerade und /) eine durch '2 A teilbare ganze rationale Zahl bedeutet, besteht die folgende Multiplicationsformel: eS(/w)=^^^, (3) AVO 1) H. -Weber, III, 23. Abschnitt, vgl. iusbesonclere S. 576-596. Ueber eiue Theorie des relativ Aberschon Zahlkörpers. l'2l iiiul J)(S) = A,S"'-' + Ä,S"'-' + + A„._,S + 1 I ganze gunzzalilige Functionen im Köi'per k' = k(^0 sind, und ferner e= ±1 oder dii, je nach der Be-cliaffenheit von r- nach dem Modal 4. Es ist A{x) = Mx-s(^^]\ = 0 ft •die TeiluiigsgleichiuKj zum Divisor /'-, deren Wnrzehi (he m Teihverte S(^) (5) sind, wo p ein vohständiges llestsystem nach ,« durchläuft, aller- dings unter der Voraussetzung, dass der Coefficient .1 in (1) ungerade und prim zu /< ist.'^ Es ist nun für unseren Zweck uncrlä^^lich, den CoefÜcienten £ in der Weber sehen Formel (3) genau zu l)estimmen, was wir dadurch erreichen, dass die Function A(>S') «lurch die Thetafunction •dargestellt wird. Ist />« eine behebige ganze Zahl von k, dann kann man setzen Hco-=c + do), J Avo rt, />, c, d ganze rationale Zahlen sind, so dass 1) Für iinseren. Zweck genügt es schon, wenn wir ein für allemal annehmen : .-1 = 1. 122 Art. 9.— T. Taka'i : = //- — ((( + d)n + ad—bc = 0, a—fi h c d—n m =:ls'fi)=z nfx = ad — hc. Für die cuDJiigirte Zahl ij. ergibt dann /i=^d—b(o, I f/.(o=^ —(■-{- a (0. J Icli .setze nun (7) (Ö) (9) Avo für den constanten Cuefticienten '^- nucli zu verfügen ist. Für diese Function ergibt sich 'In'- ^Hv + i) Y _ l'fl + /- (:.-ib,^—hG) — ^ _ \yi + h+ ah^ Nun ist naeli (G), (7), (-s) ô/>(r) ^ ' k>o weit gilt unsere Formel fin- jede ganze Zahl // vun k. Ist nun IJ. wie in (2) eine ungerade Zahl au> dem Ringe mit dem Fühi'er '1, dann ist ■m^üd. (4), (10) und UeUor eine Tlieoric «I>;r + 1)= -) so bestimmen, dass Avird. Setzen wir 17=0 und v= ,^ , so erhalten wir nacheinander 4 O.fJ. .ß.m-1 ' 2 '1.^7 /r=— ^ c^o r(7 ,/ .«" + (Z -m+-^7- Daher ist cd m -c — d — - ^, und für e in (3) erhahen wir, indem wir r = 0 setzen, cd m -(■ '-d - — ^ £ = i cder nach (U>) 6 = (-l) ' / und specieh, '■' .-f('^ + 2), (11) wenn c = 0, (4), £ = ( — 1) ^ Da nach (6) her + ((f — (? )oj — C = 0, SO folgt aus (!) "■"'+-:r. (12> 2'24 Art. 9.— T. Takagi 6 a — d — c A B 2//, wo // eiiie ganze Zahl ist, weil nach (2) h (hirch lA teilbar ist. Der in (12) angegebene Fall tritt daher ein, wenn für J=0, (4) und J = l, (S), o) so angenommen wird, dass C gerade ausfällt, was stets angeht, oder wenn für J = 5, (8) die Zald // dem Ringe mit dem Führer 1 angehört, so dass b' gerade wird; in beiden Fällen ist ■ 1 h'c ■ + e = (-l) 2 2 (13) § 32. üeber die arithmetische Natur des Teilungskörpers. Es sei co=[A, B, C] (1) eine zur Stanundiscriminante J gehörige Irrationalzahl von k, von der wir annehmen, dass A ungerade ist und C gerade, wenn J=0, (4) oder J = l,(8), sodass, wenn }(=;e((o), k'=:k[;f] gesetzt wird, nach ^ 29 (I) k' = Mc2) = K(2), wenn J = 0, (4),^ (II) k' = M(2).:.K(l), „ ^:=1, (>S), i (2) (III) k' = M(4)-Iv(4), „ J=5, (8) ) und folglich k' der llingclassenkörper fin* den JUng li mit dem Führer 2, 1, 4 im Falle (I), (II), (III) (3) ist. Ferner sei m ein l)eliebiges ungerades Ideal von k, T'(m) der Teilungskörper, welcher entsteht, wenn dem Ordnungskörper k' ein eigentlicher m**' Teilwert von S(v)=^/)(sn(u,}l) adjungirt wird, und welcher relativ Abel' seh in Bezug auf k' ist, von einem Ucbor eine Theorie des relativ Abel'schtu Zahlkörpers. ]05. Relativgrade, welclier licclistens gleich «l>(m) ist. Es handelt sicli darum, nachzuweisen, dass T'("0 auch relativ Al)ersch in Bezug auf k selbst ist, und vur allem die C'lassengruppe in k zu bestim- men, welcher T'(in) zugeordnet ist. Wir bezeichnen durchweg mit ^ eine ungerade Zahl vom Ivinge li in (3), welche ein Primideal ersten Grades von k ei-zeugt, mit Ausschluss einer endlichen Anzahl, die in iii oder in die Discriminante der m-teilungsgleichung von S(r) in k' aufgehen^ und wir setzen Dann ist nach (3), (4), § :51 wo e die in (13), § 31 angegebene Bedeutung für /^=cu bat, und die Coefficienten At, As, J^.., durch vs teilbar sind.'^ Versteht man daher unter r in (4) einen eigentlichen m*^" Teil der Periode von iS(v), so sind aV(7') und S(wv) Wurzel der m-teilungsgleichung, wenn, wie vorausgesetzt, cj nicht in m aufgeht, und es folgt eS(ujv) = S{vy, (vs). (5) Wenn nun ^^^ ein l'rimideal ersten Grades in T'O») ist, welches mit einer endlichen An zahl Ausnahme in ein w aufgeht, so muss s(vyE,s(v), m, (6; so dass nach (5) eSiwv)^S{v), m. (7) Da nach Voraussetzung '4> nicht in die Discriminante der Teilungs- gleichung aufgeht, so ist dies nur dann UKiglich. weini 1) H. Weber, 1. c. S. 594' ; vgl. auch T. Takagi, Ou a fimdauiental property of the eqnatiijn of division etc. Proceedings of the Tokyo Math. Physical Soc, Ser. 2, vol. 7. S. 414. i|0^^ Art. 9— T. Takagi : BS(ô^v)=Siv) (8) (Lb., wenn w = ], (m), e = l, \ (9) oder vs=-l, (m), e = — 1. j Umgekehrt, wenn eine Zalil ro die Bedingung (9) erfüllt, und ist ^ ein Primideal von T'("Oj ^velches in w aufgebt, dann folgt nacb (5), da (8) und somit (7) besteht, die llelation (ß). Weil aber S(:r) den Relativkörper T'(i")/k' erzeugt, und für jede Zahl « in k' SO ist für jede Zahl A von T'(in) ilemnacb ist V' ein Primideal ersten Grades in T'(in). Da ei=±l eine Congruenzbedingung für die Zahl «j nach einer Potenz von 2 als Modul bedeutet, so ist hiermit nach § 26 dargetan, •dass der Körper T'(m) relativ Abel' seil in Bezug auf k, und zwar derjenige Idealengruppe zugeordnet ist, welclie durch die Zahlen ^y- des Ringes u erzeugt wird, die der Congruenzbedingung (9) genügen: ,'/=-4-V fm\ 1 (10) Es ist nunmuhr unser Ziel, diese Idealengruppe näher zu untersuchen; wie es sich lierausstellen wird, ist der Index dersel- ben gleicli (in)/i', weiui // S'(?')/a/ ^0' entsteht: T(iu)=K(r'm). (15) Dieses merwürdige Ergebnis wollen wir noch auf einem diiecteren AVeg herleiten. ]~)a nach § 29 k[V..]-M(4), so zerfällt ein Primideal {vs) \un k, wo dann und nur dann in e = (-\) , (18) also e = l, dann und nur dann, wenn a = l (4), b' = 0 ('2), oder \-iv Diaquotetrammine C^ HgO (1) n -v \ Cis-diaquodiethyleiiediamiiie ^°JÏ,0{2)^'^J^^ ^^ cobaltic salt. Brownish red Tcoîî^S^^^e?!«"! X, b) Trans. do Greyish brown L H20(6) 'J ' /ON r/-i T^TTT \ /TT /^ \ iv TriaQ uotriammiiie cobaltic (3) [Co(NH,)3(H,0)3]X3 salt (Prepared from [Co (NHgls (N03)3]). Eeddisb violet Tetraquodiammine cobaltic (4) [Co (NH3)2 (H.jO^jXg salt, Aqueous solution only is known. ? [Co (NH3) (H,0)6]X3 Unknown. /r\ r/n /TT r\\ nv Normal cobaltous . (5)^ [Co (HaO)6jX2 g^^^_ p^Qgg ^Q^Q^^^. /^^ rno ri tt n /attt > 1Y Chloraquot^trammine (6) [Co CI H,0 (NH3),]X, cobaltic salt. Eeddish violet Chlorodiaquotriammine (7) [CoCl(H20)2(NH3)3]X., ^r^^l isomers, one is violet and the other greyish indigo. /Q\ rn nuTT n^ /ATTT Mv Chlorotriaquodiammine (8) [CoCl(H.,0)3(NH3),]X., ^^^^^^.^ ^J^_ j^^^^^g,^ ^^1^^^ /n\ rr^ /-.i tt /-» /-kits \ iv Dichloraquotriammine .(9) [Co C1,H,0 (XH3)3]X ^^^^^^^^ ^i^i^ (dichro-salt). According to the manner in which the chlorine atom- coordinate to the central cobalt atom, there are two isomers in this salt, one is green and tlie other gre3\ /irk\ rn n^ /tt /^n /xttt ^ nv Dichlorodiaquodiammine (10) [CoCl,(H,0),(NH3),]X ^oi^aitic salt Coufigutation of A(iuotri- and diauiuiiut^^Co' alt Complexes. Q 111 the coinpouiid (10) three isomers nmy be tlieoretieally produced, but only one is known, wliich is green. Among the above ten series of complex salts, the absorption spectra of (1), (2), (5) and ((>) having ah-eady been studied in aqueous sokition,^'^ I have now measured the absorption spectra of the following eight salts : ( (3) [Co ClCHaO)^ (NH3)3] SO, (Violet) (4) [Co Cb H,,0 iNHs);,] CI (Green) (5) [CoCl2(H,0)2(NH3)JSO,H (6) [Co CI (K,0;3 (NH^)^] SO^+H^O (7) [Co (H.ß\ (NHs).,] CI, Prepared from [Co (NH3)3 (N03)3]. (8) [Co (H^O), (NH3)3] CI3 Prepared from [Co Cb H^O (NH3)3j CI. In order to determine whether the coordinated chlorine atoms of these salts dissociate in water or not, I have also measured the conductivities of the following six complex salts : [Co CI (H^O), (NH3)J SO,. K,0, [Co CI H^O (NHg),] SO,, [Co CI (H,0), (NH3)3] SO,, [Co CI {^H,),] SO,, [Co CO3 (NH3),]2SO„ [Co H^O (NH3),], (S0,)3. It was found, as the result, that the chlorine atoms (or halogen atoms generally) and the radicals of the strong acids, which coordinate to the central cobalt atom, are ajDt to dissociate, when th^y are dissolved in water and are sometimes in equilibrium (1) Y. Shibata, Journ. College of Science, Imp. Univ. Tokyo Vol. XXXVII, Art. 2, 1915. 4 Art. 10.— K. Î^Iatsimo : as described below. In this investigation I was able to establish the structural fornuilae of all the cobalt complex salts, mentioned above, by the study of their absorption spectra as well as their conductivities and I have explained clearly the bi'ha\i(air in aqueous solution of such colialtanimines as have strong acid radicals in their complex nucleus. Theoretical Part. The configuration of the trinitrotriammine col>altic salt, [Co (NOa^, (NHg)^], a starting substance of many triammine cobalt complex gaits, has already been determined by Y. Shibata,^'* as follows : NH, N0„ ■ NO. NH, ' NH, NO, If this salt is treated Avith cone hydrochloric acid [Co CbH,,0(NH,)J(l (9), the so-called dichro-salt, which is of a green colour is obtained. (hi treating the dich.ro-salt with a cold mixture of 2 Vol. of str^(KH,)s] SO, is ground with strong hydrochloric acid, it changes into the grey [Co Cb H2O (NHA] CI ^'"'^ which is the isomer of the green [Co Cb H2O (NH.,)s] CI. If tlie acpieous sohition of [Co C12 H2O (NH,}3] SO^H is warmed on the water l)ath fora htlle while, tiie colour of the solution is changed into dark violet and, when it is evaporated in vacuo, a violet crystalline pi-ecii>itate, (1) Jom-n. College, Scienco, Imp. Univ. Tokyo, V. 1. XX'XVH, Art. 2, 1915. (2) Wernor : Zeit, anorg. Chem., 15, 143, 1897. C.)ufl,i;'>irati Ho SO4 HCl N'H, (Green) SO4H * Two isomers of [Co Cl (H-P« (NH3)3] SO/"'' can be derived from [Co CI, H,.0 (NHgjg] SO,H. The indigo blue coloured one (which is easil}^ obtained by the action of water upon [Co CL, HjO (NHsl^JSO^H and l>y adding an equal volume of alcohol to it) lias the following configuration H,0 XH, NH, SO4H HoO Alcoliol H.,0 NH. SOa NH3 (Indigo blue) This reaction is analogons to that of the production of the chloroxalatotriammine col)altic salt, [Co CI CA (NHg).,]/'^ from the dichrochloride. As to the configuration of tliis salt, only one form can lie tleduced theoretically, as the following equation shows : (1) Werner: Ibid. (-1 (o) Werner and Miolati : Zeit, physik. Cheui., 12, 35 1893. Art. 10. — K. Matsnno: NH, H,0' NH3 Cl + COOH COOH NHj coo. KH, COO (Indigo blue) If we assume that the two water molecules are placed in the cis-position in tlie indigo blue [Co C1(H,0), (NHg)«] SO^. they must necessarily be situated in tlie trans-position in tlie other isomer, i.e. in tlie violet [Co Cl (H^Oj^ fNH3)3] SO,. This presumption was clearly vei-itied by tlie study of the absoi-ption spectra (cf. the part on the absorption spectra). Therefore in the dichro-salt the two chlorine atoms must coordinate in the trans-position. These relations are shown as folloAVs : SH, HjO^ NHj SOiH -^ H...0 in cold (IndiiJfo blue) SO, -- ^ warmed SO4 NH. (Violet) If the indigo blue compound [Co Cl (H.,0)« (NH,)J SO, is ground up with concentrated hydrochloric acid, a grey coloured powder is obtained, analysis of which gives the formula [CoCl.HsO (NH3)3]C1"\; this substance must be considered as the isomer of the greenish dic'iro-salt of the following configuration : NH, 1 1.0 ML SO4 ^ HCl NH, Cl NH3 (Indigo b]\ii') Having come lo this conclusion, we have 110 nci'd for fuither (li N\ «lutr : Ztit. auorg. Cht-u:., 15, 144, 1897, Cunfiii'u ration of Aqtiotri- and diauiuiine Cobalt Complexes. 7 explanations to prove tlie fact tliat tlie two chlorine atoms of the dichro-salt are coordinated in the tians-position. With regard to the contigurations of the triaquotriammine cohaltic salts, we can theoretically establish two isomers: one in which all the three water molecules are coordinated in the eis and consecutiv'e positions and the other in which two of them are in the trans-position as follows : NH, n.,0 'NIL H.,0^ 'H.O NH3 (I) NH, HO. 'NH- H.,0'' NH,, (II) X« On addition of hydrochloric acid saturated at 0°C to the acetic acid solution of [Co (N02)3 (NH3)3], which is kept over night, [Co (HgO^s (NH,\] Cls ^'^ separates in the shape of reddish violet crystals. The stereochemical configuration of this salt, tliough it cannot be determined by means of ordinary chemical analysis, can be ascertained by the study of its absorption spectra. With the help of this recently well-developed spectrochemical method, I was al)le to prove that in the above triaquotriammine cobaltic salt, the three water molecules are coordinated in the eis and consecu- tive positions as indicated by (II). Accordingly, there must exist another isomer of the type (I). The position of the three ammonia molecules of the dichro salt has already been proved to be as follows : NH, (1) Werner : Ber., 39, 2678, 1906. Art. 10. — Iv. Matsuuo If the two chlorine atoms of this compouiul are rephieed hy water, we are able to produce the triaquotriammine cobaUic salt (I). Tlius, wlien the dichro-salt is dissolved in a consideralile quantity of water, acidified witli hydrocldoric acid, and evaporated in vacuo, (hu'k violet crystals separate, analysis of which satisfies the formula [Co (NH,)/(H,0)h] CI, not [Co (NH,)3 (H^O), CI] CI, HA Thus, 0.2032 gr. of the substance gives 0.1138 gr. C0SO4; 0.1353 gr. of the salt gives 0.2150 gr. AgCl. Calculated Obtaineil Co 21,3 % 21,8 % ■ ^ CI 39,31% 39,32% When the salt is heated at a temperature of 100-1 lO^'C for one liour it does not change in weight, vvliich suggests that there is no water of crystallization. The absorption spectra of this salt were compared with those of the other isomer; the great diver- gence between them indicates the différent configurations. Thus I was able to obtain one of the isomers of triaquotriam- mine cobaltic salts, verifying also both of their configurations. As the eis and consecutive triaquotriammine col>altic salt was. prepared by the action of water U})on [Co (NOjjg (NHj^]^ tlie constitution of the latter must be as follows : '' NE, NO, TJie trans-tiiaquotriammine cobaltic salt is not stabli« in water; at first the solution is bhiish violet in colour, l)ut afterwards it shows exactly the same colour as the solution of the eis and consecutive, i.e. reddish violet. The same reaction takes place in the case of the diaquodiethylenediamine cobaltic salts. Wlien the trans-salt stands for a long time in water or when it is warmed for Contii>'uratioii of A«iviotvi- ami diauiuiinu Colialt Complexes. 9 a sliort time, it produces readiW t1io st;il)lo cis->alt. Accordingly, one is much inclined to the conchisiou that the cis-aquocompounds are as a rule more stable than tlie corresponding trans-compounds in aqueous solution. Now that I have described the structures of the cobalt complex salts of tlie triammine series, I shall discuss the configu- rations of those of tlie diammine series. As tlie stereochemical structure of one of these salts, i.e. the tetranitrodiammine cobaltic salt [Co (NOa^ (NH3)2] K, a starting substance of these series, has been thoroughly investigated by Y. Shil:)ata^'^ and the late T. Maruki, no further confirmation as to it is necessary. In this salt the two ammonia molecules are coordinated in the cis- position as follows : NO., NH, -7 NH;,, ÎCO-, NO.. When [Co ÇS^O.^^ 'vNH3)2] K ^-^ is treated with a cold mixture of strong hydrochloric acid and sulphuric acid, a malachite green salt [Co CI2 (HsjO)^ (NH3)2] SO4 H is produced, and on this salt being dissolved in water acidified with H2 SO4 and evaporated in the desiccator over sulphuric acid, indigo blue crystals of [Co CI (H.^0\ (NHs)^] SOi-H.O separate out. If the latter is acted upon by strong hydrochloric acid, it changes into a greenish powder which corresponds to the formula [Co Cl2(H20)2 (^Hs)»] CI ^^\ From tliese reactions, we can now deduce tlieir constitutions as l)elow. With respect to [Co CI2 (HjOjj (NH3)2] X, we are able theoretically to derive the following three configurations : — (1) Journ. College of Science, Imp. Univ. Tokyo Vol. XLI., Art. 2, 1917. (2) Jörgensen : J. prakt. Chetu., [2] 23, 249, 1881. (3) Werner : Zeit, anorg. Cheui., 15, 16.5, 1897. Art. 10. — K. ihitsuno: NH-, NH, \n ::h, n .NUy- K + H.,S04 ^ / NO, HCl Cl / NO, 1 11,0 NO, _ H,0 (I) - H 0 NH, NH — ^ / ■ - Cl / H.,0 — Cl (11) SO^H SO.H Cl. H,0 NH, NH, Cl H,0 (III) SOiH On the action of water upon [Co iNH3)2(H,0)2 cy SO,H, we obtain the indigo blue salt [Co (NH,), (H^Ola Cl] SO,-H.,0 (('.). the configuration of which must be, as the following equations illustrate, eitlier of (A) or (P>) (below). Whichever of tlie formulae (I), (II) and (III) one may assume for the configuration of [Co (NH«).2 (H,0\ Cy SO, H, that of [Co C1(H20) 3(^113)2] SO, will be as shown by eitlier of the following two formulae (A) or (?>), thus : (1 n , ,NH, Cl U,Ü H,.0 (1) -0,H + HoO ; -^H, .. i Ji,U H,0 _ ( à) « S04 Con fif.li ration of aquotri- and diammiue Cobalt Complexes. 11 ("■^) NH, ■NH3 CI' SO4H + H,0 > ~\ ^n,o H,ü J (I) NH, H,.0. fNH, H„0 (3) NH, HO . ■NH, CI' H,0 (^) CI in) NH, C\ , 7 Î^'H= SO4H + H2O H,0 (B) NH, SO, H,0 •NHj CI H.,0 U.O H.O (III) SO, H + H2O ■^- H,,0 (B) NH, SO, 'NH. H.,0' H.,0 HO (A) Sf)^ By the study of the absorption spectra, it is confirmed that the above triaquocldorodiammine cobaltic salt, [Co(KHs\CUH20)3] SO« has the configuration shown under (A). (For full explanations -see under the section on absorption). It is naturally to be expected that some relations exist between the configuration and the colour of these chemical compounds ; examples in which the salts have analogous configurations and the same colour are shown below : r (Indigo lilm-) SO; (Indio-o blue) H,Ü (Indigo Ijlue) As we easily obtain from [Co Cl2:H20)2(NH3)2] SO,H, [CoCl(H20)3(NH3)2] SO« as above mentioned, which has the structure shown by (A), the configuration of tlie former is not as 12 Art. 10.— K. :M:i.tsuno in (II): if it were so, [Co Cl '.H^O), (NH,).,] ÖO^ sliould have the con- figuration of the type (I») as indicated l)y tlie above equation (3). However this is not in keeping with experimental results, tliere- fore it must he eitlier as shown in (I) or (III). If it were like (I), then since the possibility of producing the chlorotriaquodiammine cobaltic salts shown by (A) and (B) slîould be exactly equal the product obtained from it would be a mixture of tliese two isomers. But the compound thus obtained, i.e. [Co CI (HsO)^ (NH3)2] SO, crystaUizes homogeneously and the absorption curve tliereof suggests to us that the three water molecules are in the eis and consecutive position as shown by (A). Accordingly the above dichlorodiaquodiammine cobaltic salt has tlie constitution of the type (III), thus : — NH, . NIIj B.,0' SO4H H,0 And also from the consideration of tlie colour, we can conhrn) this configuration, comparing it with tlie compounds whicli liave analogous configurations and the same colour, thus: — (green) — \ J II, 7 / 7NH. n.o-^ / i - (jrreen) - r- NU, -] 7 / 7 NH, IIsO ' / I — 11,0 - (gr ecn) CI CI Contiguration of Aquotri- and diammine Cobalt Couiplexes. 13 From thi^j. it seems to he a rule that the cohaltammine in which the two clilorine atoms are coöidinated in tlie trans-position is green in colour. When one of tlie water molecules is replaced hy a chlorine atojn hy the direct action of concentrated hydrochloric acid upon [Co CI (H.Ojg (NHg).] SO^, a green salt is produced which corresponds to the formula [Co Cl^ (H.O)^ (NHg)^ CI. This salt is soluble in water, thereby changing its colour to violet. Werner ^'^ supposed this salt to be the isomer of the above [Co Cl^ (HsO)2 (NHg^] X (III). If tluit be correct, we are inclined to the view that the two chlorine atoms of the salt codrdinate in the cis-position as shown: H,0. SOi + HCl r H.,0 n.o NFL, H„0' CI CI In supporting this statement, this salt should be regarded as one of the excejDtions to the general rule that the cobaltammines in which the two chlorine atoms coordinate in th,e trans, are, as already described, green coloured. But is it not mere prol)al)le that this salt is identical with [Co CI, (H20),(NH3)JX sliown by (III) and that the two chlorine atoms are situated in the trans-position ? So I suggest that the formula of tlie salt may be : ci Up to this point, I liave determined tlie configurations of the isomers of [Co (NHg}^ H,0 Cy X, [Co (NH.}, (K/))^ CI] X. and [Co (NH3)3 (H^OM X3 respectively and [Co (NH,)., (H./J), CL,] X, [Co (NHg)., (H.,0)3 CI] X, and [Co (NHs),, (NO^IJ, l)ut this is not sufficient without an explanation of the study of tlie absoi'ption spectra. (1) Zeit, aaorg. Chein., 15, 172, 1897. 14 Art. 10.- Iv. Mals ino ; Experemental Part: The iVleasurement of the Absorption Spectra. Tlu' al>-orptioii spectra of these salts were measured by means of tlie quartz spectrograpJi of Adam Hilger & Co, London . The concentrations of tlie solutions were viU'ied from ]/!()() X to I/IOOO. Tiie salts used; were" specially purißed by recrystallization, the formulae of some of^which were ascertained by analysis. \\'e might go into discussions from tlie standpoint of the absorption curves, which were dia^vn by the method suggested by Hartley & Ealy. As the result, 'the eight compounds can be grouped into two classes: to the first class'.belong Fig. 1. N Cv S 5( v_ / '^ ^4x K \ \ \ \ \ \ I l(6)H,0^°^"JCl; Confii;uration of Aqnotri- and diainmine Cobalt Complexes. 15 [ v^o TT /-, cn-2 CI. (fi-. I), gi-eeii [CoC],H,0(NH3)JCl (fig. .11), violet [Co(NK,\(K,0),Cl]SO, (tig. II) and [Co(NH.;,(H2 0),]Cl3 (tig. 11) wliich are prepared from the green [Co (NH3)3 Clg HgO] CI and the second class inclndes the following four salts : _ ^^ HA2! ^'^0 ^^'^ ^^ë. ni), [Co (NH3), (H,o), cy so,H (fig. IV) [Co (NH,), (H^O),] CI, (fig. IV) which are ])repared from [Co(NH,), (NO,),] and [Co (NH,!^ (H,0), CI] SO, (fig. IV). Fig. 2. 1500 ISAO ?500 j50C 4000 Vs 1 'v..\ AT (C0CI2 H^O(NH3)3)Cl (CoCiai20)2(NH3)3')SO+-— (Co(NH3),(H20)3) CI3 fromCC0(NH3)3rt20a2)Cl In the first class as well as in the second, all the absorption curves in their own groups are remarkably similar to each other, although they differ much from those in the other class, which Art. 10— K. Matsuiu Fig. 3. 1500 ÏOO0 2500 3CO0 3500 ■«00 «00 5003 V7 ^ '\ \ \i \ \ 1 1 1 1 ! i 1 : 1 proves tliat tlieir constitutions are also identical in tlieir gruups. The structui'al fnrnmla o\ |_^^H.'0(t3)^"-J •• ^^'^''' previously (leterniined l>y Weiiier^'-* as beloAV, llie îavo water molecules of wliicli are in the trans-position thus : ILO Cl:i (1) Bcv., 40. 28;-). J'.H? Confi<;'uration of Aqiiotri- and dianuiiinelCobalt Complexes. 17 Fia:. 4. [Co(NH3)2(H20)2Cl2)S04H — (Co(Nl-r3)5(H20)3)Cl3 fron, (Co(NH-j)3(N02)3) (CoCl(H20)3(NH3)2)SOalt Complexes. 19 H.O CI. + H.O NH. H.,0^ ,NH, H,0 NH:, NH, CI3 The fact that the two water molecules of [Co(H20)2(NH3)4] CI, coordinate in tlie cis-position can easily be recognized from the absorption curve, which is very similar to that of the second class, and also by the process of formation as follows : N H3 ~ N H, ' ^?- ,NH, CI + HCl aq. >- H,0 // qI / 1 ^H, H,ü MI3 N H3 _ NH, NH, CIb When Co ^'j ]^[ Co 01. CI (1) r. Cl(6; CI is kept overnight in aqueous solu- tion, its colour changes to carmine red and then it shows the same absorption curve as that of the cis-diaquodiethylenediamine cobaltic chloride, Co j^-^ ^^^ eii^ CI.. This will be verified below. At first its two chlorine atoms are replaced by water, producing the trans-diaquodiethylenediamine cobaltic chloride, r— TT O (1) ~1 ^'°h!o(6)^"^ CL. and then it is converted into the more stable r tt'o (i^ n cis-salt, i.e. p'*' h"^oi2 ^■■"■-' ^■^•' tlius : CI3 — ^ Molecular conver- sion Cls As to the complex salts of the second class, discussions follow quite in the same way as in the case of the first class. Werner 20 Art. 10.— K. Matsimd : lias already studied the constitution of I ^^ h^O (2) ^^^^ Cl<,('^ two water molecules of which are coordinated in the cis-position, thus : , HjO / CIs The compounds belonging to this series are general!}^ more stable than those of the first class, while tlie latter seems to have a great tendency to be converted into the former. From the fact that the absorption curve of the cis-diaquodiethylenedia- niine cobaltic chloride (fig. Ill) is analogous to the curves of [Co CO3 (NHg),] CI, [Co CA (NH,),] CI and [Co CO3 ey^g] CI, we can affirm that, when tlie water molecules coordinate to a cobalt atom to form a complex salt, then unite with the central atom by tlie residual affinity of the oxygen of water as illustrated below : OH., -7 OR,. NH. >C0. NH, NH, o -c KH, The chlorotriaquodiaramine cobaltic sulphate must have the following configuration. For its absorption curve (fig. IV) is very similar to that of the cis-diaquodiethylenediamine cobaltic chloride, (1) B.T., 40, 285, 1907. Configuration or Aiiiotri- and diainiuinL" Colnilt Complexes. 21 wliich suggests that its three water iiiolecules ctxH^h^iiate in tlie eis and consecutive position, thus: — H„0 J XH, H,0* H,0 ROi As a consequence, tlie structure of [Co CI2 (HgO)^ (NH3)2] SO4H could also be determhied, as has been already mentioned. ïlie compound [Co CI2 (H20\(NH3)2] SO^H has at first a green colour in aqueous solution, but this changes gradually until its colour becomes indigo blue. This change of colour is a proof that the following substitution takes place : NH3 ci .NH3 H.O' H,0 (green) NH, SO4U n„o . .NH, ILO Cl H,,0 (indigo blue) SO4 The cui-ve of [Co (^113)3 (HsOjg] CI3 (7) is ver}^ different from that of [Co (NH3)3 (HsO)^] CI3 (8), which is the best verification that the two salts are isomeric; and the former is very stable in water. It is prepared from [Co (NH3)3 (N03)3] the configuration of the latter will be as follows : NH, KO, NH3 JiO,' NH, NO, 22 Art. 10. — K. Matsimo General Consideration of the Absorption Spectra. The aquocobaltainminos in which the water molecules co- ordinate in the cis-position tend remarkably with an increase in the number of water molecules, to produce the batliochromic phenomenon and their absorption spectra differ greatly from those of the trans-salts. All the aquocobaltammines, according to the number of the coordinated water molecules, are apt to diminish the second absorption band, which affords an additional proof that the second absorption band of these salts is, as Prof. Y. Shibata has suggested, due to tlie coordinated ammonia. Measurements of Electrolytic Conductivity. Although I have in the previous chapter demonstrated by means of the absorption apectra that in some of the chlorocobalt- ammines the chlorine atoms are replaced by water molecules in aqueous solution, the confirmation will not be satisfactory Avithout another verification. Therefore I have measured the conductivities of solutions of these salts in order to confirm the above opinion. The metliod used was that devised by Ostwald. The data are as follows : Table I. [CoCl(NH,)5]S(), [CoClH./)(NH.,)4]SO, [CoCl(H30),(NH,)3]SO, ß Ü 0 u ß u 100 200,8 200 281,4 400 2Ü8,4 800 300,4 lüOO 843,1» !oCl(H., 0)h(NH3\ 100 184,2 200 222,8 400 267. (■) 800 • 882,7 11)00 412,3 400 11)0,1 800 218,7 1()00 259,2 ;coco< ,(ßB.,\\S 100 78,83 200 87,18 400 1(2,18 800 91), 18 1600 106,2 800 259 o 1600 462 /.) ^oH/) (NH3)5],( :s 100 117 ,0 200 151,8 400 182 .2 800 217 ,9 1600 265 ,1 Configuration of Ai|uûtri- aud <;liaiumiae Cubait Complexes. 23 Where ß is the dihition, U tlie molecular conductivity. Measurements were made at the temperature of 25°C. The follow- ing table shows the variation of the conductivity with time. Table 2. [Co >H,0(NH3U(S0J3 Ö Time (Min,) U 100 0 117,0 5 122,3 10 124,3 20 129,3 30 133,6 40 137,9 50 141,3 CO 143,5 Summar}^ of the data at the dilutions of 800 ct 1600 : — Table 3. ^ i ['^^NH:)J,^°' [<^°(Nh")J,'«0.!. [Co, N'a J so. 800 I 99,13 217,9 300,4 1600 I 106,2 265,1 343,9 ^ I L^^^n^^' i^^T^n^^' [^^œ-]^«. 800 I 218,7 259,2 332,7 lüOO 1 259,2 462,9 412,3 As seen in the above tables the carbonatotetrammine cobaltic sulphate is stable in water and shows little variation of conductivity with the dilution. The molecular conductivity at the dilution of èOO- 1600 is about 100. The conductivity of the aquopentam- mine cobaltic sulphate varies more with its dilution than that of the carbonatotetrammine cobaltic salt. The dilution from 800 to 24 Art. 10. —K. Matsnno : that of 1600 causes an increase of about %)%, giving a mean value of about 250. Tlie effect of time will be seen in table 2. It increases 1\J)% during one hour. Werner and his co-workers^'^ liad previously expressed I»oth the chemical and electrochemical behaviour of the cobaltammine chlorides by the graphic represen- tation of their molecular conductivities and gave the following data: Number of ions Molecular conductivity at the dilution of 100Ô 2 Approx. 100 3 „ 250 4 „ 480 It shows that the number of ions of the cobaltammines is proportional to the conductivity of the salts. While [CoC03(IS[H3)4],SO, and [CoH^O (NH^^äJ^CSOJ, have three and five ions, as well as the molecular conductivities of about 100 and 250 respectively, the chlorocobaltammines mentioned above should have molecular conductivities of less than 100, if there occurs no transformation in water because they have two ions, judging from the molecular formula only. The results of the experiments -.are contrary to this supposition and show two or three times the required value for the molecular conductivities. They vary very much with the dilution, especially in the triammine and diammine complex salts, which have values approximating to the maximum conductivity of hydrochloric acid, i.e. 403, at a dilution of 1600. The fact that strong acid radicals such as CI or NO3 when coordinated in the cobalt complex nucleus to form a complex salt sometimes show ionic reactions, in other words, dissociate in aqueous solution, was demonstrated by Werner. For example, the purpureo cobaltammine chloride, if dissolved in water, under- goes to a certain extent the following substitution : [Co CI (NH.,)5] CI2 + HP — ^ [Co H.,0 (NH,\] CI, The above system I'eacts sometimes acidic when the coi>vei'- sion takes place, thus : (1) Wcruoi- : New Tdias of Iiior,Cl2(H,0 ,]X, [Co(NH3)2(H,0)3Cl]X„ [Co(NH3)3(N03)3], [Co (NH«), (K,0),] X,, [Co (NH3), CI H2O] X, [Co (NHä)^ C^O, . NO,] 3. — The fullowiiig two new co1)alt complex salts were produced. 1 ) Trans [Co (NB.^% (H.^0%] CI, FLO, NH, H,.0 H„0 NH, CI3 (2) [Co (NH,\, CA .NO3] NH3 n coo. ,-VH, coo-^ XO, 4. — The formula [Co (NH3)3 CA (1120)2] NO3, which was given by Werner, must be amended to [Co(NH3)3C204.N03].B[20. O. — With regard to the substitution reactions of the cobaltam- mines which have radicals of the strong acids in the complex nucleus, a clear explanation has been given by the study of absorption of light and electrolytic conductivity. In conclusion, the author tenders his sincere thanks to Prof. Y. Shil)ata for his kind guidance and suggestions. (The Laboratory of Inorganic Chemistry, the College of Science, Imperial University of Tokyo). Published March 3 Ist, 1921. JOURNAL OF THE COLLEGK OF SCIENCE, TOKYO IMPERIAL UNIVERSITY. VOL. XLI., ART. 11. The Coagulation of Arsenious Sulphide Sol by Cobaltic Complexes. By Kichimatsu MaTSUNO, Eiijaku^hi. Introduction The valency of inorganic complex ions is usually determined b}^ measuring their electric conductivities in solutions [Werner aui] MioLATi: Zs. physik. Chem., 12, 3Ö : Ï4, 506(1894)]. This method is dependent to a certain extent on the migra- tion velocity of the ions. It is well known that the valency of a coagulating ion has a great effect in determining its coagulating powcA*. Galecki [Zs. Elek. chem, 14, 7(37, (1908)] utilised this fact in determining the valency of beryllium by coagulation experi- ments with arsenious sulphide sol. Freundlich [Zs physik. Chem., 89, 504, (1912)] has shown that it is possible to follow the change in the valency of the cation of a cobaltic complex Ijy coagulation experiments with arsenious sulphide sol. But such experiments with cations of a valency greater than three have not been tried so far. It is now generally accepted that the so-called valency law of Schulze is a rough generalisation [Bancroft, J. Physic. Chem., 19, 348, (1915): Brit. As. Rep. (1918), Wo. Ostwald; Koll. Zeitsch., 26 (1920), 69]. Experiments with complex cations having a valency up to six is of interest as a test of so-called valency law. It Mill be seen from the sequel that tlie valency of the cation has a predominant effect and that the valencies determined by coagulation experiments agree perfectly with the usual formulae given to these salts. 2 Art. 11.— K. Matsuno : Experimental The limiting concentration of a salt Avhich just failed to pro- duce any perceptible change in the sol after an interval of five minutes, was taken to be the measure of its coagulating power. This method is more sensitive than other methods [Mukherjee, J. Amer. Chem. Soc, 37, 2024 (1915)]. The coagulating power of an electrolyte is dependent on the quality of the sol, the sulphide content, its age and the temperature [Mukherjee and Sen. Journ. Chem. Soc, 115, 462 (1919); 117, 350 (1920)]. Thé present investigation was carried on with the same sample of sol in order that the data can be strictly comparative. To minimise the effect of " aging ", the sol was allowed to stand for one year before the experiments were made [Compare Pauli and ]Matula, (Koll. Zeitsch, 1, 1917) who took the same precaution with ferric hydroxide sols]. The sol contained 37.6 milimoles of arsenious sulphide per litre. One c.c. of the sol was placed in each of ten carefully cleaned test-tubes of Jena Glass. Two c.c. of solution of a salt of different Table I. x^ r ,, • Limiting concentration Cobaltammine .-ß txt i\ (Eq. Mol) 1 [Co (NH3), C,0 J 01 2 [Co (NH3), (NO,), a|]3 [Co (NO,)e] 3 [Co(NH3),(N0,),g|]01 4 [Co(NH3).,(C03)],S0, 5 [Co (NH3), (NO,), g{] [Co (NH3), (NO,)J G [Co(NH3),(NO,),iJ>]Cl 7 [Co(NH3),C03]N03^H,0 Mean 10 1500 9 1500 8 1500 S lôOO 8 1500 7 1500 •7 1500 8 1500 The Coagulation of Arsenious Sulphide Sol by Cobaltic Complexes 1 2500 1 1 [Co (KHg)^ NO,] CI, 2 [Co [ls'U,% NO,] [Co (NH,)/NO,)J, 3 [Co (NHs)g SCN] CI 4 [Co(NH3)5Cl]CL 5 [Co (NH,)5 CI], (SO,H),- SO, Mean 1 [Co en,] C\ 2 [Co (NH3)s H,0], (SOJg 3H,0 3 [Co (NH3)6] CI3 4 [Co (NH^)«] (NO3), /0H\ 6 I (NH3)3Co^OH-7Co (NHs)^ CI3, H.O ■- \0H/ J 7 (NH)3 Co^OH^Co (NH3)3 | (HSO,) Mean 1 [(NH3)5 Co-NH-Co (NH3)5] CI, Mean Co '^^ Co ^NH,),' CI. 2500 1 3000 1 3000 1 3000 1 3000 1 12500 1 12500 1 15000 1 15000 1 18750 1 15000 5 240000 2 [ NH3), Co<^^g'^>Co (NH3)J CI, 4H,0 ^^ ^^^3).>Co-NH.-,-Co<,^^^ , 1 CI, ^:^ H.O - ^(NH3),J ' '240000 4 [,NH3),CoCo(NH3),]Cl, -^ 4-5 240000 1 240000 4 Art. 11. — K. Matsuno : concentrations were added t© each. During addition the tuhes were vigorously shaken to ensure thorough mixing. In this way t.ij(}_ I two sucli c/^ncentrations are obtained that the lower does not show any percepti- l)le change in five minutes whereas the liigher does. This concentration is then more carefully examined and the limiting concentra- tion determined. In Table I the limiting concentra- tions corresponding to the different salts are given. For comparison the limit- ing concentrations of a number of simple electro- lytes were determined with the same sol. The results are given in Table IV. (p. ^ 8) ■f S e Val ency The Valency Rule on Coagulation As regards the quantitative study of the effect of the valency on the coagulation of hydrosols, Whetham [Pliil. Mag., Y. 48, 474, (1899)] deduced the following equation on considerations of probabilit}^ of collision: C, : C, : C3 = K=^ : K^ : K\ where Ci, Ca and C3 are the molecular concentrations of coagulation of mono-, di- and trivalent ions respectively, K is a constant. If this be true, it must be a straight line when plotted, taking the logarithm of the concentrations of coagulation and the valency of the ions as its axes. This did not, however, liold good in the FIG 2 The Coagulation of Arsenious Sulphide Sol by Cobaltic Complexes 5 present experiment. So far as the present investigation was con- cerned, it gave a curve, as shown by Fig. 1, in place of a straight hue. Freundlich [Zs. physik. Cheni.,73, 385, (1010)] calcu- lated approximately the pre- cipitating concentration8 \)y utilizing the adsorption curve and by assuming that the neutral salts in their equiva- lent solutions are, in despite of their valencies, adsorbed in the same manner. His idea can be expressed also as follows: Since the electric charges of mono-, di- and tri- valent ions are in the ratio of 1:2:3, the amounts which should be adsorbed causing a complete coagulation should be in the ratio of 3 : 1"5 : 1, and if we plot the logarithm Lop- (Valency) ' , • •. ^• *' 01 the precipitating concentra- tions against those of the numbers , i.e. 3, I'o and 1 for mono-, di- and trivalent ions respectively it would give a straight line. It seems that this idea of Freundjich was clearly verified by the present investigation. Taking the logarithm of the valencies of the cobaltammines as abscissa and those of the limiting values (for convenience the author took the values given in the third row of the table III) measured by the method already described, as ordinate, a straight line, as shown by Fig. 2, was obtained. From th'e diagram the author was able to deduce the following equa- tions : < 3.00 1 ^ 7n S 1 S zoo 1 r». . tri- ,, ,, 4-076 ,, tetra- ., ,, 3 998 ,, nexa- ,, ,, Accordingly, putting /3 = 4 in the equation (o), the equation was deduced : Sj,= S^x— (4) Namely, the limiting value of a N-valent cobalt complex ion is equal to one N^th of that of monuvalent complex ion so far as the cobaltammines and the arsenious sulphide sol are concerned. The equation (3) could be also deduced theoretically as follows. In the adsorption isotherm, i.e. — = aC°, where « and n are constants, x, m, and C are respectively the quantity adsorbed, the The Coagulation of Arsenious Sulphide Sol l>y Cobaltic Complexes 7 quantity of the adsorbent, and the equihbriuin concentration. As a nrst approximation, we may take S , or S in the place of C of the above equation [Freundlich, loc. cit.]. If we replace S with C, "we obtain the next equation : ^=«S^ (5) And comparing the case of the N-valent, and the monovalent, we can easily deduce the following equation : ^^ = asj and ^L. ^ ,, gf x^ X " = (lf)" "-^ (f)=-| ■••••(«) According to the opinion of Freundlich, a^N _ 1 X, N Therefore the equation (6) may change to the following: (ir=^ (7) Thus it is clear that ß in the equation (o) is identical with one of the constants of the adsorption isotherm, ß has been found to be equal to 4 in the present experiments. It is quite possible in the case of the cobaltammines that the irregularity observed in coagu- lation experiments with the normal salts would be absent, as the cobaltammines used had all their co-ordination valencies satisfied, so that they had no furtlier tendency for complex formation in aqi^eous solution, the adsorbability should, therefore, be the same for ions with different valencies and hence the effect of valency is likely to be the sole factor. As show^n by the table III the limiting values calculated from the equation (4) are in fair agreement with those found by the experiment. Art. 11. — K. Matsuno : Valency Limiting value (Eq. Mol) (^Observed) or 1280 X (Calculated) Table III 1 2 3 4 5 6 1 1 - 1 1 1 187.5 3000 1.5000 53330 240000 1 80 „ 16 „ 4-5 „ — 1 „ 240000 1 1 1 1 3000 15190 47990 243000 The figures of the second row of the above table are the mean of the Umiting values obtained from seven monovalent, six divalent,, seven trivalent, four tetra valent and one haxavalent cobaltammines as shown by table I. Unfortunatel}^, as the author had not prepared a pentavalent cobaltammine, the figure is lacking. But it is very easy to predict the limiting ^-alue of the pentavalent complex ions. By using the above equation, it should be , ^nqqqq for the sanie sol. The valency effect for the normal salts. In order to compare the results obtained b}^ the cobaltammines witli tliose of the normal salts, the experiment w^as undertaken by using the following salts and the same sol. Although the salts were not widely selected, there was much deviation from the theory as shown by the table IV, V, and Fig. o. Electrolyte 1 KCL Table I\' 2 KBr 3 KNOs 4 NaCl 5 NaCgHijOs (Sod. Leuciate) Limiting value 150 10 Mean 150 8 T5Ö" 8 CuS0,5H,0 9 Zn(CH3C00)o 1 Ce,(SO,)38H20 •2 AL(S0,)3lSH,0 3 CroiSOJglSH.O 1 ZrCl, 2 Th(N03),4H.O Mean 1500 The Coagulation of A rseniuns Sulphide Sol by Cobaltic Complexes 1 BaCl,2H,0 2 NiS0,7H,,0 8 Sr(N03)2 4 CaCL 5 Co CI, OH., 0 6 FeSOjn'o 7 ZnS0,7H.,0 5000 11 15000 10 5000 1 1071 1-6 30000 1 30000 1 10000 1 24000 T. A^BLE V. Valency 1 2 3 4 Limiting value - 8 11 15000 1-6 30000 1 150 24000 or 12800 1 X 176,, 12-8 „ 10 „ 2 J 0000 Fig. 3 was drawij in tlie same manner as the Fig. 2 was graphed. 10 Art. 11. — K. MatsTino : FIGr. 3 4 000 For the monovalent ions, sodium chloride, potassium chloride, potassium bromide, potassium nitrate and sodium leuciate were used, the same limiting concentrations were obtained except the last one. For the divalent, barium chloride, nickel sulphate strontium nitrate, calcium chloride, cobaltous chloride, ferrous sulphate, zinc sulphate, copper sul- phate and zinc acetate were tested, tlie values of the last two salts were very different to those of the others, and even comparing the values of Zn" both obtained by using zinc sulphate and zinc acetate, a difference was recognized. In the case of copper sulphate it is clear that the ionisation is not so og{ d encj.) simple as we consider. namely, Cu SO4 aq. = Cu • • + SO/'. It is possibly a complicated, one. Considering tlie fact that an old aqueous solution of copper sul- phate reacts acidic, the idea of tlie following partial hydrolysis is reasonable 2CUSO4 + 2H2O = Cu,(0H)2S0,, + H,S0, But the author thinks that complex such as Cu(SOJfl;(H._,0)?/ is possible. For the trivalent, cerium sulphate, aluminium sulphate and chromic sulphate were used and the last two gave the same value, but the first one was different. As tetravalent ions zirconium cliloride and thorium nitrate were used. The limiting values ol)tained by these two salts were not only in disagreement with theiïiselveè, they were not suitable for the tetravalent ions. The Coagulation of Arseuious Sulphidi' Sol )iy Cobaltio Couiplexes XI In the case of these salts tlie IVjllowing hydrolytic reactions are most probable : Tb(NO,), + H,0 = TliO(N03\, + -2HN03 Zr CI, + H„ü = ZrOCl, + 2HC1 Chemical Changes of the Cobaltammines in their Aqueous Solutions. The chemical clianges in tlie aqueous solutions of cobaltam- mines with co-(^)rdinated radicals of strong acids were clearly explained by the author in his previous paper [This Journ., the same Vol. Art. 10]. He has been able to confirm it by the study of the limiting values as follow : (A) [Co (NH3), NO, CI] CI . H,0 If the above salt dissociates in the aqueous solution, thus, [Co(KH3)4NO,Cl]Cl aq. :|: [Co(KH,)4NO,Cl] + Cr the limiting value obtained by it will be -lorj,^ (see the chapter of the valency rule). Jjut the experimental data did not agree there- with and it gave ..^^q iii =-^ fresh solution and ,^ .^ when oO min. have elapsed. The latter figure corresponded with that of a divalent ion. It proves the truth of the author's suggestion that this kind of salt undergoes a furtlier substitution as below : [Co(KH3),IsO.Cl]Cl + H,0 ^ [Co(NH3),NO,H20]CL (B) [Co.«,g|J]]ci, [Co(NH3),g[J]]ci and [Co .n, gj! [.^]] Br. It is a well known fact that the above mentioned salts undergo colour changes when they are dissolved in water. The author proved in his previous paper [loc. cit.] the following substitutions would take place : 12 Art. 11.— K. Matsono : [Com,g^J]] Cl: (1) [Cb.;.,^;{[j]] Cl + H,0--[coen,g^Qg]]a (•2) [Co en, g^^^g]] Cl+ H,0 -^ [Co en, ^^ g]] CI3 (2) [co(NH3),gQ[J]]cLH-H,0_^ [^^°^^I^3).gjo(2)]^^^ [Co..,gjg)] Br : (1) [Coen,^\^ B. + U.ß -^ [Co en,^]^^^ Br, With these substitutions, the valency of the saUs varies also. r Cl f 1 )~i 14 The limiting concentration of Co en, ç,^) J Cl changed from tfTw) to .,^.^ during three hours and after 24 hours it gave the value , i.e. value for the trivalent. It is obvious that the tirst 15000 substitution took place within half an hour and the second within 24 hours. As to [co(NH3),^}|gj] Cl and [co m, ^J |,^]] Br, it seemed that the hrst substitution took place very rapidly, for the limiting values of both these fresh solutions Avere „-^^ and after 30 min. they gave the value ^^r^r^Q corresponding to the trivalent ion. (C J [Co (NH3),H,0 C 1] SO,, [Co (NH3),H,0 Cl] Cl Each of the above mentioned salts gave the Hmiting value of a trivalent ion, i.e. . ;.^.. .. instead of that of a divalent ion which proved the following changes occurred in their aqueous 'solutions respectivel}': The Coagulation of Arsenious Sulphide Sol by Cobaltic Complexes 18 . [CoNH3),HO,Cl]SO, + H,0 ^ rCo(NH3),(H,0)2l ^^ [Co(NH3)4R,OCl]Cl, + H,0 ^ [Co(NH,),(H,0)2]Cl3 (D) [Co(NH,)5Cl]X„ [Co(NH,X,(ILO),CL] SO,H, [Co(HN3)3(H,0)2Cl]SO, and [Co(NH,)3H,OCy Ci In the previous papei" [loc. cit.] the author discussed the equihbriam of the aqueous solution of some cobaltammines and showed that a comphcated equihbrium results. It was also confirm- ed by coagulation experiments. Freundlich [loc. cit.] expressed the same opinion regarding the salt [Co(NH3)5Cl]Clo. If in these salts the substitution occurs completely as follows : [Co(NH3)6Cl)X, + H.,0 -^ [co(NH3),H,ü] gf [Co(NH3)o(H,OXXySO,H + 2H,0-^ [co(NH,X,(H,0)4"] g J:j g- [Co(NH3\(H,0),Cl]SO, + H,0 ^ [Co(NH3)3(H,0)3] ^^^ [Co(NH3)3H,0 CL,] CI + ILO -^ [Co(NH,)3(H,0)3] CI3 the limitinti; concentrations of these salts will have a value near about 15000 since the resulting: salts are all trivalent. But the experiments show that the reactions as indicated above are not complete. [Co(NIÎ3)5Cl]CL gave ....^.. in a solution which stood 6000 I for over a night. [Co(NH3),,(H,0)oCl]SO.,H gave ^^^^^^ in a fresh solution and gave -^7777^ after three hours and there was no change '^ /500 ^ up to the next day. It shows that the following successive reactions occur: (1) [Co(NH3),(H,0),CL] SO,H + H,0 -^ [Co(NH3)2(H,0)3Cl] f^^^ (2)' [co(NH3),(H,0)3Cl] f^^^^+ tLO -: [co(NH3),(H,ü),] f^-^ [CD(NH3)3fH2 0)2Cl]S04 was ratlier stable in aqueous solution 24 Art. 11. — K. Matsuno : and gave the limiting concentration of ^^^^ instead of -.^^qq which verifies tliat there is an ecjuihbrium as indicated below: [Co(NH3)3(H.p)2Cl] SO, + H^O X [co(NH3)3(R,0)3] ^\^^ [Co(lS[H3)3H.OCL]Cl gave also the value of ^^^ in place of 15000 • (E) [Co(^^H3)3(H,0).|]CL and [Co(NH3)3(H,0}3|]C]3 If the above two salts dissociate as [Co(ISIH3^3(HyO)3]Cl3aq. ^ [Co(]S[H3)3(H._,0)3]-+3Cr in their acjueous solutions, the limiting concentrations of these salts must be -t^ttt^, but in fact both of them gave tlie value of „^^^ which is the same as those of [Co (:NH3)3(H,,0), Cl] so,, [Co(NHs)3PLO CL] CI and [Co(NH3)2(H,OX,ClJ SO,H. This is the best confirmation that the following reversible reaction takes place to a certain extent: [Co (NH3)3 (H20)3] CI3 aq. ^ [Co (NHglg (Hp)^ Cl] Cl^ aq. Summary : 1. — The relation of the valency of the cobaltammines to their coagulating power on arsenious sulphide sol has been studied and it has shown that the limiting ccncentration can be expressed hy the following formula, s; - S, X 4t where, ^^n is the limiting concentration (Eq. jMoI) of a N- valent complex ion. The equation can be deduced from Ereundhcli's absorption tlieor}-. The Coagulation cif Arsenious Sulphide Sol by Cobaltic Complexes 15 2. — The valency of many simple and complicated cobaltam- mines was determined by means of the limiting concentration. o. — r>y utilising the valency effect on the sol, the chemical changes in aqueous solutions of some cobaltainmines has been followed. The results are in agreement with those obtained from conductivity meajSurements and absorption of light. The author would like to express his sincere gratitude to Prof. Y. Shtbata and Prof. K. Ikeda for their kind advice. His best thanks are due to Dr. JNIiyasawa for his assistences. (The Laboratory of Inorganic Chemistry, College of Science, Imperial University, Tokyo.) Published march 31st, 1921. MBL WHOl LIBRARY ÜJH nL2 $ .vH-i=>?.;