be tested by persons engaged in the practical work of forecasting day by day; your Committee or any other body of scientific men can only indicate the lines on which results of value in forecasting may be looked for. The first work the Directors of the Observatories set themselves to do was to prepare the meteorological 'constants' for the positions on the summit and at the base of Ben Nevis. This has been done, based on twenty years' observations on the summit and thirteen years' at Fort William. The constants for these periods will appear in vol. iii. of the Ben Nevis observations, now in the press, and to be published during the coming winter. In your Committee's previous Reports other lines of investigation have been frequently referred to and reported on, along which researches connected with the Ben Nevis observations are being carried on by Dr. Buchan and Mr. Omond. Some of the results have a special bearing on forecasting. One or two illustrative cases may be here added. 1. The occurrence of small differences of temperature between Ben Nevis and Fort William, associated with very low humidities at Ben Nevis and great dampness at Fort William, and the relations of this state of things to the stability and continuance of an anti-cyclone, and also to thunderstorms and those heavy local rains commonly denoted as thunder-showers, have been reported on. 2. The occurrence of long-continued periods of saturation of the air at the top of Ben Nevis, as indicative of a condition of the atmosphere favourable to the development and continuance of stormy weather. 3. A marked difference in the direction of the wind on the summit from that at surrounding low-level stations. Such a difference most commonly occurs when Ben Nevis lies between a cyclone and an anticyclone, and may be indicative of the direction of movement either of the cyclone or the anti-cyclone. 4. The predictive aspects of very strong winds on the summit of Ben Nevis accompanied, notwithstanding their great force, with very low temperatures there and great differences of temperature between the summit and Fort William, and the intimate connection of the whole with cyclonic weather, have been pointed out. Recent kite observations have made us tolerably familiar with this remarkable phase of the cyclone, and to Ben Nevis we may look for important contributions of illustrative data. 5. The difference between the Ben Nevis and Fort William barometers when both are reduced to sea-level. This difference, when it amounts to several hundredths of an inch, clearly points to an abnormal condition of the air between the summit and Fort William in respect to the vertical gradient of temperature or humidity, or both. The investigation of some of the points raised in this discussion has been a chief subject of inquiry during the past eighteen months. The inquiry is a discussion of the hourly observations of pressure, temperature, humidity, sunshine, winds and rainfall at the two Observatories in their inter-relations, more especially as regards the bearings of the results on weather changes The principal point to be kept in view is the relation of the differences of temperature at the two Observatories to the differences of their sealevel pressure at the time. An illustration will explain this. During the last three days of September 1895, the sky over Scotland was clear, sunshine strong, humidity high, night temperatures unusually high, and dews heavy, with calms or light winds. On these days while at the top temperature was very high and the air clear and very dry, at Fort William, under a sky equally clear and temperature high, the air showed a large humidity, and this state of moisture extended to a height of about 2,000 feet, or nearly halfway to the summit. Thus, then, while the barometer at the top was under an atmosphere wholly anti-cyclonic, with its accompanying dry dense air, the barometer at Fort William was not so circumstanced. On the other hand, it was under the pressure of such dry dense air, above the height of 2,000 feet only, whereas from this height down to sea-level it was under the pressure of air whose humidity was large and pressure therefore much reduced. The result was that the sea-level pres sure at Fort William was 0.050 inch lower than it would have been if the dry dense air of the anti-cyclone had been continued down to Fort William. This is confirmatory of what is to be expected, that the greater density of dry air as shown in our laboratories prevails equally in the free atmosphere. The first part of the discussion is virtually finished, the chief result of which is this:-1. When the difference of mean temperature of the day is only 1200 or less, then the sea-level pressure calculated for the top of the mountain is markedly greater than at Fort William; 2. When the difference of temperature is 18°0 or greater, then the sea-level pressure for the summit is markedly lower than at Fort William. In the former case the meteorological conditions are anti-cyclonic, the weather being then clear, dry, and practically rainless; and in the latter case the conditions are cyclonic, the accompanying weather being dull, humid, and rainy. In the course of this discussion it has been marked that the reduced hourly values from day to day often indicate that the transition from the anticyclonic to the cyclonic type of weather, and vice versa, is slow, sometimes extending over several days, thus prolonging the time for the prediction of the more important weather changes. It may be remarked that the result here empirically arrived at is in accordance with the principle laid down by Dalton, that 'air charged with vapour or vaporised air is specifically lighter than when without the vapour; or, in other words, the more vapour any given quantity of atmospheric air has in it, the less is its specific gravity.' The precursor and accompaniment of the heaviest and most widespread rains is when the sea-level pressure for the summit is very greatly lower than the sea-level pressure at Fort William. This indicates the saturation of the atmosphere to a great height, while at Fort William, and, say, 2,000 feet higher, the point of saturation due to the advancing cyclone has not yet taken place. On the other hand, when this point of saturation has been reached, then the sea-level pressure for the summit shows less difference from the sea-level pressure at Fort William. The changes of pressure which occur at the two Observatories as a cyclone advances and passes on are particularly interesting and instructive. It is remarkable that comparatively few observations, when the difference of the temperature has exceeded 22°0, could be utilised in this inquiry, because in such cases high winds prevailed, resulting in 'pumping' of the barometer. These differences of temperature, rising even to 27°.0, are, however, extremely valuable for weather prediction, inasmuch as they often precede and accompany very severe storms of wind and rain. They arise from an extraordinary lowering of the temperature at the summit, This is a while at Fort William no such lowering of temperature occurs. peculiarity which kites and balloon ascents have recently familiarised us with, and it forms a prime factor in all inquiries into the theory of the cyclone, about which opinion at present is so much divided. === Report on the Theory of Point-groups.1-PART III. By FRANCES HARDCASTLE, Cambridge. § 9. 1818-1857. While Fermat and Descartes, by combining the processes of Algebra and Geometry, were evolving the foundations of that system of co-ordinates which rapidly became the common language of geometers, a contemporary mathematician, Desargues of Lyons (1593-1662), and his pupil Pascal (1623-1662) were occupied with the study of those properties of figures, in space and in the plane, which persist under the operation known as projection. And had it not been for the evil fate which caused the publications of both master and pupil to be lost, and for the oblivion into which even the memory of these writings sank for more than 180 years, it is probable that modern synthetic geometry would have been developed from the beginning side by side with analytical geometry, instead of coming into existence, as it did, a whole century and a half later than its rival. The fundamental characteristic of each-that which most distinguished both systems from the geometry of the ancients-is the same, the systematic use of the principle of projection. But it is noteworthy that, although this was present from the beginning in the structure of Cartesian co-ordinates (whereby every point of a curve is projected on to the axis), it was only after the rise of descriptive geometry under Monge (1746-1818) and Carnot (1753-1823) (who explicitly founded it upon projection from ordinary space on to the plane), that Plücker (1801–1868), by the use of homogeneous co-ordinates," really opened up the projective possibilities inherent in analytical geometry. Throughout the period now to be discussed, the projective standpoint is the one adopted by analytical as well as by synthetic geometers; the transition to the wider point of view afforded by bi-rational transformation was only effected after the ideas of the theory of functions-at that time still in its infancy—had permeated the whole domain of pure mathematics, and had influenced the theory of higher plane curves to a degree which must have been startling to the mathematicians of the early nineteenth century. 3 Among the numerous novel terms introduced by Desargues in his 'Brouillon projet d'une atteinte aux événements des rencontres d'un cône avec un plan was that of involution, and, unlike many of the others, it has survived. Starting from the definition that six points on a straight line are in involution if certain ratios can be established among the segments formed by them, Desargues proved his famous theorem that a conic and the sides of an inscribed quadrilateral determine six points in involution on any transversal. He did not, however, investigate the still Parts I. and II. appeared in the Brit. Assoc. Reports for 1900, 1902. 2 Moebius's Barycentrische Calcul was printed in 1827, and was actually the first publication in which homogeneous co-ordinates were brought forward; Plücker's paper in Crelle, vol. v. (1830), gave the first exposition of trilinear co-ordinates. Cf. Clebsch, Julius Plücker zum Gedächtniss,' Abhandlungen Göttingen, vol. xvi. (1872), pp. 1-40. 3 Discovered in De la Hire's manuscript copy by Chasles in 1845, and printed in Poudra, Euvres de Desargues (Paris, 1864), vol. i. pp. 97–230. 4 Poudra, loc. cit., vol. i. p. 101, p. 109, p. 119; also vol. ii. p. 362. F more significant fact that any conic through the four vertices of the quadrilateral cuts the transversal in a pair of points belonging to the same involution. This theorem was first published by Sturm (1803–1835) in 1826; his proof is algebraical, being derived from the equations which, ten years previously,2 had been shown by Lamé to be the necessary consequence of the simultaneous existence of three equations of the second order. He points out that the relations he thus obtains are those which establish 'cette liaison remarquable qui était nommée par Desargues involution de six points.' He afterwards mentions that the two pairs of opposite sides of a quadrilateral inscribed in a conic can be regarded as a pair of degenerate conics, and that Desargues's theorem is thus an immediate deduction from his own more general one; but he makes no statement which would lead us to suppose he saw the importance of considering what we now call a range of points in involution, viz. an infinite number of points on a straight line, such that if any two pairs are given the correspondent to a fifth point is determined by the relation called involution which holds for any six. Nor, again, is he really interested in the fact that a whole system of conics passes through the points common to two conics (although, of course, he is perfectly aware that a third conic through these points has an equation involving one linear parameter); his concern is with properties of the individual conic of the system, not with the system itself. And the same remark must be made about Lamé, although the idea of a pencil of curves is due to him that is to say, he found for the first time the equation, E+mE'=0,4 of what we now call a pencil of curves; his primary interest was with the conditions which must subsist among the coefficients of the equations of three curves, in order that they may intersect in common points, and next, in the particular properties which follow for conics; with regard to curves of higher order, to which the greater interest, when looked at as a system, attaches itself, he simply stated the equation. Gergonne (1771-1859) seems to have been the first to derive any property concerning the points of intersection of curves whose equation is of Lame's form, as a direct consequence of this form. In 1827 he thus found that if p(p+q) of the (p+q)2 points of intersection of two curves of order (p+q) lie on a curve of order p, the remaining q(p+q) points lie on a curve of order q: from which he obtained the corollary: Given two systems of m lines in the plane, if among the m2 points of intersection of the lines of one system with the lines of the other there are 2m which lie on a conic, then the m(m-2) remaining points all lie on a curve of order (m-2). Writing m=3, this is, as he points out, the theorem known as Pascal's. This proof of Pascal's theorem also appears incidentally in a long footnote to the last chapter of the first volume of Plücker's 'Analytisch-geometrische Entwicklungen,' printed in 1828; the preface is dated September 1827, later than the publication of Gergonne's paper, and it is possible that this footnote was added at the same time; this would give the priority in discovery of this particular proof to Gergonne, as well as 1 Mémoire sur les lignes du second ordre,' Gergonne's Annales, vol. xvii. pp. 173– 198. 2 Sur les intersections des lignes et des surfaces,' Gerg. Ann., vol. vii. pp. 229-240. 3 Clebsch, loc. cit., p. 17. In his Examen des différentes méthodes employées pour résoudre les problèmes de géometrie, 1818, p. 29. See Part II. of this Report, § 8 (Brit. Assoc. Report, 1902). Recherches sur quelques lois générales qui régissent les lignes et surfaces algébriques de tous les ordres,' Gerg. Ann., vol. xvii. (1827), pp. 214-252. the priority in publication which is undoubtedly his. This, however, is a very small matter: Gergonne's contribution to the elucidation of problems connected with the intersections of curves is insignificant compared with Plücker's. It was Plücker who derived from Lamé's equation of a system of curves the theorem which threw fresh light upon the so-called Cramer Paradox, which had baffled mathematicians for more than a hundred years. And it was Plücker who, simultaneously with Jacobi (18041851), first ventured upon a line of research which afterwards proved a fruitful source of theorems in the theory of point-groups-the investigation, namely, of the conditions which must exist among the co-ordinates of certain points if they are known to be the points of intersection of two curves of given (differing) orders. The problem which first led Plücker to consider the paradox was that of determining the highest degree of osculation possible between a curve of order n and one of order m. This question is treated in a footnote to an earlier chapter 2 of the work just mentioned, and its solution is made to depend upon the establishment of a new theorem, viz. that all curves of the nth order, which pass through 3 ((n)) — 2 given points intersect each other also in the same n2-((n))+2=((n-3)) points. In this passage the paradox is not explicitly mentioned; but in a paper published in the same year in Gergonne's Annales Plücker speaks of it, describing it as the fact that in certain cases two curves of the same order may cut each other in at least as many points as are required to completely determine one of them. Cramer, he continues, dans son "Introduction à l'analyse des courbes algébriques," est le premier, je crois, qui ait signalé cette espèce de paradoxe qui s'explique aisément en remarquant que, lorsqu'il est question du nombre des points nécessaires et suffisants sur un plan, pour déterminer complètement une courbe d'un degré déterminé, on sous-entend toujours que ces points sont pris au hasard, et ne sont liés entre eux par aucune relation particulière.' He establishes his new theorem in almost the same words here as in the other passage; the application is to the theory of the conjugate points of conics. The second volume of the Analytisch-geometrische Entwicklungen' was published in 1832: in this Plücker returned to the subject of the paradox," and remarked that Cramer had indicated the analytical explanation, viz. that the n2 linear equations which correspond to the n2 points of intersection of two curves of order n must, if n>3, be such that one or more, arbitrarily chosen from among them, are conditioned by those which remain; he adds that a geometrical interpretation of this explanation is needed. His own new theorem affords this geometrical interpretation, and he therefore reproduces it once more, with a proof which, when slightly elaborated, is substantially as follows: Assume ((n))-2 arbitrary points in the plane, take any two curves of order n through them, U=0, V=0, which, in general, are completely determined if we know one more point, not the same, on each. Suppose 1 See Kötter, 'Die Entwickelung der synthetischen Geometrie von Monge bis auf Staudt,' Jahresber. d. deutsch. Math.-Verein., vol. v. (1901), p. 226. Clebsch (loc. cit., p. 19) ascribes the priority to Plücker, without mentioning Gergorne. 2 P. 228. '((n)) is written throughout for } (n + 1) (n + 2). Recherches sur les courbes algébriques de tous les degrés,' Gerg. Ann. vol. xix. (1828), pp. 97-106; also Works (Leipzig, 1895) pp. 76–82. $ P. 242. |