extends much further at temperatures not particularly high than Wüllner indicates for the state of perfect formation of the continuous spectrum. The first spectrum of hydrogen, according to Wüllner, ends between Hẞ and Hy. I have seen it much further. The strong fluorescence of the glass tubes which the light causes ought to have shown long ago that the spectrum is much longer than has been indicated hitherto. The dispersion of brightness in the continuous ground of many line-spectra seems to me to correspond so much with the relative intensity of certain places in the spectrum of the first order, that I tried to get at a continuous spectrum directly from a band-spectrum by raising the temperature. In this I have succeeded very well with nitrogen; unfortunately at present there is no time for a closer investigation of hydrogen in this regard. I do not attach particular importance to a single experiment which confirms my view. The spectrum of the second order is obtained with the lowest densities in which jar-discharges still go through a narrow tube filled with air; these densities are represented by fractions of a millimetre. If, then, while constantly using jar-discharges one lets the density increase, the lines disappear and the band-spectrum appears. If the density still increases, then the brightness of the dimmer band-parts grows perceptibly faster than that of the stronger maxima; the proportion of brightness reaches unity as a limit, and successively more and more bands are replaced by evenly illuminated continuous bright parts. I could extend the continuous spectrum (in which the eye, in spite of great brightness of the whole, does not distinguish any single bands) from the red to the violet; only the most extreme bands at the refrangible end were still distinct. As the formation of continuity progressed in the direction towards the refrangible end, I do not doubt that, with a still more favourable arrangement, even the most extreme part of the spectrum could be obtained perfectly continuous. The absolute brightness of the continuous spectrum was not so great that a deception caused by excessive intensity of light was possible. If the density be then further increased, the line-spectrum again appears on a ground which is already continuously illuminated. I have convinced myself that the phenomenon I have described has no relation to fluorescence. The discharge obtains a characteristic colour when the continuous spectrum appears. While it is white for the line-spectrum, rose-coloured for the spectrum of the first order, the light becomes of a yellowish rosy tint as soon as the continuous spectrum appears; so that the eye notices the phase in question without spectral apparatus, by mere observation of the tube. Some pages back I have spoken of a yellowish rosy part in the negative half of many jars' sparks, which part gives a spectrum continuous as far as the blue: the preceding lines might contain an explanation of this phenomenon (at least with regard to the spectral peculiarity). The outside appearance of it, that in the spark there is a place of different colour and brightness, is already known in principle by observations of machine-sparks by Adams, Knoch, Dove, and others. I have seen similar phenomena with induction-sparks in dense air. The spectrum of the blue light at the cathode has often been discussed; upon another occasion I will communicate my experiences in respect to this. Here I would only like to draw attention to the fact that colour and spectrum of this light are not always identical. The changes mentioned are most striking when the negative light appears with the optical properties of the positive one. With high rarefaction the negative layers situated from the second towards the outside adopt the colour and spectrum of the positive light. With jar-discharges of greater intensity the total light near the cathode cannot be distinguished, either by its colour or by the prism, from that of the positive current. The tendency to conform itself to the magnetic curves is then still evident. But negative light can also appear far from the cathode. If tubes like those used for spectral purposes, or many brought into trade as effective articles, cylinders, balls, ellipsoids, are parted off, then each one of these parts behaves during the discharge very nearly like an independent tube having its electrodes at the two entrance-points of the current. At the negative entrancepoint light shows itself, which by its straight-lined dispersion, the capacity to cause fluorescence, and the property of conforming itself to the magnetic curves under the influence of a magnet is characterized as negative light. Its form corresponds to that light which would proceed from a cathode, the surface of which would fill the entrance-point. The diffused cloud of light which forms round such an electrode, which has more or less the shape of a point, is represented with the newly found appearance of negative light, by light almost of the colour of the positive. In its spectrum, which on the whole coincides with that of positive light, some maxima of the negative light seem certainly to be marked sharper than the corresponding wave-lengths of positive light. To that cone which, distinguished by brightness, stands perpendicular on the cathode and forms the central axis of the phenomenon, here again a cone corresponds of the colour of the negative light. I retain the details of this and other, simultaneous phenomena for a future communication. XXXIX. On Unitation.-III. The Unitates of Powers and Roots. By W. H. WALENN, Mem. Phys. Soc.* אן N former papers† it has been shown that if a, b, c,... s, t, u be the digits of a given number, and 8 be any integer less than 10, n being the number of digits in the same number, the expression n (10—8)"-1a+ (10—8)"-2b+(10—8)′′-3c+ +(10−8)t+u has the same remainder to 8 as the given number has. δ n-2 By simple substitution of 8 in this expression, the remainder to & may be found, two kinds of operation being necessary for that purpose. The first is the determination of the coefficients (10—8)”—1, (10—8)"-2, &c., multiplying each digit by its coefficient and adding the terms thus produced. The second is a repetition of this process such a number of times as will produce a single digit. This method of obtaining the remainder to 8 is called unitation. The remainder corresponding to a given number for a certain value of 8 is said to be the unitate of the number, and the divisor (8) the base of the system of unitates under consideration. The symbol Usa is used to signify the unitate of the number x to the base 8. The principles brought to bear, and the examples given in the former papers upon this arithmetical process, showed that it was useful to check calculations, to obtain remainders to a given divisor without the use of any multiple of that divisor, and to verify tables. It was further shown that negative and positive integers could be found, by the process of unitation, to represent unitates that could not be obtained by division. The nature of the operation of unitation, and its position amongst other operations, will first receive attention, as introductory to the determination of the unitates of powers and roots and to the discussion of some of their properties. Of the two kinds of operation (necessary for the complete unitation of a number) mentioned above, the first part (involving the addition of terms) is analogous to the formation of an ordinary number, in the decimal system say, another multiplier besides 10 being used; the second operation is analogous to that indicated by the symbol A" in the Calculus of Finite Differences, or to that of derivation or successive differentiation indicated by the symbol "). Unitates, however, do not bear * Communicated by the Author. + Phil. Mag. S. 4. vol. xxxvi. p. 346, and vol. xlvi. p. 36; and British Association Report for 1870, Transactions of the Sections, p. 16. upon themselves any symbol to show the number of times that the operation is repeated to produce the result. This indication might be useful in certain instances; and the results of it have yet to be worked out. Comparing the two methods at present known by which the remainders to a given divisor can be obtained, namely the operation of division and that of unitation, it will easily be perceived that obtaining remainders by division is an inverse process, whereas unitation is a direct process. Division has to be worked out by commencing with the highest or left-hand figure, and proceeding towards the smallest or right-hand figure. Unitation is begun at the right-hand figure and proceeds towards the left-hand figure. In unitating to the base 10, for instance, the unit figure is identical with the unitate; for all the powers of (10-8), and therefore the coefficients of all the terms containing that factor, become =0. These remarks respecting the operation of unitation do not relate in any way to the best and shortest method of obtaining the unitate of a given number to a given base. This was to some extent elucidated in the first article upon the subject, in which it was shown to be sometimes advantageous to commence at the left hand and reduce the work as soon as possible. For the purposes of investigation, it may frequently be desirable to retain the coefficients in the same form as the expression cited in the commencement of this article furnishes them, and to proceed with the operation as there indicated t. In practice, when necessary, the coefficients themselves may be unitated to the base 8; this either produces recurring series as coefficients, or else cancels them together with the terms to which they belong. The right-hand figure being the coefficient of the unit belonging to the given number, the following are the series for some values of 8: 8= 6 <= 9 =10 =12 1 ... ... ... ... ... 1, -1, 1, -1, 1 -8, 4, -8, 4, -2, 1 =13... .,-9, 3, -1, 9, -3, 1 Unquestionably the most useful and practical of all the systems of unitates included in the above formula (for checking *Phil. Mag. S. 4. vol. xxxvi. p. 346. † For example, in the function U,, the above formula or expression becomes 3n-1a+3n-2b+3′′-c+...+19,6837+6561m+2187n+729o+243p +81q+27r+9s+3t+u. A tube which has different widths in different parts shows sometimes, when the discharge is disruptive, the line-spectrum at the poles (which are generally introduced into the wider part of the tube); the remaining part of the wider end gives the band-spectrum; and the narrow part shows again the line-spectrum. The rotating mirror shows, as was to be expected, that the discharge is the same in all parts of the tube. The discharge of a Leyden jar in air under rather high pres sure shows sometimes a strange phenomenon. Near to the negative end of the tube appears a place which is of yellowish pink colour. The spectrum-analysis of this point gives the bands of nitrogen; the remaining part of the tube shows only the lines of nitrogen. The rotating mirror gives the image of the tube together with its pink point as one whole. (Velocity of rotation about 40 per second.) The spectrum of the pink point makes rather the impression of a continuous spectrum. The bands are wide and indistinct; if the phenomenon is at its best, only the blue and violet bands can be distinguished from the illuminated ground. I shall try further on to give an explanation of this phenomenon. I believe it to be established by the above experiments, that the different spectra have nothing to do with the mode of discharge. I could not confirm by experiment the second proposition of Wüllner-" In the disruptive discharge only a few molecules, therefore a very thin stratum of the gas, is luminous." I could obtain sparks in rarefied gas as much as several centimetres in diameter. The colour of such sparks in air is blue or pink, with the various tints which can be produced by changes in intensity and saturation of these colours. The colours of thick sparks in hydrogen are bluish white, fleshcoloured, yellowish, yellowish red, and crimson. Not only are those sparks thick which give the band-spectrum, but also those (and it is of these Wüllner was thinking) which give a line-spectrum. By introducing sparks and jars in the outer circuit, we may succeed in obtaining sparks blue in air and red in hydrogen which give line-spectra, and which filled entirely tubes of 13 centim. diameter. I shall mention subsequently observations on the negative light which refer to this point. Another assertion of Wüllner's says that the band-spectrum is always produced by a thick discharge. Part of the above can already be produced as evidence to the contrary. In capillary tubes band-spectra are seen, while line-spectra may be given out by much thicker layers of air. The induction-spark in air under ordinary pressure, which is |