matical physics and pure mathematics have given much to each other in the past and will give much to each other in the future; in doing so, they will take harmonised action in furthering the progress of knowledge. But neither science must pretend to absorb the activity of the other. It is almost an irony of circumstance that a theorem, initiated by Fourier in the treatise just mentioned, has given rise to a vast amount of discussion and attention, which, while of supreme value in the development of one branch of pure mathematics, have hitherto offered little, if anything, by way of added explanation of natural phenomena. The century that has gone has witnessed a wonderful development of pure mathematics. The bead-roll of names in that science-Gauss; Abel, Jacobi; Cauchy, Riemann, Weierstrass, Hermite; Cayley, Sylvester; Lobatchewsky, Lie-will on only the merest recollection of the work with which their names are associated show that an age has been reached where the development of human thought is deemed as worthy a scientific occupation of the human mind as the most profound study of the phenomena of the material universe. The last feature of the century that will be mentioned has been the increase in the number of subjects, apparently dissimilar from one another, which are now being made to use mathematics to some extent. Perhaps the most surprising is the application of mathematics to the domain of pure thought; this was effected by George Boole in his treatise 'Laws of Thought,' published in 1854; and though the developments have passed considerably beyond Boole's researches, his work is one of those classics that mark a new departure. Political economy, on the initiative of Cournot and Jevons, has begun to employ symbols and to develop the graphical methods; but there the present use seems to be one of suggestive record and expression, rather than of positive construction. Chemistry, in a modern spirit, is stretching out into mathematical theories; Willard Gibbs, in his memoir on the equilibrium of chemical systems, has led the way; and, though his way is a path which chemists find strewn with the thorns of analysis, his work has rendered, incidentally, a real service in co-ordinating experimental results belonging to physics and to chemistry. A new and generalised theory of statistics is being constructed; and a school has grown up which is applying them to biological phenomena. Its activity, however, has not yet met with the sympathetic goodwill of all the pure biologists; and those who remember the quality of the discussion that took place last year at Cambridge between the biometricians and some of the biologists will agree that, if the new school should languish, it will not be for want of the tonic of criticism. If I have dealt with the past history of some of the sciences with which our Section is concerned, and have chosen particular epochs in that history with the aim of concentrating your attention upon them, you will hardly expect me to plunge into the future. Being neither a prophet nor the son of a prophet, not being possessed of the knowledge which enabled Halley to don the prophet's mantle with confidence, I shall venture upon no prophecy even so cautious as Bacon's'As for the mixed mathematics I may only make this prediction, that there cannot fail to be more kinds of them as Nature grows further disclosed '-a declaration that is sage enough, though a trifle lacking in precision. Prophecy, unless based upon confident knowledge, has passed out of vogue, except perhaps in controversial politics; even in that domain, it is helpless to secure its own fulfilment. Let me rather exercise the privilege of one who is not entirely unfamiliar with the practice of geometry, and let me draw the proverbial line before indulgence in prophetic estimates. The names that have flitted through my remarks, the discoveries and the places associated with those names, definitely indicate that, notwithstanding all appearance of divergence and in spite of scattered isolation, the sum of human knowledge, which is an inheritance common to us all, grows silently, sometimes slowly, yet (as we hope) safely and surely, through the ages. You who are in South Africa have made an honourable and an honoured contribution to that growing knowledge, conspicuously in your astronomy and through a brilliant succession of astronomers. Here, not as an individual but as a representative officer of our brotherhood in the British Association, I can offer you no better wish than that you may produce some men of genius and a multitude of able workers who, by their researches in our sciences, may add to the fame of your country and contribute to the intellectual progress of the world. The following Papers were read :— 1. Observations on Atmospheric Electricity in South Africa. The observations carried out up to now consist of a series made at Bloemfontein in 1902-1903 by Mr. Lyle, of the Grey College, Bloemfontein, and a less complete series made during the same period at Cape Town by Mr. W. H. Logeman. Observations have also been carried out in Cape Town by Mr. Fincham during the present year. The instruments used were an Elster and Geitel dissipation apparatus, and-less regularly-a Kelvin portable electrometer. The paper contained the records of daily readings of temperature, humidity, pressure, and both positive and negative leaks for morning and afternoon hours. These results are used to determine the annual variation; a harmonic analysis of the results being obtained in the usual way. Cape Town is on the coast practically at sea-level. Bloemfontein is inland, and about 4,500 feet above sea-level. The magnitude of the leak is greater at Bloemfontein than at Cape Town. The positive leak, both at Cape Town and at Bloemfontein, is less than the negative in the morning; the positive leak in the afternoon in Bloemfontein is greater than the negative in the summer months, less in the winter months. In Cape Town, so far as the observations go, it seems probable that the opposite is true, that is, from June to October the positive leak is greater than the negative, and in November and December less. Mr. Lyle and Mr. Fincham had in addition observed on several days throughout the whole day. The chief results are similar to those obtained in Europe. Formation of fog leads to a diminution in leak. The rate of leak changes with the relative humidity and with the pressure. 2. Apioidal Binary Star-Systems. By ALEXANDER W. ROBERTS, D.Sc., F.R.A.S.-See Reports, p. 249. 3. On the Convergence of a Reversed Power Series. For some purposes it is useful to have a knowledge of definite limits within which a reversed series converges. The following investigation supplies such a knowledge, though the limits obtained are generally rather narrow. Start from If we apply the method of successive approximations we have, first, y λ= (1) (2) This must now be substituted in equation (2) and a second approximation obtained. The second approximation is now substituted in (2) to obtain a third. If the series thus obtained is convergent it is the reversed series wanted. Suppose, first, that b, is positive. Then the coefficients of the series obtained by successive approximation are clearly not greater than would be got if all the coefficients of powers of x in (2) were positive, i.e., not greater than the coefficients of reversed series obtained from If b, is negative, the numerical value of the coefficients is clearly not diminished if we make every sum on the right-hand side of (2) positive. Combining the two cases, we see that the coefficients of the reversed series are certainly not greater than those obtained from the solution with the negative sign before the square root being evidently the appropriate one. Now the expression for r will be expansible in a series of powers of y provided is so expansible. Values of y for which this is certainly true are given by and for these values of y the series obtained from (1) by reversion will also be convergent. It is assumed throughout that the value of y is positive; a negative value would be met by changing both sides of (1) and obtaining one or other (b,+ or b1 − ) of the cases discussed. The series 2.3 y=x+ + ... * Chrystal II. chap. xxvii. § 7. gives y236 as upper limit, it being otherwise known that the reversed series is convergent up to y = 1. gives y=236 as upper limit; the reversed series is actually convergent for all values of y. It is to be observed that no result is given for the case when any of the 's L become indefinitely great, since [√ (3, + 23)2 + 13 ̧2 − (B, + 29)] = 0, provided B =0 B1, a case which does not arise. The case excepted may be dealt with as follows: Suppose ya . . . + bmx2 + ... . (1) where b∞, and of course the series on the right of (1) is convergent for values of r within a certain region, say |x|<p. Then, if σ be any quantity such that o>p we have L | bn' = = 0, and none of the and we have a series to which the previous reasoning is applicable, and if § is expansible in a series of powers of y, then v = is also so expansible. σ The same principle can also be applied to the first case considered; thus the reversed series of (1) will certainly be convergent for values of r less than the greatest of the quantities [(Bσ + 2B, [σ") + Bo2 - (B1o +28, σ")] where σ is any quantity whatever and Blo" is the greatest of the set I am unable to find any general expression for the maximum value of this limit, owing to the difficulty arising from the fact that different values of o require different B's to be selected to give the greatest coefficient. It is to be observed that the values of the above limit for σ = 0 and for σ = co are both zero. As an example of the limits obtained we have for the case of The proof can be arranged so as to cover the case of an imaginary variable. Suppose now that =n+i, and compare two series (a) the one which would be got from (2) by successive approximation, and (b) the series got by successive approximation from where p = ni§| and B, = |b,|. 1 B1 Using the facts that the modulus of a sum is not greater than the sum of the moduli, and that the modulus of a product or quotient is equal to the product or quotient of the moduli, we see that in the parallel processes for obtaining (a) and (b) any term in (b) is not less than the modulus of the corresponding term in (a). Hence the reversed series obtained from (1) will be convergent if that obtained from (4) is convergent. By the previous work the condition is is convergent for all values of y such that (y)<√(\3, +2λ”ß„)2 +λ2”ß„2 — (λB1 + 2λ”ß„) where X is any quantity and X”ß„ is the greatest of the set |λb1| |\2b1| |X2g!• 4. On Instruments for Stereoscopic Surveying. By H. G. FOURCade. The instruments of which photographs are exhibited have been designed for the construction of topographical plans from stereoscopic photographs of the country. A preliminary account of the method was communicated to the S. A. Philosophical Society on October 2, 1901 (also 'Nature,' June 5, 1902), and a more complete Paper will shortly be published. If photographs be taken from both ends of a base, at right angles to it and under conditions ensuring perfect parallelism of the line of collimation of the camera in both positions, the co-ordinates of any point common to both pictures may be computed from the plate co-ordinates by means of the simple relations b being the base, f the focal length of the camera lens, and e the sterecscopie difference-that is, the difference of the plate a's. It is not necessary to have the ends of the base at the same height. The camera is a metal box provided with levels, a transverse telescope, and a very accurate réseau scale divided on the silvered surface of an optically plane and parallel plate of glass, which may be set in front of the sensitive film for the impression of the réseau lines or raised for the exposure of the view. Two pivots and an end contact on the frame of the réseau plate ensure the geometrically 1905. Y |