Radium rays may be inserted provisionally between Lightshock and Lamplight in the above order. Conclusions. 1. The density of the image produced on a plate by exposure to radium rays (8 and increases to a critical value and then decreases, at first rapidly and afterwards very slowly, until a time is reached when the image is totally reversed. 2. Spark-images are at first obliterated by radium rays which do not cause such a great density as that of the spark-images obliterated. 3. With prolonged exposure radium rays reverse sparkimages. I am greatly indebted to Mr. Hayles, of the Cavendish Laboratory, for carrying out many of the experiments described in this paper. XXXI. On Deflexions of the Plumb-line in India. By Major S. G. BURRARD, R.E., Superintendent of the Trigonometrical Surveys of India*. N the Philosophical Magazine for January 1904, the Rev. O. Fisher gives his reasons for thinking that the Indian pendulum results do not support my suggestion, that a chain of excessive density, many hundreds of miles long, lies hidden in the Earth's crust in Upper India. As Mr. Fisher's paper has been read by many with interest, I would ask to be allowed to explain my reasons for having omitted to refer to the pendulum results in the original paper on the subject. Those reasons may be stated as follows: 1. The correctness of the Indian pendulum results has been questioned by the highest living authorities (vide Colonel Clarke's 'Geodesy,' and Professor Helmert's report to the International Geodetic Conference, 1901). 2. The supposed hidden chain of excessive density is between 1000 and 2000 miles long; one line of pendulum stations crosses this chain at right angles. We cannot get any fair idea of the mass of a chain from the results of a single traversing line. 3. The pendulum stations on this line of observation are too far apart to allow of conclusions being formed. Mr. Fisher's conclusion that there is a marked negative variation of gravity over the whole of India, cannot be *Communicated by the Author. disputed, but when I suggested that a chain of excessive density underlay certain named places, I was using the word "excessive" relatively to the surrounding portions of the Earth's crust in India. It is a question therefore not so much of the negative variation over India compared with Europe, as one of internal variations in India itself. The pendulum observations in India show, if for the moment we exclude considerations of height, that there is a greater excess of matter in the crust under the two stations of Usira and Kalianpur than under any one of the other twenty inland stations in India. Of these two stations, Kalianpur is the one and only station situated on the line of the supposed underground chain of excessive density. Now the position of this line was deduced from the results of Astronomical observations, which are affected by horizontal attractions only. I think it is a significant coincidence that the pendulum results should show a maximum value of vertical force at the one station situated on the line derived independently from astronomical observations. Mr. Fisher thinks that the significance of this coincidence is lessened by the fact, that the same maximum value of vertical force has been observed at one other station, Usira, as well as at Kalianpur. But I do not think that the observations at Usira tell so much against the existence of a chain as the observations at Kalianpur tell in favour of it. Pendulum observations show the excess or deficiency of matter in the crust immediately under the station, but they give no clue as to the length or breadth of the matter, in which the excess or deficiency occurs. The attraction of an infinitely extended plain of rock of 1000 feet thickness is 1-56; if the plain of rock be limited in area to a circle of 7 miles radius, the observing station being at the centre, its attraction is still 18-56. The distances apart of the Central Indian pendulum stations, given in Mr. Fisher's list, are as follows: Badgaon to Ahmadpur Usira to Datairi 200 miles. Average distance apart The fact that the vertical force of gravity is the same at Usira as at Kalianpur shows that the same amount of matter underlies each; but Usira may overlie a small island of excessive density, and Kalianpur may be situated over a long wide chain of similar density. Pendulum observations at a single station do not tell us whether it overlies an island or a chain. A gravimetric survey with pendulum stations at every 20 miles is required. I have so far been excluding all considerations of height. Experience has shown that pendulum observations give as a rule a result more closely agreeing with the theoretical result, when the station of observation is at a low altitude than when it is at a high elevation. The higher the station, the greater is the deficiency of the observed force of gravity generally found to be. This phenomenon has been explained by the hypothesis that extra height is generally compensated by a corresponding subterranean deficiency of matter. Hence, when observations come to be reduced to sea-level, it is almost always found, that the largest negative variations occur at the highest stations. Now, when we come to compare Kalianpur and Usira, the table given by Mr. Fisher shows that Kalianpur is nearly 1000 feet higher than Usira; it is therefore a somewhat surprising fact, that the negative variation at Kalianpur should not be some two seconds greater than the negative variation at Usira. This consideration of heights would seem to show us, that the amount of matter in the Earth's crust at Kalianpur is considerably more than at Usira, notwithstanding that the force of gravity when reduced to sea-level has the same value at both places. If the values of heights are included in the discussion, we find that a greater amount of matter underlies Kalianpur than any other station in the plains of India. As I stated above, Kalianpur is the only pendulum station situated on the line of the chain. In the face of this coincidence, it cannot fairly be held that the pendulum results tell against the existence of a chain. Seeing that the accuracy of the old pendulum results has been questioned, and that a new series of pendulum observations is about to be commenced, I would beg all those interested in the subject to suspend judgment for the present. Dehra Dun, January 28, 1904. XXXII. On the Transfinite Cardinal Numbers of NumberClasses in General. By PHILIP E. B. JOURDAIN, B.A., Trinity College, Cambridge *. N this continuation of my paper † "On the Transfinite Cardinal Numbers of Well-ordered Aggregates," to which reference was made in § 10 of that paper, I shall begin *Communicated by the Author. † Phil. Mag. vol. vii. p. 61 (1904). by proving (§ 1) that the cardinal number of the third number-class (2) is the next greater cardinal number to 1. Cantor has already done this for 1 and No, and has not done this in detail for any other Alephs; though there are only the generalizations introduced by having to deal with series of the type of the second number-class instead of series of type w. The proof depends on the fact that ,א=א . א which is proved directly. There is no great difficulty in advancing to N,, &, and to any &y, so that the theorem a2=a is proved for any cardinal number a. This gives certain information as to the constitution of any number-class † (§ 3). According as the corresponding cardinal number has or has not an immediate predecessor, the class is built up in one of two ways from the lower number-classes. are proved, by the results in the addition and multiplication of transfinite cardinal numbers which have been hitherto obtained, to hold if à is any transfinite cardinal number (§ 4). 1. If we denote by w, the first number of the third numberclass, the whole class is formed as follows:-First after wi comes the series represented by {w1 + a}, where a takes, in order of magnitude, the values of all the numbers of the first and second number-classes. Next after *A doubt as to whether Cantor's second principle of generation can lead to numbers y beyond the second number-class, which seems to have been held by Schönflies, is, I think, settled below (§ 2). †The notation ωγ is used for the first ordinal number of the (y+2)th class. This notation (which is borrowed from Russell, cf. "The Principles of Mathematics,' Cambridge, 1903, p. 322) is necessary when we deal generally with the 7th number class. Cantor has only hitherto had to use a notation for what is here called w, he wrote . and so on. It is evident that the numbers of the first and second number-classes, arranged in order of magnitude, are ordinally similar to the series {w1+a}. We may then conclude, in a precisely similar way to that in which Cantor has proved that any part of the second number-class has a cardinal number which is either finite, or No, or that of the second number-class, that any part of the third class has a cardinal number which is either finite, or No, or N1, or that of the whole third class. Thus, we have, and have only, to prove that this last cardinal number is not N1, in order to complete the proof that the cardinal number of the third class is the next greater one to N1, and is thus properly denoted by 2. 19 For this purpose, we must prove† that if we supposed that all numbers of the third class could be arranged in a well-ordered series of cardinal number 1, we could define a number which belongs to the third class but not to this series. From this contradiction we must conclude that the cardinal number of the third class is not N. Suppose, then, that all the numbers of the third class could be arranged in a series of type w (of course, not in order of magnitude), as would be possible if the third class were of cardinal number &1. Now any series (not necessarily arranged in order of magnitude) {B} of type w1 of numbers of the third class has either a greatest number or else there is a number of the third class such that it is both the upper limit and a Limes § of Ba when the latter is arranged in *Grundlagen,' pp. 37-38; Math. Ann. xlix. pp. 228–229. 228. + Cf. Cantor, Grundlagen,' pp. 35-36; Math. Ann. xlix. pp. 227– This method, which Cantor has used to prove that c, =20, >No, Ni>No, and 24 >a; fails if we try to prove that . Since, then, c, we have some grounds for believing that c=N1 It appears to me necessary to distinguish the two different concepts of limit by two different words such as limit and Limes. These conceptions are, as is evident from their use in the general theory of ordinal types, purely ordinal, but it is in the theory of real numbers and their functions that their application is most important. Any infinite manifold of real (or complex) numbers has at least one point of condensation (Weierstrass), and each of these points (which always exist, e. y., when the elements of the manifold are real numbers), which are the points of Cantor's "first derivative," is called a Limes. Especially important among the Limites are the greatest and least Limes (which always exist, |