MISCELLANEOUS PAPERS CONNECTED WITH PHYSICAL SCIENCE BY HUMPHREY LLOYD, D.D., D.C.L., PHOTOS! OF TRINITY COLLEGE, DUBLIN ; FORMERLY PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY. REIMUXTED FROM THE TRANSACTIONS OP THE ROYAL IRISH ACADEMY ; THE REPORTS OK THE BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE ; ETC. LONDON: LONGMANS, GREEN, AND CO. 1877. BUBLIK : t ^nifrmitg $**x ' . CONTENTS. PAGE. I. On the Phenomena presented by Light in its passage along the Axes of Biaxal Crystals, 1 II. Report on the Progress and Present State of Physical Optics, . 19 III. On a new case of Interference of the Rays of Light, . . .149 IV. On the Light reflected and transmitted by Thin Plates, . .157 V. Observations of the Direction and Intensity of the Terrestrial Magnetic Force in Ireland, . . . . . . .170 VI. On the position of the Isogonal Lines in Ireland, deduced from the Observations of CAPTAIN SIR JAMES CLARKE Ross, R. N., . 217 VII. On a New Magnetical Instrument, for the measurement of the Inclination, and its Changes, 221 VIII. Remarks on the theory of the Compound Magnetic Needle, . 231 IX. On the Mean Results of Observations, 233 X. On the determination of the Horizontal Intensity of the Earth's Magnetic Force in absolute measure, ..... 246 XI. On the determination of the Total Intensity of the Earth's Mag- netic Force in absolute measure, by means of the Dip-circle, . 260 XII. On Earth-currents, and their connexion with the Diurnal Changes of the Horizontal Magnetic Needle, 271 XIII. On Earth-currents, in connexion with Magnetic Disturbances, . 295 XIV. On the probable causes of the Earth-currents, .... 299 XV. On the direct Magnetic Influence of a distant Luminary upon the Diurnal Variations of the Magnetic Force at the Earth's sur- face, 30(5 IV CONTENTS. PAGE. XVI. On the Storm of the 18th of April, 1850, . . . .312 XVII. Notes on the Meteorology of Ireland, deduced from the Observa- tions made in the year 1851, under the direction of the Royal Irish Academy, . . . 317 XVIII. The Climate of Ireland, and the Currents of the Atlantic. (Lecture delivered before the Dublin Young Men's Christian Association, October 25, 1865), 381 XIX. On the rise and progress of Mechanical Philosophy. (Intro- ductory Lecture delivered in the Philosophy School of Trinity College, in Hilary Term, 1834), 414 XX. The Applied Sciences, and the mode of Teaching them. (Preelec- tion delivered on the occasion of the Opening of the School of Engineering in the University of Dublin, November 15, 1841), 437 XXI. Address delivered at a Meeting of the Royal Irish Academy, held April 13, 1846, on the occasion of taking the Chair, . . 454 XXII. Address delivered at a Meeting of the Royal Irish Academy, held June 26, 1848, on the occasion of the presentation of the Medals awarded by the Council, 468 XXIII. Address delivered at the Opening Meeting of the British Asso- ciation for the Advancement of Science, held in Dublin, August 26, 1857, 485 NOTES. Note to Table VI., p. 289, 511 Note to Tables V. and VIII., pp. 286 and 291, . . . .511 PLATES. MAGNETIC CHART OF IRELAND . . - opposite page 216 EARTH CURRENTS. PLATES 1 and 2 . 294 METEOROLOGY OF IRELAND. PLATKS 1, 2, 3, and 4 380 MISCELLANEOUS PAPERS. I. ON THE PHENOMENA PEESENTED BY LIGHT IN ITS PASSAGE ALONG THE AXES OF BIAXAL CEYSTALS. Transactions of the Royal Irish Academy, Vol. XVII. IT is well known that when a ray of light is incident upon certain crystals, such as Iceland spar and quartz, it is in general divided into two pencils, of which one is refracted according to the known law of the sines, while the direction of the other is determined by a new and extraordinary law, first assigned by Huygens. These laws were long supposed to apply to all doubly -refracting substances ; and it was not until the subject was examined by the ablest advocate of the undulatory theory, that the problem of double refraction was solved in all its generality. Setting out from the hypothesis, that the elasticity of the vibrating medium within the crystal is unequal in three rectangular directions, Fresnel has shown that the surface of the wave is not, in general, either a sphere or spheroid, as in the Huygenian law, but a surface of the fourth order, consisting of two sheets ; and that the directions of the two refracted rays are determined by tangent planes drawn to these surfaces under known conditions. Such crystals have, in general, two optic axes, and are thence denominated bia.rnL When 'J CONICAL REFRACTION. the elasticity of the medium is the same in two of the three direc- tions, the equation of the wave-surface is resolvable into two, which represent the sphere and spheroid of the Huygenian law. The two optic axes in this case coincide ; and the law of Huygens is thus proved to be a case of a more general law, and shown to belong to uniaxal crystals only. Finally, when the elasticity is the same in all the three directions, the wave-surface becomes a sphere ; and the refraction is single, and takes place according to the ordinary law of the sines. This case comprises a few of the crystallized, and most uncrystallized substances. There are two remarkable cases, however, in this elegant and profound theory, which its author seems to have overlooked, if not to have misapprehended. In a communication made to the Academy at its last meeting, Professor Hamilton has supplied these omissions in the theory of Fresnel, and has been thus led to results in the highest degree novel and remarkable. To understand these conclusions, it may be useful to revert for a moment to the original theory of Fresnel. The general form of the wave-surface is determined by the equation (a 2 cos 2 a + tf cos 2 /3 + c 2 cos 2 -y)r 4 _ |y (ja + c 2 ) C os 2 a + & 2 ( 2 + c 2 ) cos 2 /3 4 c 2 (a 2 + ft 2 ) cos 2 7 ] r 2 + a 2 b 2 c 2 = ; in which a, j3, 7, denote the angles made by the radius- vector with the three axes, and a 2 , 6 2 , c 2 , the elasticities of the medium in these directions. If now we make cos /3 = in this equation, so as to obtain the section of the surface made by the plane of ars, the result is reducible to the form (r z - V] [( 2 cos 2 a + c 2 sin 2 a) r z - V] = 0. So that the surface intersects the plane of xz in a circle and ellipse, whose equations are r = ft, (a- cos 2 a + c 2 sin 2 a) r 2 = V. Now b, the radius of the circle, being intermediate between a and c, the semiaxes of the ellipse, it is obvious that the two curves must intersect in four points, or cusps, as represented in (fig. 1) ; and the angle which the radius-vector OP, drawn to the cusp, CONICAL REFRACTION. 3 makes with the axis of x, is found by eliminating r between the two equations, by wliich means we obtain Fig. 1. At each of the points thus determined there will be two tangents to tho plane section ; and consequently the ray OP, proceeding within the crystal to one of these points, might be supposed to be divided at emer- gence into two, whose directions are deter- mined by those of the tangents. Such seems to have been Fresnel's con- ception of this case. Professor Hamilton has shown, however, that there is a cusp at each of these points, not only in this particular section, but in every section of the wave- surface passing through the line OP ; or, in other words, that there is a conoidal cusp on the general wave-surface at the four points of intersection of the circle and ellipse ; so that there must be an infinite number of tangent planes at each of these points, and consequently a single rat/, such as OP, pro- ceeding from a point within the crystal to one of these points, must be divided into an infinite number of emergent rays, constitut- ing a conical surface.* It is evident further, that the circle and ellipse will have four common tangents, such as MN (fig. 1). The planes passing through these tangents, and parallel to the third or mean axis, are parallel to the circular sections of the surface of elastic it;/ of Fresnel's theory, or perpendicular to the optic axes. Fresnel seems to have concluded that these planes touched the wave-surface only in the two points just mentioned ; and, consequently, that a single ray, incident upon a biaxal crystal in such a manner that one of the refracted rays should coincide with an optic axis, would be divided into two, determined by the points of contact. This result, if verified by experience, would place a remarkable distinction between the phenomena of uniaxal and biaxal crystals ; but though the case was examined by M. Biot, no corresponding appearances were observed. 4 CONICAL REFRACTION. Professor Hamilton has shown that the four planes of which we have spoken touch the wave-surface, not in two points only, but in an infinite number of points, constituting each a small circle of contact, whose plane is parallel to one of the two circular sections of the surface of elasticity ; and that, consequently, a single ray of common light, incident externally in the above- mentioned direction, should be divided within the crystal into an infinite number of refracted rays, constituting a conical surface. Here, then, are two singular and unexpected consequences of the undulatory theory, not only unsupported by any facts hitherto observed, but even opposed to all the analogies derived from ex- perience. If confirmed by experiment, they would furnish new and almost convincing proofs of the truth of that theory ; and if disproved, on the other hand, it is evident that the theory must be abandoned or modified. Being naturally anxious to submit the undulatory hypothesis to this delicate test, and to establish or disprove these new results of theory, Professor Hamilton requested me to institute a series of experiments with that view. I accordingly applied myself to this interesting research with all the attention which the subject so well merited, and have fortunately succeeded in verifying both cases of conical refraction. The substance I employed in these experiments was arragonite, which is well known to be a biaxal crystal, whose axes are inclined at an angle of nearly 20. I selected it partly on account of the magnitude of its biaxal energy, and partly also because the optical elements 6f this mineral have been determined, apparently with great care, by Professor Rudberg ; and therefore the results of theory could be applied to it at once without further examination. The specimen I used was one of considerable size and purity, procured for me by Mr. Dollond, and cut with its parallel faces perpendicular to the line bisecting the optic axes. The first-mentioned species of conical refraction, it has been observed, takes place in air, when a ray of common light is trans- mitted within the crystal in the direction of the line joining two opposite cusps of the wave. If we suppose such a ray to pass in both directions out of the crystal, it is evident that it must emerge similarly at both surfaces ; consequently, the rays which are trans- mitted along this line within the crystal, and form a diverging cone at emergence at the second surface, must be incident in a CONICAL REFRACTION. Q converging cone at the first. Having therefore nearly ascertained the required direction by means of the system of rings in polarized light, I placed a lens of short focus at its focal distance from the first surface, and in such a position that the central part of the pencil might have an incidence nearly corresponding to the cusp- ray within. Then looking through the crystal at the light of a lamp placed at a considerable distance, I observed in the expected direction a point more luminous than the space immediately about it, and surrounded by something resembling a stellar radiation. Fearing that this singular appearance might have arisen from some imperfection in the crystal, I transmitted the light in the same manner through several different parts of its substance, and always with the same result. The connexion of the phenomenon with the optic axis was proved by the system of rings which appeared in the same direction, when the light was examined with a polarizing and analyzing plate. This result is of some interest in itself, independently of its connexion with theory. It has been hitherto supposed that the only method of determining experimentally the direction of the optic axes, in most doubly-refracting substances, consisted in observing the system of coloured rings, which appear around them when the incident and emergent light is polarized. Here, how- ever, we find that common, or unpolarized light, undergoes such modifications in the neighbourhood of one of the optic axes, that the apparent direction of that axis may be at once determined, and with the aid of the simplest contrivance.* But to examine the emergent cone, it was necessary to exclude the light which passed through the crystal in all but one direction. For this purpose, a plate of thin metal, having a minute aperture, was placed on the surface of the crystal next the eye, and the position of the aperture so adjusted, that the line connecting it with the luminous point on the first surface might be, as nearly as possible, in the direction of the cusp-ray. The exact adjustment to this direction was made by subsequent trial. The phenomenon * This fact is here mentioned, rather as a matter of curiosity than as one likely co be of practical value in determining the optical elements of crystals. It is to be observed, moreover, that the direction thus determined is that of the normal to the circular section of the ellipsoid of Fresnel's theory ; while the rings (there is strong reason to believe) are related to the normals to the circular sections of the surface iif elasticity. 6 CONICAL REFRACTION. which presented itself when this disposition was complete was in the highest degree curious. There appeared at first a luminous circle, with a small dark space in the centre ; and in this dark central space were two bright points, separated by a narrow and well-defined dark line. These appearances are represented in (figs, a and b). When the aperture in the plate was slightly Fig. a. Fig. shifted, the phenomena rapidly changed, assuming in succession the forms represented in (figs, c, d, e). In the first stage of its change, the central dark space became greatly enlarged, and a double sector appeared in the centre. The circle was reduced to about a quadrant, and was separated by a dark interval from the sector just mentioned. This is represented in (fig. c.) The remote sector then disappeared, and the circular arch diminished, Fig. c. Fig. d. Fig. e. as in (fig. d) ; and as the inclination of the internal ray to the cusp-ray was further increased, these two luminous portions merged gradually into the two pencils, into which a single ray is divided in the other parts of the crystal. This change is repre- sented in (fig. e). Similar observations were made without the lens, by bringing the flame of the lamp near the first surface of the crystal, and forming the converging cone by covering that surface also with a thin metallic plate, perforated with a minute aperture. In this case the line connecting the two minute apertures was adjusted as before, and the phenomena were the same as in the former CONICAL REFRACTION. 7 instance, the rays which passed along this line within the crystal forming a diverging cone at emergence. In all these experiments the emergent rays were received directly by the eye placed close to the aperture on the second surface. It was obviously desirable, however, to receive them on a screen, and thus to observe the section of the cone at different distances from its summit. After some trials, I effected this with the sun's light, the light of a lamp being too weak for the purpose. The emergent cone being made to fall on a screen of roughened glass, I was enabled to observe its sections at various distances, and therefore with all the advantages of enlargement. The light was sufficiently bright, and the appearance distinct, when the diameter of the section was between one and two inches. On examining the emergent cone with a tourmaline plate, I was surprised to observe that one radius only of the circular section* vanished in a given position of the axis of the tourmaline, and that the ray which disappeared ranged through 360 as the tourmaline plate was turned through 180. Thus it appeared that all the rays of the cone are polarized in different planes. On examining this curious phenomenon more attentively, I discovered the remarkable law, " that the angle between the planes of polarization of any two rays of the cone is half the angle between the planes containing the rays themselves and the axis" Having assured myself of the near truth of this law by experi- ment, I was naturally led to inquire how far it was in accordance with theory ; and on examining Fresnel's theory with this view, I was gratified to find that it led to the very same result. According to the known rule, the plane of polarization of any one ray of the emergent cone must bisect the angle contained by the planes passing through tho corresponding normal to the front of the wave and the two optic axes. Now, it can be easily shown that the normals to the wave, at the cusp, surround one of the optic axes, and are inclined to it all round at small angles. For the tangent of the angle which the normals to the circle and ellipse in the plane of xz make with one another is ac * These sections are not mathematically circular, the line being, in fact, one of the fourth order. 8 CONICAL REFEACTION. and it can be easily shown that the tangent of the angle which the optic axis makes with the normal to the circle, or the cusp-ray, is Now, this is about half the former, since b* = ac, nearly; and consequently the optic axis nearly bisects the angle contained by the extreme normals in the plane of xz. Hence if A and B be the intersections of the two optic axes with the sphere whose centre is at the cusp, and N the intersection of one of the normals at that point with the same (fig. 2), the angle NA C ranges through every magnitude between and 360, the arc NA being all the time very small. Let the angle NAC be denoted by a, and NPC by w, NP being the arch bisecting the angle N\ then in the triangle APN, we have or, since AN is very small, and therefore cos^jY = 1, nearly, cos o> = cos (a - i-ZV), and ta = a- ^N, nearly. But, when any side of a spherical triangle is very small in com- parison with the other two, the adjacent angles are together equal to 180 q.p. Consequently, N = a, and w = Ja, nearly. From this it appears that the angle, which the plane of polari- zation of any ray makes with the plane of the optic axes, is half the angle which the plane passing through the normal and the , near axis makes with the same plane. But this latter angle, it may be easily shown, is very nearly the same as that which the plane passing through the emergent ray and the axis of the cone makes with the plane of the optic axes. Consequently the angle, which the plane of polarization of any ray of the emergent cone CONICAL REFRACTION. 9 makes with the plane of the optic axes, is half of that which the plane containing that ray and the axis of the cone' forms with the same plane. The general phenomena being observed, it remained to ex- amine the magnitude and position of the emergent cone, and to compare the results with those furnished by theory. For this purpose I viewed the aperture in the second plate through a small telescope, which was moved in a plane nearly perpendicular to the axis of the emergent cone ; and by noting the points at which the light failed, I obtained the magnitude of the section of the cone made by that plane. The distance of this section from the crystal being then measured, the angle of the cone was obtained from the trigonometrical tables, and was found to be very nearly 6. I then placed the flame of a wax taper at the centre of the section, and removing the plate from the second surface of the crystal, found the direction of the ray reflected from the surface. A well defined mark was then placed on this line, at a considerable dis- tance, and the angular distance between the centre of the flame and the mark measured by a sextant, whose centre was brought exactly to the place of the crystal. This angle was found to be 31 56' ; and consequently the angle of emergence corresponding to the central rays of the cone was 15 58'. Now to compare these results with those of theory. It is a well-known principle of the theory of waves that the direction of a ray incident upon or emergent from a crystal, and the normal to the front of the wave, are always in the same plane perpen- dicular to the surface of incidence or emergence ; and the angles which these two lines make with the perpendicular to the surface, are connected by the known law of the sines ; the index of refrac- tion being the reciprocal of the normal velocity of the wave, or of the perpendicular upon the tangent plane. Now, at the cusp, there are an infinite number of normals to the wave, and conse- quently an infinite number of corresponding emergent rays. Of these the two rays in the plane of the optic axes form the greatest angle, and their directions are determined by those of the normals to the circle and ellipse, which constitute the section of the wave- surface in that plane. If then p' and /" denote the angles of emergence of these rays, t the angle which the normal to the ircle, or cusp-ray, makes with the perpendicular to the surface, a the angle contained by the normals to the circle and ellipse, 10 CONICAL REFRACTION. and p the perpendicular from the centre on the tangent to the ellipse at the cusp, we have sin p' = - sin t, sin p" = - sin (t - a) ; Now in arragonite, according to the determination of M. Rudberg, - = 1.5326, i = 1.6863, - = 1.6908 ; a b c and substituting these values we find - = 1.68708, a - 144'48". P These values being introduced in the first two equations, p' and p" will be determined for any given surface of emergence. In this manner Professor Hamilton has found that when t = 0, or the surface of emergence perpendicular to the cusp-ray, p' = 0, and p" = 2 56' 51". And when i = 9 56' 27", or the surface perpen- dicular to the line bisecting the optic axes, p' = 16 55' 27", and p" = 13 54' 49". Accordingly, the difference of these angles, p' - p", which is the extreme angle of the emergent cone, is in the former case 2 56' 51",* and in the latter 3 0' 38". Also, half the sum of these angles, which is the angle of emergence corres- ponding to the axis of the cone, is 15 25' 8". Comparing these with the results of observation, it will be seen that they agree nearly with respect to the mean angle of emergence, the difference amounting only to 33' ; whereas the angle of the cone determined by experiment is about double of that furnished by calculation. I also measured the angle of the cone by tracing the outline of its section on a screen of roughened glass, when the sun's light was employed instead of that of a lamp. The mean diameter of this section being then accurately ascertained, and the distance of the screen from the aperture measured, the angle was given by " It is easily shown that the sine of the angle of the cone, in this case, is generally expressed by the formula CONICAL REFRACTION. 11 the tables. Measurements taken in this manner gave for the value of the angle, 6 24', 5 56', 6 22', respectively ; and the mean of these is 6 14', which, like the former measurement, differs very little from the double of the calculated angle. The results of observation thus appeared to be at variance with those of theory in two important particulars. In the first place, the emergent rays appeared to form a solid cone, instead of a conical surface ; and in the next, the magnitude of this cone was about double of the expected magnitude. Conceiving that these discrepancies might probably be owing to the rays which are inclined to the cusp-ray at small angles, and which pass by the edge of the aperture, I determined to ascertain the fact by trying the effects of apertures of various sizes. I found accordingly that when the aperture was at all con- siderable, such as that formed by a large-sized pin, two concentric circles were seen to surround the axis, the interior of which had about double the brightness of the exterior annulus. And it was remarkable that the light of the interior circle was unpolarized, while that of the surrounding annulus was polarized according to the law already explained. When smaller apertures were used, the inner circle contracted, the breadth of the exterior annulus remaining nearly the same ; until the former was finally reduced to a point in the centre of a fainter circle. When the aperture was still further diminished, a dark space sprung up in the centre, enlarging as the aperture decreased ; until finally, with a very minute aperture, the breadth of this central space increased to about f ths of the entire diameter. The phenomena exhibited in these cases assumed the forms represented in figures (/) and (/ <>f the I{ ;/1 .SV / y< p ^ DIFFRACTION. 55 could be conceived capable of modifying the attractive force of the body, or the density of the imagined atmosphere, and without effect. The metallic wires and plates which produced the fringes were heated to redness, and cooled down below the freezing-point ; they were traversed by voltaic currents, and the charges of power- ful batteries transmitted through them ; but in whatever manner the condition of the diffracting body was varied, no change whatever was perceived either in the intensity or dimensions of the diffracted fringes.* Although the phenomena of diffraction were studied by many diligent observers! after the publication of the Optics, no material accession was made to the knowledge of their laws until the prin- ciples of the wave-theory were applied to their explanation by Young. The exterior fringes, formed without the shadows of bodies, were ascribed by Young to the interference of two portions of light, one of which passed by the body, and was more or less inflected, while the other was obliquely reflected from its edge, the latter losing half an undulation at the instant of reflexion.* The fringes formed by narrow apertures were, in like manner, sup- posed to arise from the interference of the two pencils reflected from the opposite edges ; while the interior fringes, within the shadows of narrow bodies, were accounted for by the interference of the pencils which passed on either side of the body at an insen- sible distance, and were inflected into the shadow. The observed facts closely correspond with the calculated results of this theory ; and in the case last mentioned Young proved that the phenomena admitted of no other explanation. Placing a small opaque screen on either side of the diffracting body, so as to intercept the portion of light which passed by one of its edges, the bands immediately disappeared, although the light passing by the other edge was un- modified. The same effect was produced, and by the same means, upon the crested fringes of Grimaldi, formed within the shadows of bodies having a rectangular termination. Thus the phenomena * " Sur les Causes de la Diffraction," Annales de Chimie, torn. xli. Similar ex- periments had been made some time before by Mayer, and with the same result. (jt;iqen Memoir*, vol. iv. t Maraldi (Mem. Acad. Par. 1723), Mairan (Ibid. 1738), Du Tour (JMM*tt xentcs, torn, v.), Mr. Brougham (Phil. Trans. 1796-7), and Mr. Jordan (New Obterta. tions concerning the Inflexion of Light. London, 1795). t " On the Theory of Light and Colours." TV'tf. Trans., 1802. " Experiments and Calculations relative to Physical Optics." PAH. Tram*. 180*. 56 KEPORT ON PHYSICAL OPTICS. of the fringes, or the alternations of light and darkness, were shown to be cases of the more general principle of interference ; and the connexion is now admitted by some of the warmest ad- vocates of the Newtonian theory.* The bending of the light into the shadow, or the fact of inflexion itself, was at first ascribed by Young to the refraction of an ethereal atmosphere encompassing bodies, and decreasing in density with the distance. He afterwards, however, adopted the simpler doctrine of Huygens and Grimaldi, and referred the phenomenon to the fundamental property of waves. But perhaps the most important of the labours of Young on this subject is that in which he descends into numerical details, and, taking the observations of Newton, as well as his own, calculates the differences of the lengths of the paths traversed by the two pencils, when they destroy or reinforce one another by interference. These intervals he found to constitute an arithmetical progression for the successive bands, the first term of which was the same in the same species of light, whatever be the distance at which the fringes are received, or the other conditions of the experiment. And, finally, comparing these constants with the similar intervals of the two pencils reflected by the surfaces of a thin plate, as deduced from the experiments of Newton, he found that their difference was within the limits of error to which such observa- tions are liable, and that we are warranted in concluding that the two classes of phenomena are to be referred to one simple prin- ciple.! It is true that, in these calculations, Young starts from an erroneous principle respecting the lights which form the diffracted fringes by their interference, and he has remarked some discord- ances in his results which have, no doubt, their origin in that circumstance ; but the results of the exact theory are not greatly different from that which he adopted, and the more complete analysis of Fresnel has only tended to confirm the conclusion ob- tained by Young. The important experiment of Young, on the disappearance of the fringes in the shadow of a narrow opaque body, when the light ffifflg by one of its edges was intercepted, was that which first d him to the principle of interference. An instructive variation ' Biot, Precit elementaire, vol. ii. p. 472, 3 Edit. 'Experiments and Calculations relative to Physical Optics." Phil. Trans. DIFFRACTION. 57 in this experiment was made by M. Arago. The interior fringes were found to disappear likewise, when the light passing by one of the edges was transmitted through a plate of some transparent substance ; and, by varying the thickness of the interposed plate, M. Arago discovered that the disappearance of the fringes in this case arose from their displacement, the bands being always trans- ferred to the side on which the plate was interposed. From this it followed that the light was retarded in the denser medium.* M. Arago afterwards produced the same modification in the inter- ference bands formed by two mirrors ; and the experiment, in this form, is a complete crucial instance, as applied to the two theories of light. The amount of the displacement determines the velo- city of light in the medium, and therefore the refractive index, with an accuracy unattainable by any other method. Professor Powell has suggested a very elegant modification of this experi- ment, which at once establishes the truth of the law that the velocity of light is inversely as the refractive index of the medium traversed, f The experimental laws of the diffracted fringes were next examined by MM. Biot and Pouillet. In the case of a narrow rectilinear aperture which was that chiefly studied they found that the deviations produced in the different species of simple light, or the distances of the bands from the axis of the pencil, were in all cases proportional to the lengths of the Jits, the magni- tude of the aperture remaining the same. The same analogy was preserved in different media, the deviations varying in the inverse ratio of the refractive indices of the media, or in the direct ratio of the fits.J M. Pouillet adds, that they were unable to explain these laws, having adopted the theory of emission. They are all simple consequences of the wave-theory. The interval of the fits is exactly half the length of a wave ; and the true connexion between the place of the fringes and the latter quantity had been already pointed out by Young. Mayer afterwards studied the phenomena of diffraction, but without adding any new facts to those already known. As to the * "Sur un Phenomena remarquable qui s'observe dans la Diffraction de la Lu- mierc." Annaletde Chimie, torn. i. t Phil. Jfaff., Second Series, vol. xi. p. 6. t Biot, Trait* de Physique, torn. iv. Supplement a 1'Optique. { Siemens de Physique, torn. ii. p. 437. 58 KEPORT ON PHYSICAL OPTICS. theory, he adopted that of Newton, with some modifications. With Newton, he ascribed the inflexion of light into the shadow to the operation of an attractive force ; but, unwilling to admit the ex- istence of a repulsive force, he attempted to account for deflexion by the impact of the molecules reflected from the edge against those which passed by it.* Fresnel at first adopted and developed Young's theory of dif- fraction, and found that the general laws of the fringes the dependence of their magnitude upon the length of a wave, and upon the distances of the luminous origin and of the screen were thus fully explained. It was shown that, as the position of the screen is varied, the successive points at which the same fringe is formed are not in a right line, but constitute an hyperbola ; and that when the distance of the luminous origin is lessened, the inclination of these hyperbolic branches, considered as coincident with their asymptots, augments, and the fringes dilate in breadth, f Fresnel, however, was soon dissatisfied with this theory. If the exterior bands had their origin in the interference of the direct and reflected light, their intensity should depend on the curvature of the edge ; it is found, on the contrary, that the fringes formed by the back and by the edge of a razor are precisely alike in every respect. As to the other cases of diffraction, there were many phenomena, and especially those exhibited in Newton's experi- ment with the two knife-edges, which proved that the rays grazing the edges of the body were not the only rays concerned in the production of the fringes, but that the light which passed by those edges at sensible distances was also deviated, and concurred in their formation.* Fresnel was thus led to seek a broader foundation for his theory, and the result of his investigations is given in the able memoir which was crowned by the French Academy in 1819. In this memoir the laws of diffraction are derived from the two prin- ciples to which the laws of reflexion and refraction are them- selves referred the principle of interference and the principle of Huygens. To apply these principles to the present case, Fresnel supposes the surface of the wave, when it reaches the obstacle, to * Comm. Soc. G'Mingensis Recentiores, vol. iv. p. 49. t Annales de Chimie, torn. i. p. 239. J Memoires sur la Diffraction de la Lumitre, p. 368. DIFFRACTION. 59 be subdivided into an indefinite number of equal portions, and he applies the mathematical laws of interference, unfolded in this memoir, to determine the resultant of all the elementary waves sent by them at the same instant to any point. This resultant is expressed by means of two integrals, which are to be taken within limits determined by the particular nature of the problem. Its square is the measure of the intensity of the light ; and it is found that its value has several maxima, and minima, which correspond to the intensities of the light in the bright and dark bands. The problem of diffraction was thus completely solved ; and it only remained to apply the solution to the principal cases, and to compare the results with those of observation. The cases of diffraction selected by Fresnel are : 1st. the phenomena produced by a single straight edge ; 2nd. by an aperture terminated by parallel straight edges; and 3rd. by a narrow opaque body of the same form. The agreement of observation and theory is so complete, that the computed places of the several bands seldom differ from those observed by more than the hundredth part of a millimetre, the case of diffraction by narrow apertures alone ex- cepted. The small differences between observation and theory, in this case, Fresnel ascribes to a false judgment of the eye as to the position of the centre of the dark bands, occasioned by the different intensities of the bright bands on either side, the mini- mum always appearing nearer to the brighter light than it really is. The computed places of the bands, in the first case of diffrac- tion, were found to differ from those deduced from the hypothesis of Young by a small numerical quantity, the distance of the first dark band being less in the former theory, in the ratio of '936 to unity ; but small as the difference is, the measures of Fresnel completely decide the question.* M. Poisson applied Fresnel's integral to the case of diffraction by an opaque circular disc, and arrived at the singular result, that the intensity of the light in the centre of the shadow is precisely the same as if the disc were removed. This remarkable anticipa- tion of theory has been verified by the observation of M. Arago.f Fresnel himself solved the problem in the analogous case of a circular aperture, and arrived at the result, that the intensity of the light of any simple colour, at the central spot, will be the * Hilmoire stir la Diffraction de la Lumu-re, p. 420. t /**., P- 460. 60 REPORT ON PHYSICAL OPTICS. same as that reflected by a plate of air, whose thickness bears a certain simple relation to the radius of the aperture, and to its distances from the luminous origin and from the eye. With homogeneous light, therefore, the illumination of the central spot vanishes periodically, as the distance of the eye from the aperture is varied ; and in white light it assumes in succession the most vivid and beautiful hues, coinciding with those of the reflected rings of thin plates. These interesting phenomena were observed about the same time by Sir John Herschel, and their laws deduced, independently, from observation.* With the exception of the observations now referred to, no attempt has been made to verify the theory, by comparing the intensity of the light in the fringes with that deduced from the formulae ; and indeed it is obvious that a comparison of this nature is ill calculated to afford any conclusive evidence on the question. Fresnel thought, however, that the expression for the intensity might be indirectly verified, by superposing two sets of fringes (such as the interior and exterior fringes of a narrow opaque body), by means of double refraction, and then examining the position of the new maxima and minima. This ingenious suggestion does not appear to have been acted on. The intensity of the light in the partial waves sent from each point of the primary wave, considered as a distinct centre of disturbance, will necessarily be different in different directions, depending on the angle which these directions form with the front of the original wave ; and to solve the problem of diffraction in its most general form, it would be necessary to know the law of this variation. Fresnel has shown, however, that the rays whose directions are inclined at sensible angles to the normal to the front of the primary wave, destroy one another by interference ; so that the actual effect is produced by rays indefinitely near that normal, and which therefore may be regarded as of equal inten- sity. The truth of this assumption, however, is disputed by M. Poisson. From his theory of the propagation of motion in fluid media, this mathematician inferred that the absolute velocities of the molecules are insensible in directions making finite angles with the direction of the original vibrations. He concludes, therefore, that these velocities, or the intensity of the light in the * Essay on Light, Art. 729. DIFFRACTION. 61 partial waves, cannot be regarded as sensibly equal in directions inclined to it at very small angles.* Fresnel's reply to this part of M. Poisson's theory has been already referred to. The principle of Huygens itself, which forms the basis of Fresnel's theory, though not denied by M. Poisson, is yet objected to as introducing a needless complication into the question ; and indeed it does not seem easy to understand, at first view, why each point of the pri- mary wave in this mode of composition should not give rise to a retrograde as well as to a direct wave.f An objection of a different nature has been raised against Fresnel's theory, derived from its supposed discordance with phe- nomena. It is a consequence of that theory, when applied to the case of diffraction by a narrow aperture bounded by parallel straight edges, that if a point be taken in the axis of the pencil, whose distances measured from the centre and edge of the aperture differ by half a wave, that point will be the limit within which all the interior fringes are confined ; and beyond that point the centre of the image will be always white. This result is confirmed by the previous experiments of M. Biot ; by the observations of Fresnel himself ; and by those of Professors Airy and Powell, by whom they have been since repeated. M. Biot found that the central band was dark and white alternately, to a certain distance from the aperture ; after which it was always white. He remarks that when this limit is attained, we may diminish the breadth of the aperture, and even bring its sides into actual contact, without any change in the central band, except its enlargement and consequent diminution of intensity. J Newton's celebrated experiment with the two knife-edges has been adduced in opposition to these results. Newton found that when the distance of these edges was the 400th part of an inch, the light which passed between the knives parted in the middle, * It may be necessary to state that it was part of M. Poisson's theory, that tho vibrations are normal to the wave. t See Annales de Chimie, torn. xxii. p. 270 ; and Airy's Mathematical Tracts, p. 267. } Traits de Physiqut, torn. iy. pp. 749, 760. The description of the phenomenon given by Mayer is very similar : " Prout ilia distantia acierum semper magis magisque imminuitur, fasciae adeo evanescunt, ita ut denique non nisi fascia media remaneat ; sed ad dextram atque sinistram adeo in latitudinem extensa, ut non nisi lumen langui- dum, a medio spectri initialis utrinquc instar caudse cometce sese dilatans, represents." G'dttingen Memoirs, vol. iv. p. 61. (52 REPORT ON PHYSICAL OPTICS. and left a dark space in the centre * The experiment has been repeated by Mr. Barton, and with a similar result. t These ex- periments, however, were made with curved edges; and as Professor Powell has observed, we have no ground for supposing that the phenomenon may not be modified by this change in the conditions under which it is presented. The theory of Fresnel has not been applied to the more complex problem of an aperture with curvi- linear edges, and the analytical difficulties of the problem seem to be insuperable. There seems to be some uncertainty, however, with respect to the phenomenon itself. Professor Powell repeated the experiment with edges of various curvatures, and always found that the centre was a point of relative brightness, as com- pared with other points in the line perpendicular to the length of the aperture. J As to Newton's experiment, it seems certain, as the same writer has observed, that we are not acquainted with all its conditions ; and it is apparent from many passages that the illustrious observer himself was far from being assured with respect to the real nature and circumstances of these phenomena.! But there is another essential circumstance to be taken into account, in comparing the experiments of Newton with the results of Fresnel's theory. In that theory the origin of light is supposed to be a point ; and this condition is practically fulfilled by making the light to diverge from the focus of a lens of high power, the origin of the light in that case being (by the principles of the wave-theory) the minute image of the sun in the focus. In Newton's experiments, however, the sun's light was made to pass through a hole of sensible magnitude; and in the remarkable experiment now referred to, that hole was a quarter of an inch in diameter. The problem of diffraction in this case is one of much greater complexity. It is necessary to determine the joint effect produced at any point of the diffracting aperture by the several indefinitely small portions of a wave transmitted through the ex- * Optics, Book iii., Obs. vi. and vii. t- Phil. Mag., vol. ii. p. 268. J Ibid., p. 429, &c. " The subject of the third book I have also left imperfect, not having tried all the experiments which I intended when I was about these matters, nor repeated some of those I did try until I had satisfied myself about all their circumstances. To com- municate what I have tried, and leave the rest to others for further inquiry, is all my design in publishing these papers." Optics, Advertisement 1. See also latter part of Obs. 11, Book iii. DIFFRACTION. 63 ternal hole ; and, considering each of these as a new centre of disturbance, to find their total resultant at any point of the screen on which the fringes are received. The method of solution has been pointed out by Professor Airy ; and he has shown that when the external hole is a rectangular parallelogram, and the diffract- ing aperture of the same form, and similarly placed, the law of illumination at any point of a screen will be similar to that pro- duced by a rhomboidal aperture, in Fresnel'a method of obser- vation; the dimensions and distances in the two cases being connected by certain relations.* From these investigations Pro- fessor Airy concludes that the size of the external hole could not account for the dark central shadow mentioned by Newton in the sixth observation. He has confirmed this conclusion by experiment ; and employing holes of various magnitudes, he found the central band in all cases bright. The effect recorded by New- ton is ascribed by Professor Airy to the influence of contrast on the retina. A remarkable class of phenomena arise when a lens is placed close to an aperture of any form, and the light received on a screen at its focus, or on an eye-glass at its own focal distance from it. In fact, the phenomena of diffraction are in this manner pro- duced with holes of considerable dimensions, and were observed by Sir W. Herschel, with the undiminished apertures of his great telescopes ; the stars being seen encompassed by several dark and bright rings, succeeding one another at equal intervals, when a high magnifying power was employed. But the phenomena be- come more distinct when the aperture is limited by a diaphragm of moderate size, the diameters of the rings varying inversely as those of the apertures. The effects produced by diaphragms of dif- ferent sizes and forms have been examined in much detail by Sir John Herschel and M. Arago.f The phenomena produced by minute apertures, when combined with a lens in the manner now spoken of, have been studied with much zeal and success by Fraunhofer. The most remarkable of these phenomena are those produced by a fine grating, such as may be formed by stretching a fine wire between two parallel screws of * " On the Calculation of Newton's Experiments on Diffraction." Camli torn. iii. t Ibid. I 114 REPORT ON PHYSICAL OPTICS. while in others, finally, they receive a motion of continued rotation. To the two latter cases I shall have occasion to advert hereafter. The phenomena of fixed polarization are ascribed by M. Biot to the operation of certain forces, which he denominates polarizing forces. In the case of uniaxal crystals these forces are supposed to act in the planes containing the two rays and the axis of the crystal, the ordinary polarizing force tending to arrange the axes of the molecules in the plane containing the ray and the axis, while the extraordinary polarizing force draws them towards the perpendicular plane. If the molecules were similarly circum- stanced in every respect, they would necessarily obey the stronger of these forces, and there would be but one plane of polarization. This, however, is supposed not to be the case. Owing to the different phases of their fits, at their incidence upon the crystal, the molecules are disposed to yield more readily to one or other of these forces ; so that when a polarized ray meets a double-refracting medium, some of the molecules fall under the influence of the ordinary polarizing force, and have their axes of polarization turned into the plane containing the ray and the axis of the crystal, while others are actuated by the extraordinary force, and have their axes arranged in the perpendicular plane. The number of molecules which yield to one or other of these forces, or the intensity of the two polarized rays, is supposed to depend on the angle which the plane of primitive polarization makes with the two planes just mentioned. When the plane of polarization coincides with the former, the extraordinary force has no effect, and the ray receives only the ordinary polarization ; the converse takes place when the plane of polarization coincides with the perpendicular plane. Similar suppositions were made to account for the phe- nomena of polarization in biaxal crystals. Such was the state of the theory of double refraction when the subject was taken up by Fresnel. The law of refraction, we have seen, whether in the theory of emission or in that of waves, was intimately connected with, and dependent on the law of velocities ; so that, considered as a physical question, the problem resolved itself into the determination of the latter. With the exception, however, of the reasonings of Young respecting the form of the wave-surface in a medium compressed or dilated in a given direction,* no attempt had been made to deduce the velocity of the * Quarterly Seview, vol. ii. DOUBLE REFRACTION. 115 -extraordinary ray from the principles of either theory. Indeed the general law of the velocities was itself unknown, even as an experimental fact, although an important relation between the velocities of the two pencils had been discovered by the labours of Sir David Brewster and M. Biot. But this was not all. It was evident that no physical theory of double refraction could be re- garded as complete, which did not at the same time account for the attendant phenomenon of polarization. In this branch of the subject, however, nothing had been accomplished; and all that had been said in explanation of the phenomenon of polarization did not go further than some vague speculations as to its cause. The theory of Fresnel to which I now proceed, and which not only embraces all the known phenomena, but has even outstripped observation, and predicted consequences which were afterwards fully verified, will, I am persuaded, be regarded as the finest generalization in physical science which has been made since the discovery of universal gravitation. Fresnel* sets out from the supposition that the elastic force -of the vibrating medium is, in general, different in different directions. This is, in fact, the most general supposition that can be made; and whether we suppose that the vibrating medium is the ether within the crystal, or that the molecules of the body itself partake of the vibratory movement, there will be obviously such a connexion, and mutual dependence, of the parts of the solid and those of the medium in question, that we cannot hesitate to admit for the one what has been already established on the clearest evidence for the other.f Now if a disturbance be produced in a medium so constituted, and any particle displaced from its position of rest, the resultant of the elastic forces which resist the displace- ment will not, in general, act in the direction of that displacement (as in the case of a medium uniformly elastic), and therefore will not drive the displaced particle directly back to its position of equilibrium. Fresnel has shown, however, that there are three directions at right angles to each other, in any of which, if the particles are displaced, the elastic forces do act in the direction * " Mcmoire sur la Double Refraction," Mem. Inst., torn. vii. t M. Savart has shown that the elasticity of crystals, determined by means of their sonorous vibrations, is, in general, different in different directions. The optic axis of Iceland spar is the axis of least elasticity: that of rock crystal is the axis of greatest elasticity. 1 2 116 REPORT ON PHYSICAL OPTICS. of the displacement, whatever be the nature or laws of the mole- cular action ; and the only assumption which he makes is that these three directions are parallel all throughout the crystal.* These directions Fresnel denominates axes of elasticity. He conceives that they ought also to he axes of symmetry with respect to the crystalline form ; but observes that M. Mitscherlich has noticed some crystals in which this does not hold.f If on each of these axes, and on every line diverging from the same origin, portions be taken which are as the square roots of the elastic forces in their direction, the locus of the extremities of these portions will be a surface which Fresnel calls the surface of elasticity. This surface determines the velocity of propagation of the wave, when the direction of its vibrations is given. For the velocity of undu- latory propagation in an elastic medium, being as the square root of the elastic force, must be represented by the radius- vector of the surface of elasticity in the direction of the vibrations. Now let us conceive a plane wave advancing within the crystal. By the principle of transversal vibrations the movements of the ethereal molecules are all parallel to the wave. But the motion of each displaced particle is resisted by the elastic force of the medium, and that force is, in general, oblique to the direction of the displacement. Fresnel shows, however, that the displacement may be resolved in two directions in the plane of the wave, such that the elastic force called into action by each component will be the resultant of two forces, one of which acts in the direction of the displacement itself, while the other is normal to the wave* The latter, by the principle of transversal vibrations, can produce no effect ; and the former will give rise to a wave propagated with a constant velocity. These two directions, he finds, are those of the greatest and least diameters of the section of the surface of elasticity made by the plane of the wave ; and if the original displacement be resolved into two, parallel to them, each component will give rise to a plane wave whose velocity of propagation is * This will be the case, if the homologous lines of the groups of particles are all parallel ; an arrangement at once the simplest and most natural, and which appears to be observed in most crystallized bodies. Fresnel admits, however, the possibility of other regular arrangements ; and he conceives that the phenomena of circular polari- zation in rock crystal oblige us to suppose that its molecules are arranged according to some less simple law. t See Bulletin de la, Socie'tt Phikmafhi.jne, March, 1824. DOUBLE REFRACTION. 117 represented by that diameter, and the vibrations in each wave will preserve constantly the same direction. Thus it appears that a polarized plane wave will be resolved into two within the crystal ; and these will be propagated with different velocities, and consequently follow different paths. The amplitudes of the component vibrations are as the cosines of the angles which the direction of the original vibration contains with the two fixed rectangular directions ; and as the squares of these amplitudes represent the intensities of the two pencils, the law of Malus respecting these intensities follows as an immediate consequence.* Again, the planes perpendicular to these two directions are the planes of polarization of the two pencils ; and it is easily inferred that one of them must bisect the dihedral angle contained by the two planes passing through the normal to the wave, and the normals to the circular sections of the surface of elasticity, while the other is perpendicular to it. This conclusion does not coincide mathematically with the experimental law of M. Biot : but the differences are much within the limits of the errors of observation, and the results of experiment must be regarded as confirmatory of the theory. The velocity of propagation of a plane wave in any direction being known, the form of the wave-surface diverging from any point within the crystal may be found. For if we conceive an indefinite number of plane waves, which, at the commencement of the time, all pass through the point which is considered as the centre of disturbance, the wave-surface will be that touched by all these planes at any instant. This surface is of the fourth order. Fresnel has deduced its equation, although in an indirect manner ; and he has shown that it may be geometrically constructed by means of an ellipsoid whose semiaxes are the same as those of the surface of elasticity. The form of the wave-surface being known, the directions of the two refracted rays are given by the construc- tion of Huygens. From the construction now alluded to it appears that there * Young seems to have been the first to observe that the law of the square of the cosine could be derivecTfrom the hypothesis of transversal vibrations, (Encyc. Brtt., CHROMATICS, p. 161). The subject of the experimental confirmation of this importan law has been recently brought before the French Academy by M. Arago, and he has indicated the practical results which mny be derive 1 from this law in its application to photometry. llersclicr* Essay on Lijht : French Transaeion, Suppl., p. 690. 118 REPORT ON PHYSICAL OPTICS. are two directions the normals, namely, to the two circular sec- tions of the ellipsoid, in which the velocity of the two rays is the same. These directions are called by Fresnel the optic axes r although he sometimes applies this term to the normals to the circular sections of the surface of elasticity, or the directions in which a plane wave is propagated with a single velocity. It thus appears that crystals have in general two optic axes, and can have no more. When two of the three principal elasticities are equal, the two optic axes unite, and the wave-surface resolves itself into the sphere and spheroid of revolution. Thus the form of the wave in uniaxal crystals, which Huygens assumed as the most natural, comes out as a simple corollary from the general theory of FresneL When, lastly, the three elasticities are all equal, the wave-surface becomes a sphere; the velocity is accordingly the same in all directions, and the law of refraction is reduced to the known law of Snellius. It was easily shown to follow from the general construction, that the difference of the squares of the reciprocal velocities of the two rays, in biaxal crystals, is proportional to the product of the sines of the angles which their common direction within the crystal contains with the two axes; so that the remarkable law of Sir David Brewster and M. Biot is brought under the same theory. But it appeared further, from that theory, that the velocity of neither of the rays is constant, and that the refraction of both is performed according to a new law. This conclusion was at variance with all the received notions upon the subject ; and indeed the ex- periments of M. Biot on limpid topaz* seemed to warrant his- assumption that the refraction of one of the rays followed the ordinary law of the sines. It became, therefore, a matter of much interest to decide this question by accurate experiment. This has been done by Fresnel himself by the ordinary method of prismatic refraction, as well as by the nicer means afforded by the dis- placement of the diffracted fringes ; and the result in both cases has been conclusive in favour of his theory. The numerical data afforded by the observations of M. Biot on topaz enabled Fresnel to compute, according to the principles of that theory, the velocity of the ray in different directions ; and the observed variation was- found to agree with that deduced. * Mem. Iml., torn. iii. DOUBLE REFRACTION. 119 The phenomenon of dispersion, in singly-refracting substances, proves that the elasticity of the vibrating medium varies with the length of the wave. The same thing must take place in double- refracting media, in which the elasticity is different in different directions ; and as we have no reason for supposing that the elas- ticities should vary in the same proportion in the direction of the three axes of elasticity, it will follow that in general each refractive index will have its appropriate dispersive ratio. Sir David Brew- ster first showed that this was actually the case, and that Iceland spar and other double-refracting substances had two dispersive powers.* M. Rudberg has recently examined the laws of dis- persion in double-refracting media with much care, following the accurate method of Fraunhof er. He has in this manner determined the greatest and least refractive index, corresponding to the seven principal dark lines of the spectrum, in Iceland spar and rock crystal, and the three principal indices in arragonite and topaz ; and has found, in accordance with the discovery of Sir David Brewster, that the ratio of these indices increased with the refran- gibility of the light.f The experiments of M. Budberg confirm also the fundamental position of Fresnel's theory- namely, that the velocity of a ray in a given medium is the same as long as its plane of polarization is unchanged. The angle contained by the optic axes, in biaxal crystals, is a simple function of the three principal elasticities; and if their ratio vary with the colour of the light, the inclination of the axes must likewise vary. Such a variation has been established by the observations of Sir John Herschel ; and it has been found that the inclination of the axes is greater in red than in violet light for some crystals, while in others it is less.J In the case of Rochelle salt, the angle between the optic axes of the red and violet rays * Treatise on New Philosophical Instruments, Edin. 1813. t Annales de C/iimie, torn, xlviii. For the calculation of the phenomena of double refraction in biaxal crystals, according to Fresnel's theory, it is necessary to know the three principal refractive indices, or the velocities of propagation of rays whose vibra- tions are parallel to the three axes of elasticity. Beside the researches of M. Rudberg, I do not know that we possess any other in which all those data have been directly determined. It is true that if we know the greatest and least index, and the an* contained by the optic axes, the mean index can bo deduced. But the 11 of the optic axes cannot be determined experimentally with the same prec other elements. J Phil. Trans. 1820. 120 EEPORT ON PHYSICAL OPTICS. amounts to 10. Generally the position of the three axes of elas- ticity is invariable, and the optic axes for all colours are confined to one plane ; but Sir John Herschel has lately observed that, in borax, the optic axes belonging to different colours lie in different planes ; and we are compelled to conclude that the direction of the axes of elasticity in this, and probably in many other crystals, varies with the colour. The first addition to the theory of Fresnel was made by M. Ampere. The results alluded to are contained in two short papers read to the French Academy in the year 1828, and since embodied into one, and published in the Annales de Chimie* Fresnel had arrived at the equation which belongs to all the tangent planes of the wave-surface, and had shown in what manner the equation of the surface itself might be thence deduced by differentiation and elimination. This direct process, however, he seemed to think would involve complicated and embarrassing calculations. The method which he substituted for it consisted in verifying the equation, to which he was led by reasonings not altogether rigo- rous, and proving (by calculations which he found too tedious to transcribe) that it satisfied the conditions already assigned. M. Ampere has supplied the direct demonstration, and deduced the equation of the wave-surface in the manner originally pointed out by Fresnel. From this equation he has derived also the beautiful geometrical construction given by Fresnel, and which the latter had obtained indirectly. A very concise demonstration of the same theorem, and of the other principal points of Fresnel's theory, was given not long after by Mr. M'CuHagh.f This writer has shown that both the magni- tude and direction of the resultant elastic force, called into action by any displacement, may be represented by means of an ellipsoid whose semiaxes are the three principal refractive indices of the medium ; and from this ellipsoid, by the aid of a few geometrical lemmas, he has deduced in a clear and simple manner the leading * " Memoire sur la Ddtermination'de la Surface courbe des Ondes lumineuses," &C., t t " On the Double Refraction of Light in a crystallized medium, according to the principles of Fresnel," Transactions of the Royal Irish Academy, vol. xvi. A further development of the principles of this memoir has been recently given by the author in the 17th vol. of the same Transactions, under the title Geometrical Propositions applied to the Wave-theory of Light." DOUBLE REFRACTION. 121 results arrived at by Fresnel. The axes of this ellipsoid coincide in direction with, and are inversely proportional to, the axes of Fresnel's generating ellipsoid; and Mr. M'Cullagh has demon- strated the truth of Fresnel's construction for the wave-surface, by means of a simple geometrical relation between its tangent planes and the sections of the two ellipsoids. In the third supplement to his " Essay on the Theory of Systems of Rays"* Professor Hamilton has presented that portion of Fres- nel's theory, which relates to the fundamental problem of the determination of the velocity and polarization of a plane wave, in a very elegant analytical form ; and from the velocity and direction of the wave he deduces those of the ray, and therefore the form of the wave-surface, by means of the general relations suggested by his view of mathematical optics. In this system, of which the author gave a brief sketch at the late meeting of the Association, the laws of reflexion and refraction, ordinary or extraordinary, are comprised in two fundamental ex- pressions, which state that the partial differential coefficients of the first order of a certain function taken with respect to two final coordinates in the plane which touches the reflecting or refracting surface at the point of incidence, are not altered by reflexion or refraction. The function here considered is the characteristic func- tion of the author, whose particular form may be considered as characterizing the optical system, and on whose properties, he finds, all the problems of mathematical optics may be made to depend. On the principles of the wave-theory, this function is equal to the undulatory time of propagation of light, from any one assumed point to another, in the same or in a different medium ; and the expressions just alluded to signify simply that the components of normal slowness of the wave parallel to the bounding surface, or the reciprocals of the velocities of wave-propagation resolved in the direction of that surface, are not changed by reflexion or refraction. The normal slowness of wave-propagation is, then, of fundamental importance in this theory ; and if it be represented in magnitude by a line taken in its direction, there is obtained for its expression a curved surface which, on the principles of Fresnel, is found to be a surface of two sheets, connected with the wave-surface by a remarkable relation of reciprocity. When this relation is com- * Transactions of the Royal Irish Academy, vol. xvii. 122 REPORT ON PHYSICAL OPTICS. bined with the laws of reflexion and refraction just alluded to, they lead to a very elegant construction for the reflected or refracted ray, which is, in most cases, more convenient than that of Huy- gens. Thus, when a ray proceeds from air into any crystal, we have only to construct the surfaces of wave-slowness belonging to the- two media, and having their common centre at the point of inci- . dence. Let the incident ray be then produced to meet the sphere, which represents the normal slowness of the wave in air ; and from the point of intersection let a perpendicular be drawn to the reflecting or refracting surface. This will cut the surface of slow- ness of the reflected or refracted waves in general in two points. The lines connecting these points with the centre will represent the direction and normal slowness of the waves ; while the perpen- diculars from the centre on the tangent planes at the same points will represent the direction and slowness of the rays themselves. This important curved surface presented itself also to M. Cau- chy in his able researches on the propagation of waves in elastic media, although he does not seem to have been aware of all its properties. The properties of the same surface, and its use in constructing the direction of a reflected or a refracted ray, were also discovered, independently, by Mr. M'Cullagh, who has re- cently applied them to the geometrical development of the theory of double refraction.* The relations between the surface of wave-slowness, and that of the wave, have led Professor Hamilton to the discovery of some new geometrical properties of the latter. These properties are demonstrated by means of certain transformations of the equation of the wave-surface ; and it is shown that this surface has four conoidal cusps, at the extremities of the lines of single ray-velocity, at each of which the wave is touched (not by two planes as Fres- nel supposed, but) by an infinite number forming a tangent cone of the second degree ; while, at the extremities of the lines of single wave-velocity, there axe four circles of plane contact, in every point of each of which the wave-surface is touched by a single plane. These singular properties have led Professor Hamilton to anticipate two new laws of refraction called by him conical refraction, be- cause in each case a single ray is refracted into an infinite number ''Geometrical Propositions applied to the Wave-theory of Light," Transactions of the Royal Irish Academy, vol. xvii. DOUBLE REFRACTION. 123 forming a species of cone. External conical refraction corresponds to the cusp on the wave-surface ; and takes place without, when a single internal ray coincides with either of the lines of single ray- velocity. Internal conical refraction, on the other hand, takes place within the crystal, when a single ray is incident externally at an angle corresponding to the line of single wave-velocity within. In this latter case, if the crystal be hounded by parallel planes, all the rays of the cone will emerge at the second surface parallel to the ray incident on the first, so as to form a small elliptic cylinder, whose magnitude will depend upon the angle of the cone and the thickness of the crystal. All these remarkable conclusions have been verified in the fullest manner by experi- ment.* I shall now proceed to give a brief account of the labours of M. Cauchy in this interesting department of analysis. The researches of this eminent mathematician, on the propagation of motion in elastic media, are scattered through various livraisons of the Exercices de Mathematiques ; and he has given a valuable sum- mary of the results of these investigations, as applied to the wave- theory of light, in a memoir read to the French Academy in the year 1830.f Having assigned the general equations of motion of a system of molecules, acting on one another by attracting or repelling forces which vary according to any function of the distance, M. Cauchy observes that it is not necessary to have recourse to their general integrals in order to determine the laws of undulatory propagation. It is sufficient, in fact, to determine the law of propagation of a plane wave. For if we consider a great number of plane waves inclined to one another at small angles, and which are at first superposed in the neighbourhood of the point which is considered as the origin of the disturbance, the vibrations in the elementary waves, to which each of these gives rise, may be sup- posed too small to affect the sense separately, and these waves become efficacious only by superposition. Consequently the general wave-surface will be the locus of all the points in which the elementary plane waves are superposed ; and will therefore be the * " On the Phenomena presented by Light in its passage along the axes of biaxal Crystals," Transactions of the Royal Irish Academy, vol. xvii. t " Mcmoire sur la Theorie de la Lumiere," Mem. List., torn. x. 124 EEPOET ON PHYSICAL OPTICS. surface touched by them all at any instant.* Hence the problem is reduced to the determination of the law of propagation of a plane wave. M. Cauchy then shows that a disturbance, confined originally to a given plane, will in general give rise to three pairs of plane waves parallel to the original plane, and propagated with uniform velocities, the two waves of each pair moving with equal velocities in opposite directions. The velocities of propagation of the separate pairs, he proves, may be represented by the reciprocals of the axes of a certain ellipsoid, whose form depends upon the position of the plane wave and upon the nature of the system ; and the absolute displacements of the molecules will be parallel to the directions of these axes. Accordingly, a system of plane waves, superposed at first at the point of original disturbance, will be subdivided into three corresponding systems; and these, by their superposition, will generate a curved surface of three sheets, each sheet being touched by all the plane waves of the same system. From these principles it follows that a single ray of light will be, in general, subdivided into three polarized rays; a ray being said, in this theory, to be polarized parallel to a certain line or plane, when the vibrations of the ethereal molecules are parallel to that line or plane. M. Cauchy does not state the precise physical condition on which the existence of the third ray depends. It would seem, however, that it must arise from the circumstance that the vibration normal to the wave is not absolutely insensible, or that the actual vibrations are not accurately in the plane of the wave. He states that the intensity of this ray will be in all cases very small, and that its observation therefore will be a matter of difficulty ; and he promises in a future communication to point out the means of manifesting its existence. The formulae, on which the solution of the general problem depends, may be reduced to contain nine constant coefficients depending on the law of distribution of the molecules in space. Three of these represent the pressures sustained in the natural condition of the medium by any three planes parallel to those of the three coordinates ; and these (M. Cauchy afterwards concludes) * M. Poisson does not admit the legitimacy of this conception of the wave-surface : id he thinks that an assemblage of indefinite plane waves, having a small part in common at the origin of the motion, cannot represent the initial condition of a medium disturbed at that point. DOUBLE REFRACTION. 125 vanish of themselves. When the general theory is applied to the case in which the elasticity is the same in all directions round any line parallel to one of the axes of coordinates, M. Cauchy finds that the nine coefficients are reduced to five ; and that two sheets of the wave-surface become the sphere and spheroid of the Huy- genian law, provided that the remaining constants fulfil two assigned equations of condition. In the general case, in which the elasticity is unequal in all directions, he investigates the sections of the wave-surface made by the planes of the three coordinates ; and he finds that for two sheets of that surface they are reduced to the circle and ellipse of FresneFs theory, provided that the constants fulfil three assigned equations of condition. The wave-surface itself differs a little from the surface of the fourth order obtained by Fresnel ; but is reducible to it when the excen- tricities of the ellipses just mentioned are small, as is the case in all known crystals. Thus the results obtained by M. Cauchy embrace and confirm those of Fresnel ; and the mathematical laws of the propagation of light are shown to be particular cases of the more general laws of the propagation of vibratory motion in any elastic medium composed of attracting and repelling molecules. Considered,, however, simply with reference to the theory of light, the solution given by M. Cauchy cannot, I conceive, be considered as a com- plete physical solution. In other words, the phenomena of light are not connected directly with any given physical hypothesis ; but are shown to be comprehended in the results of the general theory, in virtue of certain assumed relations among the constants which that theory involves. If, indeed, we were able to assign the precise physical meaning of these equations of condition, we should have nothing more to desire in the general theory of light ; for these equations must necessarily express the characteristic properties of the vibrating medium. In this point of view their discussion becomes a subject of the highest interest ; and it is probable that the important conclusions of which we have yet to speak may in this manner be confirmed and extended. These conclusions are contained in a memoir presented to the French Academy by M. Lame, in the spring of the present year,* * " Memoire sur les Lois dc 1'Equilibre dc 1'Ether dans les Corps dinphanes." A full account of this paper is given in the Annaki de Chimie for March. The memoir iUelf is not yet published. 126 REPORT ON PHYSICAL OPTICS. in which the author has proposed to determine the laws accord- ing to which the molecules of bodies act on those of the ether, and the molecules of the ether on one another. Setting out from the existence of transversal vibrations, as established by the fact of the non-interference of rays oppositely-polarized, the author supposes a disturbance of the ether to take place in vacuum that is, in a space devoid of all ponderable matter, and proceeds to consider what will be the result when that disturbance reaches the ether contained in a transparent body. Assuming the property of transversal vibrations noticed by Fresnel, and more explicitly stated by M. Poisson namely, that they are propagated without any attendant change of density, M. Lame then seeks the conditions to be satisfied by the function, which represents the mutual action of the molecules of the ether and those of the solid body, in order that this property may subsist. Introducing, accordingly, this principle into the partial differential equations, which express the laws of the vibratory movement generally, he arrives finally at an equation of condition, from which he concludes that " the action of ponderable matter on the ether varies in the inverse ratio of the square of the distance ; and that the elasticity of the ether itself is proportional to its density." In order to determine the sign of this action that is to say, whether it is attractive or repulsive, it is necessary to integrate the differential equations. After certain transformations of these equations tending to facilitate their examination, he obtains their^ integral in the case of a single spherical and homogeneous mole- cule of the body, around which the ether is distributed in spheri- cal shells. The conclusions deduced from this case being combined with the established fact, that the velocity of light is less in transparent bodies than in vacuum, he arrives at the result, that the mean density of the ether is less in the former, or that the action of the molecules of these bodies on those of the ether is rr-pukive. M. Lame concludes also from the examination of the same case, that the retardation of the vibratory motion, in pene- trating into a dense body, will be greater, the less the length of an undulation, so that the refraction will be greater for waves of shorter length. This he conceives to be the true explanation of the phenomenon of dispersion. M. Lame has likewise endeavoured to connect the phenomena of double refraction with an assumed constitution of the ethereal DOUBLE REFRACTION. 127 fluid. He takes the case in which the ether is supposed to be distributed round the molecules of the body in confocal ellip- soidal shells; and he concludes that a vibratory movement, propagated from vacuum into a body so constituted, will be separated at its entrance into two component movements, which will advance with different velocities. The two component vibra- tions, he finds, will be at right angles, and parallel to the lines of greatest and least curvature of the elementary ellipsoids. Thus, the bifurcation of a ray of light on entering a crystallized medium, and the opposite polarization of the two pencils, are found to be consistent with a molecular constitution such as that described. These results are of the highest interest ; and will, no doubt, receive an early examination from those engaged in the same department of analysis. Their author seems to be persuaded that his methods will lead him to the mathematical laws of other phe- nomena, which he conceives to depend, in like manner, on the motions of the ethereal fluid.* I cannot close this division of the present Report without referring to the phenomena of absorption by crystallized media, although the laws of these phenomena are as yet wholly without the pale of theory. Dr. Wollaston seems to have been the first who noticed any facts connected with this interesting subject. The absorbing properties of crystals were found to vary with the direction : certain crystals of palladium, for example, appear- ing of a deep red colour when viewed along the axis, and of a yellowish green in a transverse direction. Tourmalines were observed also to possess analogous properties.! Similar obser- vations were afterwards made by M. Cordier and the Count de Bournon. The next step of any importance in this new field of research * In a continuation of this memoir, recently read to the French Academy, M. Lame has considered particularly the mode of vibration of the particles of the ether which are disposed round the ponderable particles of body in concentric spherical shells of decreasing density. Transparent homogeneous bodies are supposed to consist of a multitude of such particles distributed uniformly In space, and at -distances incomparably greater than their diameters ; and he conceives that the waves propagated from the particles adjoining to the surface of emergence will, by their interference, give rise to phenomena resembling the fixed lines in the spectrum. Ann. Chim., torn. Ivii. t Phil. Trans. 1804. 128 REPORT ON PHYSICAL OPTICS. was made by Sir David Brewster. This philosopher observed that in some double-refracting crystals, as carbonate of barytes, the two pencils were differently coloured ;* while in others their intensity was widely different.! The unequal absorption of the two pencils is most remarkable in tourmaline, in which it was observed, nearly at the same time, by M. Biot and Dr. Seebeck ; and the former philosopher inferred from the phenomena that the more refrangible rays of the spectrum are more easily ab- sorbed by the mineral, when polarized parallel to its axis, than when perpendicularly.^ Sir David Brewster, to whom we owe the greater part of our knowledge on this subject, has shown that similar properties belong, in a greater or less degree, to most coloured crystals which possess double refraction ; and that the absorption of light by such media varies, in general, both with the colour of the light and with the position of the plane of polarization. When a ray of common light therefore enters a plate of such a crystal, the two pencils into which it is divided will be unequally absorbed, and the emergent light will be partially polarized, the differ- ence of the intensities of the oppositely-polarized portions in- creasing with the thickness of the medium traversed. But the two pencils differ, in general, in colour as well as in intensity ; and this difference, in uniaxal crystals, Sir David Brewster found to depend on the inclination of the ray to the axis, vanishing when the ray coincided with the axis, and becoming a maximum when it was perpendicular to it. A ray of common light, therefore, transmitted perpendicularly through a plate of such a crystal, will emerge coloured; and the resulting colour will, in general, vary with the inclination of the surface to the axis. Thus the phenomena of dichroism, observed by Wollas- ton and others, are reduced to the more general laws of absorp- tion. Analogous properties belong to biaxal crystals, and depend in like manner on the planes of polarization of the two pencils, and on the direction of the ray. These properties Sir David Brewster found could be modified by heat ; and were even communicated by such influences to crystals in which they did not naturally reside. * Edin. Trans., vol. vii. ^ p hi i fi. anSf 1814> I Traite de Physique, torn, iv., 313. " On the Laws which regulate the Absorption of Polarized Light by double- refracting Crystals," Phil. Trant. 1819. COLOURS OF CRYSTALLINE PLATES. 129 Notwithstanding the important labours of Sir David Brewster, much remains to be done connected with this subject. Sir John Herschel has proposed empirical formulae to represent the in- tensity of the transmitted light as dependent on its direction; and the results of the formulae present a general accordance with observed facts.* It is much to be desired that these laws should be placed beyond doubt by an extensive series of expe- riments directed to this specific object. Although the laws of absorption by crystallized media are necessarily more compli- cated than those of ordinary media, yet they bear an evident and close relation to the well-known laws of double refraction, which seems to hold out a clue to their discovery; and I feel persuaded that it is in the phenomena of dichroism that the physical theory of absorption will first take its rise, and seek its confirmation. IV. Colours of Crystalline Plat<>*. If a beam of light, polarized by reflexion, be received upon a second reflecting plate at the polarizing angle, it is wholly transmitted when the second plane of incidence is perpendicular to the first. But if between the polarizing and analyzing plates, as they are termed, there be interposed a plate of any double- refracting crystal, a portion of the light is reflected, whose quan- tity depends on the position of the interposed crystal. In order to analyze the phenomenon, the crystalline plate may be placed so as to receive the polarized beam perpendicularly, and then turned round in its own plane. It is then observed that there are two positions of the plate in which the reflected light totally disappears, just as if no crystal had been interposed. These two positions are those in which the principal and the perpendicular sections of the crystal coincide with the plane of the first reflexion. When the plate is turned round from either of these positions, the light gradually increases ; and it becomes a maximum when the principal section is inclined at an angle of 45 to the plane of the first reflexion. These phenomena were observed by Malus. The reflected light in these experiments was in all cases white. But M. Arago observed that when the interposed plate is suffi- * Essay on Light, pp. Sol, &c. K 130 REPORT ON PHYSICAL OPTICS. ciently thin such as the laminae into which mica or sulphate of lime may be readily divided by cleavage, the most gorgeous colours appear, which vary with every change of inclination of the plate to the polarized beam. When the plate is perpendicular to the transmitted pencil, and then turned round in its own plane, the tint does not change, but only varies in intensity, being a maximum when the principal section of the crystal is inclined at an angle of 45 to the plane of primitive polariza- tion, and vanishing altogether when it coincides with that plane, or is perpendicular to it. On the other hand, when the crys- tal is fixed, and the analyzing plate turned, so as to vary the inclination of the plane of the second reflexion to that of the first, the colours change in the most striking manner ; and it is found that the colour reflected, in any one position of the plane of the second reflexion, is always complementary to that reflected in the perpendicular position. The colours disappear altogether when the thickness of the crystalline plate is reduced below a certain limit.* The experimental laws of these phenomena were investigated with unwearied zeal by M. Biot.f When the light was inci- dent perpendicularly on plates of the same substance, of different thicknesses, the tints were observed to follow the same law as the colours of thin plates ; the thicknesses of the crystal at which each tint was developed in perfection being proportional to the thicknesses of the plate of air which gave the same tint in Xewton's scale. These thicknesses vary with the nature of the crystal, and are always much greater than the corresponding thicknesses of the uncrystallized plate which exhibit the same tint. Pursuing the same inquiry, afterwards, for oblique inci- dences, M. Biot found that, in uniaxal crystals, the tint de- veloped was determined by the length of the path traversed by the light within the crystal, and by the square of the sine of the angle which its direction made with the optic axis, jointly. From this law it followed, that if a crystalline plate of moderate thickness be cut perpendicularly to the axis, and a converging or diverging pencil transmitted through it, the lines of equal tint or the isochromatic lines, as they are sometimes called, will be disposed in concentric circles similar to Newton's rings. * Him. Int. 1811. t nid ^ I8i2 . COLOURS OF CRYSTALLINE PLATES. 131 This phenomenon was observed, under different circumstances, by Sir David Brewster, Dr. Wollaston, M. Biot, and M. Seebeck. The researches of M. Biot were followed by those of Sir David Brewster. In investigating the law of the tints in biaxal crys- tals, Sir David Brewster considers the optic axes as the re- sultants of others which he denominates polarizing axes. The tint developed by a single axis is taken as the measure of its polarizing force, and is assumed to vary as the square of the sine of the angle contained by the ray with it ; and when two such axes cooperate, the tint resulting from their joint action is measured by the diagonal of a parallelogram whose sides represent the tints produced by each axis separately, and whose angle is double the angle contained by the two planes passing through them and the ray. This law Sir David Brewster has verified by comparison with the observations of M. Biot on sulphate of lime, and its agreement with phenomena was complete.* When analytically developed by M. Biot, it was found to accord with the beautiful law to which he was himself conducted by analogy namely, that the tint is measured by the product of the sines of the angles which the direction of the ray within the crystal makes with the optic axes.f From this law it easily followed that the isochromatio lines, in biaxal crystals, will be lemniscates, whose poles are in the apparent direction of the optic axes.J This phenomenon was first discovered by Sir David Brewster in topaz. The law has been established in the most complete manner by Sir John Herschel ; and he has found that the constant parameter, or the product of * " On the Laws of Polarization and Double Refraction in regularly crystallized Bodies," Phil. Trans., 1818. t From the researches of M. Biot it appeared that the measure of the tint, in uniaxal crystals, observed the same law as that attributed to the difference of the squares of the velocities of the two rays in the theory of emission. The same relation was assumed to hold generally ; and thus from the law of the tints in biaxal crystals the relation of the velocities of the two pencils, noticed in the preceding section, waa inferred. I M. Biot has observed an apparent exception to this law in the dioptiilr <>f the Tyrol, in which the rings are in general unsymmetrical with respect to the two axes. One of the axes presents the ordinary phenomena ; but in the neighbourhood of the other there is a remarkable distortion of the rings near their centre, when the crystal- line plate is turned in its own plane. These distortions seemed to observe a regular law, and were the same in all the specimens examined. It may be remarked that the optic axes of this crystal are unsymmetrically placed with respect to the crystalline form, Mem. Inst., torn. x. K2 132 REPORT ON PHYSICAL OPTICS. the radii-vectores drawn from any point to the two poles, varies inversely as the thickness of the plate for different plates of the same substance, and increases from one curve to another in the same plate in the ratio of the numbers of the natural series. To account for these varied phenomena in the hypothesis of emission, M. Biot has imagined his ingenious and beautiful theory of mo'veable polarization. When a polarized ray of any simple colour enters a crystalline plate, the component molecules are supposed, in this theory, to penetrate at first to a certain depth without losing their primitive polarization ; and then to commence a series of regular oscillations round their centres of gravity, the axes of polarization being carried alternately to one side or other of the axis of the crystal, or of the perpendicular line. These oscillations being isochronous, the thickness transversed by the molecule in its motion of translation during each of them is constant, and is assumed to be equal to double the depth to which it has penetrated before commencing its vibrations. The oscillatory movement is supposed to stop, when the molecules repass into air through the second face of the crystal ; and the emergent ray has a fixed polarization, the same as if the last oscillation of the mole- cules had been actually completed at the instant of emergence. Thus a polarized ray which has traversed a thin crystalline plate is ultimately polarized either in the primitive plane, or in a plane inclined to it at an angle equal to double the angle which it forms with the principal section, according as the thickness of the crystal is an odd or an even multiple of a certain length.* The formulso deduced from these postulates are found to represent all the more obvious laws of the tints with much fidelity. This assumed difference between the effects produced by thick and thin crystals has however been completely disproved by the decisive experiments of Fresnel. When two mirrors, slightly in- clined, are placed so as to receive the incident light at the polar- izing angle, and two laminae of sulphate of lime of the same thickness are interposed one in the path of each of the reflected pencils, and so that their principal sections are inclined at angles of 45 on either side of the plane of primitive polarization, the phenomena of the interference bands prove in the clearest manner that the light emergent from each consists of two pencils polarized respectively in the principal section, and in the perpendicular * " Sur un nouveau genre d'Oscillation," &c., J/,/:. COLOURS OF CRYSTALLINE PLATES. 133 section of the crystals ; and that the results differ from those pro- duced by thick crystals only in this, that the two pencils are superposed.* The light resulting from the union of these oppositely polarized pencils has, in certain cases, the properties ascribed to it in the theory of M. Biot ; but these properties are immediate and necessary consequences of the laws of interference of polarized light, and of the theory of transversal vibrations. Let us now inquire what account the wave-theory furnishes of the same phenomena. A ray of light on entering a crystalline plate is divided into two, or, in the language of the wave-theory, a series of waves incident upon the crystal is resolved into two within it, which traverse it in different directions and with different velocities. One of these sets of waves, therefore, will lag behind the other, and they will be in different phases of vibration at emergence. When the plate is thin, the emergent waves are superposed ; and as the retardation will then amount only to a few undulations and parts of an undulation, it would appear that we have here all the conditions necessary for their interference, and the consequent production of colour. Such was the sagacious conjecture of Young. And indeed, shortly after the publication of the first researches of M. Biot on the laws of the tints for different thicknesses, it was observed by Young that these tints corresponded accurately to the interval of retardation of the two pencils ; so that they were manifestly due to interference, f This correspondence is now made out in the fullest manner. It is an easy consequence of Fresnel's theory of double refraction, that the interval of retardation of the two pencils, in traversing a crystal- line plate, is nearly proportional to the length of their path within the crystal multiplied by the product of the sines of the angles which their directions make with the two optic axes ; and as this has been found to be the general measure of the tint, it follows that the forms of the isochromatic curves the lemniscates and the circles, are all necessary consequences of the wave-theory. * See Report made to the Academy of Sciences, in 1821, on the memoir of Freanel relative to the colours of crystallized plates, Annalea de Chimie, torn. xvii. Indeed, a more obvious objection to M. Biot's theory may be drawn from the fact which he has himself observed ; namely, that the phenomena of colour may be produced by crottmg two thick plates of nearly the same thickness, although the thickness in each was sufficient to furnish two images sensibly separated, and therefore having a Jljctd polarization. f Quarterly Rn'icic, vol. xi. 134 REPORT OX PHYSICAL OPTICS. But in the first application of the principle of interference to the colours of crystalline plates there arose a difficulty to which the known laws afforded no answer. So far as this explanation went, the phenomena of interference and of colour should be produced by the crystal alone, and in common light, without either polarizing plate or analyzing plate. Such, however, is not the fact ; and the real difficulty seemed to be, not to explain how the phenomena are produced, but to show why they are not always- produced. It occurred to MM. Arago and Fresnel to inquire how far the state of polarization of the two pencils might modify the known laws of interference ; and the results of this inquiry* have happily furnished an account of the difficulty, and completed the solution of the problem. It was found that two rays of light polarized in the same plane interfere, and produce fringes, under the same circumstances as two rays of common light ; that, when the planes of polarization are inclined, the interference is diminished and the fringes decrease in intensity ; and that, finally, when the angle between these planes is a right angle, the rays no longer interfere at all. Hence the two rays which emerge from a crystal- line plate, being oppositely polarized, cannot interfere ; and, to produce the phenomena of colour in perfection, their planes of polarization must be brought to coincidence by the analyzer. The non-interference of rays oppositely polarized is a necessary result of the mechanical theory of transversal vibrations. Fresnel has shown, on the principles of that theory, that the intensity of the light resulting from the union of two such rays is constant,. and equal to the sum of the intensities of the components, what- ever be the phases of vibration in which they meet. But though the intensity of the light does not vary with the phase of the component vibrations, the character of the resulting vibration will. It appears from theory that two rectilinear and rectangular vibra- tions compound a single vibration, which will be also rectilinear when the phases of the component vibrations differ by an exact number of semiundulations ; that, in all other cases, the resulting vibration will be elliptic ; and that the ellipse will become a circk, when the component vibrations have equal amplitudes, and the difference of their phases is an odd multiple of a quarter of a wave. These results of theory have been completely confirmed " Mcmoire s.ir 1' Action qu 5 b s Rayons de la Lumicre polarisee exercent les uns sur lea autref," ^nnales c'e C/n'mic, ton-, x. COLOURS OF CRYSTALLINE PLATES. llj-3 by experiment. When a polarized beam diverging from a lumi- nous origin is transmitted through two rhomboids of Iceland spar of equal thickness, having their principal sections inclined 45 on either side of the plane of primitive polarization, the emergent light will diverge as if from two near points, and the two portions will be oppositely polarized. It was found by Fresnel and Arago that the light resulting from the union of these pencils was plane- polarized, circularly polarized, or e lliptically polarized, according to the difference of the paths traversed when they met. Here, then, we have an account of the facts which seem to have suggested the theory of moveable polarization ; and we learu moreover that they are but particular cases of the general result. The light arising from the union of the ordinary and extraordinary pencils, which emerge from the crystalline plate, will be polarized in tl^e primitive plane, or in a plane inclined to it at an angle equal to double the angle which it makes with the principal section, according as the interval of retardation of the two pencils is an even or an odd multiple of half a wave. In all other cases the thickness of the crystal having any other value than those which exactly answer to these intervals the resulting light will be ellip- tically polarized. The ellipse will become a circle, and the .light will appear to be completely depolarized, when the two pencils are of equal intensity, and the interval of retardation is an odd multiple of a quarter of a wave. Here, then, is suggested an easy method of putting the theory of moveable polarization to the test. If a plate of sulphate of lime, whose thickness corresponds to such an interval, be placed in a beam of polarized light of some simple colour, so that its principal section is inclined at an angle of 45 to the plane of primitive polarization, the emergent light should, according to the theory of waves, be circularly polarized ; and the two pencils into which it is divided by the analyzing rhomb should not vary in intensity during its revolution. According to tho theory of moveable polarization, on the other hand, the light should be plane-polarized; and one of the images should vanish when the principal section of the rhomb coincided either with the primitive plane, or with the plane perpendicular to it. This experi- tnentum crucis was tried by MM. Fresnel and Arago, and the result was just as had been predicted by the wave-theory.* * Anr.ales dc Chimie, torn. \vii. 136 KEPOET ON PHYSICAL OPTICS. In the prosecution of their researches on the laws of interference of polarized light, MM. Fresnel and Arago discovered further that two oppositely polarized rajs will not interfere, even when their planes of polarization are brought to coincidence, unless they belong to a pencil the whole of which was originally polarized in one plane ; and that, in the interference of rays which have under- gone double refraction, half an undulation must be supposed to be lost or gained, in passing from the ordinary to the extraordinary system. The latter principle is a beautiful and simple consequence of the theory of transversal vibrations. When a vibration in any given direction is resolved into two at right angles, and each of these again into a second pair, in two fixed directions which are also perpendicular, it will easily appear that, of the four com- ponents into which the original vibration is thus resolved, the two in one of the final directions conspire, while those in the other are opposed. The tint produced by the interference of the former, therefore, corresponds to the actual difference of routes of the two polarized rays in the plate ; while that arising from the latter is that due to the same difference augmented or diminished by half an undulation. The former of the two laws now mentioned explains the office of the polarizing plate in these phenomena. To account mechanically for the fact of the non-interference of the two pencils, when the light incident upon the crystal is unpolarized, it is necessary to consider such light as a rapid succession of systems of waves polarized in all azimuths ; so that if any two planes be assumed at right angles, there will be an equal quantity of light actually polarized in each. Each of these portions, when resolved into two within the crystal, and these afterwards reduced to the same plane of polarization by the analyzing plate, wiU exhibit the phenomena of interference. But the interval of retardation differs by half a wave in the two cases ; the tints produced therefore will be com- plementary, and the light resulting from their union will be of a uniform whiteness. We are obliged to admit, therefore, that common light consists of a rapid succession of systems of waves, in each of which the vibrations are different. But the phenomena of interference (which are exhibited by common light) compel us also to admit, as Professor Airy has observed,* that the vibrations do not change * Mathematical Tracts, p. 407. COLOURS OF CRYSTALLINE PLATES. 137 continuously ; and that in each system of waves there are probably several hundred successive vibrations, which are all similar, although the vibrations of one system bear no relation to those of another, and the different systems succeed one another with such rapidity as to obliterate all trace of polarization. This per sattum transition from one system of waves, to another in which the vibrations are wholly different, seems a complication in the machinery of light, for which the elegant simplicity of the parts better known has not prepared us ; and I cannot but indulge the hope that the hypothesis, which now stands as the representative of experimental laws, may be found to merge in some simpler physical principle. The laws of interference of polarized light have thus supplied the defective link in the explanation of the colours of crystalline plates first suggested by Young. The magnitudes of the resolved vibrations are known, when the planes of polarization of the two pencils are given with respect to the plane of primitive polari- zation, and the plane of analyzation ; and as the laws of double refraction enable us to find the interval of retardation of these pencils, we have all the data necessary for the computation of the intensity and colour of the light resulting from their interference. This computation has been given by Fresnel, not only for a single plate, but likewise for two plates superposed ;* and the theory has been since more fully developed by Professor Airy.f The results are found to be, in all cases, in exact accordance with the observed facts ; and all the circumstances of the coloured rings in uniaxal and biaxal crystals are completely explained. The form of the rings, or isochromatic curves, depends upon the interval of retardation alone ; and the value of this interval had been deduced but approximately. Mr. M'Cullagh has recently given a general and exact method for its calculation, and for the determination of the forms of the rings for any plate of a double- refracting crystal bounded by parallel planes. This method is made to depend upon the properties of the surface of wave-slowness, of which I have spoken in another place ; and it is found that if the incident ray be produced to meet the sphere (which is the surface of wave-slowness for air), and through the point of inter- * Annales de Chimic, torn. xvii. t Cambridge Tra>tsactions, 1831, and Math. Tracts. 138 REPORT ON PHYSICAL OPTICS. section a perpendicular be drawn to the refracting surface, meeting- the two sheets of the surface of wave-slowness for the crystal, the interval of retardation of the rays at emergence will be measured by the thickness of the crystal multiplied by the difference of the corresponding ordinates.* By the aid of an expressive notation for the path of a ray, the author has extended his conclusions to the case of a ray which has undergone any number of internal reflexions. If the double-refracting energy of the crystal were the same for the light of every colour, the colours of the rings should follow exactly the Newtonian scale of tints, and their magnitudes should observe the same laws as those of the rings formed between two object-glasses. This is the case in carbonate of lime, beryl, and some other crystals ; and in these, therefore, the colours are similar to those of thin plates. But many remarkable deviations from this law have been observed by Sir John Herschel and Sir David Brewster. Thus, in the common uniaxal apophyllite, it waa observed by the former writer, the diameters of the rings are very nearly the same for all the colours of the spectrum ; so that the rings of different colours are superposed, and form a succession alternately black and white, which may be traced through a great number of orders, f In this remarkable case, then, the double- refracting energy of the crystal varies, very nearly, in the sub- duplicate ratio of the lengths of the waves for the rays of different colours. A very remarkable case of the inversion of the New- tonian scale of tints was observed by Sir John Herschel in some rare varieties of the same mineral. The diameters of the rings, instead of contracting as the refrangibility increases, enlarge, and actually become infinite for rays of mean refrangibility. Having passed through infinity, they again acquire a finite value; and diminish as the refrangibility increases up to the extremity of the spectrum. Here, then, for rays of a certain mean refrangibility the crystal is singly refractive ; and as the double refraction changes its character in passing through zero, the crystal is positive for the If yi , y , y. represent the corresponding ordinates of the sphere, and of the two sheets of the surface of wave-slowness for the medium, and the thickness of the stal, (y. - y,.), (y. _ y ,.) ^11 be the retardations of the two refracted waves at emergence, and (y. - yj will be the interval between them.-" Geometrical Propo- sitions applied to the Wave-theory of Light," Tram. Royal Irish Academy, vol. xviL t Phil. Tram., 1820. COLOURS OF CRYSTALLINE PLATES. 139 rays of one end of the spectrum, and negative for those of the other.* This singular phenomenon is accounted for on the prin- ciples of Fresnel's theory by supposing that the elasticity increases, with the length of the wave, faster in the direction of the axis of the crystal than in the perpendicular direction ; so that the difference of these elasticities is positive for the rays of one end of the spec- trum, negative for those of the other, and vanishes at some inter- mediate point. In biaxal crystals similar deviations take place in the magni- tude of the lemniscates corresponding to the different simple colours. But there is here another source of irregularity which is not found in uniaxal crystals. The optic axes vary, in general, with the colour ; so that the lemniscates differ also in the position of their poles, and the colours are not the same in different parts of the same ring. Where the optic axes belonging to different colours are in different planes, as Sir John Herschel has observed to be the case in borax, the irregularity produced in the coloured curves is yet more striking. In all the preceding cases, the laws of double refraction are dependent only on the direction, and are the same all throughout the mass. It is otherwise, however, in many crystals, such as analcime and some varieties of apophyllite. The complicated arrangement of the coloured bands which these substances display in polarized light proves them to consist of several distinct por- tions, possessing different optical properties ; and the phenomena indicate relations among the molecular forces, and principles of aggregation, of which it is difficult in some cases even to form a conception. These remarkable phenomena, and their laws, were discovered by Sir David Brewster.f When a polarized ray traverses a plate of Iceland spar, beryl, or almost any other uniaxal crystal, in the direction of its axis, it suffers no change of any kind ; so that when the emergent ray is analyzed by a double-refracting prism, the two pencils into which it is divided are colourless, and one of them vanishes when the principal section of the prism is parallel or perpendicular to * Cambridge Trans. 1821. Similar properties have boon observed by the same author in other crystals, as hyposulphate O f limo and vesuvian. From th tints exhibited by the latter substance it appears that the most refra images is the least dispersed. t Edin. Trans., vols. ix. & x. 140 KEPORT ON PHYSICAL OPTICS. the plane of primitive polarization. But when a ray passes in the same manner through a plate of rock-crystal, the phenomena are very different. Two images are given in every position of the prism ; and these images are of complementary colours, while the colours change in the most beautiful manner as the prism is turned round in its cell. These phenomena indicate that the plane of polarization has been changed, and differently for the different rays of the spectrum. They were first observed by M. Arago ; and he has given an account of his observations in his memoir on the colours of crystalline plates, read to the Institute in the year 1811. The subject was then taken up by M. Biot, in a paper published in the Memoires de Vlnstitut in the year 1812 ; and the analysis of the phenomenon was completed in a second memoir read in the year 1818*. When a polarized ray of any simple colour passes through a plate of rock-crystal in the direction of the optic axis, it is still polarized after emergence ; but its plane of polarization is changed. The angle through which the plane is made to re- volve varies with the colour of the light, and with the thickness of the plate, being proportional to that thickness divided by the square of the length of the fit or wave. In some crystals the plane of polarization revolves from left to right, while in others it is turned in an opposite direction ; and the crystals themselves are denominated right-handed or left-handed, according as they produce one or other of these effects. When two plates are superposed, the effect produced is, very nearly, the same as that due to a single plate whose thickness is the sum or difference of the thicknesses of the two plates, according as they are of the same or of opposite denominations. This curious distinction between plates cut from different crystals has been connected by Sir John Herschel with a corre- sponding diversity in the crystalline form. The ordinary form of the crystal of quartz is the six-sided prism terminated by the six- sided pyramid. The solid angles formed at the junction of the pyramid and the prism are sometimes replaced by small secondary planes, which in the same crystal lean all in the same direction ; and it is found that when that direction is to the right (the apex of " Memoire sur les rotations que certaines substances impriment aux axes de polarisation des rayons iumineux." COLOUES OF CRYSTALLINE PLATES. 141 the pyramid "being uppermost), the crystal is right-handed, and that, on the contrary, it is left-handed when the planes lean in the opposite way.* Sir David Brewster has shown that the amethyst, or violet quartz, is actually composed of alternate layers of right- handed and left-handed quartz. It is to the cropping out of the edges of these layers that the undulating appearance peculiar to the fracture of this mineral is owing. The structure itself is dis- played in the most beautiful manner in polarized light, f Some liquids, and even gases, have been found by MM. Biot and Seebeck to possess the same property as quartz, though in a much feebler degree, and to impress a rotation on the plane of polarization of the intromitted ray, which is proportional to the thickness of the substance traversed. These liquids do not lose their rotatory power by dilution with other liquids not possessing the property. They retain it even when raised to the state of vapour ; and, in general, the rotatory power is independent of the mode of aggregation, provided the molecular constitution is un- changed. Lastly, when two or more liquids possessing this pro- perty are mixed together, the rotation produced by the mixture is the sum of the rotations produced by the ingredients, in thicknesses proportional to the volumes in which they are combined. From these and other facts, M. Biot concludes that the property of rota- tory polarization is inherent in the ultimate particles of bodies, and does not depend on their mutual distance or arrangement.* On the other hand, quartz is found to lose the property when deprived of its crystalline structure. Thus, Sir John Herschel observed that quartz held in solution by potash did not possess the property : and the same thing has been remarked by Sir David Brewster with respect to fused quartz. The phenomena of rotatory polarization in rock-crystal M. Biot ascribed to a continued rotation of the molecules of light round their centres of gravity, produced by the operation of some unknown forces. Fresnel has proved that they arise from the * Cambridge Trans., vol. i. t &* Trans., vol. ix. | M. Biot has recently extended his researches on this subject to a great variety of substances, Annales du Museum d 'Histoire Naturelle, om. ii. In a memoir read to tho French Academy last year he has applied the laws of circular polarization to tho ana- lysis of the process of vegetation in the grasses : and he has shown, in general, the importance of the indications drawn from these phenomena in the researches of organic chemistry. Inttitut, Nos. 1 & 9. 142 REPORT ON PHYSICAL OPTICS. interference of two circularly polarized pencils which are propagated along the axis with unequal velocities, one revolving from left to right, and the other in the opposite direction. A plane-polarized ray, in fact, is equivalent to two circularly polarized rays of half the' intensity, in one of which the vibrations are from left to right, and in the other in the opposite direction. When a plane-polar- ized ray, therefore, is incident perpendicularly upon a plate of rock-crystal, cut perpendicularly to the axis, it may be resolved into two such circularly polarized rays. These are supposed to be transmitted with different velocities ; so that when they assume a common velocity at emergence, one of them is in advance of the other. They then compound a single ray polarized in a single plane ; and this plane, it can be shown, is removed from the plane of primitive polarization through an angle proportional to the interval of retardation of the two pencils, and therefore measured by the thickness of the crystal. But this interval varies also with the colour of the light ; and we are obliged to suppose that it is the same for a given number of waves, whatever be their length, so that, for a given thickness of the crystal, it varies inversely as the length of the wave. From this supposition it will follow that the deviation of the plane of polarization of the emergent ray is inversely as the square of that length, agreeably to the experi- mental results of M. Biot.* The laws of rotatory polarization were thus completely ex- plained ; and it only remained to prove the truth of the hypo- thesis, that two circularly polarized pencils, whose vibrations are in opposite directions, will actually be transmitted along the axis of quartz with different velocities. This supposition is easily put to the test -of experiment, since such a difference of velocities must give rise to a difference of refraction, when the surface of emer- gence is oblique to the direction of the ray. According to the hypothesis, therefore, a plane-polarized ray, transmitted through a prism of rock-crystal in the direction of the optic axis, should undergo double refraction at emergence ; and the two pencils into which it is divided should be circularly polarized. This has been completely verified by Fresnel, by an achromatic combination of right-handed and left-handed prisms arranged so as to double the separation ; and he has shown that the two pencils are neither com- * Annale* de Chimie, torn, xxviii. p. 147. COLOURS OF CRYSTALLINE PLATES. 143 mon nor plane-polarized light, but possess all the characters which are impressed upon a polarized ray by two total reflexions from glass at an angle of about 50. The refraction of quartz, then, in the direction of its optic axis, is wholly different from that of every other known crystal. In other directions, the two pencils into which a single ray is divided were supposed to obey the ordinary laws, and to be plane-polarized in opposite planes. This supposition has been overturned by Pro- fessor Airy ;* and it has been shown that the two pencils in quartz are, each of them, elliptically polarized, the elliptical vibrations of the two rays being in opposite directions, and the greater axes of the ellipses being in the principal plane, and perpendicular to it, respectively. The ratio of the axes, in these ellipses, is the same in the two rays, f but varies with their inclination to the optic axis, being a ratio of equality when the direction of the ray coincides with the axis, and increasing indefinitely with their incli- nation to that line according to some unknown law. As to the course of the refracted rays, Professor Airy finds that it is still determined by the Huygenian law ; but that the sphere and spheroid, which determine the velocity and direction of the two rays, do not touch, as in all other known uniaxal crystals, the latter surface being contained entirely within the former. This position is certainly a startling one. The two sheets of the wave-surface being thus absolutely separated, there is a complete interruption of continuity in passing from the velocity of one ray to that of the other ; a result which does not hold in any other case with which we are acquainted. It is however necessary to the explanation of the phenomena ; for the interval of retardation does not vanish with the inclination of the ray to the axis. Professor Airy has given an elaborate calcu- lation, founded on these hypotheses, of the forms of the rings, &c., displayed by quartz in plane-polarized and circularly polarized * "On the Nature of the Light in the two rays produced by the double refraction of Quartz." Cambridge Transactions, 1831. t In the Supplement to this paper Professor Airy has explained a highly ingenious method of determining experimentally the relation between the ellipticity and the dirvc- tion of either of the rays. This method depends upon a remarkable effect which he had been led to expect from theory : namely, a sudden change of half an undulation in the interval of retardation, and therefore a change of half an ordrr in the rings when the im-idftit light is elliptically polarized. From the results of some experiments conducted in this method, Professor Airy seems to think that the ratio of the axes in the ordinary ray approaches more nearly to one of equality than in the extraordinary ray. 144 REPORT ON PHYSICAL OPTICS. light and in any position of the analyzing plate; and he has found the most striking agreement between the results of calcu- lation and those of observation. We yet want a mechanical theory which will account for the peculiar form of the wave-surface just alluded to. Fresnel seems to have thought that the difference of the velocities of the two rays in the direction of the axis might be physically explained by an helicoidal arrangement of the molecules of the vibrating medium, which will have different properties according as the helices are right-handed or left-handed. But this hypothesis can hardly be supposed to apply to the case of fluids, in which the property of circular polarization is independent of direction ; and we are driven to confess that, with respect to these import- ant laws, physical theory is as yet wholly at fault. The singular relation betweent he interval of retardation and the length of the wave seems to afford the only clue to the unravelling of this- difficulty. The phenomena of depolarization, and of colour, impressed by double-refracting substances upon the transmitted light, are, we have seen, the necessary results of the interference of the two pencils into which the light is divided within them. These properties, then, enable us to discover the existence, and to- trace the laws, of double refraction, even in substances in which the separation of the two pencils is too minute to be directly observed. By such means the important discovery has been made, that a double-refracting structure may be communicated to bodies which do not naturally possess it, by mechanical compression and dilatation. Sir David Brewster observed that when pressure was applied to the opposite faces of a parallelepiped of glass, it deve- loped a tint in polarized light, like a plate of a double-refracting crystal ; and the tint descended in the scale as the pressure was augmented. Singly-refracting crystals, such as muriate of soda and fluor-spar, acquired the properties of double refraction by the same means.* All this is in perfect accordance with the wave-theory. Owing to the connexion of the vibrating medium with the solid in which it is contained, its elasticity is rendered unequal in differ- ent directions by the effect of compression, the maximum and minimum corresponding to the directions of greatest and least * Phil Trrms. 1815 and 1816. COLOUES OF CRYSTALLINE PLATES. 145 pressure. Accordingly, the vibrations of the ray on entering the plate are resolved into two in these rectangular directions, and these are propagated with unequal velocities ; the colour developed is determined by the interval of retardation. These results of theory were experimentally confirmed by Fresnel, by the method of inter- ferences ; and it was found that the velocity with which a ray traversed the glass was greater or less, according as it was polarized parallel or perpendicular to the axis of compression. The bifurcation of the ray at oblique incidences is a necessary consequence of this difference of velocities ; but this was also shown by Fresnel by direct experi- ment. A series of glass prisms were placed together with their refracting angles alternately in opposite directions, and the ends of the alternate prisms powerfully pressed by screws. A ray trans- mitted through the combination was found to be divided into two oppositely polarized.* The opposite effects of compression and dilatation may be seen in a thick plate of glass which is bent by an external force. When this body is interposed between the polarizing and analyz- ing plates, so as to form an angle of 45 with the plane of primitive polarization, two sets of coloured bands are seen sepa- rated by a neutral line ; and these vanish altogether when the com- pressing force is withdrawn. By crossing the glass with a plate of mica or sulphate of lime, Sir David Brewster found that the parts towards the convex, or dilated side of the neutral line, had acquired a positive double-refracting structure, and those at the concave, or compressed side, a negative one.f The intimate con- nexion between the double-refracting property, and the internal state of the body as to condensation or rarefaction, is likewise proved by the curious observation of M. Biot, that glass, when in a state of sonorous vibration, possesses the power of depolarizing the light. In these cases of induced double refraction, the phenomena are related to the form of the entire mass ; and are essentially different from those produced by regular crystals, in which the law of elas- ticity and of double refraction depends solely on the direction, and is the same in all parts of the substance. Sir David Brewster has lately succeeded in communicating a regular double-refracting structure to a mixture of resin and white wax, by pressing it into Annalcs dc Chimit, torn. xx. 1 rhil. Tnms. 1310. 146 REPORT ON PHYSICAL OPTICS. a thin film between two plates of glass. This film had a single axis of double refraction at every point in the direction of the axis of pressure ; and the tint developed depended solely on the in- clination of the ray to this line. Sir David Brewster has drawn from this phenomenon some highly interesting conclusions respect- ing the origin of double refraction in regular crystals. He men- tions several facts which seem to prove that this property is not inherent in the molecules themselves ; and he conceives that it is developed by the unequal pressure caused by the forces of aggre- gation, which are in general different in the direction of three rectangular axes. Thus the double-refracting properties and the crystalline form are referred to the same agency.* Sir David Brewster and Dr. Seebeck had before observed the phenomena arising from unequal condensation and rare faction in the case of uncrystallized bodies unequally heated. These pheno- mena may be studied by applying a bar of hot iron to the edge of a rectangular plate of glass, and placing it in the polarizing ap- paratus, so that the heated edge may form an angle of 45 with the plane of primitive polarization. At the end of some time the whole surface of the plate is observed to be covered with coloured bands, the parts near the opposite edges having acquired a positive double-refracting structure, and those near the centre a negative one. The effects are reversed when a plate of glass uniformly heated is rapidly cooled at one of its edges ; and all the appear- ances vanish when the glass acquires the same temperature throughout.! These phenomena maybe endlessly varied by vary- ing the form of the glass to which the heat is applied. If now, by any means, the glass be arrested in one of these transient states, it will have acquired a permanent double-refracting structure. This has been accomplished by M. Seebeck by raising the glass to a red heat, and then cooling it rapidly at the edges. As the outer parts, which are thus more condensed, assume a fixed form in cool- ing, the interior parts must accommodate themselves to that form, and therefore retain a state of unequal density. The law of the change of density, and therefore the double-refracting structure, will depend on the external form ; and M. Seebeck found, accord- ingly, that the coloured bands and patches which such bodies display in polarized light, assume a regular arrangement which PA*ZVi..l880. t PAH. ZVM. 1816. COLOURS OF CRYSTALLINE PLATES. 147 yaries with the shape of the mass.* The laws of these pheno- mena have been completely analyzed by Sir David Brewster ; and he has shown that the colours are those of crystallized plates, the direction of the axes, however, being different in different parts of the substance. As the double-refracting structure is communicated to bodies which do not possess it naturally, by mechanical compression or unequal temperature, so, by the use of the same means, that structure may be altered in the bodies in which it already resides. Thus Sir David Brewster and M. Biot have found that the double refraction of regular crystals may be altered, and the tints they display made to rise or descend in the scale, by simple pressure. But the changes induced by heat are yet more remarkable. Pro- fessor Mitscherlich discovered the important fact that, in general, heat dilates crystals differently in different directions, and so alters their form ; and their double-refracting properties have been found to undergo a corresponding change. Thus Iceland spar is dilated by heat in the direction of its axis ; while it actually contracts by a small amount in directions perpendicular to it. The angles of the primitive form thus vary, the rhomboid becoming less obtuse,f and approaching the form of the cube. M. Mitscherlich, accord- ingly, conjectured that its double-refracting energy must in these circumstances be diminished ; and the conjecture was fully verified by experiment. This inquiry has been followed up by M. Eud- berg ; and the effects of heat on the refractive indices of double- refracting crystals examined by the direct method of prismatic refraction. In conformity with the observations of M. Mitscher- lich, it was found that the extraordinary index in Iceland spar increased considerably with the temperature, while the ordinary index underwent little or no change. In rock crystal, on the other hand, both indices diminished as the temperature augmented, and nearly by the same amount. In arragonite a similar effect was produced on the three principal refractive indices, the least * The experiments of M. Seebeck are recorded in Sehweigger't Journal, 1814. The depolarizing property of unannealed glass seems to have been first noticed by M. Arago ; and was afterwards studied by Sir David Brewster in glass which had been melted and cooled in water. Phil. Trans. 1814. t A change of temperature, from the freezing to the boiling-point, produced a <-hange of 8' in the dihedral angles at the extremity of the axis.-2?wtf. Soc. March, 1824. L2 148 EEPOET ON PHYSICAL OPTICS. index, however, undergoing the smallest proportionate diminu- tion.* The inclination of the optic axes, in biaxal crystals, is a simple function of the elasticities of the vibrating medium in the direc- tion of three rectangular axes ; and the plane of the optic axes is that of the greatest and least elasticities. If, then, these three principal elasticities be altered by heat in different proportions, the inclination of the axes will likewise vary ; and if, in the course of this change, the difference between the greatest elasticity and the mean, or between the mean and the least, should vanish and after- wards change sign, the two axes will collapse into one, and finally open out in a plane perpendicular to their former plane. All these variations have been actually observed. Professor Mitscherlich found that, in sulphate of lime, the angle between the axes (which is about 60 at the ordinary temperature) diminishes on the applica- tion of heat ; that, as the temperature increases, these axes approach until they unite ; and that, on a still further augmentation of heat, they again separate and open out in a perpendicular plane. The primitive form of the crystal undergoes a corresponding change, the dilatation being greater in one direction than in another at right angles to it. Sir David Brewster has observed an analo- gous, and even yet more remarkable property, in glauberite. At the freezing temperature this crystal has two axes for all the rays of the spectrum, the inclination of the axes being greatest in red light, and least in violet. As the temperature rises, the two axes approach, and those of different colours unite in succession ; and at the ordinary temperature of the atmosphere, the crystal possesses the singular property of being unlaxal for violet light and biaxal for red. When the heat is further increased, the axes which have united open out in order, and in a plane at right angles to that in which they formerly lay ; and at a temperature much below that of boiling water, the planes of the axes for all colours are perpen- dicular to their first position, f The inclination of the optic axes in topaz, on the other hand, augments with the increase of tem- perature ; and the variation, M. Marx has observed, is much greater in the coloured than in the colourless varieties of this mineral. + * Phil. Mag., Third Series, vol. i. 409. t Edin. Trans., vol. xi. ; and Phil. Mag., Third Series, vol. i. 417. J Jahrb. do- Chanic, vol. ix. III. OK A NEW CASE OF INTERFERENCE OF THE RAYS OF LIGHT. Transactions of the Royal Irish Academy, Vol. XVII. THE experiment of Fresnel, on the interference of the lights pro- ceeding from the same origin, and reflected by two mirrors inclined at a very obtuse angle, has been justly regarded as one of the most important in the whole range of physical optics. The principle of interference itself had, indeed, been stated broadly by Young, and supported by the evidence of phenomena, which, to the unbiassed inquirer, left little to desire. All these phenomena, however, admitted of other possible explanations ; and the advo- cates of the corpuscular theory of light had recourse to these, rather than admit the truth of a law which afforded such strong support to the undulatory theory. In most of these phenomena, the light was in part intercepted by an obstacle, and it was con- ceived that, in passing by the edge, the molecular action, which might be supposed to exist between the particles of the body and those of light, was sufficient to account for the facts observed. But, in Fresnel's experiment, the two lights which interfere are regularly reflected by the surfaces of the mirrors, according to the ordinary laws, and are divested of every extraneous circum- stance which could, by possibility, be supposed to influence the result. This experiment, accordingly, has materially changed the character of the controversy respecting the nature of light ; and the advocates of the Newtonian theory, of the present day, are forced to admit the principle thus rigidly established, and labour only to show how the theory and that principle may be recon- ciled. 150 ON A NEW CASE OF INTEKFEEENCE While examining this important experiment the adjustment of which is a matter of some delicacy it occurred to me that the fact of direct interference might be shown in a yet simpler man- ner, by the mutual action of direct and reflected light. An inter- ference of this kind was assumed by Young to account for some of the phenomena of diffraction ; but Fresnel showed that the explanation was incomplete, and that the phenomena in question were caused merely by the interference of the secondary waves, reflexion playing no part in their production. Under these cir- cumstances it is somewhat strange that the fact of the interference of direct and reflected lights should not have been itself submitted to the test of experiment ; especially as the character of this interfer- ence, if it were found to exist, might be expected to throw some light upon the laws of reflexion itself. The theory of such interference is easily deduced from the general principles. Let light proceeding from a single luminous origin fall upon a reflecting surface, at an incidence of nearly 90 : a screen placed at the other side of the reflector will be illumi- nated, throughout a certain extent, by both direct and reflected lights ; and, if the difference of the paths traversed by these lights amounts only to a small multiple of the length of an undulation, the two lights will form fringes by their interference. Let the intensities of the direct and reflected lights be denoted by 2 and a' 2 , and that of the resulting light by A~ ; then, by the theory of the composition of coexisting vibrations, we have A* = a~ + 2 oof cos 2ir and V denoting the lengths of the paths traversed by the two waves, from their origin to any given point, and X the length of an undulation. The intensity of the resulting light will be a maximum, and equal to (a + a') 2 , at these points for which cos 2 TT = + l , or y _ g = 2n It will be a minimum, and equal to (a - a') 2 , when r OF THE RAYS OF LIGHT. 151 n being any number of the natural series 0, 1, 2, 3, &c. Bright fringes, therefore, will be formed at all the points included in the former equation, and dark ones at the points corresponding to the latter. A B A. M Let OP be the reflector; OM the screen placed in contact with it, and perpendicular to its plane ; and let A be the luminous origin, and A its reflected image at an equal distance below the line OB. Then, if M be any point whose illumination is required, 8 = AM, X = AM. Now if AB be denoted by p, BO by d, and OM by v, it is obvious that & = ffl + ( p _ ^ %* = d 2 +(p + z) 2 . Hence, approximately, and therefore the angle AOB being denoted by a. Hence the general expres- sion of the intensity of the light, at any point M, is / #\ A* = a 2 + 2aa cos UTT tan a v + a*. Again, substituting for S' - S its value just found, we see that the successive fringes will be formed at the distances given by the formula x = \ m X cotan a ; in which m is any number of the natural series, its even values giving the places of the bright fringes, and its odd values those of the dark ones. Accordingly, the bright fringes are formed at t distances 0, 21, 4/, &o., and the dark ones at the distances inter- mediate, /, 3/, 5/, &c., / being equal to A cotan a : the successive fringes, therefore, are equidistant. It is obvious that the angle 152 ON A NEW CASE OF INTERFERENCE a must be very small, or the incidence very oblique, in order that the fringes should have any sensible breadth. We have hitherto assumed that the light has undergone no change by reflexion, excepting the change of direction. Let us now suppose that the phase of the vibration is accelerated, and let us examine the effect produced in the position of the fringes. Let the amount of this acceleration be denoted by the angle fnr ; then the differences of the phases will be So that the successive fringes will be formed at the points for which m being any number of the natural series. But we have already found that X - S = 2 tan ax; so that the points in question are given by the formula, x = | (m + /x) A cotan a ; the even values of m corresponding to the bright fringes, and the odd values to the dark ones. It is evident from this that the magnitude of the fringes will be unaltered ; and that the only effect of the acceleration is to push the entire system from the edge, the amount of the shifting being equal to juA cotan a. In order to submit these results to the test of trial, I employed the apparatus consisting of two moveable metallic plates, which is of so much use in experiments of interference. The plates being closed, so as to form a narrow horizontal aperture, the flame of a lamp was placed behind ; and the light thus diverging from the aperture was received, at the distance of about three feet, on a piece of black glass truly polished, and also horizontal. This reflector was then adjusted, so that its plane might pass a little below the aperture ; or, in other words, that the light might be incident upon it at an angle of nearly 90. It is evident, then, that the light thus obliquely reflected will meet the direct light diverging from the aperture under a very small angle, and with a difference in the lengths of their paths which is capable of indefi- nite diminution. The two lights, therefore, are in a condition to interfere ; and I found, accordingly, that when they were received upon an eyepiece, placed at a short distance from the reflector, a very OF THE RAYS OF LIGHT. 153 beautiful system of bands was visible, in every respect similar to one- half of the system formed by the two mirrors in Fresnel's experiment. The first band was a bright one, and colourless. This was succeeded by a very sharply defined black band ; then followed a coloured bright band, and so on alternately. Under favourable circumstances I could easily count seven alternations ; the breadth of the bands being, as far as the eye could judge, the same throughout the series, and increasing with the obliquity of the reflected beam. The first dark band was of intense blackness. The darkness of the succeeding bands was less intense, as they were of higher orders ; and after three or four orders, they were completely obliterated by the closing in of the bright bands. At the same time the colouration of the bright bands increased with the order of the band; until, after six or seven alternations, the colours of different orders became superimposed, and the bands were thus lost in a diffused light of nearly uniform intensity. All these circumstances are similar to those observed in Fresnel's experiment, and correspond exactly with the results of theory. These bands are most perfectly defined when the eyepiece is close to the reflector. Their breadth and colouration increased with the distance of the eyepiece, but remained of a finite and very sensible magnitude, when the latter was brought into actual contact with the edge a circumstance which distinguishes them altogether from the diffracted fringes formed on the boundary of the shadow. These fringes appear to me to possess some interest in a theo- retical point of view, independently of that which attaches to them as illustrations of an important general law. Depending on the interference of two lights, one of which proceeds directly from the luminous origin, while the other has undergone re- flexion, they would seem to afford the means of detecting any difference which might exist in their condition when they meet, and therefore of tracing the modifications produced by reflexion. There are two circumstances which chiefly demand our atten- tion in the case of reflected light namely : 1st, tlic amplitude of the vibration, on which the intensity of the light depends ; and 2ndly, the phase. The facts before us seem, to a certain extent, to bear on both these points. The reasonings of Fresnel, with respect to the intensity of reflected light, are partly of an analogical nature, and very far 154: ON A NEW CASE OF INTERFERENCE indeed from being strictly demonstrative. Still, however, they have led to conclusions fully borne out by experience, and of the most interesting kind ; and we can hardly refuse our assent to doctrines which bear with them such characters of truth. The formula which Fresnel has obtained for the intensity of reflected light has not received any direct confirmation from experiment, except in the case of a few observations made by M. Arago. It results from this formula that the intensity of the reflected light must be equal to that of the incident, or the ichole of the light reflected, at the limiting incidence of 90. Fresnel himself notices this consequence, and adds that we should doubtless find it to be experimentally true, if we could reach this limit. Now the present experiment affords the means of examining this conclusion, and seems fully to establish it. We have already alluded to the intense blackness of the first dark bar, in the phenomena now described. As far as the eye can judge, the intensity of the light is absolutely nothing at the points corresponding to this bar ; and . as the intensity of the light in the dark bands is generally ex- pressed by the formula (a - a') 2 , we are forced to admit that a = d y or that the intensities of the direct and reflected lights are equal at this extreme incidence. With respect to the effect of reflexion upon the phase of vibra- tion, there seems to be some uncertainty in the theory. The phenomena of thin plates compel us to admit that half an undula- tion is either lost or gained, by the wave reflected from the first or second surface ; so that half an undulation must be added to, or subtracted from, the difference in the lengths of the paths traversed by the two waves. That such an effect should take place is in the highest degree probable from theoretical considerations. The light in the one case is reflected from the surface of a denser, in the other from that of a rarer medium ; and the mechanical laws, on which Fresnel has founded the doctrine of reflexion, lead us to the conclusion that the displacements of the ethereal particles, in the moment after reflexion, must be of opposite signs in the two cases. This difference in the phase of the vibration is equi- valent to a difference of half an undulation in the length of the path. But it does not seem to be clearly understood to which surface we are to attribute this physical change in the condition of the ray. Dr. Young, indeed, who was the first to state this law, says. OF THE RAYS OF LIGHT. 155 expressly that where " light has been reflected at the surface of the rarer medium, it must be supposed to be retarded one-half of the appropriate interval." I cannot avoid thinking that the very analogy by which he himself illustrates this point, and still more the reasonings of Fresnel on the subject, lead to an opposite conclusion, and tend to ascribe the effect which is found to take place to reflexion at the surface of the denser medium. In fact, it would appear from Fresnel's conclusions, that the sign of the vibratory movement is in all cases changed by reflexion at the surface of the denser medium, the angle of incidence exceeding the polarizing angle ; and it can readily be shown that this change of sign is equivalent to the addition of TT to the phase. The present case of interference seems to support this view. It follows, as we have seen, from theory, that if the light under- goes no change of phase by reflexion, the distances of the succes- sive dark fringes from the edge of the shadow will be as the odd numbers, 1, 3, 5, &c. ; so that the distance of the first dark band from the edge will be half the interval between each succeeding pair of dark bands. But it appears, on the contrary, from the phenomena, that the distance is as far as the eye can judge exactly equal to the succeeding intervals ; or that the bands are all shifted from the edge by the amount of half an interval The phenomena, therefore, require us to suppose that the phase of the reflected wave is accelerated, and that the amount of this accele- ration is exactly half a phase, or TT. For the general expression for the shifting of the bands is ^X cotan a ; and as this is found to be equal to A cotan a, it follows that /x = 1, or the acceleration equal to TT. It appears then that when light is reflected at the surface of a denser medium, the wave at the limiting incidence at least gains half an undulation at the instant of reflexion. In order to satisfy myself more fully of the effects of reflexion upon the phase, I repeated the experiment with polarized light. The light was polarized, before it reached the aperture in the screen, by transmission through a good tourmaline; and the fringes were observed in various positions of the plane of polariza- tion with respect to the plane of reflexion. I could detect no sensible difference in the position of the fringes under all these changes of circumstance ; and, in particular, the distance of the first dark band from the edge of the shadow seemed, as before, to be precisely equal to the intervals of the succeeding 156 ON A NEW CASE OF INTERFERENCE. bands, whether the light was polarized in the plane of incidence, or in the plane perpendicular to it. This result seems to be just what might be expected from Fresnel's theory of reflexion. From this theory it appears that if + a be the coefficient of the displacement of the incident ray, or the amplitude of the vibration, and t and f the angles of incidence and refraction, the coefficients of displacement of the reflected ray will be sin (t - tan (i - ?} -a-^r. =, or +a _ x . , tan ( + i'} according as the plane of polarization coincides with the plane of reflexion, or is perpendicular to it. Now the former quantity is always negative, so long as i is greater than *', or the ray incident on the surface of the denser medium. Under the same circum- stances, the latter quantity is positive or negative, according as i + i' is less or greater than 90, or the angle of incidence below or above the polarizing angle. For very oblique reflexion, then, both displacements are negative ; and, therefore, whether the plane of polarization coincides with, or is perpendicular to the plane of reflexion, the wave will undergo a change of half a phase at the instant of reflexion. From Sir David Brewster's important researches on the nature of metallic reflexion, it appears that a plane-polarized ray, which is incident upon a metallic reflector, becomes elliptically-polarized after reflexion ; a result which indicates a difference in the phases of the two resolved vibrations. But it appears further, from the same researches, that this difference of phase varies with the inci- dence, and vanishes altogether at the extreme incidences ; so that at the limiting incidence of 90, there is either no alteration in the phase of vibration, whether parallel or perpendicular to the plane of reflexion, or that alteration is the same for the two vibrations. From some observation of the fringes produced by the interference of direct light with that reflected from speculum metal, I conclude that the former is the case. IY. ON THE LIGHT REFLECTED AND TRANSMITTED BY THIN PLATES. Transactions of the Royal Irish Academy, Vol. XXIV. THE problem of thin plates lias been completely solved for the cases in which the incident light is polarized in either of the two principal planes, and therefore also for the case of unpolarized light, which, it is well known, may be regarded as composed of two equal pencils, polarized respectively in these two planes. There is no difficulty in perceiving in what manner the theory may be extended to light polarized in any plane ; but the results become more complicated, from the necessity of including the con- sideration of phase. When the incident light is polarized in the plane of incidence, or in the perpendicular plane, we have only to seek the magnitude of the resultant vibration, after the successive divisions which it undergoes by reflexion and refraction at the two surfaces of the plate. But when the light is polarized in any other plane, the incident vibration must be resolved into two, in the two principal planes ; and for each of these components we must know the phases, as well as the magnitudes of the resultant vibrations, before we can estimate their joint effect. These phases are in general different, and therefore the resulting light is ellipti- caHy-polarized. The preceding, and some other results of theory, were pointed out by the author many years ago ; and the general formulir, in which they are included, were at the time laid before tho Academy.* In the present communication it is proposed to develop these consequences ; and in particular, to deduce the law * fromiliiiffs of the Koi/nl Irish Academy, June 13, 1842. 158 ON THE LIGHT REFLECTED -according to which the elliptic polarization varies, both with the thickness of the plate and with the incidence. It is assumed in this investigation, that in the reflexion and refraction of light at the surface of a transparent medium, the phases of the incident, reflected, and refracted vibrations coincide at the refracting surface. This assumption is that made by Fresnel. Its theoretical truth is indeed now disproved by the more complete analysis of Cauchy, and by the experimental labours of M. Jamin; but its deviation from the phenomena is exceedingly small, except within a small range of incidence in the neighbour- hood of the angle of polarization, and the conclusions based upon it are therefore sensibly in accordance with fact, except for the same incidences. It is unnecessary to add, after what has been just stated, that the elliptic polarization here considered is altogether dictinct, both in its origin and in its laws, from that produced by reflexion at a single surface. Let us suppose that the incident light is polarized either in the plane of incidence, or in the perpendicular plane. Let u and u denote the ratios of the reflected and refracted vibrations to the incident vibration at the first surface of the plate, and for light falling upon it from without; (u) and (u'} the corresponding ratios for light proceeding in the opposite direction ; and u 2 the 'ratio of the reflected to the incident vibration at the second surface. Then, the amplitude of the incident vibration being unity, the amplitudes of the vibrations which emerge at the first surface, after one reflexion at the second, = u f u 2 (u) ; after three internal reflexions, = ' 2 (u} 2 (u'} = 1st portion x (u) i^ ; after five internal reflexions, = 2nd portion x (u) w 2 , &c. But (u} = -u, u' (11} = 1 - u 2 * These amplitudes, accordingly, form a series in geometric pro- gression, whose first term is u' u z (u') = w 2 (1 - w 2 ), and whose common ratio is (u) u 2 = - u u z . But the interval of retardation, after one internal reflexion, * The truth of these relations is evident from the known formulae of Fresnel. It has heen deduced independently, from very simple general principles, by Professor Stokes, and has been shown by him to hold even in the case of change of phase. See an interesting paper " On the perfect blackness of the central spot in Newton's Hinge." Cambridge and Lublin Mathematical Journal, 1849. AND TRANSMITTED BY THIN PLATES. 159 = 2 r cos 6', T being the thickness of the plate, and & the angle of refraction ; and the corresponding difference of phase is 4"" n, a = -r- T cos a ; A A being the length of an undulation. The difference of phase, after three internal reflexions, is 2a ; after five internal reflexions, 3a ; and so on. Hence, if ^ denote the phase of the vibration reflected at the first surface, at the instant of reflexion, - a will be the phase of the portion which emerges there after one internal reflexion ; $ - 2a, after three, &c. And the sum of all the inter- nally-reflected vibrations will be u-i (1 - u~) [sin (< - a) - uu 2 sin ($ - 2a) + u % u sin (0 - 3a) - &c.] . in which, it can be easily shown, the quantity within the brackets is equal to sin (0 - a) + uu z sin 1 + 2ww 2 cos a + w 2 M 2 2 * Adding u sin $, the vibration reflected externally at the first surface of the plate, the sum of all the reflected vibrations is ,. sin (0 - a) + uu-i sin 6 u sin d> + 2 (1 - w 2 ) ^ - ,-V ' 1 + 2uu 2 cos a + wW Let this quantity be put under the form P sin + Q cos ^. Then we find cos o + uu t . P = U + U 2 (1 - U) ; 1 + 2uui COS a + W - w 2 (1 - w 2 ) sin o 1 + 2miz cos a + ifu*' IVherefore, the intensity of the resulting light is* t/ 2 + 2m/ 2 cos a +t/^ t * "l+2tt,COSa + V ,nd its phase, ;//, will be given by the formula, -_Q = M 2 (1 - M 2 ) sin a n ^ ~ P ~ u (1 + w./j + ih (1 + ") cos a' * This expression for the intensity has been alieady obtained by Mr. Airy "On the Phenomena of Newton's Rings, when formed between two transparent substances of different refractive powers." Camb. Tram., 1832. 150 ON THE LIGHT REFLECTED Substituting 1 - sin 2 ^ for cos a, the expression for the intensity becomes (u + # 2 ) 2 - 4ww 2 sin 2 ~ (1 + ww 2 ) 2 - 4w* 2 sin 2 When the media are the same on the two sides of the plate, 2 = - u, and the foregoing formula is reduced to the known one, 4w 2 sin 2 I 1 = - w 2 ) 2 + 4w 2 sin 2 ^ The greatest and least values of I, in the general formula, cor- respond to sin a = 0, or a = mir. When a = 2mir, the expression is reduced to \1 + uuz When a = (2m + 1) TT, it becomes / / u - u 2 V \i - wJ * When u and t* 2 have the same sign, the former value is a maximum,. and the latter a minimum ; the rings consequently begin from a bright centre. On the other hand, when u and 2 have opposite signs, the former value is a minimum, and the latter a maximum ; and the rings begin from a dark centre. When the incident light is polarized in the plane of incidence? the signs of u and w 2 are determined solely by the relative refrac- tive densities of the plate, and of the two media which border it on either side. When the refractive density of the plate is greater or less than those of both media, M and u z are of opposite signs, and the rings are dark-centred. On the other hand, when the refractive density of the plate is intermediate to those of the two media on either side, u and u z have the same sign, and the rings ar& bright-centred. This inversion of the phenomenon was observed by Young. AND TRANSMITTED BY THIN PLATES. 1G1 When the incident light is polarized perpendicularly to the plane of incidence, the signs of u and w 2 depend on the incidence, as well as on the relative refractive densities of the media. When the angle of incidence is less than the polarizing angle at either .sur- face, the sign of u, or of 2 , is the same as in the case of light polarized in the plane of incidence ; when it is greater, the sign is opposite. Accordingly, when the incidences at both surfaces are less than their respective polarizing angles, the character of the rings will be the same as for light polarized in the plane of inci- dence. And the same thing is true when the incidences at both surfaces are greater than their respective polarizing angles ; for in this case the signs of u and w 2 are changed simultaneously. But when the incidence at one surface is less, and at the other greater, than their respective polarizing angles, one of the quantities, u and w 2 , changes sign, and the other does not ; and the character of the rings will be the opposite to that presented when the light is polarized in the plane of incidence. It follows, then, that when the refractive density of the plate is greater or less than those of the two media, the rings will be dark -centred at the lower and higher incidences, and bright-centred at the intermediate incidences. This phenomenon was observed by Arago, and explained by Mr. Airy. On the other hand, when the refractive density of the plate is intermediate to those of the two media (the light being still polarized in the perpendicular plane), the rings will be bright-centred at the lower and higher incidences, and dark-centred at the intermediate incidences. This latter case has not been hitherto noticed. At the polarizing angles of either surface, u = 0, or u = 0, and the rings vanish. 7. Now let the incident light be polarized in any plane, inclined at the angle y to the plane of incidence ; and let the amplitude of the incident vibration (= 1) be resolved into two, cos y and sin 7, in the plane of incidence and in the perpendicular plane, respec- tively. Let v and i\ denote the values of u and v, in the case of light polarized in the plane of incidence ; w and r 2 the correspond- ing values for light polarized perpendicularly ; then the intensities of the light in the two reflected pencils are E* + 2cf, cos Q + t ./ ., r u? + 2M-tf> cos a + f^ h)2 ~ l + - + rY CCS ~ 7 ' " 1 + Stfw, cos a 4 tftf 126 ON THE LIGHT REFLECTED And the intensity of the reflected light will be the sum of the in- tensities of the two component portions. When the obliquity of the incident pencil is so small, that the squares and products of v, v w, iff*, may be neglected in com- parison with unity, the intensity of the reflected light will be, nearly, (p 2 + 2m, cos a + 2 2 ) cos 2 7 + (tf + 2ww, cos y + w?} sin 2 7. This expression is independent of the phase, and therefore the intensity is constant for a given incidence, when W Z w, cos 2 7 + vcw-i sin 2 7 = 0, or tan 2 7 = --- . It will be easily seen, on substituting for v, v z , w, iff z , their well- known values, that the value of tan 7 will be real, and therefore the disappearance of the rings possible, only when the angles of incidence at the two surfaces of the plate are, in the one case greater, and in the other less, than the respective polarizing angles.* This is the explanation of the phenomenon observed by Sir David Brewster. Again, since the values of v and v z are, in general, different from those of w and w z , it follows that the phases of the two com- ponent vibrations are unequal, and consequently that the result- ing reflected light is elliptically-polarized. This consequence of the wave-theory does not appear to have been noticed by observers. 8. We may now proceed to consider more particularly the case in which the media are the same on the two sides of the plate. In this case, v z , = - v, w z = - w, and the general expressions for the phases of the two polarized pencils are reduced to The difference of phase is given by the formula tan (,/, - v) = ^ ~ sin 1 - p* + w 2 ) cos a + vW It follows from this, that i// - x varies with a, and therefore with * See the paper above referred to, Proceedings, vol. ii., p. 268. AND TRANSMITTED BY THIN PLATES. 16^3 the thickness of the plate ; and that, in the phenomena of the rings, it will go through all its values within the limits of a single ring. 9. The difference, ^ - \> generally vanishes, when sin o = 0,* Accordingly, the difference of phase is nothing both at the bright and at the dark rings ; and the light at the former is plane-polar' izt'd. The difference of phase is a maximum, when tf + w z cos a = ^ - ; 1 + 2 M? 2 and denoting by A the maximum value of i// - \, we have i* - v? __ V/KI-^HI-V)}' 10. In order to express a and A in terms of the angle of incidence, let us make -- g- = p; and substituting in Fresnel's formulas, we find sin (9 - &} up - 1 tan(0-y) = /a-p. ~ sin (0 + 9) ~ W + 1' " tan (0 + tf) ~ /i + p'' H being the refractive index for light entering the plate from the surrounding medium. Making these substitutions, the thickness of the plate corresponding to the maximum difference of phase is determined by the formula, COS a , + fj.- i y Or, since p 2 = 1 + (1 - ^} tan 2 0, Accordingly, the value of cot ^ increases, from (p + p"*) at a perpendicular incidence, to infinity when the incidence is most * When 6 = 90, we have v* = !,; = !; and the expression for ten ty - X ) becomes -, when a = Zmir. It appears from the following, that the difference of phase is a maximum in this case. M 2 164 ON THE LIGHT REFLECTED oblique ; and, in a plate of varying thickness, the points of maxi- mum difference of phase commence near the middle of an interval, and approach indefinitely to the dark rings as the incidence approaches to 90. The tangent of the maximum difference of phase is ( - 1) (p 2 - 1) tan A = or, putting for p 2 its value, When n is greater than unity, or the plate denser than the surrounding medium, A = 0, when = 0. As 6 increases, A in- creases continuously; until, when approaches to 90, A approaches indefinitely to the value, This is its greatest value. For ordinary flint glass, and for the extreme red ray, ju = 1'60, and A = 26. When fi is less than unity, or the plate rarer than the sur- rounding medium, A = 0, when = 0, as before. And A increases as increases, up to the limiting incidence, for which 8 = sin~>> and cot A = 2 ^ . The greatest difference of phase, therefore, is the same in hoth cases, and in both corresponds to the extreme incidence at which light is admitted into the plate.* 11. Again, the media being the same on the two sides of the plate, the expressions for the intensities of the two portions of the reflected light, polarized respectively in the plane of incidence and in the perpendicular plane, become * It is evident that cot A = 0, and therefore A = 90", when the incidence has either of the values given by the formulas tan 1 = , and tan 2 = u 2 * ** l-M 2 l-/i*' hoth of which are real when the plate is rarer than the surrounding medium. Theee values, however, are both greater than the limiting incidence, and the light dc.es net enter the plate. When the incidence is inteimediate to the two preceding values, A is imaginary. AND TRANSMITTED BY THIN PLATES. 105 sin 2 - cos 2 y u? sin 2 | sin 2 y 1 - "2v~ cos a + 1 <4> ~ 1 - 2iv- cos a + ic 1 ' Let the ratio of the corresponding amplitudes be denoted by tan y' ; then w Ifl - 2c 2 - / -- *v\i - 1 - c 2 cos a + tan y = tan y - ' The angle 7' will be the azimuth of the plane of polarization, when the reflected light is plane-polarized. When the thickness of the plate is that corresponding to the maximum difference of phase, or cos a = ^ - - , the ratio of the 1 + V amplitudes becomes tc tan 7 = tan T,- The angles A and 7' being known, the character of the elliptic polarization is completely determined. 12. We may now proceed to examine the intensity and phase of the transmitted light. It will be easily seen, by following the same reasoning as before, that the transmitted light consists of an indefinite number of portions, which emerge at the second surface of the plate after 0, 2, 4, &c., internal reflexions ; and that the amplitudes of their vibrations form a series in geometric progression, whose first term is wV 2 , and whose common ratio is - w 2 , in which u and u 2 denote, as before, the ratios of the amplitudes of the reflected to the incident vibrations, at the first and second surfaces of the plate respectively, and u' and w' 2 the corresponding ratios for the re- fracted vibrations. It is likewise evident that, if $ denote the phase of the first portion, $ - a will be that of the second, - 2u that of the third, &e., in which 4;r a = - T cos u , A as before. Consequently the resulting vibration, which is the sum of all these, is liu't !sin

- a) + u'u-i sin (^ - 2a) - &c.j sin d> + HH-, sin (^ + a) " M X 2 1 + Ktftfc cos a 166 ON THE LIGHT KEFLECTED Revolving this, we find 1 + uu z cos a = W " a _ , , + W' * 1 1 + 2MW, COS a + W + 2WM 2 COS a + tt'tt, Hence the intensity of the transmitted light is r and the phase is given by the formula ., Q - wwa sin o tan ;// = - -g = -= --- . P 1 + UU Z COS a 13. When u and u, have the same sign, the greatest and least intensities correspond to a = (2m + 1) TT, and a = 2imr. They are, respectively, , and 1 - When u and M 2 have opposite signs, the greatest and least values of /' correspond to a = 2mtr, and a = (2m + I)TT. Hence the bright rings of the transmitted system occur at thicknesses which produce the dark rings of the reflected system, and vice versa. These values never vanish, since u' and M' Z are never evanes- cent ; and there are no Hack rings in the transmitted light. 14. When the incident light is polarized perpendicularly to the plane of incidence, and when the coefficients u and u 2 are both positive, or both negative, the preceding values of the greatest and least intensities remain unaltered. In other words, the rings exhibit the same character, whether the angle of incidence is less than the polarizing angle at both surfaces, or greater. But when one of these quantities is positive, and the other negative, i.e. when the incidence at one surface is less than its polarizing angle, and at the other greater, the preceding formulas are transposed, and the bright rings are changed into dark ones, and nee versa. When u = 0, or u z = 0, i.e. when the light is incident on either surface at its polarizing angle, the preceding values are both re- duced to (wV 2 ) 2 , and there is no variation of intensity caused by interference. 15. It may be easily shown from Fresnel's formulas, that AND TRANSMITTED BY THIN PLATES. 167 When the media are the same on the two sides of the plate, 2 ' = 0, and u z = - u. In this case, therefore, u'tt 2 ' = 1 - 2 , and the gene- ral expression for the intensity becomes (1 - ')' I' I - 2'U? COS a + U* ' Comparing this with the expression for the intensity of the reflected light (5), we learn that /+/' = !; or that the intensities of the reflected and transmitted lights are complementary, the maximum of the former corresponding to the minimum of the latter, and vice versa. This law explains the rela- tions of the reflected and transmitted rings observed by Newton and Arago. The greatest and least intensities are 1 - tt'V 1 + U-) ' 1 and Accordingly, the intensity of the light in the bright rings is equal to that of the incident light. 16. When the incident light is polarized in any plane, inclined at the angle y to the plane of incidence, the transmitted light is composed of two portions, polarized in the plane of incidence, and in the perpendicular plane, respectively. The phases of these two portions are given by the formulas, - W 2 sin a , ww-t sin a tan w = - , tan v = -: - ; l + ta*oea 1 + ww 2 cos a and as these are in general unequal, the light will be elliptically- polarized. The difference of phase is given by the formula f (r X. ) ~ j + ^p a Hence the ellipticity varies with a, and therefore with the thick- ness of the plate. The difference, \J/ - x'> will be a maximum, when cos a = Substituting in the preceding formula, and denoting the greatest difference of phase by A', we have ON THE LIGHT REFLECTED tan A' = - 17. The intensities of the two component pencils are (tV cos 7 ) 8 j, = (^V 2 sin T ) 2 _ ^ " 1 + 2w , cos a + rV 1 + 2'Wa cos a + ftr a 8 ' And denoting the ratio of the amplitudes by tan 7', as before, llflDz V 1 + 2W 2 COS a + V^'V-i tan 7 ' = tan 7 7^1 + 2tnc z cos a + w When the thickness of the plate is that corresponding to the maxi- mum difference of phase, this expression is reduced to 18. When the media are the same on the two sides of the plate, ih = - v, ? a = - w, and the preceding formulas for the maxi- mum difference of phase are reduced to r + w , cos a = , tan A = which are identical with those already obtained for the reflected light. Accordingly, the difference of phase of the two portions of the polarized beam is the same in the reflected and in the transmitted pencils. In the same case the ratio of the amplitudes is tan 7 ' = tan 7 In resuming the theory of the rings in polarized light, I was under the impression that the knowledge of the subject was still confined to the general principles laid down by myself, many years ago. Since the foregoing paper was read, however, I have learned that the problem has been discussed by M. Jamin, in an interest- ing memoir published in the Annales de Chimie in 1822. In this memoir the author has confined himself to the case in which the media are the same on the two sides of the plate. In another respect, he has treated the problem more generally than it has been considered in the foregoing paper, having based it upon the theory of Cauchy, instead of that of Fresnel. lie has thus been AND TRANSMITTED BY THIN PLATES. 169 led to detect some curious phenomena of the rings, when the light is polarized perpendicularly to the plane of incidence, which had escaped the notice of preceding observers. In applying his theory, however, to light polarized in any plane, M. Jamin disregards, as insensible, the additional terms which depend upon the Cauchian phase. His formulas, thus simplified, should therefore accord with those obtained in the present paper for a plate bordered on both sides by the same medium. Such, however, is not the case : the results, in fact, are entirely at variance, M. Jamin's formulas giving plane-polarized light at the extreme incidence, where (according to the theory above given) the departure from plane polarization is greatest. I believe that this discrepancy is accounted for by an error, into which M. Jamin seems to have fallen, in computing the maximum difference of phase of the two components of the light, which are polarized respectively in the two principal planes. V. OBSERVATIONS ON THE DIRECTION AND INTENSITY OF THE TERRESTRIAL MAGNETIC FORCE IN IRE- LAND, MADE BY THE REV. HUMPHREY LLOYD, M.A., F.R.S., &c.; BY CAPTAIN EDWARD SABINE, R.A., F.R.S., &c.; AND BY CAPTAIN JAMES CLARKE ROSS, R.N., F.R.S., &c. Fifth Report of the British Association for the Advancement of Science. THE observations which form the subject of the present communi- cation were made during the years 1834 and 1835, in compliance with the recommendation of the British Association urged in the first and second Reports of its proceedings. Their main object has been to determine the direction of the lines of magnetic dip and intensity in Ireland, and to make a small, but it was hoped exact, addition to our knowledge of the laws of distribution of the earth's magnetism. The observations are threefold : first, observa- tions of the horizontal part of the earth's magnetic force, as de- termined by the time of vibration of a needle suspended horizontally, after the method of Professor Hansteen ; secondly, observations of dip, made in the usual manner ; and thirdly, observations oi dip and intensity at the same time, and with the same instrument, according to the method adopted by Professor Lloyd, and already submitted to the Association.* I. HORIZONTAL INTENSITY. The instruments employed in the first series of observations were constructed after the model of that of Professor Hansteen. The needles are cylinders 2| inches long, and -13 of an inch in diameter, suspended by a few filaments of the silkworm's thread. * Fourth Report, p. 557. Transactions of the Royal Irish Academy, fW. XJ'IL TERRESTRIAL MAGNETIC FORCE IN IRELAND. 171 They are inclosed in a small rectangular box, supported upon levelling screws, and having a tubular pillar screwed on at top for the silk suspension. At the bottom of the box is a divided circle, for the purpose of noting the arc of vibration : the temperature is observed by means of a small thermometer inclosed in such a manner as to avoid contact with the bottom and sides of the box. Before the commencement of the observations, the bottom of the box is to be rendered truly horizontal by means of the levelling screws on which it rests, and of a small spirit level with which it is furnished. The needle being then suspended so as to hang near the bottom, its deviation, if any, from the horizontal position will be detected by its inclination to the surface. It is then to be slightly moved to one side or other in the brass stirrup by which it is supported, until it hangs truly parallel to the lower surface of the box ; and when this adjustment is once accurately made, no further alteration will be required, unless the change of dip be considerable. When an observation is to be made, the needle is raised or lowered by a small roller to which the silk suspension is attached, so that it may hang about midway between the upper and lower surfaces of the box. It is then drawn aside from the magnetic meridian through an arc of 25 or 30, by a piece of brass wire inserted in the side of the box, and is allowed to oscillate. The registry of the oscillations is commenced when the amplitude of the vibration on either side of the meridian is reduced to 20, and it is continued during 360 vibrations ; 'the moment of the comple- tion of every 10th vibration during that interval being noted by a chronometer. The amplitude of the final arc, or of the arc of the 360th vibration, is also observed ; and the temperature of the air in the box, as indicated by the interior thermometer, is noted at the beginning and end of the observation. It is obvious that in this manner seven intervals of time are obtained, each corresponding to 300 vibrations, viz. the interval between the Oth and 300th vibration, between the 10th and 310th, &o., and between the 60th and 360th ; and the mean of these is taken as the result. But to this result several corrections must be applied. 1. The time as shown by the chronometer is to be corrected for rate; and accordingly the chronometer's rate must be determined from time to time by comparison with a good timekeeper, or by 172 OBSERVATIONS OF THE TERRESTRIAL astronomical observations. In the present series the rate was observed at the commencement and end of each group of observa- tions by the former and easier method. The amount of the correction due to rate is in most cases very small, the correction in the time of 100 vibrations corresponding to a daily rate of 2 s being less than O s> 01 with the slowest of the needles employed. 2. Professor Hansteen has applied a correction for the arc of vibration, so as to reduce the time to that corresponding to in- finitely small arcs. The correction is investigated on the same principles as that usually applied to pendulum observations. It is however more complicated in its form ; for, instead of a single series of vibrations (as in the case of the pendulum), we have here seven distinct series, each commencing from a different arc. The principle, however, seems hardly applicable in the present instance. It is assumed that the successive arcs of vibration decrease in geometric progression, as they must necessarily do if the resistance of the air be proportional to the velocity. This is found to hold good in the vibrations of the pendulum when the arcs are very small ; but it is by no means true when they are so considerable as those in which the horizontal magnetic pendulum is made to vibrate. Where, however, the vibrations commence from the same arc, and the terminal arc does not much vary, the correction itself may perhaps be disregarded. In the following observations, in which the initial arc was 20, the 360th or terminal arc was generally 2, and was in all cases included between the limits 1 and 4. In such cases, then, the correction must be, nearly, a constant quantity ; its application to the observed times is there- fore nearly equivalent to their multiplication by a constant co- efficient, and the ratio of the times (with which alone we are concerned in this class of observations) remains unaltered. For these reasons no attempt has been made to introduce a correction for the arcs in the following results ; but the terminal arcs are given, so as to put the reader in possession of all the circumstances of the observation. 3. By far the most important correction is that due to tem- perature. If T' be the observed time of 100 vibrations correspond- ing to the actual temperature t', and T the corrected time cor- responding to the standard temperature t, the correction is T- T' = T' t-f;. MAGNETIC FORCE IN IRELAND. 173 a being a constant coefficient whose value is to be determined experimentally for each needle. The following observations were made with the cylinders L (r/), L (b), in order to determine the value of the coefficient a for each. The apparatus being inclosed in a large glass bell, the time of 100 vibrations of cylinder L (a), commencing with the arc of 10, was observed at the mean temperature of the room, and when the air of the bell was heated artificially from below, by means of a spirit lamp. The final arc varied between 4 and 5. The observations with cylinder L (b), were made in the bell without the apparatus. In this case no means were taken to observe with any accuracy the arc of vibration ; and in order to reduce as much as possible any error arising from this source, the observations were continued in each instance until the arcs were reduced to the smallest appreci- able, and the mean of the last five intervals of 100 vibrations then taken as the result. The chronometer's rate varied from + 8 '6 to + l s> 4 per diem, and had therefore no appreciable influence on the results. Cylinder L (a). Cylinder L (b). Date. Hour. Time. Temp. h. m. s. March 4, 1 42 242-64 54-5 1 55 242-82 54-5 8, 3 46 242-30 58-8 4 4 242-24 58-0 Mean 242-50 56-5 1, 3 44 243-96 82-5 4 10 243-96 83-1 [ 8, 2 6 243-94 74-5 Mean 243-95 80-0 Date. Hour. Time. Temp. h. m s. March 19.* 1 11 7 293-48 53-6 11 40 292-52 55-0 2 42 294-00 60-3 3 5 293-76 59-5 Mean 293-44 57-1 12 50 295-24 80-0 1 16 295-20 83-3 Mean 295-22 81-6 The constant coefficient sought is to be calculated from the formula * A series of observations had been made with this cylinder, in the same manner and on the same days as those with cyl. L (a) ; but the results were unsatuf some of them indicating an increase of force with increased temperature. 1 dictory results have been noticed by many observers, and are usually at! disturbing effects of current* of air, determined by inequality of 174 OBSERVATIONS OF THE TEERESTRIAL in which t and t' are the two temperatures, and T and T' the corresponding times of vibration. We find Cyl. L (a). T = 242 8 -50, T - T' = 1 8 '45, t - t' = 23-5 a = -000254 Cyl. L (J). T' = 293 s -44, T - T' = 1"78, t-f= 24-5 a = -000248. It is to be observed that these cylinders were made at the same time, and were therefore probably tempered to the same degree ; and to this circumstance we may, with much probability, ascribe the close agreement in the values of the constant which determines the effects of temperature upon the force of the needle. No observations were made to determine directly the effects of temperature upon the other needles employed in the course of these observations; and, in correcting the results obtained with them, the coefficient employed by M. Hansteen, viz., -00017, has been that adopted. The standard temperature (t), to which all the results contained in the following pages are reduced, is 60 Fahr. 4. All that we know of the diurnal variations of the intensity of the horizontal force is due to M. Hansteen and Professor Christie. These writers agree in fixing the hours of minimum intensity at 10^- A. M. The intensity then increases, and attains its maximum, according to Professor Christie, at about 7| p. M. The amount of this maximum is 1*0024 in summer, the minimum intensity being unity; but this amount, as well as the hour of its occurrence, changes with the season. Of the law according to which the force varies between its two limiting values, we know nothing ; and it is therefore impossible, in the present state of our knowledge, to apply a correction for these variations. It was proposed to evade this difficulty, in the ensuing observations, by observing at a fixed hour. To this limitation, however, it was found impracticable to adhere, and the results still remain uncertain by the amount of the diurnal change. 5. The variations of the magnetic force give rise to another and still graver class of errors. The least experience in obser- vations of this nature is enough to prove that the horizontal in- tensity is, from some cause or other, subject to irregular fluctuations ; and these fluctuations, like those of the barometer in our climates, are much more considerable than the regular horary changes. It seems probable that these variations in the intensity of the hori- MAGNETIC FORCE IN IRELAND. 175 zontal force are, like those in its direction, not local phenomena, but occur at the same time at places widely separated. To elimi- nate them from our results, therefore, it would suffice to have a regular series of observations made at some fixed station, con- temporaneous with those made at the different stations ; and, if these be not very remote, we may assume that the variation of the observed force at each from its mean amount is the same as that observed at the same time at the fixed station. Unhappily these means of freeing the results from the admixture of what may be called accidental phenomena have not been attended to in the following, or indeed in any similar series of observations, and there is reason to believe that the errors due to this cause are the largest in amount of any by which the present series is affected. The amount of these fluctuations, from day to day, may be judged of from the following specimen of a series of observations such as that alluded to, commenced by Captain Sabine in the month of June, 1835. The apparatus in which the needle was vibrated was unmoved during the continuation of the series, and the needle remained permanently suspended. The height of the barometer was noted, as well as the temperature ; the hour of observation was, nearly, 10 A. M. Time 0/200 Vibrations of a Standard Needle. Date. Barom. Therm. Time. June 15, Inch. 30-260 66-0 1139-60 16, 30-268 64-5 1138-20 17, 30-188 62-0 1138-27 18, 30-188 61-0 1135-87 19, 30-200 59-0 1136-67 20, 30-080 62-0 1135-93 21, 62-0 1137-67 22, 29-640 61-0 1138-53 23, 29-682 59-0 1138-40 24, 29-360 57-5 1137-80 25, 29-850 57-0 1136-87 26, 29-580 57-0 1136-70 27, 30-165 57-0 1137-13 29, 30-120 59-0 1137-80 30, 29-820 58-0 1137-73 July 2, 29-900 62-0 1137-30 4, 29-850 60-5 111*71 5, 29-650 60-0 1135K>;5 6, 30-000 58-5 1136-60 7, 29-850 59-0 1137-87 176 OBSERVATIONS OF THE TERRESTRIAL The mean time of 200 vibrations, deduced from these results, is 1137"'66 at the temperature 60'6. But the time observed on the 18th of June is 1135"'87 at 61-0 ; so that on this day the rate of the needle was less than the mean by 1 s - 79, a difference which corresponds to an increase of '003 in the horizontal force. The observation of the 5th of July exhibits a difference somewhat greater on the other side. 6. The last source of error which requires to be noticed under this head is the change of the magnetic condition of the needles employed. Independently of the derangements of magnetic equilibrium induced by the presence of iron, or other disturbing causes, it is well known that most needles lose something of their original force. This loss is greatest at first; and the needle, if originally well tempered and then magnetized to saturation, is usually found to arrive at a nearly settled state in about a year. Most of the cylinders employed in the following observations seem to have reached that condition ; and the changes of mag- netic state which they have exhibited are, except in the case of Cyl. S (b), unimportant. In order to detect any such changes, and to correct for them if they arise, it is necessary to observe at the place chosen as the base of reference, at the termination of each series of observations, as well as at their commencement. If it is then found that the needle has lost any small portion of its force, or if the time of vibration has augmented, the amount of the correction due to each result may be found by assuming the change to have been regular, or proportional to the time elapsed. When the loss is very small, however (as was the case in the obser- vations which form the subject of this paper), the correction may be disregarded, provided we take as the time of vibration at the base of reference the mean of the times observed at the commence- ment and end of the series. The needles used in the present series are the cylinders L (a], L (ft), made by Dollond, and belonging to Mr. Lloyd ; cylinder S (b) belonging to Captain Sabine, and cylinders E (e) and B, (d) in the possession of Captain James Boss. All the circumstances of the observations are given in the annexed Table. The first, second, and third columns contain time place, day of the month, and hour of the observation. In the fourth column is set down the observed time of 100 vibrations, or the immediate result of obser- vation divided by 3. The fifth column contains the terminal an; MAGNETIC FORCE IN IRELAND. 177 the initial arc being in all cases 20. In the sixth column is given the chronometer's rate ; in the seventh the temperature ; and in the eighth the deduced or corrected time. The hour, set down in the third column as the hour of observation , is the mean of the com- mencement and end ; and the recorded temperature is also the mean of those observed at the beginning and end of the observation. It will appear from the preceding that the corrections employed in deducing the corrected from the observed times are those due to temperature and to the rate of the chronometer. In addition to the observations which follow there were others of an earlier date, made for the purpose of comparing the hori- zontal intensity at Dublin and Limerick, the two stations with which all the other places in Ireland have been immediately com- pared. In the observations alluded to, the rate of cylinder S (b] was observed in the Philosophy School, Trinity College ; and the local attraction of the building was determined by subsequent comparisons of the force there with that in the garden of Trinity College, the place which was afterwards selected for all the Dublin observations. These earlier comparisons, as well as some other imperfect ones obtained previously to the autumn of 1834 with two other cylinders, have not been included in the annexed Tables ; partly because the needles employed do not seem to be as trust- worthy as the rest, but chiefly because of the uncertainties of the double comparison which they involve. 178 OBSERVATIONS OF THE TERRESTRIAL TABLE I. Time of Vibration of Cylinder S (b)*. Place,! Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. m. 8. s. s. Limerick, . July 23, 1 44 400-23 *=w 0-0 66-0 399-82 >) >> 2 14 400-33 P 8- 67-0 399-85 25, 12 20 401-19 EL<3 62-5 401-02 12 50 400-77 CO 61-5 400-67 "Mean, 400-63 __ 64-2 400-34 London, . . Aug. 20, 11 22 390-14 3-0 -0-3 65-0 389-81 11 46 390-08 3-0 66-2 389-67 > 5> 12 44 390-34 2-5 66-0 389-94 , 21, 10 58 389-98 2-5 62-7 389-80 11 29 389-88 3-0 63-0 389-68 , 12 389-70 2-8 62-5 389-53 > 12 31 389-78 3-0 62-0 389-65 , 22, 10 47 390-18 2-5 63-0 389 '98 11 17 389-93 3-0 62-5 389-76 > 11 47 389-82 3-0 61-0 389-75 > >? 12 33 389-76 3-0 61-5 389-66 i 23, 10 57 389-76 2-5 60-5 389-73 > 11 37 389-78 3-0 60-5 389-75 > ,, 12 13 389-37 3-0 60-5 389-34 , ,, 12 45 389-32 3-0 61-0 389-25 27, 10 27 389-50 2-8 ' 57-0 389-70 , )? 11 2 389-33 2-5 57-5 389-50 , ,, 11 36 389-30 2-5 56.5 389-53 , 12 13 389-43 3-0 57-0 389-63 , 12 49 389-43 3-0 57-0 389-63 Limerick, Ballybunan, . Glengariff, . Killarney, LimericK, Tulla, . . . Templemore, Mean, Sept. 9, 16, 27, Oct. 4, 9, M 12, >, 17, 12 17 5 20 11 20 11 40 12 43 8 50 8 36 389-74 403-70 404-29 402-55 403-93 404-56 404-16 395-82 I Between 2 and 3. 0-0 61-2 61-0 62-0 66-0 65-5 53-5 52-0 51-jQ 389-66 403-63 404-15 402-14 403-56 405-01 404-70 396-43 * The observations in London were made by Captain James Ross ; all the others with this needle by Captain Sabine. t Limerick, garden at Somerville, one mile from town. London, Regent's Park. Ballybunan, field adjoining the innGlengarifl; Mr. E cles' 8 garden.-Killarney, Mueros8 demesne.-Tulla, Kiltanon, Mr. Molony's garden.- Templemore, Sir B. Gardens grounds; local attraction suspected. -Clonmel, Darling Hill, garden adjoin- ing the house. Fermoy, field near the river. MAGNETIC FORCE IN IRELAND. 179 TABLE I. (Continued.) Place. Date. Hour. Time. Arc. Rate. Temp. Corn Time. h. m. s. 8. g Clomnel, . . Oct. 19, ! 9 402-16 ^W 0-0 55-0 402 : 50 Limerick,. . ,, 29, 12 6 404-16 1 55-5 404-47 Fermoy, . . Dec. 2, 3 45 399-93 s-i 48-0 400-75 Limerick,. . ,, 10, ! 2 45 402-67 ?3 42-0 403-89 London, . . July 4,1 2 43 402-36 1-0 70-0 401-68 3 14 402-38 1-5 71-0 401-63 ,, 5, 9 54 401-84 1-0 62-0 401-70 10 21 401-73 1-5 64-0 401-46 | 11 13 401-02 2-0 60-0 401-02 55 55 11 46 401-50 1-5 62-0 401-36 55 55 4 48 401-74 1-5 64-0 401-47 6, 9 53 401-36 2-0 65-0 401-02 10 30 401-96 2-0 - 68-0 401-42 55 7, 9 32 400-96 2-0 60-0 400-96 5> 55 10 6 401-09 1-5 62-0 400-95 10 43 401-53 1-5 64-0 401-26 Mean. 401-62 64-3 401-33 Limerick, . . July 27,- 4 413-54 -12- 76-2 412-41 ,5 28, 11 58 412-83 66-8 412-35 Mean, 413-18 71-5 412-38 Time of Vibration of Cylinder L (a)* Place.t Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. m. s. s. 8. Limerick, . . Sept. 9, 1 23 243-26 0-0 60-0 243-26 Ballybunan, . 17, 10 35 243-87 63-0 243-69 Glengariif, .[ ' 21, 12 242-29 68-0 241-80 Killarney, . \ Oct. 4, 12 15 242-46 66-0 242-09 Limerick,. . 8, 1 14 243-16 4-0 f!5- 62-0 243-00 Dublin, . . ,, 10, 2 31 243-71 3-0 f 1- 55-2 244-00 99 99 2 50 243-81 3-5 55-0 244-11 H, 2 4 243-52 3-5 59-0 243-ijS Mean. 243-68 56-4 243-90 Armagh, , . : ,, 14^ 2 22 246-23 3-5 49-5 246-88 15. 2 16 246-30 4-0 ,-,0-7 246-87 Mean, 246-26 1 50-1 246-88 * The first five observations, and those of December 19, 21, 23, 1835, were made by Captain Sabine. All the others with this needle by Mr. Lloyd. t Dublin, Provost's garden, Trinity College. Armagh, grounds of the obser- eld near the barracks. Strabcme, an open field adjoining the 180 OBSERVATIONS OF THE TERRESTRIAL TABLE I. (Continued.} Place. Date. Hour. Time. Arc. Rate. Temp. Corr. Time. Cam, . . . Oct. 21, h. m. 2 57 247-49 2-5 s. + 2-0 50-2 a. 248-10 Strabane, . . 23, 1 22 248-00 2-5 51-8 248-51 Enniskillcn, . ,, 24, 3 45 247-09 3-5 38-5 248-42 Dublin, . . 25, 3 24 243-14 4-0 45-7 244-01 >, 28, 3 243-39 3-5 53-0 243-82 > 3 26 243-49 4-5 53-0 243-92 Mean, 243-34 . 50-6 243-92 Dublin, Aug. 19, 2 39 243-81 3-0 + 7-3 72-0 243-06 Markree, . . . 21, 2 35 249-30 3-5 69-2 248-71 Ballina, . . 22, 3 2 249-90 2-0 . . 71-5 249-16 Belmullet, . 24, 1 50 250-40 1-5 69-5 249-79 Achill Ferry, 25, 12 249-35 2-0 69-2 248-76 Leenan, . " . ,, 26, 3 58 248-19 2-0 . 59-8 248-18 Oughterard, . 27, 3 16 246-06 4-0 . 57-5 246-19 Ennis, . . . 28, 3 42 244-44 2-0 . 75-2 243-49 Limerick, . . 29, 2 52 243-45 3.-5 . 67-8 242-96 Cork, . . . ,, 31, 1 46 241-50 2-5 . 69-5 240-90 Waterford, . Sept. 1, 1 55 242-48 2-0 . 69-0 241-92 Broadway, . 2, 3 15 241-24 2-0 . 66-5 240-83 3 35 241-26 2-5 66-2 240-87 Mean, 241-25 66-4 240-85 Rathdrum, . Sept. 3, 4 8 243-47 3-5 . 63-0 243-27 Dublin, . . 12, 5 5 243-70 4-0 ^6-0 57-8 243-81 14, 2 29 243-92 4-0 60-2 243-89 15, 2 36 243-14 2-5 63-8 242-89 >> 3 2 243-73 3-0 63-5 243-50 Mean, 243-62 61-3 243-52 London, . . Sept. 19, 11 48 236-51 3-0 68-6 235-98 ,, ?) 2 44 236-65 2-0 69-4 236-07 i, 22, 12 43 236-63 3-5 . 72-2 235-89 Mean, ~~ 236-60 i 70-1 235-98 town. Enniskillen, field near town. Markree, demesne of the castle, part surrounded nib tall trees. Ballina, open field near town ; no shelter from sun. Belmullet, on the beach, at extremity of Broadhaven. Achill Ferry, on the beach, near ferry. Leenan, field at extremity of Killery harbour.-Oughterard, in a wood near the menEmus, open field near town. Limerick, garden at Somerville. Cork, demesne e banks of river, between Cork and Blackrock. Waterford, demesne adjoining Denver, side opposite town.-Broadway, open field.-Rathdrum, demesne of Avon- dale ; deep wood London, Westbourne Green, Harrow Road. MAGNETIC FORCE IN IRELAND. 181 TABLE I. (Continued.} Place. Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. in. 8. s. 8. London, . . Oct. 23, 1 18 235-09 3-0 + 7-8 52-4 235-52 24, 11 45 235-05 3-0 54-5 235-35 ? J ?J 2 23 235-12 3-5 55-0 235-39 Mean, 235-09 54-0 235-42 Dublin, . . Xov. o, 1 46 243-18 3-0 54-2 243-51 ,, 6, 12 21 243-35 3-0 48-0 244-06 5 j M 1 55 243-37 3-5 48-7 244-04 Mean, 243-30 50-3 243-87 Limerick, . Dec. 19, 11 40 243-03 + 4-0 45-5 243-90 21, 12 19 242-36 36-0 243-81 ii 23, 11 17 242-68 33-0 244-32 Mean, 242-69 38-2 244-01 Dublin, . . Dec. 29, 1 31 243-22 4-0 -f 5-5 47-6 243-96 1 52 243-18 4-0 47-0 243-95 Mean, 243-20 47-3 243-95 Dublin, . . Jan. 11, 11 21 242-67 3-0 33-8 244-24 >> >i 11 44 242-48 4-0 34-0 244-03 ii 12 > 10 12 242-86 2-0 33-8 244-43 Mean, 242-67 33-9 244-23 Time of Vibration of Cylinder L (b)* Place. t Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. m. 8. s. 8. Limerick, . Sept. 9, 2 14 292-55 _ 0-0 60-2 292-54 Ballybunan, . >i l"t 10 3 292-25 62-0 292-10 Glengariff, . K'ilhirney, . Limerick, ,, 27, Oct. 4, 8. 12 24 12 38 1 38 293*16 292-15 292-51 4-0 + 15- 68-0 68-0 (52-0 292-57 291-57 292-31 ' 2 o 292-92 4-0 82*0 292-72 "Mean, 292*72 152-u 292-52 Dublin, . . Oct. 11, 1 34 292-H2 1-5 + 1-0 62-5 292-63 2 50 292-73 2-5 292-84 Mean, 292-78 MM 292*74 C:.i-l in-ford,. Aniia-h, . . Oct. 13, ,: 14, i, 13, 1 45 1 M 1 45 294*60 29-VN9 29.V70 3-0 2-5 3-0 .VK-, 51-0 52-fi 294-69 296*66 2! I'-'-'' Mean, 296*80 51-8 296-40 * First six observations, and those of 21st and 23rd of D"ecember, 1835, were made by Captain Sabine : all the others with this needle by Mr. Lloyd. t Carlingford, open field east of town. Coleraine, demesne adjoining th spot near river surrounded by tall trees ; local attraction apparently due to basalt. 182 OBSERVATIONS OF THE TERRESTRIAL TABLE I. (Continued.} Place. Date. Hour. Time. Arc. Rate. Temp. Corfl Time. h. m. s. 8. s. Coleraine, . Oct. 18, 4 3 293-64 3-0 + 2-0 47-5 294-55 >i 20, 12 44 294-61 3-0 . 57-6 294-78 Mean, 294-12 52-5 294-66 Cam, . . . Oct. 21, 2 28 297-07 2-0 . 51-2 297-71 Strabane, 23, 12 57 297-60 2-5 51-5 298-22 Enniskillen, . 24, 3 14 296-50 3-0 42-0 297-83 Dublin, . . 25, 4 292-70 3-0 __ 46-8 293-66 i> 28, 2 36 293-14 3-5 54-0 293-57 Mean, 292-92 _ 50-4 293-62 Dublin, . . Aug. 19, 3 3 293-70 2-0 + 7-3 72-0 292-80 Markree, . . 21, 12 45 300-14 2-0 66-8 299-60 ) J 5 j 1 57 300-84 2-0 . . 72-0 299-91 Mean, 300-49 69-4 299-75 Ballina, . . Aug. 22, 2 29 301-75 ins. 75-7 300-53 Belmullet, . 24, 12 26 302-23 1-0 65*5 301-78 1 15 301-98 1-0 69-0 301-27 "Mean, 302-10 67-2 301-52 Achill Ferry, ! Aug. 25 j 1 35 300-61 1-5 68-0 299-98 Leenan, . 1 ,, 26, 3 32 298-65 1-5 59-5 298-66 Oughterarcl, 27, 2 53 296-34 3-0 58-2 296-44 Ennis, . . 28, 3 18 294-61 1-5 74-5 293-52 Limerick, . i, 29, 2 26 292-97 2-5 67-8 292-38 j Cork, . . . Waterford, . ii 31, Sept. 1, 1 11 1 29 289-58 292-13 l-o 1-5 68-2 68-2 288-97 291-51 Broadway, . Rathdrum, . 2, ,, 3, 2 46 3 42 290-21 292-95 1-5 3-0 67-5 63-5 289-64 292-67 Dublin, . . ii 12, 4 41 293-16 3-0 + 6-0 58-2 293-27 ii 14, 2 3 293-47 3-5 61-0 293-38 15, 2 9 293-62 2-5 64-5 293-27 Mean, 293-42 61'2 293-31 London, . . Sept. 19, 11 20 284-46 2-0 . 69-0 283-80 > 3 19 284-51 2-5 . 66-2 284-05 ,i 22, 12 17 285-03 3-0 71-8 284-17 London, . . Mean, Oct. 23, 12 53 284-67 282-65 2-5 + 7-8 69-0 52-2 284-01 283-17 24, 12 12 282-76 2-5 52-8 283-24 12 33 282-84 3-0 . 53-0 283-31 ,, ,, 1 57 282-34 3-0 54-2 282-72 Dublin, . . Mean, Nov. 5, 1 19 282-65 292-61 2-5 53-0 55-0 283-11 292-95 ,, 6, 11 55 292-59 2-0 . 48-8 293-38 "nf " 2 23 292-67 3-0 48-4 293-49 Mean, 292-62 50-7 293-27 MAGNETIC FORCE IN IRELAND. TABLE I. (Continued.} 183 Place. Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. m. 8. 8. 8. Limerick, . Dec. 21, 1 6 291-10 + 4-0 36-0 292-84 i. 1 44 291-14 36-0 292-88 ,, 23, 11 53 291-49 33-2 293-44 Mean, . 291-24 35-1 293-05 Dublin, . . Dec. 29, 12 44 292-42 2-5 + 5-5 49-7 293-15 99 9) 1 8 292-19 2-5 47-8 293-06 Mean, 292-30 48-8 293-10 Dublin, . . Jan. 11, 10 55 291-85 2-0 35-2 293-64 n 12, 10 37 291-81 2-0 32-5 293-80 Mean, 291-83 33-8 293-72 Time of Vibration of Cylinder E (c}* Place, t Date. Hour. Time. Arc. Rate. Temp. Corr. Time. London, . . July 8, h. m. 7 34 440-08 1-5 8. + 1-4 58-0 8. 440-22 10 9 440-57 1-5 60-0 440-56 12 20 441-03 2-0 64-0 440-72 4 17 441-33 2-0 66-0 440-ss n 9 ', 11 26 441-93 . 57-0 442-15 2 15 441-55 61-0 441-46 6 10 441-06 61-0 440-97 ' 10, 11 7 441-76 5 64-0 441-45 4 47 441-78 2 62-0 441-62 ! 14. 10 41 441-31 5 58-0 441-45 i 13, 6 51 440-97 5 64-0 441-42 11 11 443-26 66-0 442-81 ' " 2 31 443-40 5 70-0 442-64 8 24 440-86 5 55-0 441-J.i r, is, 7 39 441-19 1-0 52-0 441-78 11 35 442-68 1-0 64-0 HJ-:;T 8 2 441-53 1-0 57-0 441-75 18, 10 14 443-32 1-0 65-0 442-93 II 19 > 7 35 441-00 1-5 48-0 441-89 Mean, 441-61 60-1 441-59 Limerick, July 27, 3 57 453-31 1-5 + 1-2 r,s-.-, 452-65 28, 12 39 454-60 2-0 60-0 I.', I -.-,! 1 7 454-06 1-5 59-0 404*18 ,, ,, i, 1 40 454-18 2-0 59-0 4.'. I -J.i * Observed by Captain James Rosa. t London, Westbourne Green, Harrow Road. Limerick, garden at Somerrille. Dublin, Provost's Garden, Trinity College. Markree, demesne of Castle ; spot sur- rounded by lofty trees. 184 OBSERVATIONS OF THE TERRESTRIAL TABLE I. (Continued.} 1 i Prtw Place. Date. Hour. Time. Arc. Rate. Temp. L-ori. Time. Limerick, . July 28, h. m. 2 19 2 49 s. 453-84 453-70 1-8 2-2 s. + 1-2 60-0 59-0 453-83 453-77 " 3 22 453-82 2-0 60-0 453-81 !' 29 11 454-00 2-2 56-0 454-30 11 31 453-88 2-2 57-0 454-10 12 454-04 2-0 59-0 454-11 "Mean, 453-94 59-8 453-95 Dublin, . . Aug. 16, 7 21 7 51 453-90 453-69 2-0 2-0 56-0 56-0 454-20 453-99 " 8 21 453-77 2-0 57-0 453-99 "Mean, 453-79 _ 56-3 454-06 Markree, . , Aug. 19, 8 3 10 464-39 465-17 1-8 2-0 54-0 58-0 464-85 465-32 10 30 465-02 2-0 58-0 465-17 "Mean, 464-86 56-7 465-11 London, . . Aug. 30, 9 41 440-72 2-5 60-0 440-71 11 10 10 441-52 2-0 61-0 441-43 11 10 38 441-72 2-0 61-0 441-63 11 31, 9 33 440-86 2-0 55-0 441-23 11 9 441-41 2-0 58-0 441-55 99 99 12 37 442-04 2-0 58-0 442-18 Mean, 441-38 58-5 441-46 Time of Vibration of Cylinder E (d)* Place. Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. m. 8. 8. s. London, . . July 19, 9 44 437-75 1-0 + 1-4 58-0 437-89 99 99 10 30 438-10 1-0 59-0 438-17 11 44 438-39 1-2 62-0 438-23 99 9J 1 31 438-30 1-0 64-0 438-00 3 6 439-18 1-0 66-0 438-72 ," 20, 7 19 438-55 1-0 60-0 438-54 9 54 439-26 1-0 61-0 439-18 9) M 10 39 440-06 1-0 . 63-0 439-82 Mean, 438-70 61-6 438-57 Limerick, . July 29, 1 18 449-17 2-0 + 1-2 60-0 449-16 ,, ,, 1 50 450-05 2-0 61-0 449-96 2 41 449-16 2-0 62-0 449-00 Observed by Captain James Ross. MAGNETIC FORCE IN IRELAND. 185 TABLE I. (Continued.} Place. Date. Hour. Time. Arc. Rate. Temp. Corr. Time. h. m. 8. s. S. Limerick, July 30, 10 41 449-89 2-0 -f 1-2 55-0 450-27 >> 11 10 449-89 2-0 56-0 450-19 )> 11 40 449-80 1-5 56-0 450-10 > 12 14 449-77 1-5 57-0 449-99 31, 11 29 449-70 2-0 54-0 450-15 > 11 58 449-76 2-0 56-0 450-06 ,, ,, 12 28 449-91 2-0 57-0 450-13 5> 12 58 449-92 2-0 58-0 450-06 Mean, 449-73 - 57-5 449-92 Dublin, . . Aug. 14, 6 36 451-82 2-5 54-0 452-27 99 9> 7 5 451-62 3-0 . 54-0 452-07 ?) 99 7 35 451-61 3-0 55-0 451-99 Mean, 451-68 54-3 452-11 Markree, Aug. 19, 11 40 462-19 2-5 _ 59-0 462-26 12 10 462-00 2-5 59-0 462-07 20 8 37 462-54 2-5 54-0 463-00 Mean, 462-24 57-3 462-44 London, . . Aug. 28, 7 16 437-91 2-0 53-0 438-43 > 7 56 438-42 2-5 56-0 438-71 9 48 439-01 2-5 60-0 439-00 29, 9 31 439-43 2-0 60-0 439-42 99 99 10 13 439-52 2-0 60-0 439-51 99 99 10 42 439-60 2-5 63-0 439-36 Mean, 438-98 58-7 439-07 The computed results of the preceding observations are given in Table II. The first and second columns contain the place and the date of the observations ; the third the designation of the needle em- ployed ; in the fourth column is given the mean time of 100 vibra- tions, corrected for temperature and for the rate of the chronometer ; and the fifth and sixth columns contain the computed values of the horizontal intensity the numbers in the fifth column being the ratios of the horizontal intensity at the place of observation to that at the station of the observer, and those in the sixth being the ratios of the same force to that at London, to which place all the observations are ultimately referred. If T denote the reduced time of 100 vibrations at any place, and T' that at the station with which it is immediately compared, 186 OBSERVATIONS OF THE TERRESTRIAL and if h and h f be the horizontal intensities at the two places, the numbers of the fifth column are computed from the formula Again, if A, denote the horizontal intensity in London, the ratio - will be determined in the same manner ; and, multiplying by *' // it the numbers in the fifth column, we obtain the values of , or the ratios of the horizontal intensity at the places of observation to that at London, as given in the sixth and last column. The stations with which all the other places in Ireland are immediately compared are Dublin and Limerick ; and it will at once appear that, as the ratios of the horizontal force at these stations to that at London enter as factors in all the final results, much accuracy is required in their determination. For this pur- pose we have three distinct series of observations. In the first and second the intensities of the horizontal force in Dublin and Lime- rick are directly compared with that in London ; and in the third these intensities are compared together. The results of these com- parisons, given in Table II., are here put together, so as to be seen at one view. I. Horizontal intensity in Dublin, the horizontal intensity in London being unity. July, Aug., 1835, . . Cyl. R (c). . . Int. = -9456 ii ,, . . R (d). . . -9421 September, 1835, . . L (a). . . -9390 , L (6). . . 9376 Oct., Nov., 1835, . . L (a). . . '9319 >i >i ii . . - L (6). . . -9319 Mean = -9380 II. Horizontal intensity in Limerick, the horizontal intensity in London being unity. July, Aug., Sept., 1834, Cyl. S (i). . . Int. = -9396 July, 1835, .... _ -9470 July, August, 1835, . _ R ( c ). . . _ .946! >i ii - R (rf); . . -9513 Mean = -9460 MAGNETIC FORCE IN IRELAND. 187 III' Horizontal intensity in Limerick, the horizontal intensity in Dublin being unity. Sept., Oct., 1834, . . Cyl. L(a). . . Int. = 1-0064 . - L (5). . . 1-0044 July, Aug., 1835, . . - E (c). . . 1-0005 ?> . - R (d). . . 1-0098 Aug., Sept., 1835, . ' . - L (a). . . 1-0027 M . - L (5). . . 1-0047 Nov., Dec., 1835, . . - L (a). . . 1-0001 > . - L (6). . . 1-0021 Mean = 1-0038 If then x and y denote the horizontal intensities in Dublin and Limerick, that in London being unity, observation gives x = "9380, y = -9460, | = 1-0038, and it is required to determine the most probable values of x and y. To generalize this problem, let the mean results of observation be a, b, c, and let their weights be A, B, C respectively ; so that we have x - a - 0, weight = A, y - b = 0, J0, a and b being approximate values of x and y, let their true values be x = a -\- x, y = b + S y, and let - = c . Then y - - *-^j- = c + a~ l (S y - c' Sar), the squares a x a + Sx and higher powers of the quantities S x and 8 y being neglected ; so that the preceding equations may be written I x = 0, 8 y = 0, rr 1 (8y - e,S*0 + ^ - c = 0. These three equations are to be combined by the method of least squares. "We find in this manner A$x- Cc/r l (a~ l $y - e, i 401-33 441-59 1-0000 1-0000 Aug. 30, 31, . . July 19, 20, . . R( ') 441-46 438-57 1-0000 Limerick, . Aug. 28, 29, . . July 27, 28, . . 27, 29, . . I 7 \ 439-07 112-.-JS 408*90 l-(MM).-, 9470 9461 Dublin, . . \ 29-31, . Aug. 10, . . . 14, ... R R R i] :,' 449-92 454 -(Mi 452-11 1-0098 1-0000 1-0000 '.I.'; l:> 9456 !)1-J1 190 OBSERVATIONS OF THE TERRESTRIAL TABLE II. (Continued.} Place. Date. Cyl. Time. Intensity. Intensity. Lond. = 1. s. Markree, . Aug. 19, 1835, R (c) 465-11 9531 9012 19,20, . . R (d) 462-44 9558 9005 Dublin, . . Aug. 19, . . . L(c) 243-06 1-0000 9395 Sept. 12-15, . . 243-52 Aug. 19, ... L (&) 292-80 1-0000 9395 Sept. 12-15, . . 293-31 Markree, Aug. 21, . . . L "} 248-71 9569 8986 5) ) j L i,\ 299-75 9559 Ballina, . . 22, ... L /) 249-16 9534 8946 >? 3 J L V 300-53 9509 Belmullet, . ,, 24, ... L i 249-79 9486 8894 L '>) 301-52 9447 Acbill Ferry, " 25, ... L 248-76 9565 8977 Leenan, . . 26, ... L b) L (a) 299-98 248-18 9544 9610 9038 L(ft 298-66 9629 Oughterard, 27, ... L( / 246-19 9766 9179 L ( 296-44 9773 Ennis, . . ,',' 28, ... L / 243-49 9984 9373 L 293-52 9969 Limerick, . ,',' 29, ... L / 242-96 1-0027 9430 L 'i 292-38 1-0047 Cork, . . . " 31, . . . L 240-90 1-0199 9582 Waterford, . Se'pt. 'l, . . -. L L ( 288-97* 241-92 1-0285* 1-0114 9498 Broadway, . " 2, . . . L L / 291-51 240-85 1-0107 1-0204 9602 Rathdrum, . ;; 3, ... L L i i 289-64 243-27 1-0238 1-0002 9409 Dublin, . . 11 12-15, L L > /' 292-67 243-52 1-0027 9390 London, . . L( 19-22, . . L > i 293-31 235-98 9376 1-0000 London, . . L Oct. 23, 24, .. L > i 284-01 235-42 1-0000 1-0000 Dublin, . . Nov. 5,' 6, . . M LJ ( > i 283-11 243-87 1-0000 9319 Dublin, . . Nov., Dec., Jan., L L < > t 293-27 244-02 1-0000 9319 Limerick, . Dec. 10,'ai, 23,' L i L < > i 293-36 244-01 1-0000 1-0001 " " L I ) 293-05 1-0021 e number in the last column. ; the result has been therefore MAGNETIC FORCE IN IRELAND. 191 II. Dip and Intensity. All the observations with dipping needles are comprised in the two Tables which follow. The first (Tab. III.) contains the results obtained with needles of the ordinary construction, and used ex- clusively for the determination of the dip. In the first, second, and third columns are given the place , day of the month, and hour of observation. The fourth column contains the observed inclination (the mean of the usual 8 readings) when the marked end of the needle is a north pole ; the fifth contains the similar result of ob- servation with the poles reversed ; and the sixth is the mean of these angles, or the resulting dip. The needles employed are Needle L (1) constructed by Robinson, and Needle S (1) made by Dollond ; the latter of these is 11 J inches in length, the former 4^ inches. Table IV. contains the observations made for the purpose of determining the dip and intensity at the same time ; the latter ele- ment being deduced from the direction in which the needle rests under the combined influence of magnetism and gravity, while the former is inferred from the position assumed under the influence of the earth's magnetism alone. Each of these angles of direction is deduced from the usual eight readings, all the reversals being made just as in the ordinary mode of observing the dip, the re- versal of the poles of the needle excepted. These angles are given in the fifth and sixth columns of the table ; is the angle which the needle makes with the horizon when unloaded, and the in- clination when a small weight is attached to the southern arm at a fixed distance from the centre. The temperature is noted at the commencement and end of each observation, with the view of correcting the value of the force ; and the mean temperature is set down in the fourth column of the table. The needles employed in these observations are of the same dimensions as those used for the determinaiion of the dip alone, and are adapted to the same divided circles. Three small holes are drilled close to each other on each arm, at a distance from the centre about two thirds of its length ; and much care has been bestowed to make them coincide accurately with the axis of form of the needle. The weight is a small cylinder of brass, which is inserted in one of the holes on the southern arm, the diameter of the cylinder corresponding accurately to that of the hole. This weight is so adjusted as to bring the needle into a position neajrly at right angles to the line of the dip, that being the position in which the resulting value of the force will be leasf affected by the friction of the axle on its supports. 192 OBSERVATIONS OF THE TERRESTRIAL TABLE III. Observations of Dip. Needle L (1).* Place. Date. Hour. N. Pole. S. Pole. Dip. h. m. Limerick, . July, 1834, 70 59-4 71 l'-8 71 0-6 ?j j? 70 58-0 70 52-6 70 55-3 71 3-6 70 59-8 71 1-7 ' j> 71 1-6 71 3-2 1 71 2-4 70 56-0 70 58-4 70 57-2 Mean, '.' . . 70 59-7 70 59-2 70 59-5 Dublin, . . Aug. 7, . . 70 48-2 70 55-0 70 51-6 8, . . 70 55-9 70 59-4 70 57-6 9, . . 70 55-5 70 53-1 70 54-3 19, . 70 47-7 70 51-3 70 49-5 Sept. 22, . . 70 47-4 71 4-5 70 56-0 23, . . 70 48-5 70 59-0 70 53-8 Mean, . . . 70 50-5 70 57-1 70 53-8 Carlingford, Oct. 13, . . 12'20 71 2-0 71 30-6 71 16-3 Armagh, 14, . . 12 30 71 27-2 71 33-1 71 30-2 .* 15, . . 12 10 71 30-6 71 34-8 71 32-7 Mean, . . , 71 28-9 71 34-0 71 31-5 Coleraine,t . Oct. 20, . . 11 25 71 11-8 71 19-4 71 15-6 Cam, . . . 21, . 12 45 71 48-2 71 47-5 71 47-8 Strabane, . 23, . . 11 40 71 47-2 71 56-0 71 51-6 Enniskillen, 24, . . 2 28 71 48-5 71 47-5 71 48-0 Fermoy, . . Dec. 2, . . 70 28-1 70 44-5 70 36-3 Markree, Aug. 21, 1835, 12 71 54-6 71 51-2 71 52-9 " M 4 45 71 55-6 71 53-1 71 54-4 Mean, . . . 71 55-1 71 52-2 71 53-6 Ballina, . . Belmullet, . Aug. 22, . . 24, . . 4'l5 10 45 71 58-8 71 59-7 72 5-0 72 5-7 72 1-9 72 2-7 Achill Ferry, 25, . . 10 71 52-2 71 56-6 71 54-4 Galway, . * . 23, . . 7 50 71 20-4 71 23-4 71 21-9 Ennis," . . >> 4 20 70 55-6 71 7-4 71 1-5 Limerick, . it 29, . . 1 10 70 54-0 70 49-8 70 51-9 Cork, . . . Waterford, . Broadway, . Gorey,. . . Rathdrum, . Dublin, , t 31, . . Sept. 1, . . 2, . . 3, . . M 4 12 12 8 1 30 8 5 2 30 1 30 70 26-5 70 40-5 70 14-8 70 45-0 70 39-9 *TA AAA 70 32-1 70 34-7 70 24-0 70 41-8 70 42-3 *7/\ /iQ.fi 70 29-3 70 37-6 70 19-4 70 43-4 70 41-1 Tfk AG..^ ,, 5, . 1 50 |U ''4: 70 53-0 /(J 49'U 70 58-2 i(J 4o / 70 55-6 ( 7, . . 3 35 70 56-0 70 52-5 70 54-2 9, . . 2 45 70 55-7 70 53-1 70 54-4 14, . . 12 35 70 55-9 70 57-5 70 56-7 ,, 15, . . 11 55 70 55-8 70 50-8 70 53-3 Mean, . . . 70 53-5 70 53-5 70 53-5 * The observations in Limerick (July, 1834) and that in Fermoy (Dec. 2), were made by Captain Sabine : all the other observations with this needle by Mr. Lloyd, t Evident local disturbance at this place : Rock, basalt. MAGNETIC FORCE IN IRELAND. 193 TABLE III. (Continued] . Observations of Dip Needle S (1).* Place. Date. Hour. N. Pole. S. Pole. Dip. Limerick, . Aug. 1, 1834, 71 38-8 70 24-3 71 1-6 16t, - 70 48-5 71 22-3 71 5-4 Mean, 71 13-6 70 53-3 71 3-5 Glengariff, . Sept. 27, . 70 52-3 71 9-0 71 0-6 28, . 70 50-4 71 14-2 71 2-3 Mean, 70 51-4 71 11-6 71 1-5 Killarney, . Oct. 4, . 71 5-6 71 3-4 71 4-5 Tulla, . . ,., 12, . 71 16-2 71 15-4 71 15-8 TABLE IV. Observations of Dip and Intensity. Needle L (4) J. Place $. Date. Hour. Temp. (CO (0 Limerick, . June, 21, 1834 h. m. el-o - 6 55-0 July, 22, . 65-0 _ 7 10-6 28, . 65-0 _ 6 21-3 Mean 64-7 _ 6 49-0 London, . . Aug. 28, . 2'35 70-0 69 8'-0 _ 11 53-2 29, . 12 42 68-5 69 8-5 - 12 7-4 1 18 67-7 69 5-6 _ 12 26-1 " " Mean 68-7 69 7-4 _ 12 8-9 Dublin, . . Sept. 22, . 23, . 2 is 2 45 61-0 62-5 71 2-2 70 53 -8 _ 8 5-6 - 7 37-0 26, . 2 40 66-2 _ 7 59-6 29, . 2 40 62-5 70 44 -8 _ 8 9-5 Mean, 63-0 70 53 -6 - 7 57-9 Carlingford, Oct. 13, . . 12 52 61-2 71 20-6 _ 5 29-4 All the observations with Needle S (1) were made by Captain Sabine. t The needle was rubbed on a hone in the interval between the observations (Aug. 1 and 16) ; the marked end most. JThe observations in Limerick (June, July 1834) were made by Captain Subine: all the other observations with this needle by Mr. Lloyd. London, Sir James South's observatory, Kensington. Limerick, garden :tt Somervillo. Ballybunan, in the field in front of Captain Raymond's Lodge. Valentin. O 194 OBSERVATIONS OF THE TERRESTRIAL TABLE TV .(Continued.} Observations of Dip and Intensity. Needle Hi (4). Place. Date. Hour. Temp. e. Armagh, Oct. 14, . . h. m. 1 5 48-8 71 19-4 - 7 7-4 15, . . 12 38 52-0 71 33-2 - 7 15-2 Mean, 50-4 71 26-3 - 7 11-3 Coleraine, 20, . . 12" 2 56-3 71 12-2 - 8 19-2 Cam, . 21, . . 1 28 49-5 71 49-6 - 5 4-8 Strabane, 23, . . 12 18 48-8 71 39-4 - 5 59-1 Dublin, . 25, . . 3 15 47-0 70 54-1 - 8 53-9 Dublin, . Aug. 19, 1835, 1 13 71-5 70 51-6 - 13 4-9 Markree, 21, . . 3 50 67-0 71 55-6 - 11 2-6 Ballina, . 22, . . 3 50 66-5 71 51-8 - 11 15-2 Belmullet, 24, . . 11 15 65-5 71 57-5 - 10 56-2 Achill Ferry 25, . . 10 37 62-0 71 53-2 - 10 49-2 Galway, . 28, . . 8 18 59-0 71 17-4 - 11 9-1 Ennis jj ' ' 4 45 67-2 70 59-1 - 12 1-6 Limerick, 29, . . 1 40 69-2 70 47-5 - 12 32-0 Cork, . . 31, . . 12 30 68-5 70 33-2 - 13 18-5 Waterford, Sept. 1, . . 12 35 66-2 70 38-8 - 13 38-6 Broadway, 2, . . 2 66-8 70 31-6 - 13 33-5 Gorey, . 3, . . 8 30 60-0 70 43-1 - 14 1-1 Rathdrum, 3 64-7 70 40-8 - 13 58-0 Dublin, . Sept. 4, '. '. 1 56 71-8 70 43-6 - 12 29-4 5, . . 2 19 65-5 70 52-8 - 13 15-4 7, . . 4 8 70-0 70 52-2 - 13 20-8 9, 3 5 70 46-2 14, . . 1 5 70 53-4 15, . . 12 25 62-0 70 55-0 - 13 10-5 London, Mean, Sept. 19, .. 1 67-3 70 50-5 69 5-8 - 13 4-0 1 10 68-0 69 7-4 - 16 54-2 22, . . 11 32 70-8 69 12-4 - 17 16-3 1 45 70-0 69 13-6 - 16 49-4 London, . . Mean, Oct. 23, . . 11 33 69-6 50-5 69 9-8 69 10-6 - 17 0-0 - 16 32-0 2 25 51-6 69 2-2 - 16 35-4 24, . . 1 16 53-8 69 6-0 - 16 44-4 Dublin, . . Mean, Nov. 5, . . 12 22 52-0 56-2 69 6-3 70 49-6 - 16 37-3 - 12 54-6 2 32 52-8 70 45-8 - 12 54-5 " 6 \,' ' 1 15 49-0 70 53-9 - 12 36-6 Mean, 52-7 70 49-8 - 12 48-6 em the sea beach at the Foot."-Dingl e , on the sea beach at Lord Ventry's.-Tulla, Kiltanon, Mr. Molony's demesne.- Youghal, in the garden of the "Devonshire Anns inn. N MAGNETIC FORCE IN IRELAND. 195 TABLE IV . (Continued.) Observations of Dip and Intensity. Needle S (2) *. Place. Date. Hour. Temp. < e. h. m. Limerick,* . JuJy, 1835 63-0 71 16-9 - 15 9-0 Ballybunan, Nov. 8, 52-0 71 29-1 - 13 56-3 Valentia, . 12, 47-0 71 15-0 - 14 37-3 Dingle,. . 18, 43-0 71 17-7 - 13 45-8 Tulla, . Dec. 10, 47-0 71 36-5 - 14 46-0 Limerick, 26, 27, 45-0 71 14-6 - 15 29-6 Youghal, 29, . 47-0 70 49-0 - 16 0-5 Limerick, Jan. 4, 1836, 52-5 - 15 23-7 When the observations of dip made at the same station with different needles are compared together, it will be found that they are by no means in accordance. Thus the dip at Limerick in November, 1833, deduced from four observations with a needle on Meyer's principle, was 71 ll'-7, while the mean of five observations with needle L (1) at the same place and in the following year was only 70 59'-5, differing from the former by 12'. When from this difference the amount of the annual change is deducted, the re- mainder appears to be greater than can be fairly ascribed to the errors of observation. But these discrepancies in the results given by different needles have been placed in the strongest light by the recent observations of Captain James Ross in London. In these observations, which were undertaken with the view of determining the amount of the annual decrease of dip at London, eight different needles were employed, and from eight to ten observations were made with each, the result of each separate observation being a mean of eighty readings. The results were as follow : Needle B (1), Admiralty, ... (10 obs.) Needle L (1), (9 obs.) Needle S (1), (8 obs.) dip 69 r-5 69 6-3 69 11-3 * The observations in Limerick, July, 1835, made by Captains Sabine and Ross ; ill the remaining observations with this needle by Captain Sabine. o2 196 OBSERVATIONS OF THE TERRESTRIAL Needle J, (10 obs.) . . .dip. = 6916'-1 Needle E, (10 obs.) ... 69 18 -9 Meyer's needle, (8 obs.) ... 59 19 -6 Needle B (2), Admiralty, ... (10 obs.) ... 69 21 -8 Needle P, (8 obs.) ... 69 42 '6 Mean dip = 69 17 '3 Thus it appears that there is a difference amounting to 41' in the results of two of the needles used ; and that this difference is very far beyond the limits of the errors of observation will appear from the fact that the extreme difference in the partial results with one of these needles B (1) does not amount to four minutes and a half, while with the other (P) the extreme difference is only two minutes. In fact, it so happens that these very needles which differ most widely in their mean results, are those in which the accordance of the partial results is most complete. Of the eight results obtained with needle (P), there is one only which differs from the mean of the eight by a single minute ; and yet the mean of all the observations with this needle differs by more than 20' from the mean of any of the others, while its excess above the mean of the entire series amounts to 25'. These differences cannot be ascribed to any partial magnetism in the apparatus ; for three of the needles (J, P, and E) were of the same dimensions, and were used with the same circle, and yet their results, as we see, are widely discordant. "We must seek, then, in the needles themselves the cause of these perplexing discrepancies, and we are forced to conclude that there may exist, even in the best needles, some source of constant error which remains uncor- rected by the various reversals usually made; and that accordingly no repetition of observations with a needle so circumstanced can furnish even an approximation to the absolute dip. If this error be due to the incomplete adjustment of the needle (such as devi- ation of centre of gravity from the axle, &c.), its magnitude will be a function of the dip, and of the force, which may be assumed to be constant where the variations of these elements are not consider- able. Hence, to determine its amount for any particular needle, it is necessary to make a careful series of observations with it at some station for which the dip has been accurately determined (from the mean of several needles) ; and the difference will be MAGNETIC FORCE IN IRELAND. . 197 a constant correction, to be applied to all future results within certain limits. It fortunately happens that the two ordinary needles used in the present series of observations in Ireland were among those employed by Captain James Eoss in London ; so that their cor- rections may be considered to be accurately known. The mean difference of the values of the dip as given by needles L (1) and L (4) having been well determined by observations elsewhere, the results obtained with the latter needle in London may be grouped with those of the former. Thus, the mean of seven observations made with needle L (4), September and October, 1835, when reduced to needle L (1), is 69 9'8. If then we combine this with the direct result of the nine observations with needle L (1), viz. 69 6'3 (allowing double weight to each of the latter observa- tions on account of the double number of readings), we find 69 7''3 as the mean value of the dip deduced from sixteen ob- servations with the two needles, and reduced to needle L (1) as the standard. Comparing this with the mean result of the eight needles, the correction of needle L (1) is found to be + 10'*0. For the other needles employed in Ireland, we have Needle S (1), . . . . correction = + 6'-0. Meyer's needle, ........ 2''3. With respect to Meyer's needle, however, it is to be observed that as the angles from which the dip is deduced differ in general very widely, and as these angles are usually varied in different obser- vations with the same needle, there is a presumption, at least, that every constant error will be removed by repetition, and that the differences of the separate results from the absolute dip will be equal on the positive and on the negative side. This seems to be confirmed by the amount of the final difference in the present in- stance, which does not appear to be larger than may be fairly ascribed to the errors of observation. It seems bettor therefore to regard this needle as subject to no constant error. The degree of confidence to which these determinations are entitled may now be estimated, by applying the corrections so obtained to the observations made with these needles at Limerick in 1833 and 1834, the only other station at which they have been all employed. The observations in the former year are reduced to the latter, assuming the annual decrease of dip in Ireland to 198 OBSERVATIONS OF THE TERRESTRIAL be 3'. The very close agreement of the results must of course be regarded as in a great measure accidental. Observed Dip. Corrected Dip. Needle L (1), July, 1834, .. 7059'-5 . . . 719'-5 Needle S (1), Aug., 1834, . . 71 3-5 ... 71 9 -5 Meyer's needle, Nov., 1833, . 7111-7 ...719-7 "We have hitherto spoken only of the needles whose poles are changed in each observation, and which are used exclusively for the determination of the dip. The necessity of a correction in the results obtained with the other needles, whose poles are unaltered, is obvious. By reason of the deviation of the centre of gravity of the needle from the axle, the weight of the needle itself has in all cases a certain moment acting with or against the directive force. Let , as before, be the inclination of the needle to the horizon when unloaded, and the corresponding angle when the weight is attached, and let p denote the ratio of the moment of the needle itself to that of the added weight ; then the dip, S, will be given by the equations* (1) in which t is the correction sought. The constant coefficient p in the expression for this correction will be known when the corresponding values of the angles , , and are known at some one station. Its value, in the case of Needle IV., has been thus found to be 'O0205.t It will easily appear, from the second of the preceding formulas, that when the coefficient is so small as that just assigned, the variations in the values of e, resulting from moderate changes in the angles on which it depends, will be inconsiderable. In the observations in Ireland, for example, the entire change in the amount of the correction is a small fraction of a minute. In this and other similar cases, therefore, the correction may be regarded as constant; and its value maybe inferred from any series of simul- taneous observations made with the needle to be corrected, and * Transactions of the Royal Irish Academy, Vol. XVIL, p. 450. . 451. MAGNETIC FORCE IN IRELAND. 199 with some other whose correction is already known. In this manner it has been found that the mean difference of the results of Needles L (1) and L (4) is & - 4 = + l'-5 ; being somewhat smaller than that assigned above. But if 8 denote the absolute dip, we have already found that 8 - Si = + 10'- ; and adding these differences, the correction of Needle L (4) is 8 - S 4 = + ll'-S. The correction of Needle S (2) is inferred from the observations made with that needle in Limerick, as given in Table IV. Limerick, July, 1835, . . dip = 71 16'-9 Dec., . . 71 14-6 Mean, . . 71 15 -8. This mean corresponds, in time, to the middle of October, 1835. But the true dip in Limerick (July, August, 1834) was found to be 71 9'-5 ; and when reduced to October, 1835 (assuming the annual decrease to be 3'), it is 71 6'*0. The correction of the needle is therefore - 9' '8. The corrections of the needles being determined, we may now proceed to deduce the values of the absolute dip at the several places at which observations have been made. These values are given in the following table (Table V.). In the first and second columns are written the place and date of the observation. The third and fourth columns contain the corrected values of the dip, deduced from the results of Tables III. and IV. by the application of the corrections now explained ; and the last column contains the mean dip inferred from the two preceding. In taking this mean, double weight has been assigned to the results obtained with needles of the ordinary construction, the number of readings with these needles being double of that made with the needles whose poles are unaltered. 200 OBSEKVATIONS OF THE TERRESTRIAL TABLE V. Dip. Final Results. Place. Date. Dip (1). Dip (2). Mean. London, . . . August, 1834, . 69 18-9 69 18-9 Dublin, . . . Aug., Sept. 71 3-8 71 5-1 71 4-1 Limerick, . July, Aug. . 71 9-5 . 71 9-5 Glengariff, . Sept., 27, 28, 71 7-5 . 71 7-5 Killarney, . Oct., 4, . . 71 10-5 . . 71 10-5 TuUa, . . 12, . . 71 21-8 . 71 21-8 Carlingford, Armagh, 13, . . 14, 15, . 71 26-3 71 41-5 71 32-1 71 37-8 71 28-2 71 40-3 Coleraine, . 20, . . 71 25-6 71 23-7 71 25-0 Cam, . . 21, . . 71 57-8 72 1-1 71 58-9 Strabane, . 23, . . 72 1-6 71 50-9 71 58-0 Enniskillen, 24^ , . 71 58-0 71 58-0 Fermoy, London, . . Dec., 2, . . Sept., Oct., 1835, 70 46-3 69 16-3 69 19-8 70 46-3 69 17-3 Dublin, . . Sept., 4-15, . 71 3-5 71 2-0 71 3-0 Markree, . Aug., 21,. . 72 3-6 72 7-1 72 4-8 Ballina, 22, . . 72 11-9 72 3-3 72 9-0 BelmuUet, . 24, . . 72 12-7 72 9-0 72 11-5 Achill Ferry, 25, . . 72 4-4 72 4-7 72 4-5 Galway, . . 28, . . 71 31-9 71 28-9 71 30-9 Ennis, . . 71 11-5 71 10-6 71 11-2 Limerick, . >, 29, . . 71 1-9 70 59-0 71 0-9 Cork, . . . 31, . . 70 39-3 70 44-7 70 41-1 Waterford, . Sept., 1, . . 70 47-6 70 50-3 70 48-5 Broadway, . 2, . . 70 29-4 70 43-1 70 34-0 Gorey, . . 3, . . 70 53-4 70 54-6 70 53-8 Rathdrum, . 70 51-1 70 52-3 70 51-5 Dublin, . . Nov., 5, 6, . 71 1-3 71 1-3 Limerick, . July, Dec., . . 71 6-0 71 6'0 Ballybunan, Nov., 8, ... 71 19-3 71 19-3 Valentia, . 12, ... 71 5-2 71 5-2 18, 71 7*9 71 7'9 Tulla, . . Youghal, . Dec., 10, . . . 29, ... . . . 71 26-7 70 39-2 71 26-7 70 39-2 The dip being known, the intensity will be given by the formula sin (S - 0) = cos 0, (2) in which /3 is constant, and $ the measure of the force exerted by the earth on the needle. This force, however, varies with the temperature to which the needle is exposed ; and it is necessary to determine the amount of this variation before we can know the relative values of the terrestrial magnetic force at different stations. MAGNETIC FORCE IN IRELAND. 201 Let t, then, be the observed, and t' the standard temperature, and let #' be the value of corresponding to the latter ; then ^ ' = a ({>' (t t'), (3) in which a is a constant to be determined by observation. For Needle L (4) it has been found thaf a = -00016. But we may proceed in another way, which will perhaps be found convenient in practice. We may correct the observed value of 9 by subtracting the change due to temperature ; or, in other words, we may reduce the value of 9 to that corresponding to the standard temperature, and to the standard condition of the needle. For this purpose it is only necessary to find the relation between the corresponding changes of and 9. Differentiating, therefore, the equation (2) with respect to these variables, and dividing the result by the equation itself, we find d cos 8 sin 1' d 9 ~ cos 9 sin (8 - 9)' d 9 being expressed in minutes. Now it is easy to see that the variations of the second member of this equation, arising from changes in the angles 8 and 9 on which it depends, will be in- considerable for the limited extent of those changes in Ireland. Assuming it to be constant, therefore, its value will be given when we know the corresponding values of 8 and 9 at some one station. Thus, at Dublin, September, 1835, it was found that 3 = 71 3'-0, 9 = - 13 O'-O ; from which we find the value of this constant to be -00010. But, since d = ^ - q>' = - a $' (t - t'}, the first member of the equa- tion is -~ ---00016^; (/ (/ U 17 so that the correction is finally & - 9 = + 1-6 (t - *. * It is obvious that the coefficient in this correction might hare been determined directly, by observing the angles and 9' corresponding to very unequal tenipciuturr.- It did not seem safe, however, to subject the apparatus to the action of high artificial heat, and the thermo-electric currents induced by inequality of temperature would in all probability havo sensibly affected the results. 202 OBSERVATIONS OF THE TERRESTRIAL Now if 0x, 8,, and 0j be the values of $, 8, and at the station with which the rest are compared, we have sin (8 - 0) = |3 cos 0, 0o sin (8 - ) = /3 cos ; cos sin (8 - ) m and dividing - = C08 Oo gin ( g _ 0) > which expresses the ratio of the force at the two stations in a form suited to logarithmic calculation. The following table gives the results of this computation. It contains the place and date of observation ; the angle corrected for temperature ; and the total intensity at each station, compared in the first instance with Dublin or Limerick, and in the second with London. For the intensity of the magnetic force at Dublin we have the three following determinations : * * *' Aug., Sept., 1834, . . Intensity = 1-0194 Sept., 1835, .... 1-0213 Oct., Nov., 1835, . . 1-0211 Mean, . . 1-0206. The intensity at Limerick, compared with London, is observed to be 1-0262 ; and the intensity at the same place, compared with Dublin, is 1-0030. Accordingly, for the determination of the values of the total force at Dublin and Limerick, observation furnishes us with three results, in the two former of which the intensities at these two stations are directly compared with that at London, while in the third they are compared together. To s infer from these data, therefore, the most probable values of the force at the two stations, we must proceed precisely as in the ana- logous problem respecting the horizontal intensities, and we have only to substitute in the formulas already given,* for a, b, c, &c., their particular values. We have then a = 1-0206, b = 1-0262, e = 1'0030, f, = - = 1-0055, c - c = -0025. (t And since the comparison of Dublin and London is the mean of three distinct comparisons, while each of the other two results is * Page 188. MAGNETIC FORCE IN IRELAND. 203 inferred from one only, the weights may be assumed as follow : A = 3, 5 = 1, (7=1. Substituting these values, therefore, in the formulas alluded to, we find x = a + B x = 1-0210, y = b + $y = 1-0251. The numbers in the fifth column of the table are deduced from those in the fourth, by multiplying by one or other of these numbers, according as the force at the station in question has been compared in the first instance with that at Dublin, or with that at Limerick. TABLE VI. Intensity. Needle L (4). Place. Date. Angle. Intensity (1). Intensity (2). London, . . Aug., 1834, . . - 11 55-0 1-0000 Dublin, . . Sept., .... - 7 53-1 . 1-0194 Limerick, . June, July, . . - 6 41-5 . 1-0262 Dublin, . . Sept., Oct., . . - 8 33-9 1-0000 1-0210 Carlingford,* Armagh, Oct., 13, . . 14, 15 . . - 5 27-5 - 7 26-7 1-0166 1-0044 1-0379 1-0255 Coleraine, . 20, . . - 8 25-1 9997 1-0207 Cam, . . 21, . . - 5 21-6 1-0151 1-0364 Strabane, . 23, . . - 6 17-0 1-0100 1-0312 Dublin, . . Aug., Sept., 1835, - 12 49-4 1-0000 1-0210 Markree, . . Aug., 21, . . - 10 51-4 1-0091 1-0303 Ballina, . . 22, . . - 11 4-8 1-0077 1-0289 Belmullet, . 24, . . - 10 47-4 1-0093 1-0305 Achill Ferry, Galway, . . 25, . . 28, . . - 10 46-0 - 11 10-7 1-0096 1-0086 1-0308 1-0298 Ennis, . . - 11 50-1 1-0055 1-0266 Limerick, . 29, . . - 12 17-3 1-0030 1-0241 Cork, . . 31, . . - 13 4-9 9992 1-0202 Waterford, . Sept., 1, . . - 13 28-7 9966 1-0175 Broadway, . Gorey, . 2, . . 3, . . - 13 22-6 - 14 1-1 9976 9933 1-0185 1-0142 Rathdrum, . - 13 50-5 9944 1-0153 London, . . Sept., 19-22, . . - 16 44-6 . . . 1-0000 Dublin, . . 4-15, . . - 12 52-3 . . . 1-0213 London, . . Oct., 23, 24, . . - \9 50-1 . . . 1-0000 Dublin, . . Nov., 5, 6, . . - 13 0* ' ' 1-0211 Evident local distm-bance at these two stations. The district about Carlingford is intersected with trap dykes; Coleraino lies within the basaltic field of the North of Ireland. 204 OBSERVATIONS OP THE TERRESTRIAL TABLE VI. {Continued.} Intensity. Needle S (2). Place. Date. Angle. Intensity (1). Intensity (2). Limerick, . Ballybunan, Valentia, July, Dec., 1835, Nov. 8, ii 12 - 15 19-3 - 13 56-3 - 14 37-3 1-0000 1-0084 1-0047 1-0251 1-0337 1-0299 Dingle, . i, 18, - 13 45-8 1-0097 1-0350 Tulla, . Dec. 10, - 14 46-0 1-0035 1-0287 Limerick, Dec., Jan., - 15 26-6 1-0000 1-0251 Youghal, Dec. 29, - 16 0-5 9971 1-0221 HL.~Itodytutmic and Isoclinal Lines. On a review of the preceding results of observation, it will be seen that they exhibit much irregularity. The errors of observa- tion in which we are to include the effects of the unsteadiness of the magnetic state of the needles employed, as well as the various other uncertainties arising from the imperfections of our methods of observing have, of course, their share in these discrepancies ; but they are by no means sufficient to explain the whole. The action of the earth on the magnetic needle is itself subject to irre- gularities, temporary as well as local ; and it is to these that the observed anomalies must, in great part, be ascribed. To the variations of the former kind we have already referred. The direction and intensity of the terrestrial magnetic force, at a given place, are subject to fluctuations, or irregular oscillations round their mean state, the cause of which is as yet little under- stood ; and it is only by means of simultaneous observations, made at some fixed station within the limits of the district through which these effects take place, that we can hope to ascertain their amount, and to correct for them. Of the local disturbing causes some are sufficiently obvious. Thus the needle is in general affected by the vicinity of basaltic rocks, owing to the quantity of iron they contain ; and instances have been observed in which these rocks were even found to possess magnetic polarity.* But there seem to be grounds for believing * A remarkable case of this kind has been noticed at Fair Head, on the north coast of Ireland. The magnetic polarity of one of the columns which compose this wonder- ful facade is said to be so strong as to invert the position of the compass ncedje, when the poles of the same name are made to approach. MAGNETIC FORCE IN IRELAND. 205 that disturbing actions of a local nature are exerted on a much larger scale. Whether the earth's magnetic force be an inherent property, and the resultant of the forces of all its parts, or whether it be simply the effect of thermo-electric currents produced by the heating action of the sun, the result must in either case be greatly modified by the configuration of a country, and by the nature of its superficial strata. If this view be just, the greatest irregularities should prevail in those parts of the earth in which the uniformity of surface is broken by hill and valley, and where the strata have been rent and contorted by the uplifting of moun- tain chains. In Ireland, accordingly, we should expect to find much greater anomalies in the direction and intensity of the mag- netic force than in the plains of central Europe ; and it must be, consequently, in the same degree more difficult to arrive at general results. The only mode of escaping from these difficulties was to eeek the general result of the entire series of observations, as to the po- sition of the isodynamic and isoclinal lines ; and to combine the partial results in such a manner that their deviations whether local, temporary, or casual should have the least influence on the final conclusion. Such is the object of the following computations. Let A and fi denote the latitude and longitude of any place at which an observation has been made, Ao and fi the latitude and longitude of the station which is chosen as the origin of the coordinates ; then the position of the former place may be fixed with reference to the latter in terms of these quan- tities. For let P be the pole of the earth, M and the two places, PM and PO their meridians, and MQ a great circle passing through Jfand per- pendicular to PO. It is obvious that the position of M will be determined by the rectangular sphe- rical coordinates, OQ and QJ/. Now in the right-angled triangle MPQ, we have tan PQ = tan PM cos P, sin MQ = sin PM sin P ; or, denoting the co-ordinates OQ and QM by a and 0, cot (A + a) = cot A cos (n - no] (A) sin /3 = cos A sin (ju - /*,). 206 OBSERVATIONS OF THE TERRESTRIAL When ju - juo is so small as it is within the limits of the present district of observation, we may take sin GU - /i ) = M - Mo> cos (ju - jtio) = 1, sin |3 = 0, and the preceding equations become |3 = (ft - jUo) COS X. This simplification is obviously equivalent to the substitution of the parallel of latitude for the perpendicular to the meridian. Now let us conceive any line to pass through 0, making the angle u with the meridian ; then, in the same order of approxima- tion, the perpendicular from the point M upon that line will be p = |3 cos u - a sin u ; and, substituting for a and /3 their values just obtained, p = (n - no) cos X cos u - (A - Ao) sin u. (C) It is easy to see in what manner this result may be applied in obtaining equations of condition from the data furnished by obser- vation. The increase of the force, or of the dip, may (throughout the limited area of the present district of observation) be assumed to be proportional to the distance, measured in a direction perpen- dicular to the line of equal force, or of equal dip. Accordingly, if u be the angle which the line of equal horizontal intensity passing through makes with the meridian of the place, the difference of the intensities at the two stations will be proportional to p, or h - h<> = rp ; h and A being the horizontal intensities at the two stations, and r a constant coefficient which determines the rate of increase. Sub- stituting, then, for p its value (C), and making r cos u = x, r sin u = ?/, (D) we have (ju - ju ) cos \ x - (\ - Ao) y = h - h Q . (E) The equations of condition deduced from the observations of total intensity, and of dip, will be of a similar form ; and the coefficients of the unknown quantities, in the first member of the equations, will be the same. MAGNETIC FORCE IN IRELAND. 207 The station chosen for the origin of the co-ordinates is Dublin, and it is obvious that there will be as many equations of condition a there are other places of observation. The coefficients of these equations are given in the following table. The first, second, and third columns contain the place of observation, its latitude and its longitude*. The numbers in the fourth and fifth columns are the differences of latitude and longitude (estimated in minutes #1 lati- tude) of the place of observation and Dublin, or the values of (A - Ao) and (/u - ju ) cos A ; and the numbers in the three remain- ing columns are the corresponding differences of dip, of horizontal intensity, and of total intensity, which form the second members of the equations. The dip having been observed at Dublin in each of the two years (1834 and 1835), the differences of dip are obtained by subtracting that belonging to the year in which the observation was made at the other station. TABLE VII. Place. w to (A - Ao) (M-/*O) cos A (8 - o) (A -AC) (/-/o) Cam, . . 55 15 7 15 + 114 + 34 f 54-9 - -030 + 0154 Coleraine, . 55 8 6 40 + 107 + 14 + 21-0 - -009 -0003 Strabane, 54 49 7 28 + 88 + 42 + 54-0 - -033 + -0102 Enniskillen, 54 21 7 38 + 60 + 48 + 54-0 - -031 Armagh, 54 21 6 39 + 60 + 14 -1- 36-3 - -021 + 0045 Belmullet, . 54 20 9 50 + 59 + 125 -1- 68-5 - -050 + -0095 Markree, . 54 14 8 28 + 53 -1- 77 + 61-8 - -039 + 0093 Ballina, . . 54 10 9 3 + 49 + 98 + 66-0 - -045 + 0079 Carlingford, 54 2 6 11 + 41 - 3 + 24-2 - -010 + -0169 Achill Ferry, 54 9 51 + 39 + 127 + 61-5 - -042 + 0098 Leenane, . . 53 41 9 40 + 20 + 121 - -036 Oughterard, Dublin, . . 53 27 53 21 9 18 6 15 + 6 - + 109 + + 0-0 - -022 + -000 - -oooo Galway, . . 53 17 8 51 - 4 + 93 + 27-9 1 -(IONS Rathdrum, . 52 55 6 12 - 26 - 2 - 11-5 + -001 -0057 Tulla.f . . 52 53 8 41 - 28 + 88 + 17-8 + -003 + 0077 Ennis, . . 52 52 8 54 - 29 + 96 + 8-2 - -002 + 0056 Limerick, t . 52 40 8 36 - 41 + 85 + 5-0 + -005 + -0041 * The latitudes and longitudes of some of the more important stations have been kindly furnished by the officers of the Ordnance Survey. The remainder hare been taken from Arrowsmith's map of Ireland. t Observations made in the year 1835 give Tulla 5 - 5o = + 23'7 ; Limerick, 5 - 5 = - 2'-l. 208 OBSERVATIONS OF THE TERRESTRIAL TABLE VII. (Continued.) Place. w w (A-Xo) (M-MO) COS A (8 - So) (A-Ao) (f-fo) Goreyi* . 52 41 6 15 - 40 + - 9-2 - -0068 Clonmel, 52 20 7 41 _ 61 + 53 4 '013 BaUybunan, Broadway, Waterford, 52 35 52 14 52 12 9 34 6 20 7 6 46 67 _ 69 4 121 + 3 + 31 + 16-3 - 29-0 - 14-5 + -004 + -021 + -010 4 -0127 - -0025 - -0035 Dingle, . 52 6 10 20 _ 75 + 150 4 4-9 4 '0140 Killarney, 52 3 9 31 - 78 4 121 4 6-5 + -Oil Fennoy, . 52 1 8 34 - 80 + 86 - 17-7 + -019 Valentia, 51 56 10 12 - 85 4 146 4 2-2 4 -0089 Cork, . 51 54 8 28 - 87 + 83 _ 21-9 + -019 - -0008 Youghal, 51 53 7 51 - 88 4- 59 _ 23-8 4 -0011 Glengariff, 51 44 9 33 - 97 4 123 + 3-5 4 'Oil The equations of condition (E) are of the first dimension with respect to the two unknown quantities they contain, and may be written ax + by = c ; (F) in which the values of a, b, and c (or of (p - /u ) cos A, A - A , and h - h ), are given in the preceding table. In order to deduce the most probable values of the two unknown quantities, these equa- tions must be combined by the method of least squares. Accord- ingly multiplying equation (F) by the coefficient of a? (a), and by the weight (w) of the determination which it represents, and adding the results, we have and, performing the same operation with respect to the coefficient of the other unknown quantity, 8 (icab) x 4 8 ( 8 (wbc) . These are the two final equations which, by elimination, will fur- nish the most probable values of the quantities sought. Let the values of x and y, obtained from these equations, be A and B ; then substituting in (D), r cos u = A, r sin n - B ; MAGNETIC FORCE IN IRELAND. 209 and dividing, we have tan u = -. ; (H) .A. by which the direction of the isodjnamic line is determined. Again, squaring and adding, B*', (I) which gives the rate of increase of the force in the* normal direction. The lines of absolute intensity, and of dip, will be obtained by a similar process, the only difference being in the values of the second members of the equation (F) . Before we can apply these formulas to the investigation of the lines of horizontal intensity, it is necessary to assign the weights due to each equation of condition, or to the determination which it involves. We shall assume, accordingly, that the weights of the values of (h - /*), recorded in the preceding Table, are measured by the number of separate comparisons from which they have been deduced,* and we shall have, on this principle, Limerick, . . . weight = 12, Markree, ... 3, Armagh, ... 2, the weights of each of the other determinations being represented by unity. The values of a, b, and c being <:iven in Table VII., we may now proceed to calculate the coefficients of the equations (Q-). "VVe find, (?ra 2 ) = 257431, S(wab) = + 49169, (#) = 116509, S(tcac) = - 23-598, S(icbc) = + 35-644. And the equations are 257431 x + 49169 y = - 23-598, 49169 x + 116509 y = 4- 35-644 ; * For a more correct method of estimating the weights of observed results, th< reader is referred to the Report on the Magnetic Isoclinal and iBodynamic lines in t!i- British Islands, Eighth Report of the British Attociation for the Adfattcemt*tofSctrcr, p. 95, &c. P 210 OBSERVATIONS OF THE TERRESTRIAL from which we have, by elimination, x = - -0001633 = A ; y = + -0003748 = B. Finally, substituting these values in equations (H, I) tan u = - 2-2952, u = - 66 28', r = - -000409.* The positive branches of the axes of co-ordinates having been assumed to be those which stretch to the north and to the west, it follows that the lines of equal horizontal intensity lie to the east of north, making an angle of 66^ nearly with the meridian of Dub- lin. The horizontal intensity decreases as we proceed northward, the decrease being equal to the distance traversed in a direction perpendicular to these lines (estimated in geographical miles, or minutes of latitude) multiplied by the coefficient '000409. The lines are laid down in the accompanying chart for differences of O'Ol in the value of the intensity, the corresponding intervals of distance being 24-4 geographical miles. On a comparison of the separate determinations with the result- ing lines, it will be observed that the intensities in the northern group are greater than those due to their position, those of the western group less, and those of the south-western again greater. These deviations may, in part, arise from the inexactness of the assumption with which we set out in the computation of the lines, and from the sensible deviation of those lines from parallelism. But they are probably owing in a much greater degree to the dis- turbing causes to which we have already alluded. The separate results composing each of these groups were for the most part ob- tained about the same time, and they are therefore probably affected in the same manner, and nearly in the same amount, by the irre- gular fluctuations in the direction and intensity of the resultant magnetic force. Of these, the changes in the direction of the force are by far the most influential. The relation between the corre- sponding changes in the dip and in the horizontal intensity is ex- pressed by the formula dh y = - tang sin 1'rfg; * The inclination of the lines of equal horizontal intensity, deduced with the aid of additional observations, is = - 62 40', the rate of increase in the normal direction remaining unaltered: Eighth Report, p. 174. MAGNETIC FORCE IN IRELAND. 211 (IS being expressed in minutes. Hence when 8 = 71 0', the change of the horizontal intensity, , corresponding to a change of one minute of dip, is - '00084 ; and for a variation of 12' in the dip, the corresponding variation of the horizontal force is -01. In deducing the lines of dip from observation, it seems advis- able to separate the results of the two years. For the weights we shall assume Limerick (1834), .... weight = 5, Armagh, .... = 2; the weights of each of the other determinations being unity. Making the computations for the year 1834, we obtain the fol- lowing results : S(urf) = 86660, S(icab) = + 36129, S(wb z ) = 64303, S(wac) = + 11043, 8(wbc) = - 18633. The final equations accordingly are 86660* + 36129y = + 11043; 36129* + 64303y = - 18633. From which we deduce a? = + -3228 = cos t>, y = - '4705 = s sin v ; in which v denotes the angle which the isoclinal line makes with the meridian of Dublin, and s the coefficient which determines the rate of increase of the dip in the perpendicular direction. Dividing, squaring and adding, we find tan v - - 1-458, t? = - 55 33', - -571. The following are the results of calculation for the year 1835 : S (a 2 ) - 149922, 8(ab) = + 31821, S(V) - S (ac) = + 32157, S (be) = - 18372 ; 212 OBSERVATIONS OF THE TERRESTRIAL so that the final equations are 149922o? + 31821 y = + 32157; 31821* + 55339y = - 18372. From these we deduce x = +'3250, y=- -5196, tan v = - 1-599, v = - 57 59', s = -613. It would appear, then, that the angle which the isoclinal lines in Ireland make with the meridian is on the increase a result which is in conformity with the general progress of these lines, as inferred from a comparison of recent observations with those of an earlier date.* For the mean of the two years, x = + -3239, y=- -4950, tan *? = - 1-528, = -5648', s = -592. The lines in the annexed chart are deduced from these last results ; and it appears from them that the interval of the lines corresponding to a difference of half a degree of dip is 50-7 geo- graphical miles. The lines of dip and of horizontal intensity being known, the lines of total intensity may be deduced. For if / denote the total intensity, h its horizontal component, and S the dip, as before, and differentiating, and dividing by the equation itself, /-' df=h~ l dh + tan S sin 1'dS. (I.) Now, if the values of * and y for the lines of dip and of horizontal intensity be denoted by ar (8) , ae w and y w , y (h] , and if x (f] * This is further confirmed by the results obtained with the help of additional obser- vations. See Eighth Report of the British Association for the Advancement of Science, p. 118. MAGNETIC FORCE IN IRELAND. 213 and yy) be the corresponding quantities for the lines of total intensity, (II.) df=ax (f) -by (f] , in which a = (fjt - /K O ) cos A, b = A - A ; /* and A being the longitude and latitude of any assumed station, and /i and A those of Dub- lin. Substituting these values in (I.), it becomes tan S sin But as a and 6 are entirely independent, their coefficients must be separately equal, and we have - l x = k~ l xh + tan S sin tan 5 sin l'y (6 ) ; so that the values of #(/) and^(/) are found when those of #(*), &(&), y (h }, ya) are known. Let the second members of equations (IV.) be denoted, for abridgment, by P and Q, then X( f ] = tcOBW =/P, 2/(/) = ^ sin ; =/Q ; in which w is the incKnation of the line of total intensity to the meridian, and t the coefficient which determines the rate of increase. Dividing the latter by the former, there is tan w = -p . (V.) And squaring and adding, t=/SJ I Tw. (Vi.) From the preceding formulas it appears that the direction of the isodynamic line at any point is dependent on the values of /i and of c at that point, so that these lines will not be parallel, even though tlie lines of dip and of horizontal intensity should be so. The devia- tions, however, will not be considerable within the limits of Ire- 214 OBSERVATIONS OF THE TERRESTRIAL land ; and for our present purpose it will be enough to seek the mean direction of the lines, and the mean rate of increase in the direction perpendicular to them. We must therefore employ in the preceding formulas the values of /, h, and 8, corresponding to the mean point of the island, or the point whose latitude and lon- gitude are 53 25' and 7 55',* and for which therefore Now it has been already found that se w = - -0001633, a? (8) - + -3239, y (k ) = + -0003748, y (8) = - -4950 ; and substituting these values in the formulas 8 - 8 = (jU - fJLo) COS X(j) - (X - Xo)y(S), h - h = (ju - juo) cos X#(A) - (X - X ) y(A), we find 8 - 8 = 21'-4, h-h Q = - -0113. Consequently, 8 = 7124'-4, A = -9282,/= 1-0295. We have now the numerical values of all the quantities which enter the formulas P = Jr l x(h) + tan 8 sin l'a?( 8 ), Q = h~ l y( h ) + tan 8 sin ry(s) ; and we find on substitution, P = + -0001042, Q = - -0000242. Introducing these values in (V.) and (VI.), tan w = - -2322, w = - 13 4', t = -0001102. These results, however, are not entitled to much confidence. An attentive consideration of the formulas (IV.) and (V.) will show that the direction of the resultant isodynamic lines will vary * This point corresponds, almost exactly, to the town of Athlonc. MAGNETIC FORCE IN IRELAND. 215 very widely with moderate variations in the values of X( h ^ xy w , (5 ), i/(s), on which it depends ; or, in other words, that a small error in the position of the lines of dip, or of horizontal intensity, will entail a very great one in that of the lines of total force. Thus, if we were to take for the lines of dip those inferred from the observa- tions of the year 1835 alone, we should find P = + -0001051. Q = - -0000455, tan w = - -4329, w = - 23 25' ; a result differing by more than 10 from the former. In these latitudes, therefore, very great accuracy is necessary in the deter- mination of the lines of dip and of horizontal force before we can make, in this manner, even an approximation to the direction of the lines of total force. For these reasons the results of the direct method, to which we now proceed, seem to be deserving of more confidence. In the calculation of the isodynamic lines from the results of observation by the statical method, we shall take the number of observations at each station to represent the weight of the result ; we have in this manner Limerick, .... weight = 4, Armagh, ..." =2, Youghal, .... =2; the weight of each of the other determinations being unity. The following are the results of the computation : S(wa*) = 178390, S(waV) = + 38216, 8(wb v ) = 96066, 8(wac) = + 13-3229, S(wbc) = - 2'2448. The final equations therefore are 178390* + 38216^ = + 13-3229, 38216* + 96066y = - 2-2448 ; from which we obtain, by elimination, x = + -00008711, y = - -00005802. 216 OBSERVATIONS OF THE MAGNETIC FORCE IN IRELAND. Consequently, tan w = - '6661, w = - 33 40', t = -0001047.* The lines of total intensity thus deduced are laid down in the annexed chart for differences of '005, these differences correspond- ing to intervals of 47'6 geographical miles. It will be seen that their direction diverges widely from that of the lines of dip ; and although the position of the two classes of lines may need further correction, it does not seem likely that such correction will have the effect of diminishing, at least by any considerable amount, the divergence. * The mean inclination of the isodynamic lines deduced with the help of additional observations is u = - 35 0', the rate of increase in the perpendicular direction remain- ing unaltered. Eighth Rpeort of the British Association for the Advancement of Science, p. 185. rERRESTRIAL MAGNETIC FORCE IN IRELAND. +3.KrnjS!< VI. ON THE POSITION OF THE ISOGONAL LINES IN IRE- LAND, DEDUCED PROM THE OBSERVATIONS OF CAPTAIN SIR JAMES CLARK ROSS, R. N. From the Proceedings of the Royal Irish Academy, 1850. IN the year 1835 I laid before the British Association, then as- sembled in Dublin, a Eeport on the Direction and Intensity of the Terrestrial Magnetic Force in Ireland, based upon observations made by Lieut.-Colonel Sabine, Sir. James C. Boss, and myself. In these observations Mr. Eobert Were Fox and Professor Phillips afterwards took part ; and the survey was subsequently extended to the whole of the British Islands. The details of this extended survey are given in a Memoir on the Magnetic Isoclinal and Iso- dynamic lines in the British Islands, drawn up chiefly by Lieut.- Colonel Sabine.* The observations contained in these Reports are limited to the Magnetic Inclination and Intensity. Observations of the Declina- tion, as well as of the other two elements, were indeed made by Sir James Boss ; but they have only lately been given to the public in a Memoir by Lieut.-Colonel Sabine, on the lines of Magnetic Decimation in the Atlantic, f In this Memoir, the observations referred to are combined with a large mass of other materials, and the position of the isogonal lines inferred from the whole by a graphical process. The Irish portion of these observations is, how- ever, so distinct, and so complete in itself, that it seemed to me desirable that they should be discussed by the same method which had already been applied to the observations of the other two ele- * Eighth Report of the British Association for the Advancement ofScimet. t Philosophical Transactions, 1849, Part ii. 218 ON THE ISOGONAL LINES IN IRELAND. ments, in the reports above referred to ; such a discussion serving to complete the Magnetic Survey, so far as Ireland is concerned, and to furnish a formula for the Magnetic Declination at any point in the island whose position is known. The following is the mode of doing this : If S denote the magnetic declination at any place ; S that at some near station which is taken as the origin of co-ordinates ; and x and y the actual distances (in geographical miles) between them, measured on the parallel of latitude and on the meridian, respectively or the co-ordinates of position of the former station referred to the latter as an origin ; the relation of these quantities is expressed approximately by the equation in which M and N represent the increase of declination correspond- ing to each geographical mile of distance in the two directions. If A and /j. denote the latitude and longitude of the former station, A and [i those of the latter, y = A-A , x = OU-M O ) cos A. It is evident, that if X and y be treated as variable, S being constant, the preceding equation is that of the locus of all the points of given declination. It is that of a right line, making the angle with the meridian, and the increase of declination corresponding to each geographical mile of distance, in a direction perpendicular to this line, is It is evident then that, to obtain the values J/"and N, obser- vation must give the values of the declination at three, or more, stations. The observations of Sir James Boss were taken at twelve stations, well distributed throughout the island ; and as they were all made during the months of October and November, 1838, no correction is required to reduce them to a common epoch. For convenience of reference, they are here extracted from Colonel ON THE ISOGONAL LINES IN IRELAND. 219 Sabine's Memoir, together with the longitudes and latitudes of the places of observation. Station. \ M S Valentia, 51 56' 10 17' 28 42' Killarney, 52 2 9 30 28 11 Westport, 53 48 9 29 29 9 Limerick, 52 40 8 36 28 3 Cork, . 51 54 8 28 27 44 Markree, 54 14 8 28 29 15 Shannon Harbour 53 14 7 53 28 3 Edgeworthstown, 53 42 7 33 28 8 Londonderry, . Waterford, . . 54 59 52 15 7 19 7 8 28 47 26 44 Armagh, . . 54 21 6 39 28 8 Dublin, . . . 53 21 6 15 27 35 Taking Dublin as the origin of co-ordinates, and substituting the values of A - A , /* - n , and S - S , given by this Table, in the equation above given, we obtain eleven equations of condition, from which the values of M and N are obtained by the method of least squares. They are the following : Jf=0'-690, ^ We may now test the accuracy of these numbers, by employing the formula to calculate the values of the declination at each of the eleven stations. The result of this calculation gives, at Waterford, a difference between the observed and calculated values amounting to 34', an amount which far exceeds the probable error of obser- vation. This difference is, therefore, probably due to some local irregularity of the magnetic force. But, whatever be its cause, it is obvious that it tends to vitiate the general result ; and that a nearer approximation to the values of M and N will be obtained by excluding that observation from the computation. We thus obtain, from the remaining ten equations, And substituting these values, we find ang. (tang = - ~ ) = - 37 25' ; 0'-867. 220 ON THE ISOGONAL LINES IN IRELAND. Accordingly, the isogonal lines in Ireland lie to the east of north, making an angle of 37 25' with the meridian of Dublin ; and the declination increases as we proceed in the north-westerly direction, the increase being 52" for each geographical mile, in a direction perpendicular to these lines.* Finally, the declination at any point of the island, whose longi- tude and latitude are known, is given by the formula S - S = 0'-527 (X - X ) + 0'-689 fc - ,,) cos X ; the declination at Dublin, <5 , being supposed known. Or, if we substitute for cos X the value corresponding to the mean latitude (\-53ir), S-S =0'-527 (X -X ) +0'-412 (^-^J. The mean declination at Dublin, for the year 1850, is 26 29' west ; and as the yearly value of the secular change of the declina- tion is - 6' - 06, the mean declination, in any not very remote future year, will be given by the formula S = 2629'-6'-06x ; n being the number of years, counted from the present. If greater accuracy be desired, the diurnal and annual variations of the de- clination, corresponding to the time of the day and of the year, must be added. . * This result agrees very closely with Colonel Sabine's map of the isogonal lines in the Atlantic, as to the direction of the lines, but gives a more rapid rate of increase. ' VII. ON A NEW MAGNETICAL INSTRUMENT, FOR THE MEASUREMENT OF THE INCLINATION, AND ITS CHANGES. From the Proceedings of the Royal Irish Academy, 1842. IF a soft iron bar, perfectly devoid of magnetic polarity, be held in a vertical position, it immediately becomes a temporary magnet under the inducing action of the earth's magnetic force, the lower extremity becoming a north pole, and the upper a south pole. Ac- cordingly, if a freely- suspended horizontal magnet, whose dimen- sions are small in comparison with those[of the bar, be situated near, in a plane passing through one of these 'poles, it will be deflected from the magnetic meridian. The deflecting force is the induced force of the bar, which may be regarded as proportional to the energy of the inducing cause, i. e. to the vertical component of the earth's force ; while the counteracting force is the horizontal com- ponent of the same force, acting directly on the magnet itself, to bring it back to the magnetic meridian. Thus the magnet will take up a position of equilibrium, under the action of these oppos- ing forces; and this position will serve to determine the ratio which subsists between them. When the right line connecting the centre of the horizontal magnet, and the acting pole of the bar, is perpendicular to the magnetic meridian, the tangent of the angle of deflection will measure the ratio of the two forces, and will therefore be proportional to the tangent of the. magnetic inclination. Accordingly, by observing the changes of position of the horizontal magnet, so circumstanced, we can infer those of the inclination itself. But the iron bar may have (and generally will have) a certain portion of permanent magnetism, which will concur with the in- 222 ON A NEW MAGNETICAL -INSTRUMENT. duced magnetism in producing the deflection ; and it becomes necessary to institute the observations in such a manner as to be able to eliminate the effects of this extraneous cause. For this purpose we have only to invert the bar, so that the acting pole, which was uppermost in one part of the observation, shall be lower- most in the other. The induced polarity will, under these circum- stances, be opposite in the two cases ; and the acting force will in one case be the sum of the induced and permanent forces, and in the other their difference. Let X and T denote the horizontal and vertical components of the earth's magnetic force, M the intensity of the permanent mag- netism in the acting pole, and m the magnetic moment of the sus- pended magnet. The intensity of the induced magnetism is, by hypothesis, equal to *T, k being an unknown constant ; and when this is of the same name as the permanent magnetism, the intensity of the acting force, at the unit of distance, is kY+ M. Accordingly, the moment of this force to turn the suspended mag- net is (kY+ M) mr cos u, u being the angle of deflection, and r a constant depending on the distance ; or, making, for abridgment, kr = p,Mr = q, (p Y + q] m cos u. But this deflecting force is resisted by the earth's horizontal force, the moment of which to turn the magnet is Xm sin u ; and the magnet will rest when these moments are equal. Hence the equation of equilibrium is (1) By the same reasoning it will appear, that when the induced and permanent magnetisms are of contrary names, there is pY- q = Xt&nu'; (2) in which u' is the new angle of deflection when the bar is inverted. Adding these equations together, and observing that Y = X tan 6, ON A NEW MAGNETICAL INSTRUMENT. 223 being the inclination, we have 2p tan = tan u + tan u '. (3) This equation would furnish at once the inclination sought, provided we knew the value of the constant k. In order to deter- mine it, we have only to place the iron bar horizontally in the magnetic meridian, its acting pole remaining in the same place as before, but pointing alternately to the north and south. The in- ducing force is, in this case, the horizontal component of the earth's magnetic force ; and it will be readily seen that the equations of equilibrium are similar to (1) and (2), substituting X for Y. If therefore v and v denote the angles of deflection in these positions, we have 2p = tan + tan if ; (4) and dividing (3) by this, tan u + tan u' /K \ tan 9 = x - - -- r . (5) tan v + tan v Thus, from the deflections produced in these four positions of the bar, we obtain the inclination. In order to determine the changes of the inclination, it is not necessary to observe the deflections in the horizontal position of the bar. Let equation (1) be differentiated, X, Y, and u being all variable, and let the resulting equation be divided by (3). We thus obtain the following equation, from which p and q are both eliminated : A F 2Aw 2tanw AX _ Y ~ COS*M (tan u + tan u} tan u + tan u'' X ' But from the relation Y = X tan 0, we have AF AX A0 Y~~^~ + sin 0cos0 ; and substituting A0 cos?/ At/ + v BJn(i*-tQ AX ^ snT20 " cos M sin (u + ') sin (w + n) X ' The second term' of the right-hand member of this equation con- tains a correction required for th- sii.iuli:inoous changes of the horizontal intensity ; but this correction will be generally small, 224 ON A NEW MAGNETICAL INSTRUMENT. and, when the bar has no permanent magnetism, will vanish alto- gether. In this latter case, in fact, it appears from (1) and (2) that u' = u ; so that the preceding equation is reduced to A-5L|A. (7) We must remember that, since the angle u in the preceding for- mulas is the deviation of the suspended magnet from the position which it would assume under the action of the earth alone, its changes are the differences between the observed changes of position of the suspended magnet and the corresponding changes of decli- nation. Let a denote the deviation of the suspended magnet, measured from some fixed line, and a' the corresponding angle when the iron bar is removed ; then ., u = a - a', AM = Aa - Aa'. But Aa = &w, Aa' = k'n' ; in which n denotes the number of divi- sions of the scale of the instrument corresponding to the angle Aa, n the number corresponding to the angle A a' as shown by the de- clinometer, and k and k f the arc- values of a single division in each instrument. Hence Aw - 1m - k'n. (8) I now proceed to the construction of the apparatus employed in these measurements. The magnet is cylindrical ; its length is three inches, and dia- meter one-fourth of an inch. A mirror is attached to the stirrup by which it is suspended, by means of which the varying position of the magnet may be observed with a telescope at a distance, after the method of Gauss. This mirror is of course vertical ; and it has a motion round a vertical axis, by means of which it may be ad- justed to any desired position of the observing telescope. The mirror is circular, and is three-fourths of an inch in diameter. The moveable part, the stirrup to which it is attached, has the form of a cross ; and it is rendered vertical by means of three screws, near the ^xtremities of three of the arms of the cross, the heads of which project and hold it. The mirror is maintained in contact with these heads by springs at the back. The box is octagonal : the interval between the opposite sides is four mches, and that between the top and bottom two inches. The ON A NEW MAGNETICAL INSTRUMENT. 225 top and bottom, and the connecting pillars, are formed of gun- metal ; the eight sides are closed by moveable pieces, three of which are of glass, and the rest of ebony. To the top of the box is attached an upright tube of glass, eight inches in length, which incloses the suspension thread. The suspension apparatus at the top of the tube is of the usual construction ; the circular piece to which it is attached has a movement of rotation, and its outer sur- face is graduated to 5, for the purpose of determining the effect of torsion of the suspension thread. The base of the instrument is a circle of gun-metal, six inches in diameter, graduated on the edge. The box is connected with this circle by a short conical stem, forming the axis of a second plate, which revolves upon the fixed one. This moveable plate carries two verniers, by which the angle of rotation may be read off to minutes. Two tubular arms, slightly inclined to one an- other, are attached to this plate ; and their other extremities are connected by a cross-piece, which carries a short scale at a distance of eighteen inches from the mirror. This part of the apparatus is employed in determining the total angles of deflection. The soft iron bar is a cylinder, twelve inches long, and three- fourths of an inch in diameter. One of its extremities is inclosed in a hollow cylinder of brass, connected with a horizontal pivot which revolves in a fixed socket. The axis of this pivot being in the line passing through the centre of the suspended magnet, and perpendicular to the magnetic meridian, it is obvious that the bar has a movement of rotation in the plane of the magnetic meridian itself. The distance of the axis of the bar from the centre of the magnet is about five inches ; and it is so placed that the induced pole is in the direction of the axis of the pivot, and thus remains fixed during the movement of the bar. The changes of position of the suspended magnet are observed at a distance by means of a fixed telescope and scale. The scale, whose divisions are reflected by the mirror, is attached above the telescope to a support near the eye-end. Having explained the principle of this instrument, and given the details of its construction, it remains only that I should describe the observations made for the purpose of testing its performance. I shall pass over for the present those which relate to the absolute inclination, because they have yielded results which can be regarded only as approximations to the truth, and I have not succeeded as Q 226 ON A NEW MAGNETICAL INSTKUMENT. yet in tracing the errors to their source. It is manifest, however, that an instrument may be a good differential instrument, while it is incapable of yielding absolute results; and there are special reasons why this should be the case with the apparatus now under consideration. Accordingly its failure in the latter respect, even though established, would furnish no ground for despairing of its success in the former. It is obvious that the apparatus is wholly free from the errors belonging to magnetical instruments moving on a fixed axle ; and the only doubt of its performance must relate to the changes of induced magnetism in the iron bar. Thus it might be questioned, before trial, whether such a bar receives in all cases an amount of free magnetism proportional to the inducing force ; whether, again, the minutest changes in the latter are accompanied by corresponding changes in the former; and whether, lastly, the changes thus produced are instantaneous, or, at least, demand no appreciable time for their development. In the first experiments which I made, for the purpose of de- termining these questions, the induced magnetism of the iron bar was altered by means of a permanent magnet, placed in the same right line with the bar, and at a known distance from it. The effect produced upon the position of the suspended magnet being observed, the distance was altered by a known amount, and a new observation taken ; and so on, at many different distances. Then, the law of action of the inducing magnet being known, we may calculate the changes of deflection of the suspended magnet, on the supposition that the changes of the induced force of the bar are proportional to those of the inducing action, and then compare them with the changes of deflection observed. The calculated and observed results of many series of observations, taken in this manner, were found to accord as nearly as the accuracy of the observations themselves allowed. In making this comparison, however, it is necessary to take into account the effect of the direct action of the fixed magnet upon the suspended one. The axis of the former magnet being not far from the vertical passing through the centre of the latter, its action upon it and upon the iron bar follow, nearly, the same law ; so that its direct effects upon the position of the suspended magnet are, very nearly, proportional to those which it produces through the medium of the induced force of the bar. On this principle the ON A NEW MAGNETICAL INSTRUMENT. 227 observed results may be cleared, appf oximately, of those parts of the changes which are foreign to the question. Still it must be admitted that such a complication of the results tends to weaken their evidence; and it was therefore desirable to obtain further proof, in a manner less exceptionable. The object being to alter the inducing action according to a known law, and to observe the changes of the induced force, as shown by the position of the suspended magnet, it is manifest that it may be attained by simply varying the angle which the iron bar makes with the direction of the earth's magnetic force, the distance of its pole from the suspended magnet remaining unchanged. In fact, it will be seen, by pursuing the same reasoning as before, that if R denote the total force of the earth, and ^ the angle which the bar makes with its direction, the equation of equilibrium of the suspended magnet is pit, cos i// + q = X tan u, the line connecting the pole of the bar with the centre of the sus- pended magnet being, as before, perpendicular to the magnetic meridian. Hence, if the bar be devoid of permanent magnetism (or q = 0), and if the forces R and X remain unchanged during the experiments, we have tan u = a cos ^, being a constant. In order to observe whether the deflections "of the suspended magnet obeyed this law, a small divided circle was attached to the piece upon which the iron bar moved, in such a manner that the axis of the pivot passed through its centre. The circle being fixed, and the bar connected with the moveable arm carrying the vernier, we have the means of determining the angle through which it is moved. The plane of the motion coinciding with the magnetic meridian, the inclination of the bar to the vertical was altered by 5 between the successive observations of the position of the sus- pended magnet. The following Tables contain the results of two such series of observations. The first column of each gives the in- clination of the bar to the vertical; the second, its inclination' ( Bar removed, . . 47 35, mean = 16 58 Inclination to Vertical. * Reading of Scale. Angular Differences. Observed. u Calculated. Difference. + 150' + 10 + 50 34 10' 29 10 24 10 19 10 2-6 14-4 24-8 33-2 -2' l'-S -1 14-8 - 33-4 -0 1456'-2 15 43 -2 16 24 -6 16 58 -0 1457'-8 15 45 -0 16 25 -2 -l'-6 -1 -8 -0-6 - 5 14 10 40-0 i + 27-1 17 25-1 17 23 -3 + 1 -8 ON A NEW MAGNETICAL INSTRUMENT. 229 In the preceding observations a telescope of law power was employed, and the arc-value of a single division of the scale (which was at the distance of eighteen inches from the mirror) was 3'-98. The differences of the observed and calculated results, therefore, do not in general exceed the amount which may be fairly ascribed to errors of observation ; and the accordance is sufficient to establish the fact, that the changes of the induced force of the bar are, within the observed limits, proportional to those of the inducing action. It is important to note also that the changes of the induced force, produced artificially in these experiments, are much greater than any which are likely to arise from the variations of the vertical component of the earth's magnetic force, and therefore that the experiments may be regarded as severe tests of the performance of the instrument. The preceding observations further showed, that the changes in the inducing force were imtantly followed by their effeets upon the suspended magnet; so that the changes of induced force required no appreciable time for their development. It remained only to ascertain, in a somewhat fuller manner, how far the bar was susceptible of minute magnetic changes, from very small variations of the acting force. For this purpose, a series of readings of the scale was taken, the inclination of the bar to the vertical being altered by half a degree between the consecutive readings. The mean difference of the successive readings was found to agree, very exactly, with the calculated difference ; while the partial differences deviated from the mean by an amount not exceeding the limits of error of observation. It may be presumed, therefore, that the changes of the induced force in the iron bar are continuous ; and, accordingly, that the sensibility of the instrument is only limited by the optical power, which is applied to observe the changes of position of the suspended magnet.* In the experiments above described, the arc- value of the divisions of the scale was nearly 4' : with the modifications since introduced into the reading part of the apparatus, the scale * Against this conclusion is the fact, that considerable changes in the induced force of the bar seem to be attended with some permanent changes of polarity ; and it may be presumed that the same thing will take place, in a proportionate degree, with tho minute changes induced by tho variations of the earth's force. It remains for future examination to determine how far such permanent changes, if they occur, may impair the accuracy of the results. 230 ON A NEW MAGNETICAL INSTRUMENT. divisions have nearly the same value as in the instrument for the measurement of the declination, so that the readings may be made with certainty to less than the tenth of a minute. The present value of the inclination in Dublin is about 70 50'; and the mean deflection produced by the iron bar in its actual position being about 19, it follows that the changes of inclination are inferred with the same degree of precision, very nearly, as the observed changes of angle. The last test to which the instrument was subjected was, to employ it for some time in the regular observation of inclination changes, for which it is destined, and to ascertain how far the mean results of the observations of successive weeks agreed in exhibiting the law of the diurnal variation. The instrument was accordingly observed for five successive weeks, every second hour during the day and night, and the means calculated, omitting those days in which the series was broken by changes of adjustment during experiment. The curves now laid before the Academy represent the projected results of the observations of each of these weeks, together with that of the mean of the whole. An inspec- tion of them is sufficient to show that the curves of the separate weeks accord with one another, and with the mean, as nearly as can be expected in the results of such limited series, the dis- cordances being only such as are due to the known irregularities in the direction of the earth's magnetic force. Till. REMAKES ON THE THEORY OF THE COMPOUND MAGNETIC NEEDLE. Proceedings of the Royal Irish Academy, 1848. WHEN two magnetic needles are united by a fixed vertical axis passing through their centres, and perpendicular to both, the moment of the force exerted by the earth upon them is the sum of the moments which it exerts upon each needle separately, and is, therefore, X (M sin u + M' sin M') ; in which M and M' denote the magnetic moments of the two needles, u and u the angles which their magnetic axes make with the magnetic meridian, and X the horizontal component of the earth's magnetic force. In the state of equilibrium this moment is nothing ; so that if u and U Q ' denote the corresponding values of u and u' t there is M sintfo + -3/'sinw'o = 0. (1) Consequently, if two lines be taken from any point of the vertical axis, parallel to the magnetic axes of the two needles, and pro- portional to their magnetic moments, M and M', the diagonal of the parallelogram constructed upon them must lie in the magnetic meridian, when the compound needle is at rest. Again, if we substitute u = t( + *', ' = '< + ''> i Q * ue general expression of the statical moment, it becomes, in virtue of (1), X(M cos + JT cos ') sin /. Hence the compound needle is acted upon as a single needle, 332 ON THE THEORY OF COMPOUND MAGNETIC NEEDLE. whose magnetic axis lies in the direction of the diagonal of the parallelogram above mentioned, and whose magnetic moment is fj. = M cos MO + M' GOSU'O. (2) Accordingly, the diagonal of the parallelogram already referred to will represent in magnitude the magnetic moment of the compound needle. For, if the equations (1) and (2) be squared, and added together, and the angle contained by the magnetic axes of the two needles, u' - u , be denoted by a, we have t? = M 2 + 2MM' cos a + M '\ (3) In the case of the astatic needle, a = 180 - S, being a very small angle, and cos a = - cos 8 = - 1 + \ S 2 , q .p. whence V = (H-M f y + MM'&. (4) Accordingly, when M - M ' is not a very small quantity, the second term may be neglected in comparison with the first, and /u = H-M', nearly. On the other hand, when M - M' = 0, we have n = MS. Eeturning to (1), and substituting for u' its value U Q + a, we have to*- g-*" ; (5) - + COSa by which the position of the needle with respect to the magnetic meridian, when at rest, is determined. In the case of the astatic needle, the preceding equation becomes tan w - -^fySsinl'. (6) From this we learn, 1. That the tangent of the angle of deviation of the astatic needle from the magnetic meridian varies, cceteris paribus, as the angle, S, contained by the magnetic axes of the two component needles. 2. That however small that angle be, provided it be of finite magnitude, the tangent of the deviation may be rendered as great as we please, and therefore the deviation be made to approach to 90 as nearly as we please, by diminishing the difference of the moments of the two needles. IX. ON THE MEAN RESULTS OF OBSERVATIONS. Transactions of the Royal Irish Academy. Vol. XXII. 1. THE problem in which it is sought to determine the daily mean values of the atmospheric temperature or pressure, from a limited number of observed values, is one of fundamental importance in meteorology ; and, accordingly, many solutions of it have been proposed by meteorologists. These solutions are derived, for the most part, from the known laws of the diurnal variation of these elements. Many of them are accordingly applicable only to the particular cases considered ; while for others, which are really general in their nature, that generality is not established. It is the object of the following investigation to supply this deficiency, and to show in what manner the daily and yearly means may be obtained in all the periodical functions with which we are concerned in magnetism and meteorology. 2. It is known that the mean value of any magnetical or meteorological element, for any day, may be obtained, approxi- mately, by taking the arithmetical mean of any number of equidistant observed values ; the degree of approximation, of course, increasing with the number. A somewhat more exact mean may be deduced, as has been shown by Cotes and Kramp, by combining the equidistant observed values in a different manner ; and Gauss has given a method, whereby the values of the integral, [*" Udx, may be obtained with still greater accuracy J- from the observed values of the ordinate, U, corresponding to certain definite abscissae.* But in the case of periodical functions, * Commentationes Socutatis Reyia Scicntiaruin Goltinyensis, torn. iii. 234 ON THE MEAN RESULTS OF OBSERVATIONS. it will appear from what follows that the refinement of Cotes is unnecessary; and, in the case under consideration, there are practical reasons of another kind for adhering to the method of equidistant observations, and which, therefore, deprive us of the advantages of Gauss's method. 3. Any periodical function U, of the variable x, may be represented by the series U = A n + Ai sin (x + ai) + A 2 sin (2x + a 2 ) + A 3 sin (3a? + o s ) + &c., in which the first term, A , is the mean value of the ordinate 7", and is expressed by the equation ^o = This is the quantity whose value is sought in the present inves- tigation. It is obvious that the values of U return again in the same order and magnitude when * becomes x + 2ir ; so that if x = at, the period is represented by -^. If then 2* be divided into n equal parts, so that the abscissae of the points of division are x f X + ~n' # + , &c., x + -, the sum of the corresponding ordinates will be -4 2 2 sin + -4 3 S sin jsf x + ~\ + a 3 J + &c., in which denotes any one of the series of integer numbers, from to n - 1 inclusive. The multiplier of A m , in the general term of this series, is , K* - sin (mx 4 a m ] S cos + cos (mx + a m ) S sin ?^ But, when m is not a multiple of n, ON THE MEAN RESULTS OF OBSERVATIONS. 235 and, therefore, the preceding term vanishes. When m is a multiple of , 2/W7T . S cos - = n, S sin -- = ;* n n and accordingly the term is reduced to Hence, all the terms of the series vanish, excepting those in which m = kn, k being any number of the natural series, and - S ( U] = Ao + A n sin (nx + a) + A 2n sin (2nx + a 3 ,) + &c. That is, the arithmetical mean of the n equidistant ordinates is equal to the sum of the terms of the original series of the order k)i> whatever be the value of x. The original series for V being always convergent, the derived series, which expresses the value of - 2 ( U) t will be much more so ; and, when the number n is sufficiently great, we may neglect all the terms after the first. Hence, approximately, The error of this result will be expressed by the second term * These results are easily established. The roots of the equation y" - 1 = 0, being comprised in the formula cos + V(- 1) sin, the m'* power of any one of these n w roots is cos 1- V ( 1) sin ; and the sum of the m lh powers of the roots is limit ,. . 2imir 2 cos + v(-l) 2 sin . Now, when m is not a multiple of , this sum = 0, and therefore 2tmir . limtr 2 cos =0, 2 sin = 0, n n as above. When m is a multiple of n, the sum of the m"' powers of the roots = n, and therefore 2i;ir . 2wir 2 cos = ii, 2 sin = 0. 236 ON THE MEAN RESULTS OF OBSERVATIONS. of the series, -4 w sin (nz + a n ), the succeeding terms being, for the same reason, disregarded in comparison ; and accordingly the limit of error will be A n . Thus, when the period in question is a day, we learn that the daily mean value of the observed element will be given by the mean of two equidistant observed values, nearly, when A* and the higher coefficients are negligible ; by the mean of three, when A 3 and the higher coefficients are negligible ; and so on. 4. The coefficient A 2 is small in the series which expresses the diurnal variation of temperature ; and, consequently, the curve which represents the course of this variation is, nearly, the curve of sines. In this case, then, the mean of the temperatures at any tico equidistant or homonymous hours is, nearly, the mean temperature of the day. The same thing holds with respect to the annual variation of temperature ; and the mean of the temperatures of any two equidistant months is, nearly, the mean temperature of the year. These facts have been long known to meteorologists. 5. The coefficient A 3 is small in all the periodical functions with which we are concerned in magnetism and meteorology; and, therefore, the daily and yearly mean values of these functions will be given, approximately, by the mean of any three equidistant observed values. In order to establish this, as regards the daily means, I have calculated the coefficients of the equations which express the laws of the mean diurnal variation of the temperature, the atmospheric pressure, and the magnetic declination, as deduced from the observations made at the Magnetical Observatory of Dublin during the year 1843. The observations were taken every alternate hour during both day and night; and the numbers employed in the calculation are the yearly mean results corre- sponding to the several hours. The origin of the abscissse is taken at midnight. 6. The following is the equation of the diurnal variation of temperature : U- A, = + 3-60 sin ( * + 239-0) + 0-70 sin (2x + 67-2) + -26 sin (3a? + 73 -5) + -03 sin (4x + 102 -7) + -14 sin (5* + 258 -6) + -09 sin (6* + 180 -0). Hence, the error committed in taking the mean of the temperatures ON THE MEAN RESULTS OF OBSERVATIONS. 237 at any two equidistant hours as the mean temperature of the da}% is expressed nearly by the term 0'70sin(2#+67-2); and, consequently, cannot exceed 0'70. To obtain the pairs of honionymous hours, whose mean temperature corresponds most nearly with that of the day, we have only to make sin (2* + 67 -2) = 0, which gives for x the values x = 56-4, x = 146-4, corresponding to the times t = 3 A 46"', t = 9* 46"'. Accordingly, the best pairs of homonymous hours, so far as this problem is concerned, are 3* 46'" A. M. and 3 7 ' 46"' p. M., or 9*46 M A.M. and 9* 46 m P.M. The error committed, in taking the mean of the temperatures at any three equidistant hours as the mean temperature .of the day, is, very nearly, andean not, therefore, exceed 0> 26. The best hours are those in which the angle, in the preceding expression, is equal to 180 or 360. The corresponding values of x are x = 35-5, x = 95-5 ; whence t = 2 h 22'", t = 6 A 22'". Accordingly, the best hours of observation are 2 h 22"' A. M., 10 A 22'" A. M., 6* 22'" P. M. ; and Q k 22'" A.M., 2 h 22'" P.M., 10'' 22'" P.M. By taking the mean of any four equidistant observed values, the limit of error will, of course, be less. Its amount, which is the coefficient of the fourth term of the preceding formula, is only 0-03 ; and, accordingly, the mean temperature of the day is inferred from the temperatures observed at any four equidistant hours with as much precision as can be desired. 238 ON THE MEAN RESULTS OF OBSERVATIONS. 7. The law of the diurnal variation of the atmospheric pressure is contained in the following equation : U-A, = + -0024 sin ( # + 244-3) + -0089 sin (2x + 144-4) + -0008 sin (3a? + 27 -9) + -0006 sin (4as + 78 -5) + -0001 sin (5x + 228 7) + -0002 sin (fa + 180 -0). The second term in this formula being the principal one, the mean of the pressures observed at any two equidistant hours, so far from approaching the mean daily pressure, may recede from it by the greatest possible amount within the limits of the diurnal variation. The error committed, in taking the mean of the pressures observed at three equidistant hours as the mean daily pressure, is, very nearly, + 0008 sin (3tf + 27'9), and cannot therefore exceed '0008. It is needless to inquire into the least value of this quantity, which is in all cases less than the probable error. 8. The law of the diurnal variation of the magnetic declination is expressed by the equation U- A, = + 3'-29 sin ( * + 657) + 2'-08 sin (2* + 224-5) + 0'-63 sin (3s + 71 7) + 0'-30 sin (40 + 237 -5) + 0'-13 sin (5* +114 7), the coefficient of the last term being evanescent. Hence the error to which we are liable, in taking the mean of the decimations observed at any three equidistant hours as the mean of the day, is, very nearly, + 0'-63sin(3tf + 717), and cannot exceed 0'63. This term vanishes, and the mean of the three observed values will deviate from the true daily mean by an amount less than the errors of observation, when * = 36-l, or x = 96-l ; that is, when t = W 25*, or t = Q h 25. Accordingly, the best three hours of observation, for the elimination of the diurnal variation of the declination, are 2* 25* A.M., 10* 25* A. M., 6* 25* P.M.; 625*A.M., 2*25* P.M., 10* 25* P.M.; ON THE MEAN EESULTS OF OBSERVATIONS. 239 which coincide, almost exactly, with the best hours for the deter- mination of the mean temperature. By taking the mean of the declinations observed at any four equidistant hours, as the mean of the day, the limit of error is reduced to 0'-30. 9. It appears from the preceding, that any three equidistant observations are sufficient to give the daily mean values (and, therefore, also the monthly and yearly mean values) for each of these elements, with nearly the requisite precision ; and that, by a suitable choice of the hours, the degree of accuracy may be augmented as much as we please. But, in determining the par- ticular hours for a continuous system of observations, this should not be made the primary ground of selection. The error of the daily means being in all cases reduced within narrow limits by the method already explained, we should choose the particular hours which correspond nearly to the maxima and minima of the observed elements, so as to obtain also the daily ranges. This con- dition will be fulfilled in the case of the magnetic decimation, very nearly, by the hours 6 A.M., 2 P.M., 10 P.M. ; which will, moreover, give nearly the maximum and minimum of temperature, and of the tension of vapour, together with the maxi- mum pressure of the gaseous atmosphere* And, if we add the intermediate hours, 10 A. M. and 6 p. M., we shall have, nearly, the principal maxima and minima of the two other magnetic elements. Accordingly, for a limited system of magnetical and meteorological observations, at places for which the epochs of maxima and minima do not differ much from those at Dublin, the best hours of observation appear to be 6 A.M., 10 A.M., 2 P.M., 6 P.M., 10 P.M. The conditions of the problem are altered, if at any place the laws of the diurnal variation have been already obtained from a more extended system of observations. In this case the mean of the day may be inferred from observations taken at any hours whatever, * The ternary combination above proposed possesses the further advantage of coin- ciding, nearly, with one of those deduced above, aa the most favourable for the deter- mination of the mean temperature and mean declination. The errors of the resulting means are found by making x = 90 in the third terms of the general formulas ; and we thus find the error of temperature = - 0-07, while that of the declination = - 0'-20. 240 ON THE MEAN EESULTS OF OBSERVATIONS. by the addition of a known correction ; and the hours of observa- tion should therefore be chosen chiefly, if not exclusively, with reference to the diurnal range of the observed elements. 10. The next question which presents itself for consideration, with respect to the daily means, is one which affects more nearly the reduction of the observations hitherto made at Dublin. In the extended system prescribed by the Council of the Royal Society in 1839, and followed at the Magnetical Observatory of Dublin during the four years commencing with 1840, observations were directed to be taken twelve times, at equal intervals, throughout the day namely, at the even hours of Gottingen mean time. In a system of observations so frequent, and extending over so con- siderable a time, blanks must unavoidably occur ; and the question which presents itself here is in what way are the daily means ta be deduced in such a case ? It has been shown that the effect of the regular diurnal variation may be nearly eliminated, and the mean of the day obtained, by taking the mean of three equidistant observed values. For the elimination of the irregular changes, however, the number of ob- servations combined should be as great as possible; and in the case of the magnetic elements, in which these changes are often very considerable, this condition is an important one. Now it is obvious that the twelve results of any day may be resolved into two groups of six equidistant results, or into three groups of four, or into four groups of three. Hence, when one result is wanting in the day, the mean may be inferred either from one group of six results, from two groups of four, or from three of three. The last of these combinations, containing nine separate results, is, of course, to be preferred. When two results are wanting, the mean may be inferred from one group of four re- sults, or from two groups of three ; of which the latter combination, containing six results, is to be preferred. When three results are wanting, the mean of the day can be inferred (in general) only from one group of three ; and when more than three are wanting, that mean cannot be generally obtained. 11. What has been said above applies to the irregular changes of short period such, especially, as those to which the magnetic elements are subject. But there are also irregular changes of longer duration (as, for example, those produced in the atmo- spheric pressure by the passage of the greater aerial waves), which ON THE MEAN KESULTS OF OBSERVATIONS. 241 complicate the problem, inasmuch as a different process is required for their elimination. In the reduction of the magnetical and meteorological obser- vations made at the Observatory of Dublin, the civil day is adopted ; and the observations being made at the odd hours of Dublin mean time, very nearly, the epoch of the mean of all the twelve results is mean noon. But in the case of deficient observations, the epoch of the mean, inferred from the remaining observations, may deviate one or more hours from noon ; and its amount, therefore (as com- pared with the mean reduced to noon), is affected by an error equal to the change which the observed element undergoes in that time. In the case of the atmospheric pressure, this error is often very considerable, and much exceeds that due to the changes of whose elimination we have hitherto spoken. The law of the changes here referred to being unknown, we can only deal with them on the assumption that their course is uniform throughout the space of a day ; and this assumption will, probably, seldom err much from the truth. Upon this principle, the effect of the irregular change will be eliminated by taking the mean of two or more results equidistant from noon (that is, the mean of a forenoon and afternoon result corresponding to the hours x and 12 - x y or any combination of such means) ; and we have only to consider in what manner this process can be combined with the elimination of the regular diurnal change. Let the mean of the four equidistant observed values commenc- ing with the n th hour be denoted, for brevity, by IV ; then the epochs of the means IVi, IV 3 , IV 5 , are 10 A.M., noon, and 2 P.M., respectively ; so that the two conditions are satisfied by the com- binations | (TV, + IV.), and IV 3 . In like manner, the means of any three equidistant observed values being denoted by III,,, the epochs of the means IIIi, Ills, Ills* III 7 , are 9 A. M., 11 A.M., 1 P. M., and 3 P.M. respectively ; so that both conditions are satisfied by the combinations i (III, + III 7 ), and | (III 3 + IH 5 ). 12. When, from the number and disposition of the blanks, none of these combinations can be had, and therefore both changes (regu- lar and irregular) cannot be eliminated, we must attend chiefly to 242 ON THE MEAN RESULTS OF OBSERVATIONS, that which is greater in amount. For the purpose of comparing their magnitude, I have taken the differences of the successive daily means, for the decimation, the atmospheric pressure, and the temperature, as deduced from the observations of the year 1843; and have calculated the square-root of the mean of the squares of these differences. The results, which may be taken as the measures of the irregular changes from day to day, are the following : Mean Fluctuation from Day to Day. Magnetic decimation, . . . Fluctuation = 1/-04. Atmospheric pressure, ... =0 -214. Atmospheric temperature, . . = 3'07. Similarly, if we take the differences of the yearly means corre- sponding to the successive hours of observation, and combine them in the same way, we obtain the mean two-hourly fluctuations, arising from the regular diurnal change. These numbers are the following : Mean Fluctuation in two Hours. Magnetic declination, . . . Fluctuation = 2' f 04. Atmospheric pressure, ... =0 -0065. Atmospheric temperature, . . 1'46. These numbers, compared with the twelfth part of the former, serve to measure the relative magnitude of the regular and irregu- lar changes to which the elements are subject in the same time. We thus find that, in the case of the magnetic declination, the irregular change (which is less than -^th part of the regular) may be safely neglected ; and we have only to attend to the diurnal changes, and to the irregular changes of short period. The daily means are, therefore, to be deduced from one of the combinations of Art. 10, giving the preference to that which contains the great- est number of individual results. In the case of the atmospheric temperature, the irregular change (which is less than one-fifth part of the regular) is small ; and we must attend chiefly to the latter. The mean of the day is, there- fore, to be inferred from one of the combinations of Art. 10, giving the preference to those of Art. 11, whoe epoch is noon. In the case of the atmospheric pressure, on the contrary, the irregular change (which is triple the regular) is the more import- ant. The mean of the day is, therefore, to be deduced from any ON THE MEAN RESULTS OF OBSERVATIONS. 243 combination whose epoch is noon, giving, however, the preference to one of those of Art. 11, in which the diurnal change is also eliminated. 13. I now proceed to consider the reduction of the monthly means, in the case of deficient observations. For the purpose of determining the regular diurnal variation of any magnetical or meteorological element, it is necessary to take the mean of an adequate number of separate results corresponding to each hour of observation, so as to eliminate the irregular and accidental changes. The results usually so combined are those of each month. Their number is, in general, sufficient for the purpose above mentioned ; while, on the other hand, the course of the diurnal change is sufficiently different from one month to the next, to demand a separate determination. But in the case of deficient observations, the monthly means of the results corresponding to each hour will not exhibit, in general, the true course of the diurnal change without a correction. If a result be wanting at one hour of a day, in which all the results are much above the mean, it is obvious that the monthly mean corresponding to that hour will be too small, as compared with the means of the other hours ; while, on the other hand, it will be too great when all the results of the day in question are beloic the mean. The error will be greater, the greater the variation of the element observed from day to day. In the case of the atmospheric pressure, it is so considerable, that the uncorrected monthly means afford no approximation to the law of the diurnal change, in the case of deficient observations. The remedy which first suggests itself, in such a case, is to omit all the results of a day in which one or more are wanting. This process is inartificial and unsatisfactory. The weight of the mean is diminished in the proportion of the number of observa- tions combined ; and it is therefore important to employ all the observed results in its deduction, provided we can obtain a correction. Such a correction is easily found. 14. Let x denote the observed value of any element, at any hour on any day ; and let a denote its mean value for that day ; then x = a + K, in which is the magnitude of the diurnal variation corresponding to the hour in question. Let there be n days of observation to be R2 244 ON THE MEAN RESULTS OF OBSERVATIONS. combined ; then, summing the n results, dividing by , and denoting the mean values by x, , and f , Now, at any particular hour of any day, let one of the results be wanting ; and let a' denote the mean for that day ; summing the n - 1 ^results, Sn-i* = S n a-a + Sn-it. And dividing by n - 1, 8a -a f a- a' a + r- = - , n-1 n-1 whence a'-a x + = - a n-1 The correction, therefore, is + j. Similarly, if ^> results be wanting, we find Set -pa . x + = a + c, -|i in which Sa' denotes the sum of the means of the days on which the deficiencies occur. Hence, the correction to be applied to the observed mean, x, deduced from the n -p values, is + =". n-p 15. The preceding correction depends, as might have been anticipated, on the difference of the daily means, for the days of deficient observations, and the mean daily mean. "With the view of determining its probable amount, I have taken the differences between the mean of each day, and the mean of the month, for the declination, the atmospheric pressure, and temperature, as deduced from the observations of the year 1843 ; and have calculated the square root of the mean of the squares of these differences, or the values of the expression li L =Ll. The values of this quantity, which may be denominated the mean daily error, are the following : ON THE MEAN RESULTS OF OBSERVATIONS. 245 Mean Daily Error. Magnetic declination, . . Daily error = 0'-95. Atmospheric pressure, . . =0 '301. Atmospheric temperature, . = 4'25. Now, the mean value of n in each month (Sundays being omitted) is 26. Hence the mean correction, in the case of a single deficient observation, is, for the magnetic declination, 0' - 04 ; for the atmospheric pressure, 0'012 ; and for the temperature, 0-17. In the case of the two meteorological elements, and especially in that of the atmospheric pressure, the correction is too considerable to be overlooked ; in the case of the magnetic declination, and probably also in that of the other magnetic elements, it may be disregarded. X. ON THE DETERMINATION OF THE HORIZONTAL INTENSITY OF THE EARTH'S MAGNETIC FORCE IN ABSOLUTE MEASURE. Transactions of the Royal Irish Academy. Vol. XXL THE attention of mathematicians and experimentalists has been, for some time past, directed to the means of determining the intensity of the earth's magnetic force in absolute measure. These means consist, it is well known, in observing the time of vibration of a freely-suspended horizontal magnet, under the influence of the earth alone, and then employing the same magnet to act upon another, which is also freely suspended, and noting the effects of its action combined with that of the earth. From the former of these observations we deduce the product of the horizontal component of the earth's magnetic force into the moment of free magnetism of the first magnet, and from the latter, the ratio of the same quantities ; and, the product and the ratio being thus known, the two factors are absolutely determined. The former part of this process involving no difficulty which may not be overcome by due care in observing, we shall confine our attention, in the present communication, to the latter. Two methods have been proposed for this second observation, one by Poisson, and the other by Gauss. The method of Poisson consisted in observing the time of vibration of the second magnet, under the combined action of the first and of the earth, the acting magnet having its axis in the magnetic meridian passing through the centre of the suspended magnet. In the method of Gauss, which is now universally adopted, we observe the position of equilibrium of the second magnet, resulting from the action of the same forces. The acting magnet being placed transversely with INTENSITY OF THE EAETH's MAGNETIC FORCE. 247 respect to the suspended one, the latter is deflected from the meridian, and the amount of this deflection serves to determine the ratio of the deflecting force to the earth's force. The position chosen by Gauss for the deflecting magnet is that in which its axis is in the right line passing through the centre of the suspended magnet, and perpendicular to the magnetic meridian, in which case the tangent of the angle of deflection is equal to the ratio of the two forces. From this ratio it remains to deduce that of the magnetic moment of the deflecting bar to the earth's force. The difficulty of this process arises from the form of the expression of the force of the deflecting bar. This force being expressed by a series descending according to the negative odd powers of the distance, with unknown coefficients, it is evident that observation must furnish as many equations of condition, corresponding to different distances, as there are terms of sensible magnitude in the series ; and from these equations the unknown quantities are to be deduced by elimination. Now, the greater the number of unknown quantities thus eliminated, the greater will be the influence of the errors of observation on the final result ; and if, on the other hand, the distance between the magnets be taken so great, that all the terms of the series after the first may be insensible, the angle of deflection becomes very small, and the errors in its observed value bear a large proportion to the whole. It fortunately happens, that at moderate distances (distances not less than four times the length of the magnets) all the terms beyond the second may be neglected. The expression for the tangent of the angle of deflection is thus reduced to two terms, one of which contains the inverse cube of the distance, and the other the inverse fifth power ; that is, if u denote the angle of deflection, and D the distance, in which Q and # are unknown coefficients, the former of which is double of the ratio sought. Accordingly, the method recom- mended by Gauss consists in observing the angles of deflection, u and u', at two different distances, D and IX, and inferring the coeflicient Q by elimination between the two resulting equations of condition. 248 ON THE DETERMINATION OF THE HORIZONTAL The object of the present Paper is to point out the means by which the quantity sought may be obtained, without elimination, from the results of observation at one distance only ; and thus not only the labour of observation be diminished, but (which is of more importance) the accuracy of the result increased. Before entering on this, however, it will be expedient to ascertain the amount of the probable error in the received method. The coefficient of the first term, obtained by elimination between the two equations of condition above alluded to, is The distances being greater than four times the length of the magnets, the angles of deflection are small, and there is, approxi- mately, t&mi = ^(tan^, tan*/ = w'tanl', u and u' being expressed in minutes; and making If = qD, the preceding expression becomes The probable errors of u and a' are equal ; and, by a well-known theorem of the calculus of probabilities, the probable error of Q is ^TT AM ' In determining the ratio of this error to the quantity itself, we may observe that there is, approximately, 21510 258 ON THE DETERMINATION OF THE HORIZONTAL V. Magnet away, Scale reading = 495-1 498-3. !(.-*,) u D 3 tan n 368-46 5 54,' 52" -5 20233 211-99 3 26 8 20261 133-41 2 10 6 20292 89-39 1 27 16 20313 63-00 1 1 32 20390 45-99 44 55 -5 20420 1-25 1-50 1-75 2-00 2-25 2-50 These results verify the conclusions to which we have arrived above. The values of the function D 3 tan u are constant for all distances in the first three series, the differences in the resulting values being less than the probable errors of observation ; and, consequently, the coefficient of the inverse fifth power of the distance is insensible. In the fourth and fifth series, on the other hand, in which the lengths of the magnets are equal, the values of this function form an increasing series, as D increases; and therefore, in this case, the coefficient of the inverse fifth power of the distance has a sensible negative value. We may further employ these results to test the accuracy of our conclusions, by deducing from them the values of the two coefficients, in the expression for the tangent of the angle of deflection, and comparing their ratio with that furnished by theory. It is needless to make this computation for the numbers of the first three series ; for it is manifest from the results, that the second coefficient is insensible, as theoretically it should be. From the results of IV. we deduce, by the method of least squares, Q = -2148, hQ = - -0017, h = - -0078. We obtain, in like manner, from the results of V., Q = -2037, hQ = - -0022, h = - -0110. And the mean of the resulting values is - '0094. Now, in these two series, the length of each of the magnets was three inches ; that is, I = I' = -125, the half lengths being expressed in feet. Substituting these values in the expression for h, it becomes h = - -0094, agreeing exactly with the mean of the experimental values. INTENSITY OF THE EARTH'S MAGNETIC FORCE. 259 We are therefore justified in concluding that, in the case of small magnets, the ratio of the two coefficients may be inferred j), jj denoting the actual inclination of the needle to the horizon. This moment is opposed by that of the added weight, or by Wr, W being the weight, and r the radius of the pulley by which it acts ;f and the equation of equilibrium is therefore m ( Y cos ri - X cos a sin >j) = irr. When the needle is removed, in the second part of the process, * froecedinffs of the Royal Iruh Academy, January 24, 1848. t It is here supposed that the weight is attached to a fine thread pasiii light pulley, whose centre is on the axis of the cylindrical axle of the needle, in manner proposed by Mr. Fox. If the weight he attached to the s< needle, at a fixed point, its moment is icr COST;. 262 OX THE DETEKMINATION OF THE TOTAL and applied to deflect another substituted in its place, the moment of its force to turn the latter is mm 17', in which m' is the moment of free magnetism of the second needle, and U a function of the distance of the centres of the two needles, and of certain integrals depending on the distribution of free magnetism in them. The moment of the earth's magnetic force, opposed to this, is of the form already assigned, in which we have- only to substitute ', r/, and a', for m, 11, and a. Hence the equa- tion of equilibrium is Y cos i)' - X cos of sin r{ = m U ; (2) the quantity m' disappearing from the result. The magnetic moment of the deflecting needle, m, is eliminated from equations (1) and (2) by multiplication ; and we thus obtain a single re- lation between the intensity of the earth's magnetic force, the observed angles a, TJ, a', rj', and the quantities ', r, and TJ. Hence the magnetic intensity will be determined when these are known. There are three obvious cases of these formulas, each of which suggests a different method for the determination of the terrestrial magnetic intensity. 1. When the planes in which the needles move coincide with the magnetic meridian, or a = 0, and a = 0, the left-hand members of (1) and (2) are reduced to mR sin (0 - ij), R sin (0 - r{) ; R denoting the total force, and the inclination. "Wherefore, by multiplication, we have R 2 sin (0 - n) sin (0 - ,,') = Uwr. (3) 2. When the planes in which the needles move are per- pendicular to the magnetic meridian, or a - 90, and a = 90, the left-hand members of (1) and (2) become, respectively, w*Fcosj, FCOSTJ' ; whence Y- cos j cos i{ = Uicr. (4) 3. Finally, the equilibrium may be produced, in both cases, by turning the instrument in azimuth until the free needle stands ver- tically. In this case >, = 90, r,' = 90, and the left-hand members become - mX cos a, - X cos a ; whence X- cos a cos a = Vur. (5) Thus we may apply this principle to the determination of the total intensity, or to that of either of its two components. INTENSITY OF THE EARTH'S MAGNETIC FORCE. 263 In comparing the foregoing methods, it is to be observed that the third fails when the inclination approaches to 90, on account of the magnitude of the error of R resulting from a given error of 0, when the total force is deduced from its horizontal component. In like manner, and for the same reason, the second method fails in the vicinity of the magnetic equator, or line of no inclination. The first alone is applicable at all parts of the earth's surface, and I proceed to consider it more in detail. The observed angles, rj and ?/, are liable to error, the friction of the needles on their supports causing them to rest in positions slightly different .from those due to the acting forces. The pro- bable errors of TJ and /, due to this cause, vary with the angles themselves. To determine their magnitude in any case, we have mR sin (0 - TJ) = F, F being the moment of the deflecting force ; and when friction is taken into account, mR sin (0 - rj + Arj) = F + f; f denoting the moment of friction, and rj - Arj the new angle of equilibrium. Developing the latter equation, and subtracting the former, mR cos (0 - rj) AJ/ = ./' ; the angle A) being expressed in parts of radius. Hence, cos (0 - ri) A j is constant with a given instrument, and at a given point of the Earth's surface. To find the probable error of the force corresponding to the error of the observed angle, we must differentiate the equation of equilibrium, mR sin u = F, with respect to R and w, where u = - r j ; and we have A R sin u + R cos u A = 0. But i / \ u 5 I 1 ? 1 ~ 1v tj, and rj, being the observed angles of inclination under ill opposite actions of the deflecting force. Hence the probable error of u is - 1 Au = i -y/Arj'i + ATJ".. = , ' ' v/x, 264 ON THE DETERMINATION OF THE TOTAL since Ar/ 3 = Arn. Accordingly, the second term of the preceding equation becomes R cos u Aj = - ; and we have &R = We learn, then, that the probable error of the force varies inversely as the sine of the angle of deflection ; and that it is therefore requisite for accuracy that this angle should be con- siderable. There is no difficulty in augmenting the angle of deflection as much as we please in the first part of the process, in which the magnet is deflected by a weight ; but in the second the case is different, and with the slender needles to be employed as deflectors, a large deflection can only be obtained by placing the deflecting needle at a very short distance from the moveable one. The most convenient arrangement appears to be to attach the deflecting needle to the moveable arm of the divided circle which carries the verniers, and at right angles to the wires of the microscopes.* So attached, it must always be rendered perpen- dicular to the deflected needle in the course of the observation, although in a different plane. The form of the function denoted by 77, in this position, is easily found. Let the distances of any points of the axes of the deflecting and deflected magnets from their respective centres be denoted by r and r', and let /u and // denote the quantities of free magnetism at these points, contained in the slices perpendicular * To obtain the value of AS by observation, we must substitute for / its value given above. But when ij = 6, or when the needle is undeflected, / = mltAO ; wherefore V 2 sin u ' In the instrument with which I made trial of this method, the length of the needles was 3J inches, and the angle of deflection produced, in the position of the deflecting needle here described, was 24 10'. But the probable error of a single reading of the inclination, obtained by repetition the needle being lifted off the agate planes between the successive readings-was l'-6 ; and if four readings (which is a very usual number) e taken, the probable error of their mem will be one-half of this. In this case, therefore, A0 = 0'-8 ; and the probable error of the deduced force, computed by the preceding formula, is AS - -0004 Jt. to the axes whose thicknesses are dr and dr. Then the force exerted by the former upon the latter is MI drdr ~* ' s denoting their mutual distance. The portion of this force contained in the plane of the deflected magnet, and perpendicular to its axis, is and the moment of this force to turn the magnet is obtained by multiplying by r. But s 2 = D 2 + r 2 + r\ T) being the distances of the centres of the two magnets; and accordingly the total moment of the acting forces is fir dr . f/r'dr nurdr.f, (D 2 + r Expanding the denominator, and making, for abridgment, = i)h, &c., = ' a , &c., in which the integrals are to be taken between the limits = + /, r =T 9 I and I' being half the lengths of the two magnets, this becomes -i/r -j o K JJ - " 3 mm' - (m 3 m r + mm' 3}-- + -^-7 (rn^ti + 2m 3 m\+mm f ,}^ + &c. j , or mm'U, in which 15 Now it is to be observed, that the variations of the ratios ^ ^ & c arising from the variations of v, are of a lower order m m of magnitude than that of m, and may be disregarded in then- effect upon the value of U* On the supposition that the quantity * This circumstance was first pointed out by Dr. Lament. 266 ON THE DETERMINATION OF THE TOTAL of free magnetism, at any point of a magnet, is proportional to the distance from the centre, or that /u = AT, we have * = A-/ 3 , z 3 = | A-/ 3 , w 5 - 1 M\ and when & becomes k + SA-, these values will all be altered proportionally, and consequently the ratios '-, , &c., will be absolutely unchanged ; and the same thing is manifestly true, if the quantity of free magnetism be supposed to vary as any simple power of the distance, whether integer or fractional. This is a point of considerable importance in reference to the method now proposed. For it follows that, at a given distance between the two needles, the function V may be regarded as constant; and therefore that, even when U is unknown, the value of R will be relatively determined, by a process which is inde- pendent of the changes induced by time in the magnetic moments of the needles employed. Accordingly, if the value of the force be found at any one place, by any independent means, it will be absolutely known at all ; and it is only necessary that the observer should include in his series an observation at some base-station, at which the absolute value of the force is determined simultaneously by the ordinary method. I now proceed to show, however, that the value of the constant V may be found by deflection, by the instrument itself, and without any subsidiary apparatus ; and that the method may therefore be rendered rigorously absolute. It is obvious that the ordinary process is inapplicable in this case, owing to the large number of terms which acquire a sensible value, in the value of the function U, and the consequent difficulty and uncertainty of the elimination : moreover, the position which has been adopted for the deflecting needle will not admit of the required alteration of distance. Now here I premise, that it is not necessary that the usual deflection distance should be one of the series employed in deducing the coefficients of the inverse powers of the distance in the value of U: it is not even requisite that the relative positions of the two magnets should be the same in the two cases. For if the value of the corresponding function be found, for any other position, and at any distance, that of U will be known by a comparison of the deflections produced. Accordingly, 1 propose INTENSITY OF THE EARTH'S MAGNETIC FORCE. 267 to determine, in the first place, the value of the corresponding function in a different relative position of the two magnets, and by means of deflections at the usual distances; and thence to conclude that of U in the position of the magnets here employed. In using the dip-circle for this purpose, it will be found most convenient to adopt the third of the methods above described, in which the equilibrium is produced by turning the instrument in azimuth until the deflected magnet becomes vertical ; for in this case the deflecting magnet is always horizontal, and can be placed in the usual position with respect to the deflected magnet without difficulty. For this purpose the apparatus is provided with a gun- metal bar, the middle of which is broad, and has a rectangular aperture which enables it to pass over the box containing the deflected magnet : this bar rests horizontally on two supports fixed outside the box, on the level of the agate planes. The deflecting magnet is to be placed on this support at different known distances, and on each side of the deflected magnet, its axis being in the plane in which the latter moves ; and the apparatus is to be turned in azimuth until the deflected needle is vertical. In this case equation (2) becomes - X cos a = m V ; in which V is of the form where n = --- - - , q = -- ----- / - 6 --- ,,i m m m m 8 m /., -HI r, ,. flaw's 45 ;;/.-, - 3- , q = 3 -- 15 ----- / + - 6 --- 7> Let F,, F-, F 3 , &c., denote the values of V corresponding to the distances D,, A, -D 3 , &o. ; and a,, a 2 , a ;i , &c., the corresponding azimuths observed ; then it is obvious that V-. cos ! - Fi cos a, = 0, F 3 cos ai - F, cos a a = 0, & from which equations the values of the coefficients j>, J . obtained by elimination in the usual manner. IIen- / is com- pletely determined. Now let the deflecting magnet be removed from the horizontal bar, and placed in its ordinary position between the microscopes ; and let the observation be it-peat ed, tin- tnrfniment being turned 268 ON THE DETERMINATION OF THE TOTAL in azimuth until the deflected needle is vertical. Then, if a,, denote the corresponding azimuth, we have - X cos a y = m U ; whence there is U = F cosa . cos a Thus the uncertainty of the result, arising from the smallness of the angle of deflection at the usual distances, is removed from the regular series of observations, and is thrown upon the determina- tion of the constant, which may be made at leisure, whenever convenient, and may be repeated as often as is required for accuracy. In speaking of U as constant, and independent of the changes of the magnetic moments of the needles, I have expressly limited the statement to those small and regular losses of magnetism which occur in time. It would not be safe to extend the assump- tion to the case of the larger changes brought about abruptly by concussion, or by any other accidental cause; and still less to those in which the magnetic distribution of the needles is altered by contact with, or proximity to, magnetic bodies. In such cases (the occurrence of which is easily detected), the value of the constant U should be re-determined. It may be useful to add a few words respecting the order of the observations. The apparatus should be furnished with three needles, all of the same size, viz., 3^ inches in length. One of these (which we shall denote by the letter A) is to be employed in observations of inclination : the other two, B and 0, are to be used in the observations of intensity, B being the loaded needle, which is also used as a deflector, and C the deflected needle. The two latter needles should not have their poles reversed, nor be touched with a magnet. The observations may be conveniently taken in the following order : 1. Needle A is to be placed on the agate planes, the other two needles being removed, and a complete observation of inclination taken in the four usual positions of the needle and limb, and with the poles of the needle direct and inverted. INTENSITY OF THE EARTH'S MAGNETIC FORCE. 260 2. Needle A is then to be removed, and the loaded needle, B, substituted ; and its inclination to the horizon, TJ, is to be observed in the four positions of the needle and limb. The deviation of this needle from the position due to the earth's magnetic force alone is u = 9 - j, the angle 7j being positive when measured at the same side of the horizontal line with 0, and negative in the contrary case. 3. Needle B is now to be removed from the agate planes to its supports between the microscopes, and needle C substituted ; and the inclination of the latter to the horizon is then to be observed in one position of the needle and limb. The observation is to be repeated with the north end of needle B turned in the opposite direction, by the revolution of the moveable arms which cany the microscopes ; half the difference of the readings in the two- positions is the angle of deflection, u'. The total intensity is given by the formula sm u sin u which is fitted, without any artifice, for logarithmic computation.* In strictness a correction is required for the effect of the change of temperature of needle B, in the two observations in which it is employed ; but as one of these observations may be made to follow the other quickly, and as the needle may be placed in both in nearly the same circumstances, the correction may generally be disregarded. It will be necessary, however, that this needle, when employed as a deflector, should be protected from the heat of the observer's body by a small case of glass or of metal. The method here proposed appears to offer the following advantages to the travelling observer : 1. It is applicable, with equal accuracy, at all parts of the globe. 2. It dispenses with the employment of a separate instrument * If the weight be attached to the loaded needle at a fixed point, the formula becomes 270 ON THE DETERMINATION OF THE TOTAL INTENSITY. for the determination of the magnetic intensity, and with the separate adjustments required in erecting it. 3. The constants to be determined the magnitude of the added weight, and the radius of the pulley by which it acts, can be ascertained with more ease and certainty than those which are required in the method of vibrations, and are less liable to subsequent change. 4. The observations themselves are less varied in character than the usual ones, and may be completed in a shorter time. XII. OX EARTH CURRENTS, AND THEIR CONNEXION WITH THE DIURNAL CHANGES OF THE HORIZONTAL MAGNETIC NEEDLE. Transactions of the Royal Irish Academy. Vol. XXIV. 1. WHEN the discovery of Oersted had made known the con- nexion which subsists between magnetism and current electricity, the idea occurred to many that the magnetism of the earth or, at least, its diurnal fluctuation, was the result of electric currents traversing its crust. This idea gained much force from the fact, soon after discovered by Seebeck, that electric currents are generated when heat is applied to a circuit composed of different metals ; and it was supposed that the phenomena were thus traceable to the thermal agency of the sun, operating in succes- sion upon the conducting substances of which the earth's crust is composed. The most explicit statement, and chief support of this hypo- thesis, is contained in a memoir by Professor Christie, published in the Philosophical Transactions for 1827. In this memoir it is maintained that the phenomena of the diurnal variation of the horizonal magnetic needle correspond with those which are pro- duced experimentally, by the application of heat to a globe composed of two metals ; and the author was even led by his experiments to anticipate the fact, at that time unknown, that at opposite sides of the equator the poles of the needle, having the same name as the latitude, are always deflected in the same direction. I propose, in the present memoir, to examine this theory apart from the hypothesis as to the origin of the currents ; and to show, 272 ON EARTH CURRENTS, AND THEIR CONNEXION that the diurnal variation of the horizontal magnetic needle is due to electric currents traversing the earth's crust. 2. The first and most important step towards the result above referred to has been made by Mr. Barlow. In a Paper* com- municated by him to the Royal Society in the year 1848, he established the important fact, that a wire, whose extremities are connected with the earth at two distant points, is unceasingly traversed by electric currents, the intensity of which varies with the azimuth of the line joining the points of contact with the ground. The direction of these currents was proved to be the same at both extremities of the same wire, and was shown to depend on the relative positions of the Earth-connexions, while it was wholly independent of the course followed by the wire itself. The currents cease altogether when either of the contacts with the earth is interrupted. From these facts Mr. Barlow concluded that " the currents are terrestrial, of which a portion is conveyed along the wire, and rendered visible by the multiplying action of the coil of the galvanometer." Mr. Barlow further observed that, apart from sudden and occasional changes, the general direction of the needle of the galvanometer appeared to exhibit some regularity. He was thus led to institute a series of observations for fourteen days and nights simultaneously on two telegraphic wires, one connecting Derby and Eugby, and the other connecting Derby and Birmingham, the positions of the needles in both circuits being recorded every five minutes, day and night. From these observations he con- cluded " 1. That the path described by the needle consisted of a regular diurnal motion, subject to disturbances of greater or less magnitude. " 2. That this motion is due to electric currents passing from the northern to the southern extremities of the telegraph wires, and returning in the opposite direction. " 3. That, exclusive of the irregular disturbances, the currents flowed in a southerly direction from about 8 or 9 A.M. until the evening, and in a northerly direction during the remainder of the twenty- four hours." * " On the spontaneons Electrical Currents observed in the wires of the Electric Telegraph." Phil. Trans., 1848. VARIATIONS OF THE MAGNETIC NEEDLE. 273 Mr. Barlow was thus led to examine whether any relation subsisted between these movements and the daily changes of the horizontal magnetic needle. And having made, for this purpose, a series of simultaneous observations with a delicate declinometer, he came to the conclusion that, although generally the currents flow southwards during that part of the day in which the variation of the horizontal needle is tcesterly, and northwards when the vari- ation is easterly, " yet simultaneous observations showed no simi- larity in the paths described by the magnetic needle and the galvanometer." An examination of the galvanometrio observations above re- ferred to led me, some time since, to an opposite conclusion ; and at the last meeting of the British Association I stated my convic- tion, founded on these observations, that the earth-currents, whose continuous flow Mr. Barlow has the merit of establishing, would eventually explain the variations of terrestrial magnetism, both periodic and irregular. I now proceed to state some of the grounds of this conviction ; and in the present paper I hope to show that the diurnal changes of the earth-currents correspond with those of the horizontal component of the earth's magnetic force. 4. Before proceeding to the detailed examination of this .corre- spondence, it is necessary to advert to a difference between the things compared, which will account for the method employed in the com- parison. The cause of this difference will be adverted to hereafter. When we examine the curves,* in which Mr. Barlow has re- presented the course of the galvanometric deflections caused by the earth-currents, we observe that the regularity of that course is con- tinually interrupted by rapid reciprocating movements, in which the needle oscillates from one side to the other of the zero, alter- nately. These movements are very different in magnitude at different times: thus, May 29 was a day of considerable movement ; May 25, one of comparative rest. These movements are similar to those of the magnetometers, with which we are familiar ; but they are much more rapid, and bear a larger proportion to the regular changes. Among the days of observation whose results are given in this diagram, it fortunately happens that one, viz., the day commencing May 28, 10 p. M. (Gdttingen time), was a " term-day " of the magnetic system, during which the magneto- meters were observed every five minutes at the Greenwich Obser- * Philowphical Transaction*, 1848, Diagram, No. 1. T 274 ON EARTH-CURRENTS, AND THE DIURNAL vatory, and consequently at the same intervals of time as the needles of the galvanometers in Mr. Barlow's observations. I find, upon an examination of the two records, that there were on that day, on the average, 5-1 alternations of the galvanometer needles in the hour, the corresponding number of alternations of the magneto- meters being only 3-1. The numbers are very nearly the same for the two galvanometers of the two lines of telegraph, as well as for the two magnetometers of the Greenwich Observatory. And the disproportion in the magnitude of the deflections, as compared with the amount of the regular daily changes, appears to be even more considerable. The frequency and the magnitude of the deflections may both be taken into account, by adding together the alternate changes, with- out regard to sign, and dividing the sum by the regular daily changes. I have selected for this calculation the observations made during the six hours commencing at 3 A. M. on May 29, that being a period of comparative disturbance. The sum of the changes of the galvanometer needle during that period, on the Derby and Rugby line, was equivalent to 571 divisions of the instrument, the mean daily range for the entire week being 11 "4 divisions: and the ratio = 50. The corresponding ratio, for the galvanometer of the Derby and Birmingham line, is somewhat smaller. The sum of the changes of the Greenwich declinometer during the same period was only 57 minutes, the mean daily range being 12*4 minutes. In like manner, the sum of the changes of the horizontal force (in parts of the whole) was -0158, the mean daily range being -0034. The ratio is accordingly the same for the two magnetic elements, and its amount is 4'6, or less than one-tenth of the corresponding ratio in the case of the galvanometric changes. We learn there- fore that the rapid changes of the earth- currents are much greater, in proportion to the regular daily changes, than the corresponding movements of the magnetometers. The physical interpretation of the foregoing facts will be con- sidered in the sequel. For the present, I shall merely notice the consequence which appears to flow immediately from them namely, that little or no accordance is to be expected, in comparing the individual remits of the two classes of measures. I have verified this conclusion by a comparison of Mr. Barlow's diagram, for the period above referred to, with the corresponding Greenwich observations. VARIATIONS OF THE MAGNETIC NEEDLE. 275 5. For the foregoing reasons, the magnetometric observations are here compared, not with the single corresponding deflections of the galvanometer needles, but with the means of several. The observations instituted by Mr. Barlow on the two lines of telegraph were taken at intervals of five minutes, and were continued for fourteen consecutive days, commencing May 17, 1847. In the Tables given in his Paper, the individual results are grouped into means corresponding to the beginning of each hour ; so that each number of the Table is the result of twelve distinct readings. The means so obtained are, notwithstanding, more affected by irregularities than the single readings of the magnetometers cor- responding to the same epochs ; and I have found it advisable to group the results still further, by taking the mean of each pair of succeeding hourly means. The following Tables, accordingly, contain the means of the results belonging to the hours, 1 A.M. and 2 A.M., 3 A.M. and 4 A.M., &c. ; they therefore correspond to the epochs, P 30 m , 3 h 30 m , &c., each number being the mean of twenty-four single readings. The positive numbers indicate currents proceeding towards Derby ; and the negative, currents in the contrary direction. It is only necessary to observe further, that of the fourteen days of observation, four, viz., May 17, 23, 24, 30, are very incomplete, and are therefore omitted altogether in this place : TABLE I. Intensity of the Currents traversing the Wire connecting Derby and Rugby. A.M. MAY. I'- :jo m 3 1 ' 30 m 5 h 30 m 7 h 30 m 9 h 30 m J.l h 30 18 + 1-9 -3-5 -1-7 4- 1-8 4-4-9 - 1-4 1!) + 4-6 + 1-4 -0-8 4- 4-5 -0-5 - 3-S *>() m - 1 ,-, - 2-< 21 + 0-6 + 0-3 + 1-2 4 0-2 41-1 - 2-4 I'l + 3-2 + 2-5 41-0 + 2-8 -2-3 - 5-2 25 -1-3 43-6 46-5 + 1-8 -0-6 - 3-2 26 + l-s + 5-0 + 3-1 + 2-2 -1-7 - M 27 -0-2 + 0-6 45-0 + 4-7 + 1-9 - 2-5 28 -2-1 -1- 1-4 + 2-4 + 10-3 .V! - -1-1 29 -2-9 4 0-1 -0-4 + 2-8 -2-1 - 12-2 Means 4 0-9 4 1-3 4 1-7 4 3-.J 40-5 - !_' 276 ON EAETH-CUERENTS, AND THE DIUENAL TABLF, I. continued. \ P.M. li>30> 3 h 30 m 5 h 30 m 7 h 30 9 h 30 ll'> 30 m 18 - 7-5 -8-3 -8-8 - 1-5 + 3-5 + 6-1 19 - 0-1 -0-4 . , . . . . 20 - 2-7 - 1-9 -2-1 -8-6 + 1-3 + 5'5 21 - 2-2 -2-4 -2-0 -5-2 -3-4 + 2-6 22 - 1-8 -1-2 -1-7 -2-2 -2-3 -1-1 25 - 2-7 + 0-8 + 0-6 o-o -1-0 + 1-4 26 - 3-9 + 0-7 + 4-4 + 3-1 + 1-8 -0-1 27 - 4-3 -0-5 + 0-6 -4-7 + 6-5 + 4-7 28 - 7-8 -0-9 -2-9 + 4-2 + 3-5 + 1-9 29 - 10-6 -6-6 -6-9 -1-0 -0-5 -3-5 Means - 4-4 - 2-1 -2-1 | -1-7 + 1-0 + 1-9 TABLE II. Intensity of the Currents traversing the Wire connecting Derby and Birminylunn* lh 30* 3>>30 m 5 h 30 ra 71, 30 m 9 h 30 m ll''30' 18 + 0-3 -9-7 -5-7 + 5-7 + 6-6 - 10-0 19 + 3-6 + 6-0 -7-0 + 9-7 -0-6 - 10-3 20 + 8-7 -2-6 -1-1 + 7-2 + 1-1 - 8-5 21 + 2-1 + 1-7 + 5-0 + 1-5 + 7-6 - 9-0 22 + 9-7 + 8-5 + 5-2 + 9-9 -6-0 -17-5 25 + 1-0 + 2-4 + 3-0 + 0-4 -0-6 - 3-1 26 + 1-8 + 3-4 + 1-2 + 0-8 -1-6 - 4-7 27 -0-8 + 0-3 + 2-3 + 2-4 + 0-5 i - 2-5 28 -1-7 + 0-4 + 1-4 + 8-4 + 5-1 - 3-7 29 -2-4 -1-1 + 1-4 + 7-8 + 1-4 - 5-7 Means + 2-2 -t-0-9 + 0-6 + 5-4 + 1-4 - 7-5 i P. M. MAT. Ih 30 m 311 30 m 5 h 30 7 h 30 9 h 30 m II 1 ' 30 18 -17-4 -21-1 -10-6 - 4-2 + 3-7 + 5-6 19 - 9-2 - 10-8 -18-2 -13-0 + 2-4 + 9-0 20 - 11-6 - 9-9 - 6-1 -10-0 + 2-9 + 11-6 21 -11-3 -12-8 - 7-0 -16-7 -5-2 + 6-4 22 - 9-3 - 7-3 - 8-9 - 9-3 -8-6 - 2-2 25 - 3-2 - 0-6 -- 1-1 - 1-5 - 1-7 + 1-0 26 - 5-5 - 3-6 + 2-3 + 0-3 + 0-1 - 2-1 27 - o-() - 1-8 - 2-1 - 7-7 + 4-4 + 4-2 28 ; - 8-0 - 2-9 - 6-1 -i- 1-4 + 0-7 + 1-6 29 - 4-2 j - 2-6 - 3-0 + 2-5 + 2-2 + 0-8 Means - 8-5 - "-3 -6-1 -5-8 +0-1 + 3-6 VARIATIONS OF THE MAGNETIC NEEDLE. 277 The means corresponding to each epoch are given in the lowest lines of the preceding Tables. On examining them, it will be seen that the law of the diurnal changes in the force and direction of the currents is very systematic. In both lines the current flows southwards from ll b 30 m A.M. to 7 h 30 m P.M. inclusive; and northwards at the remaining epochs. The maximum of the southerly current occurs at l h 30 m P.M., and that of the northerly current at 7 h 30 m A.M. 6. Let us now compare these results with those deducible from the diurnal changes of the magnetic declination and horizontal force, on the assumption that the forces which produce the latter are due to electric currents traversing the upper strata of the earth in a horizontal direction. Let rj denote the disturbing force by which the north pole of the magnet is urged to the eastward of its mean position ; , that by which it is impelled northward. Then, on the assumption above* referred to, the force of the current in the magnetic meridian, flowing northward, = aij, a being an unknown constant ; and that of the current perpendicular to the magnetic meridian, flowing eastward, = - . Hence, the force of the current in any direction, making the angle e with the magnetic meridian measured to the east of north, is / = a (n cose -K sins). The quantities r? and % are given, in terms of the horizontal component of the earth's magnetic force, by the readings of the two horizontal magnetometers. For it is evident that X being the horizontal component of the earth's magnetic force, uiid \ft the magnetic declination ; and = X - X = $X. 7. Now, to reduce the preceding formula into numbers, we have e = a - 1//, in which a is the azimuth of the line connecting the two stations, measured from the true meridian eastward. The observations of Sir James Boss, at Derby, give = - 22 25' ; 278 ON EARTH-CURRENTS, AND THE DIURNAL and for the line connecting Derby with Bugby, = - 13 7', = + 9 18'. Introducing these values in the expression for /, (85T\ 000287^-0-16-^-). A / The following Table gives the values of the quantity within the brackets in this formula, for the days on which the force of the currents was observed, and for the even hours of Gottingeu $X mean time, the unit being 10 ^ 00 . The values of >20 ra | 5 h 20 m 7 h 20 m 9 h 20 m Il h 20 m 18 + 13-8 + 10-7 + 12-0 + 9-4 + 2-6 -14-2 19 - 2-9 + 0-5 + 4-3 + 9-5 - 0-3 -21-6 20 + 12-6 + 9-1 + 21-7 + 16-5 + 7-0 - 17-7 21 - 0-8 + 6-8 + 7-9 + 14-7 4- 9-5 , - 12-8 22 + 7-2 + 14-1 + 7-8 + 14-9 + 11-8 -13-1 25 + 4-7 + 7-9 + 19-2 + 20-8 4- 10-6 - 5-1 26 + 14-7 + 16-5 + 16-5 + 17-2 + 0-5 -19-7 27 - 0-1 + 2-8 + 13-7 + 19-0 + 11-3 -10-9 . 28 - 2-6 + 12-0 + 9-8 + 23-1 + 8-1 - 7-7 ' 29 + 4-0 + 7-1 + 28-3 - 3-8 -f 2-9 -17-8 Means + 5-1 + 8-8 + 14-1 + 14-1 4-6-4 -14-1 P. M. MAY. l>>20 m 3 h 20 m i 5 h 20 ra j 7 h 20 m 9 h 20 m Il h 20 ra 18 -27-4 -20-5 - 2-9 -5-9 -4-2 - 3-5 19 - 24-2 - 9-0 - 1-5 -3-9 -0-3 + 5-5 20 -17-7 - 4-8 - 2-7 + 1-5 + 1-3 4- 1-7 21 -19-8 - 8-6 + 0-9 + 7-9 + 1-6 + 3-8 22 -22-6 - 6-0 - 3-0 -1-5 + 1-2 + 1M 25 - 16-2 - 8-4 - 3-8 -0-8 -0-5 + 8-1 26 - 29-4 - 19-3 | - 11-0 -1-9 -0-9 + 3-6 27 -19-6 - 9-6 - 1-2 -5-7 -1-9 + 12-9 28 -20-1 -10-3 - 5-0 -0-7 + 2-6 + 3-1 29 -23-0 -13-2 - 8-6 -7-1 -1-4 + 2-7 Means -22-0 -11-0 -3-9 -1-8 -0-3 4 4-n 8. For the line connecting Derby and Birmingham, we have = 4- 33 27', = 4- 55 52' ; and, introducing these values, the formula becomes g Y 0001C3 fy - 0-83 ~ 280 ON EARTH-CURRENTS, AND THE DIURNAL The values of the quantity within the brackets, for the same days and hours, are given in the following Table : TABLE IV. Calculated Values of the Intensity of the Earth- Currents, in the Line connecting Derby and Birmingham. A. M. MAT. l>>20 m 3 h 20 m 5 h 20 7 h 20-0 9 h 20 m ll h 20 18 + 13-8 + 11-1 + 13-4 + 25-4 + 24-2 + 8-9 19 - 7-5 - 6-3 - 3-4 + 8-2 + 17-7 - 1-3 20 + 11-7 - 0-3 + 12-4 + 10-4 + 25-2 + 2-5 21 + 1-6 + 4-5 + 10-0 + 18-5 + 24-5 + 4-7 22 + 0-5 + 6-3 + 3-7 + 5-1 + 23-4 - 3-9 25 + 1-3 + 4-2 + 11-8 + 14-4 + 12-1 + 4-2 26 + 3-8 + 10-5 + 10-4 + 12-9 + 3-0 - 9-1 27 + 2-4 + 1-8 + 8-1 + 17-2 + 12-7 4 1-3 28 - 9-0 + 8-7 + 3-7 + 23-5 + 27-8 + 16-1 29 + 3-8 + 8-2 + 8-1 + 25-8 + 23-5 -10-0 Means + 2-2 + 4-9 + 7-8 + 16-1 + 19-4 + 1-3 p. M. MAT. li>20 m 3 h 20 m 5 h 20 m 7 h 20 m 9 h 20 m Il h 20 m 18 -14-5 -18-0 -17-1 -11-4 - 10-9 - 9-3 19 -19-0 - 7-8 - 6-3 - 8-0 - 2-4 + 1-1 20 - 7-9 - 4-3 -17-6 -13-9 - 4-7 + 5-5 21 -16-0 - 16-3 - 8-0 - 6-4 - 2-9 - 3-3 22 -18-7 -11-9 - 5-3 - 3-2 - 1-4 - 0-5 25 -12-1 -11-3 -ll-l -10-2 - 6-2 - 2-3 26 -23-3 -19-7 -15-8 - 8-3 + 2-5 + 2-0 27 -14-8 - 9-1 - 5-8 -23-9 - 14-4 + 10-6 28 - 6-8 - 9-2 -12-2 - 4-0 + 2-5 + 3-7 29 -17-6 - 10-8 -16-1 -15-2 - 1-6 - 2-6 Means - 15-1 -11-8 -11-5 -10-5 -4-0 + 0-5 9. The mean results corresponding to the several hours, in these four Tables, are projected in curves in Plate I., figs. 1 and 2. As we are concerned only with relative values in both cases, the ranges of the observed and computed results have been previously VAKIATIONS OF THE MAGNETIC NEEDLE. 281 equalized, by multiplying the former by constant coefficients. The curves of both lines exhibit a general resemblance to the course of the diurnal variation of the declination ; but the influence of the horizontal force is also very evident, especially in the afternoon branch of the curve, where its effect is to retard the return from the minimum at 1 P. M. This effect is much greater in the Derby and Birmingham line than in that of Derby and Eugby, the azimuth of the former line, measured from the magnetic meridian, being much greater than that of the latter. The agreement of the calculated with the observed curves is probably as close as could be expected in the results of so short a series ; and we seem entitled to conclude that the diurnal movements of the two mag- netometers are justly accounted for by electric currents traversing the upper strata of the earth. 10. Upon a closer examination of the two sets of curves, how- ever, there are found some differences to which it is necessary to advert. In the first place, the turning points of the calculated curves are generally later than those of the observed, by about one hour. Thus, in the Derby and Birmingham line, the maximum of the observed force occurs about 8 h O m A.M., and the minimum about 12 h 40 m r. M. The corresponding times for the calculated force are 9 h O m A. M., and l h 40 m P. M., nearly. We shall presently find grounds for believing that time may possibly be required, in order that the current may produce its full magnetic effect. 11. Another discrepancy is, that the calculated curve is, for the most part, above the observed, especially in the Derby and Birmingham line. This will be evident if the two curves be referred to the same axis of abscissae. It is probably to be accounted for by the fact, that the zero from which the magnetic deflections are measured is not the true one, corresponding to the absence of deflecting force. As we have no means of determining the latter, we are accustomed to take the mean position for the entire day, or the mean of the readings taken at equal intervals, as the point from which the deflections are measured. But there is reason to believe that this is not the true position of rest, corre- sponding to the absence of all disturbing force. The comparative quiescence of the magnets, during the early hours of the morning, seems to indicate that they are then near their true positions of equilibrium ; and this indication is confirmed by the galvanometric curves, the zero-line, which corresponds to the absence of all current, 282 ON EARTH-CURRENTS, AND THE DIURNAL dividing the area of the diurnal curve unequally, and being nearer to the night observations than to those of the day. From the last line in Tables I. and III., we find that the mean of all the daily observations of the galvanometers is neyatire on both lines, or below the true zero. On the other hand, the mean of the niyhi observations is positive, or above the zero ; and the same thing is true of that portion of the night (from 1 A. M. to 5 A. M.) during which the magnetic changes are smallest. From these facts it would seem to follow that the true zero of the magnetometric observations lies between the mean of the day and the mean of the hours of magnetic repose. This conclusion, however, it must be remembered, is derived from the observations of a single fort- night only; and, in the absence of fuller knowledge, we should not be justified in changing the origin usually employed. Thus magnetometric observations furnish merely differential results; and we are ignorant even of the relative values of the effects, and therefore unable to compare them accurately with their physical causes, whether real or supposed. It is true that, if the galvanometricandthe magnetometric results were completely iden- tified, the zero of the latter could readily be obtained by their com- parison. For if rj and be the values of rj and ,* corresponding to the true zero, measured from the mean of the day, it is plain that, for the hours at which the current changes sign on any line, ijo cos i - sin E = rj cos e - % sin t. If therefore these hours be known for two lines, which differ con- siderably in azimuth, and also the corresponding values of j and , we should have two equations for the determination of the two unknown quantities, i and . It is obvious, however, that this process cannot be legitimately employed in the comparison and identification itself. 12. In order to see how far the correspondence, which we have found in the mean results, is traceable in separate days, I projected in curves the observed and calculated values of the intensity of the currents on the Derby and Birmingham line, as given in Tables II. and IV., for the five days commencing May 25. The accordance of the two curves is very remarkable, every alter- nation in direction in one of them having its counterpart in the other. 13. It remains to say a few words of the manner in which the VARIATIONS OF THE MAGNETIC NEEDLE. 283 electric currents may be supposed to operate in producing the magnetic effects. An electric current, traversing the earth's crust in a horizontal direction, may affect a horizontal magnetic needle above its surface in two ways. For the current may either act directly upon the needle, according to the known laws of electro-magnetic action ; or it may induce temporary magnetism in the earth itself, which will thus affect the needle differently from before. I believe, with Dr. Lament, that the former hypothesis is inadmissible, at least as regards the principal part of the observed effect. In addition to the reason assigned for this by Dr. Lamont, I may adduce the known similarity in the course of the magnetic changes over considerable portions of the earth's surface, a similarity incom- patible with the supposition that the magnet is directly acted on, to any great extent, by the subjacent current. We must suppose, therefore, that the earth is acted on, to a considerable depth, by the wave of currents which sweep over its surface, and which alter by induction the magnetism of the subjacent mass, and that the effect produced upon the freely suspended magnet at its surface is the result of this induced change. On this supposition, the mag- netic phenomena, whose laws we are considering, are the indirect effects (not of the subjacent current merely, but) of the entire wave traversing an extended portion of the earth's surface. We can thus understand the cause of the similarity in the more rapid magnetic changes to which we have adverted ; and we may even frame some idea of the depth acted on by the superficial current, from the geographical limits of the phenomena. Upon this supposition, also, we are prepared to expect differences in the laws of the observed and computed currents, such as have been above noticed. For the galvanometric measures belong only to currents at the place of observation ; while the magnetic changes are, by hypothesis, the mean results of currents occupying a con- siderable portion of the earth's surface. Hence, also, it follows that the complete identification of the two classes of phenomena can only be made by the help of simultaneous observations of earth-currents at numerous points in an extended district. 14. Before concluding this part of the subject, I must refer briefly to the previous investigations of Dr. Lamont connected vvith it, so far as they have been yet made public. In a letter, dated July 29, 1SG1, which was read by the 284 ON EARTH-CURRENTS, AND THE DIURNAL Astronomer Boyal at the last meeting of the British Association, Dr. Lament stated that he had found " that electric currents, or (as they may be more properly termed) electric wares, varying in direction and intensity, are constantly passing at the surface of the earth, and that these waves correspond perfectly with the variations of terrestrial magnetism." The correspondence here referred to seems to relate to the epochs of the two classes of changes, and not to their amount. For in the latest account of Dr. Lament's researches, of which I am aware, and which is con- tained in a letter to Professor De la Eive, dated October 30, 1861, the writer states that the galvanometer indicates, not the earth-current itself, but its momentary changes. In fact the needle of the galva- nometer, in his observations, appears to have been affected only during a rapid increase or decrease of the earth-current, and to return to zero when the current became equable. It is obvious that, if such were the case generally, there could be no corre- spondence, such as has been pointed out in the preceding pages, between the magnitude of the magnetic changes and the deflections of the galvanometer. It is stated, in fact, by Dr. Lamont, that in such circumstances the galvanometer does not vary much from its mean position, even when the magnetometers have changed con- siderably; and he expresses his doubt whether the constant part of the action of the earth-current can be observed at all with our present means. Dr. Lamont ascribes the singular effect above described to the double conductor, the current moving (as he believes) at first in the wires, and afterwards diffusing itself in the earth below. I venture to suggest that it may be due to some disturbing cause, which operates more powerfully (in relation to the principal effect) in short than in long wires. No effect of a similar kind appears to have occurred in the observations of Mr. Barlow, which were made upon telegraphic wires of great length. And, on the other hand, it is a fact well known to telegraphists, that the currents, produced by chemical action upon the terminal plates, interfere with the primary current much more in short, than in long wires. In confirmation of the same view, I may mention the fact observed at many points of the earth during the remarkable aurora of August and September, 1859, namely, that the proper earth- current was strongest, caten's paribus, in the longest lines of telegraph. VARIATIONS OF THE MAGNETIC NEEDLE. 285 15. The connexion of the diurnal changes of the horizontal needle with earth-currents being assumed, we may reason from the former to the latter, and infer the laws which govern the diurnal changes of the earth-currents from the known laws of the related phenomena. We have already seen that the intensity of the current in the magnetic meridian (flowing northward) = ai\ ; and that of the current perpendicular to the magnetic meridian (flowing east- ward] = - . Hence, if p denote the intensity of the resultant current, and the angle which it makes with the magnetic meridian, measured to the east of north in which r\ = Xty sin 1', and = $X, X being the horizontal component of the earth's magnetic force, and ip the magnetic declination*. The azimuth of the direction of the current, measured from the true meridian in the same direction, is Substituting in these formulas the mean values of cty and $X at Dublin, for summer, winter, and for the entire year, we obtain the daily changes of p and o>, given in the following Table. The values of Si/ and $X were observed at Dublin during the four years 1840-43, twelve times in the day, namely, at the alternate hours. The values corresponding to the intermediate hours are obtained from them by the usual formulas of interpolation. * The preceding combination of the daily changes of the declination and horizontal intensity, as well as the graphical representation of the results, seems to have been first employed by Professor Hansteen, to represent the laws of the disturbing force, to which these changes are due. The only difference between the two representations is, that the azimuth of the disturbing force at any hour is 90 greater than that of the earth- current. Hence, if the diagrams of Plate II. be turned 90 forward in azimuth, th. y represent the daily changes of the horizontal disturbing force. 286 ON EARTH-CURRENTS, AND THE DIURNAL TABLE Y. HOUR. AZIMUTH. Summer. Winter. Year. Summer. 1 A.M. 2 324 338 353 360 336 346 20-9 21-2 3 342 353 345 24-7 4 342 327 338 29-6 5 344 301 332 35-5 6 350 288 336 42-1 7 3 303 352 49-4 8 22 351 16 56-0 9 45 37 43 61-6 10 72 81 75 66-5 11 100 115 104 71-7 Noon 124 134 128 77-3 1 P.M. 143 149 145 78-0 2 158 162 160 70-6 3 174 174 174 57-1 4 196 191 194 46-6 5 223 218 221 45-9 6 241 247 243 50-8 7 252 279 258 50-5 8 261 311 277 43-8 9 273 327 297 35-8 10 285 333 309 31-0 11 296 336 315 27-8 12 308 342 324 24-0 INTENSITY. "Winter. Year. 16-0 18-1 13-6 17-1 12-2 18-4 12-5 20-9 13-9 23-3 14-3 25-1 12-5 27-8 13-6 34-1 19-1 39-3 26-4 46-6 39-7 54-6 49-8 63-3 51-5 64-4 46-3 58-1 35-5 46-6 23-3 34-8 15-7 30-3 14-3 32-4 15-3 34-5 21-6 29-9 28-9 29-9 31-0 28-2 26-4 25-8 20-2 21-2 16. The numbers in the fourth and seventh columns of the foregoing Table are graphically represented in the annexed diagram (Plate II., fig. 1), which accordingly exhibits the law of the changes for the entire year, the radius- vector of the curve measuring the intensity of the current, and the angle which it makes with the meridian, its azimuth. The corresponding hours are indicated on the perimeter of the curve, the afternoon hours being distinguished by brackets. It will be seen that in the early hours of the morning, namely, from 1 A. M. to 6 A. M., inclusive, the direction of the current changes little ; its mean azimuth for that period is N. 21 W. At 7 A. M. the current begins to move eastward, and its direction is due north at about 7 h 15 m . At 10 h 25 m A.M., the azimuth of the resultant current is 90, or its direction is to the cast; at 3 h 15 m P.M., it becomes south; and at VARIATIONS OF THE MAGNETIC NEEDLE. 287 7 h 35 m P. M., it is west. Finally, after midnight it reaches its stationary position to the west of north. The order of the changes is similar in summer and in winter. The principal differences lie in the times of reaching the azimuths and 270, the direction of the current being in the former azimuth about 1^ hours earlier in summer than in winter, and in the latter about two hours later. The time of the afternoon meri- dian passage is nearly the same at the two seasons. The intensity of the current is greatest in the south and east. The maximum intensity occurs at 1 P.M., the azimuth of the current being then S. 35 E. There is a secondary maximum about 7 P.M., preceded by p, secondary minimum about 5 P.M. A curious inversion, or fold of the curve, takes place during the night, between 10 P.M. and 6 A.M. The nocturnal minimum occurs between 1 A.M. and 2 A.M. ; and the direction of the current is then nearly opposite to that of the maximum at 1 p. M. 17. In the foregoing deductions, it is to be borne in mind, in conformity with the remarks of Art. 13, that the current referred to is not that actually subjacent to the place of observation, but the resultant, for that place, of all the currents occupying a consider- able portion of the earth's surface. The reasoning, in fact, relates to the magnetic effects of the currents, rather than to the currents themselves, and the immediate subject of calculation is the intensity of the magnetic disturbing force, and the normal to its direction. It is further to be remembered, that the conclusions are affected by the uncertainty to which we have before adverted (Art. 11), re- specting the zero, or origin, from which the magnetic deflections are to be measured. It is probable, however, that this uncer- tainty does not materially affect the results, except at those hours at which the magnetic variations are small, and when, of course, any given change in the amount of the deflections bears a much larger proportion to the whole. 18. Although the laws of the diurnal changes of the magnetic elements at different points of the globe have much in common, they present nevertheless marked differences. Corresponding dif- ferences must, therefore, be expected in the diurnal changes of the earth-currents at different places ; and, consequently, the general laws to which they are subject can only be known by a comparison of the results at many places widely distributed over the globe. I have accordingly thought that it would reward the labour to make 288 ON EAETH CURRENTS, AND THE DIURNAL a calculation, similar to the foregoing, for places at which hourly, or two-hourly, observations of the two magnetic elements have been made for any considerable period. The results of these- calculations are given in the following Tables. Table YI. contains the names of the places, together with their geographical co-ordinates, and magnetic elements, the positive sign denoting north latitude, east longitude, and easterly declination. The years in the sixth column of the Table are those of the hourly observations which have been employed in the calculation. The absolute values of the magnetic declination and intensity, in the fourth and fifth columns, correspond to the mean time of the period. The observations were taken, in all cases, at the full hours of Gottingen mean time ; but the corresponding local time is indicated in the last column of the Table, the numbers being the minutes to be added to, or subtracted from, the Tabular hours. Tables VII. and VIII. contain the calculated values of the azimuth and intensity of the current, at the several hours of obser- vation, for each of the foregoing stations.* * The diurnal changes of declination and horizontal intensity, employed in the cal- culation, are contained in the following works : 1. " Magnetical and Meteorological Observations made at the Royal Observatory, Greenwich (1844-47)." 2. " General Results of the Makerstoun Observations." Edin. Trans., vol. ix. 3. "Resultate des Magnetischen Observatoriums in 'M.vnchen."AbhandhO3>O3 QOoooooooooooooD tr- ~ t- t O J * T J 1 H OSOOi-IOOOOSkOO t COCMOCOOiTtlaO eo o oo -" I o ooi-ii-HTfii-ieooo-* 1 II 1 1 + -t- 1 eo co i- t- CM >O CM ^ - s g 1 + 1 1 Longitude. ^.(Mt-HT^OOCOOS O >-H rH CO -< 1 + + + + + + 1 1 + 1 + o ^kOGOOOiCOCO^H koio-^io^oo^'O 3 i S S + 4- 1 1 1 " % f in 1 I 1 '1 1 1 1 1 iS ,'si H . .. IB PI . o . ; - * : 1 ; i i 1111 1 1 j w & fi c 290 ON EAETH-CUERENTS, AND THE DIUENAL .3 .* . .3 i-ii-ii-<^0 >0 1-1 CO t- i-< rt< CO CD r- Ci (M -O 1 GC ; I co cc I (M (M 0 00 W >0 C5 fO CO CM ^-t ip CO ^ n m oi o > i o o ca OD >o H c i- (?i cs c H < o s T- CM 0 CO CM .-H .-H CM CM . Then e* = u- - 2us cos a + s 2 ; and if s be so small in comparison with u that the squares and higher powers of - may be neglected, T 3 = M~ 3 ( 1 + COS (i) ). V w / Again, if a, j3, 7 denote the angles contained by the axis of the magnet with the three axes of coordinates, x - s cos a, y = s cos /3, s = s cos 7. Substituting these values in the expressions for the components of the force above given, integrating, and observing that J /ucfe = 0, we have, for the components of the total force exerted by the mag- net on the magnetic element, Mmf u 3 \ Mmf n o -3- cosw - co"s/3 in which we have put, for abridgement, M = J /i&. The angle w is connected with a, |3, 7 by the relation u cos = a cos a f b cos /3 + c cos 7. Now let the point (a, b, c) be on the earth's surface, and let us x2 308 ON THE DIRECT MAGNETIC INFLUENCE OF THE SUN suppose, for simplicity, that the acting magnet is in the plane of the equator. Let that plane be taken as the plane of (#, y\ and the line connecting the centre of the magnet and that of the earth as the axis of x. Then, if the distance of the acting magnet be considerable, relatively to the earth's radius, b and c are small in comparison with a, and we may neglect the small quantities of the b z while in that of the moon it is about yV ; and the magnitude of the semidiurnal inequality should bear to that of the diurnal the ratios designated by these small fractions. The facts are altogether opposed to this result. The coefficient of the solar-diurnal inequality of the declination at Dublin, in the mean of the entire year, is 3'- 52, while that of the semidiurnal is 2'-13, nearly two-thirds of the former. In the case of the lunar- diurnal variation, the semidiurnal inequality exceeds the diurnal. XVI. ON THE STORM OF THE 18m OF APRIL, 1850. Proceedings of the Royal Irish Academy, 1850. HAVING watched attentively the progress of the late storm, from a favourable position, and collected some facts relative to it from the records of the Observatory, and from other sources, I avail myself of the present opportunity to lay them before the Academy. The phenomena were of a nature so unusual (I may say unex- ampled) in these climates, that it is desirable that some notice of them, however imperfect, should be placed on record; and the present summary of facts is offered, chiefly in the hope that it may serve as a nucleus to a more complete one. I shall limit myself, mainly, to those which have an immediate scientific bearing. The morning of the 18th was fine in Dublin, with bright sun- shine, light cirrous clouds being scattered loosely over the sky ; at ten o'clock these became diffused, and the sky was evenly, but lightly overcast. From the tracings of the self -registering anemometer, erected in Trinity College, it appears that on the 17th, and during the morning of the 18th, the wind blew gently from the south-west. Towards noon, on the latter day, it gradually veered to the south, and continued at that point until the arrival of the storm. This veering of the wind, however, appears to have been confined to the lower current ; the direction of the upper current, as estimated by the motion of the clouds, appeared to be nearly south-west. The first indications of the approach of the storm were observed soon after three o'clock. Massive cumuli were seen in the western and south-western portions of the horizon. These became denser as they approached, until they formed a mass of an ash-grey ON THE STORM OF THE 18TH OF APRIL, 1850. 313 colour, projected on a sky of a paler tint, while the rugged outliers from the mass, of the peculiar form which indicates a high degree of electrical tension, showed plainly that a storm was approaching. About half-past three o'clock it burst forth. The flashes of light- ning (generally forked) succeeded one another with rapidity, and at length the roar of the thunder seemed continuous. Some persons who observed the 'phenomena from a distance were able to dis- tinguish the two strata of oppositely electrical clouds, and to see the electrical discharges passing between them. Hitherto the wind was light, and there was that peculiar close- ness in the air which is the result of high temperature and excessive humidity. Shortly before four o'clock the rain commenced; this was followed almost immediately by discharges of hail, and at four, P. M., the terrific tornado, which was the grand and peculiar feature of this storm, reached us. This gale, which appears to have been a true whirlwind, first sprung up from the south-east, driving the hail before it impetu- ously. It then suddenly, and apparently in an instant, shifted to the point of the compass diametrically opposite, and blew with increased violence from the north-west. The noise about this time of the shifting of the wind was terrific, and arose (as is conjectured respecting similar tropical phenomena) from the confused conflict of hail in the air. The size of the hailstones, as well as the vehemence of the gale, appeared to be greater during the second phase of the storm than the first. These masses, many of which were as large as a pigeon's egg, were formed of a nucleus of snow or sleet, surrounded by transparent ice, and this again was suc- ceeded by an opaque white layer, followed by a second coating of ice. In some of them I counted five alternations. In less than ten minutes the tornado had passed. The wind returned to a gentle breeze from the south-west, the clouds dis- persed, and the weather became beautiful. All the phenomena, the direction of the gale perpendicular to that in which the storm- cloud was advancing, and the sudden reversal of that direction, seem to prove that it was a tornado, whose centre passed directly over the place of observation. It is evident, on comparing the direction of the wind when the whirl first reached this part of the town, with that of the progressive motion of the vortex itself, that its rotatory motion was retrograde, or in an opposite direction to that of the hands of a watch with its face upward. It is deserving 314 ON THE STORM OF THE 18TH OF APRIL, 1850. of notice also, that in the northern hemisphere this is the invariable direction of the cyclones, or great revolving storms, to which the attention of meteorologists has been directed by Colonel Eeid and Mr. Eedfleld. The late storm was, however, different from a cyclone, both in the dimensions of the vortex and in the causes from which it originated. The horizontal section of the cyclone where it meets the earth is often 500 miles in diameter ; and the vortex is supposed to be the effect of two crossing currents of air, which generate a movement of rotation. In the tornado (to which species the late storm belonged) the vortex is of much smaller dimensions, and is produced by rapidly ascending currents of air, caused by the heating of a limited portion of the earth's surface under the action of the sun's rays. In the temperate zones, accordingly, it is never produced in winter. The evidence relating to the direction of the gale, and its changes, as it passed over the College Park, is very complete and satisfactory. In the park, and garden adjoining, nineteen trees were rooted up and prostrated, eleven of them being trees of large size. Of these ten have fallen from the south-east, or under the action of the first half of the gale, and nine from the north-west. Their bearings have been accurately taken ; and the general result is, that the mean direction of the south-east gale, as indicated by that of the trees, is S. 56 E., and that of the north-west gale N. 53 W. I believe that these results are even more accurate than those furnished by the anemometer ; and they prove that in this locality the direction of the wind was exactly reversed, and, there- fore, that the centre of the vortex passed over the College. A remarkable circumstance connected with the direction of the fallen trees is their great uniformity, the individual directions seldom differing more than 10 from the mean. This is an indi- rect evidence of the great violence of the gale; and it proves, moreover, that the transition from the south-east to the north-west wind was immediate. There is greater regularity in the direction of the trees fallen from the north-west than in those which have been blown down from the opposite quarter. This may have arisen partly from the greater violence of the gale in the former direction ; but it is partly also due to the circumstance that the trees which fell from the north-west are generally larger, and in a less inclosed portion of the ground. It may be mentioned also, that the trees which fell from the north-west generally lie to the ON THE STORM OF THE 18TH OF APRIL, 1850. 315 southward of the others ; there are, however, two large trees in the garden lying side by side, but in directions diametrically opposed. It has been stated that in the College Park the shifting of the wind amounted to 180 ; and it has been inferred that the centre of the vortex passed over that spot. From what has been said as to the nature of % the phenomenon, it will follow that in other localities, over which the vortex did not pass centrally, the wind must have shifted through different points of the compass, and through angles smaller in proportion to their distances from the centre. Thus, on the southern side of the line described by the centre of the vortex, the change of the wind should be from south to west, and on the northern side of the same line from east to north. We are not yet in possession of facts which bear upon this point ; but from the limited dimensions of the vortex, and the consequent smallness of the distance necessary to produce such a variation, it is probable that evidence bearing upon it may be obtained. I shall only observe that, in seeking and comparing such evidence, care must be taken not to confound eddies arising from local obstructions with the general direction of the current. The hours of observation at the Magnetical Observatory are 7 A. M., 10, 1 P. M., 4, 7, 10. The observations of the barometer, and of the dry and wet thermometers, made at these hours on the day of the storm, are the following : Hour. Barometer. Dry Therm. Wet Therm. 7A.M. 29-944 49-5 47-4 10 29-952 54-7 50-5 IP.M. 29-964 58-6 52-0 4 29-930 56-0 52-3 7 29-944 52-6 52-0 10 29-936 51-0 49-6 The fall of rain, and melted hail, in Trinity College during the storm amounted to 0'596 of an inch ; but it is probable that much of the hail was driven out of the receiver of the gauge by the wind. It will be seen that the barometric fluctuation is small. It is stated, however, that a sudden and considerable fall of the barometer took place shortly before the storm. From the obser- 316 ON THE STORM OF THE 18TH OF APRIL, 1850. vations above given, at 1 P. M. and 4 P. M., it will be seen that the barometric equilibrium, if so disturbed, was soon restored. I have collected, from the newspapers and other sources, sucli information as I could obtain respecting the area of the city visited by the gale, but it is as yet incomplete. It appears, however, that the diameter of the vortex was not very different from the length of the city from north to south ; the gale having been limited by the Circular-road in these two directions. Hail fell, however, abundantly beyond the limits of the gale. Thus, at the gardens of the Eoyal Dublin Society, at Grlasnevin, the damage done by the hail was very great ; but it was limited to the roofs of the houses, the hail having fallen perpendicularly. The amount of the rain and melted hail registered there was 1'7 inches in 35 minutes. Further information is wanting to enable us to determine exactly the progressive movement of the centre of the vortex. We are informed by the newspapers that a storm similar to that which visited Dublin, although not so severe, took place at Mullingar, about an hour and a half previously. If this be the same storm, the direction of the progressive movement must have been nearly from west to east, and its velocity about thirty miles an hour. This direction accords with that given by the observed limits of the storm on the northern and southern sides of the city; but it seems to have been modified, at the surface of the earth, by the lower current. The velocity of the rotatory movement was, of course, vastly greater than that of the progressive ; but we have no direct measure of its amount. The damage done in Dublin has been principally in the destruc- tion of glass caused by the hail; but many chimneys have been thrown down, and many roofs dismantled, by the gale. The estimated amount of the loss sustained, as ascertained by the Metropolitan Police, is 27,800. Many houses were struck by the lightning; but, happily, there was no loss of life from that cause. There seemed to have been a disturbance of electrical equi- librium, accompanied by rain, in many remote parts of Ireland on XVII. NOTES ON THE METEOROLOGY OF IRELAND, DE- DUCED FROM THE OBSERVATIONS MADE IN THE YEAR 1851, UNDER THE DIRECTION OF THE ROYAL IRISH ACADEMY. Transactions of the Itoyal Irish Academy, Vol. XXII. THE science of meteorology is, perhaps more than any other, dependent upon co-operation and upon method. Individual ob- servers may investigate successfully certain detached meteorolo- gical problems, such as the laws of the diurnal and annual changes of temperature, pressure, and humidity, at a given place ; but little progress can be made in climatology, or in the knowledge of the greater movements of the atmosphere, and their relation to the non-periodic variations of temperature and pressure, without the co-operation of many observers distributed over a large area, and acting upon a common plan. For this task the voluntary association of individuals is insuffi- cient. However zealous such persons may be, it is not possible to bind them to that uniformity of system without which little can be effectively done. Observations taken at different hours, or by dif- ferent methods, can never be compared satisfactorily; and any comparison will involve an amount of labour in the processes of reduction which may render them impracticable. In addition to this, certain rules of observation are imposed by the conditions of some of the great problems of meteorology ; and no co-operation in which these. rules are deviated from can contribute to their solution. For these and other reasons it is desirable that, in every country, such observations should be provided for by the Government, and placed under the direction of one of its official departments. And 318 ON THE METEOROLOGY OF IRELAND. there can be no doubt of the services which meteorology, properly studied, may be made to contribute to those interests which it is the duty of every Government to promote. The health of man, the operations of agriculture by which he procures his food, and many other of his material interests, are dependent upon climat- ological relations, which must be known and studied before they can be applied. Every one acknowledges the fact, that the salu- brity of a district, and its adaptation (or the reverse) to particular human constitutions, is intimately connected with its meteorological conditions. And the same thing is true of all organized beings, and especially of those which are subservient to the uses of man. Thus, the question of the naturalization of exotic plants is, mainly, a me- teorological problem, dependent upon the climatological relations of the region to which the plant is indigenous, and of that to which it is to be transferred ; and the importance of obtaining accurate data for its solution will be recognised, when it is borne in mind that, in Europe, most of the plants useful to man belong to this class, and that those hitherto acclimatized probably bear a very small proportion to the whole. Lastly, the processes of cultivation, to which these vegetables are to be subjected, are also connected in an intimate manner with meteorological knowledge. We may instance this connexion in the operations of irrigation, and of drainage, both of which are dependent upon the knowledge of the amount of rain-fall in the district to be operated on. It is true that meteorological science has been hitherto compa- ratively barren in such applications ; and the fact itself, with many persons, would be accepted as evidence that abstract and practical knowledge are wholly separate and unconnected. But, when pro- perly understood, it leads to a different conclusion. Superficial knowledge in this science can indeed yield but few practical results ; and those by whom such results have been hitherto sought have expected to find them at the surface. There are indeed cases such, for example, as the one last referred to in which the connexion between meteorological science and its applications is obvious and simple, and in which, accordingly, that connexion has been traced and made use of. But in general it is otherwise. In a subject so complex as the laws which govern the aerial enve- lope of the earth, and where so many causes are in operation, prac- tical applications can be obtained only from mature theoretical knowledge. Thus, it may be shown that the knowledge of the ON THE METEOROLOGY OF IRELAND. 319 phenomena of temperature, requisite for the determination of the possible geographical limits of a single species of plants, is by no means inconsiderable ;* and when to this we add the consideration of the various other agencies which are at work in the atmosphere, all influencing vegetable life, it is plain that we are not in a condition to deduce any useful result connected with the distribution of species, until we have mastered a much larger amount of theoretical know- ledge than is usually brought to bear in such deductions. It would seem, therefore, to be the duty of the Government of every civilized state to provide the statistical data which have so many important bearings upon the material welfare of the people, and in the form best fitted for their discussion and examination. And to the lover of truth itself, for its own sake, the fulfilment of this duty would, fortunately, supply the wants of science in the most complete and satisfactory manner. In many countries, accordingly, provision has been made by their respective Governments for the collection and discussion of meteorological data upon a uniform and well-digested plan. The Government of Prussia appears to have taken the lead in this im- portant labour. Its example has been followed by those of Russia, Austria, Bavaria, and Belgium ; and the names of Dove, Kupffer, Kreil, Lamont, and Quetelet, to whom the superintendence of these observations has been intrusted, afford the surest warrant of their successful prosecution. f But perhaps the most important under- taking of this nature is the recent organization of a system of meteorological observations at sea by the Government of the United States. There are, at the present time, nearly 1000 masters of ships, belonging to the navy and merchant services of * For each plant there is a lower limit of temperature, helow which it will cease to vegetate; while, in order that it may blossom and bear fruit, it must receive, botwtvn the two seasons of this minimum temperature, a certain amount of heat beyond this limit which is constant for each species. It is upon this integral of effective heat, as has been shown by De Candolle, that the existence of the species depends. For informa- tion on this and other subjects connected with the applications of meteorology, see tin- interesting introduction, by M. Martins, to the Annuaire Meteorologiqtie de la Frann: t The results of many of these series have been already published. Professor Dove has published the results of the observations made in Prussia in the years 1848 ami 1849. The observations made at the Hn--> Tviitorirs h;iv- b.M-n pultlishrd from time to time by M. Kupffer, in the Recuril -K (t/wn-Hfinimfnit,;* il<, - The results of the Bavarian observations have hrcu j, p ivm by Mr. L-miniit. in t!, tnitt-n o 1 ir- es 1 1 1 9 op CM CM 1 1 H- CM 9 CM 1 t-H CO 1 1 00 1 1 00 CD rH CM 1 1 O CM * ^ ^ CO O co CM CD rH rH CM o co CM *" b I O 1 I O rH O O + O 1 1 O I O 1 b -h + O 1 >0 o 9 ,-1 CM * -H CM eo 9 eo 9 9 CM rH CD O cp op CM CO cp o eo O ^ rH CM _, CD e IO CD ^ CO co eo >O CD CO >O o <* CM ~ Tfl O co PH 9 M 9 * 9 9 CO CO op -* 9 O *O CD *0 eo 9 CO o o * *" . CM TH ^ 05 r- CO rH 00 CO 9 9 9 op "- 1 ! -f~ ~f~ + * 5 J eo ^ CM O5 I O CM 00 rH c-i eo s 9 CM i-H O 9 op o rH 3 rH CM co 9 ,_, rH CO CC OS ^ 00 CM CM CM CO *~ CM 1 eo rH 1 1 O CM o 1 CM 1 I 1-1 ** rH 1 1 CM 1 ~ ? t*. CD CM O5 05 r^ CM 05 ^ eo CO rH 9 5 ' CM 1 1 1 1 ' I 1 1 ! 1 I | ' oo- CO I- T* 3 ^ TH CM CM ^ 05 t- ,_, eo 1-1 1 1 1 co t- 1 O 1 1 1 1 ^ & 1 1 CM * ep 2 1 1 i- CM 3 CO 1 I 9 1 5 T 9 CO CM 1 I CM I op * rH CO >O 1 1 2 J"! J s- 1 n 1! t < J tT Decembe ,1 s a H ^ Winter, . ON THE METEOROLOGY OF IRELAND. 327 From the preceding Table we obtain the following corrections, which are to be applied to the means of the observed temperatures at 9 A.M. and 9 P.M., in order to reduce them to the mean of the day : April, . . corr. = + 0'l May, . * : +0-1 June, . . -0-1 July, . . +0-1 August, . 0-0 + 0-2 October, . . corr. = + 0'5 November, . + '7 December, . + '6 January, . . + '7 February, . +0'6 March, . . + -5 It hence appears that the correction is nearly constant through- out the summer, and throughout the winter months, respectively. The mean summer correction is + 0'l ; the mean winter correction + 0-6. Mean Monthly Temperatures. The mean temperatures have been obtained, at all but three of the stations, from the observations at 9 A.M. and 9 P.M., by the application of the preceding corrections. At Markree the observations were taken at 10 A.M. and 10 P.M. ; and the reducing numbers are therefore somewhat different, and smaller in amount. At Portarlington and Athy the observations were taken but once in the day, namely, at 9 A. M. ; and at these stations, accordingly, the mean temperatures are inferred from the maximum and minimum temperatures, as given by the self-regis- tering thermometers. The formula employed is that of Kromtz, viz. : mean temp. = min + a (max. - mi>i.) The mean value of the coefficient,* as deduced from the observa- tions at the observatories of Armagh, Markree, and Dublin, is a = 0-41. The following Table contains the resulting values of the mean temperature for the several months of the year 1851 : The coefficient in Ksemtz's formula appears to vary considerably at different places, both in its -mean amount, and in the law of its variation from month to month. At Armagh and Markree its greatest value is in December, and ite least in July ; at Dublin, it is the reverse. I have taken above the mean of the yearly values for the three stations. 328 ON THE METEOROLOGY OP IRELAND. rH co oo CO CM CN t CO co T* eo 05 CD CO ,_, O O5 -TJH IQ CO 00 5 t^ 00 **< S S3 S3 r- eo <# co CM CO CN oo eo CO CO ,H _ CM O 9 a to CD co co Ol r- eo rH >0 O * !5 CD * 0 00 rH rt 1 3 3 3 3 O CO * CO rH iO *O >o >o 01 o S S S . IO 00 CO O eo cq ,_ -* 05 t. CO _ rH t- co 02 1 8 8 kO t- iO O 8 8 8 8 CO g oo >o 05 00 3 O5 to | oo eo oo r~ CO rH 05 t. 00 t^ GO CO 00 I 00 05 to o co s 5 i 8 CM CM co *? 9 rH rH rH D CO CM eo N '0 ^ CO CM J, a 1 8 r~- oo >0 to t~ oo 8 8 8 5 8 8 S rH O CO CD 8 si eo oo 00 CO CM CM ^ O 00 CO O5 eo eo CM CM 1 85 >o iO CD CD O o o 2 o *c> 3 8 f~ oo oo iO ^O 5 o j. eo os co a 8 3 3 CM CM *O *C s S3 5 S S eo Oi ?o to co r- l i 5 CO CO 00 1 1 CO s 9 II 05 t" 05 CO CO rH >0 05 4ft CM 3 00 eo CM 00 TH ^ ^ ^ CO to CM eo c^ co * o rH TH Tt< CO CD 1 - . eo oo rH 05 t- O CM CM CO CO 00 >0 00 CM o CM rH 5 5 53 $ 5! >O CO 5 Tt< CO . O CO oo oo T* CO * eo co CO ^ CO CM Tf< CO f * rH 0 rrl Station. Portrush, . Buncrana, Donaghadee, Killybegs, 1o t c 2 o i i 1 ^ i* o ^ ^ & " 1 Courtown, . r& 1 Dunmore, Cahirciveen, 1 ON THE METEOROLOGY OF IRELAND. 329 Before we proceed to discuss the mean temperatures in the several months of the year 1851, it is important that we should know the absolute mean temperatures at some one station, and thereby the deviations from the means in the several months of the year in question. Over a tract of country so limited as Ireland, these deviations, will not differ much in different localities ; and therefore, knowing them for one station, we are enabled to reduce the results of the single year, with probably sufficient exactness, to their absolute mean values at all the rest. The absolute mean temperatures of the several months are known, at Dublin, by means of the series of observations made during twelve years at the Magnetical Observatory. The monthly mean temperatures, deduced from that series, are given in the fol- lowing Table. From the year 1840 to 1843, inclusive, the daily means are those of twelve equidistant hours ; from 1844 to 1850, inclusive, they are inferred from the temperatures observed at 10 A.M. and 10 P.M. ; and in 1851, from those at 9 A.M. and 9 p. M. In the last line of the Table are given the deviations of the monthly means in 1851, from the mean monthly means, as deduced from the twelve years. It will be seen from the Table, that the temperature in the months of January, February, and October, 1851, was higher than the average temperature, while, in November, it was considerably lower. The mean temperature of the entire year was only 0-3 above the average. The depression of temperature in the month of November is a remarkable case of those non-periodic fluctuations to which the at- tention of meteorologists has been drawn by Professor Dove. This fluctuation appears to have proceeded from north-east to south- west, and to have been nearly obliterated when it reached the western coast of the island. At the northern and eastern stations the unusual cold began on the 24th day of the month ; at the southern and western it commenced on the 26th and 27th. It reached its maximum about the 30th, and ceased about the 3rd of December. When we compare the mean temperatures of November and De- cember at Killough, Dublin, Courtown, and Dunmore, on the eastern coast, with those at Killybegs, Westport, Kilrush, and Cahirciveen, on the western, we observe that the temperature of November is less than that of December by 3"'3 at the former stations, while the defect is only 0"-6 at the latter. 330 ON THE METEOROLOGY OF IRELAND. 1 O^OOOOiC5i-HOOOO CO S o eo i: 1 OOOOOOCNI>eOt~->Ot-CDCD eo O CO p 3 CO O OeoTtOOC 05 "f ? OS CM 1 * + . 7o OOOOOOOO3OO S kO | 1 "-o*oco>o>ocococococo S i a i-s *OiO-*t^C5COt^>OTHO '-..... 05 co o O>OCO>OiO>OCO oo 3 QO O . ^CNOeooscOTticooscooo >0 TjH O ^>OCOi-H(Mi-irtlcOCDCOt-< (M >0 p s 'S Oi "? ( ? l rT l 9 < ^ | '* < ^' o p i ?9 fe^cot-ooococoiocoos I 9 ^ ^ "* ^ ' i i O-Hcoopost--ooeocot-Tt< ^coOTt<(Noocoeoeo4ieo o *? '? CO . t^t~OpOSTto i (g & ft ON THE METEOROLOGY OF IRELAND. 331 Upon a comparison of the mean yearly temperatures of the several stations, we observe that those of the inland stations are in defect, as compared with the corresponding coast stations. ThQs the mean temperature of Armagh (48'6) is less than that of Donaghadee by 1, and less than that of Killough by 1'6. The mean temperature of Markree (48'2) is less than that of Killybegs by 2'6, and than that of Westport by 3'5. The mean temperatures of Portarlington and Athy (47'3 and 48 0< 4) are in like manner in defect, when compared with those of Dublin and Courtown, and by an intermediate amount. I shall return to this subject hereafter, and merely notice it at present for the purpose of observing that no satisfactory conclusion can be drawn as to the dependence of temperature upon geographical position, unless the inland and coast stations be compared separately. Confining ourselves for the present to the coast stations, which are the most numerous and the most widely distributed, we observe that there is an increase of mean annual temperature in proceeding from north to south of the island, the mean temperature of Portrush and Buncrana being 49*0, and that of Dunmore, which is nearly on the intermediate meridian. 51'6. Similarly there is an increase of temperature in proceeding from east to west, the mean tempera- ture of Killough and Dublin being 50'2, and that of Westport, which is nearly on the intermediate parallel, 51 0< 7. But for an accurate determination of the rate of increase of temperature in the two directions, it is necessary to combine the results by the method of least squares. For this purpose let t denote the observed mean temperature of any month, at any given station ; T the probable temperature of the same month at an assumed central station; and let the distances (in geographical miles) of the former from the latter, measured on the meridian and parallel of latitude to the north and west, respectively, be denoted by y and x ; then, if V and U be the increase of tempe- rature corresponding to a single mile in each direction, t=T+Ux+ Vy. There will be a similar equation for each station ; and combining them by the method of least squares, we shall obtain the most probable values of the unknown quantities T, U, and V. The simplest mode of employing this method in the present instance is to take, as the arbitrary central station, that whose 332 ON THE METEOROLOGY OF IRELAND. latitude and longitude are the arithmetical means of the latitudes and longitudes of the stations of observation. The resulting equa- tions are thus reduced to the following : (a?) 4 FS (xy] = S (a*), S (yf). For the reason already stated, I shall employ in this calculation only the results obtained at the coast stations. These are, in the order of latitude, Portrush, Buncrana, Donaghadee, Killybegs, Killough, "Westport, Dublin, Courtown, Kilrush, Dunmore, Cahir- civeen, Castletownsend. The mean latitude and longitude of these stations are 53-29', and 7'39' respectively. And we find S (ar 2 ) = 39094, S (xy) = - 22569, Z (f) = 65811. Substituting and eliminating between the second and third equa- tions, we obtain J7= -0000319 S (arf) + -0000109 S (yf) ; F= -0000109 S (xt) + -0000189 S (yt). By these formulas the values of T, U, and F, for each month are calculated. They are given in the following Table. The values of U and F being known, the positions of the iso- thermal lines are determined. The inclination of the isothermal lines to the meridian, measured from north to west, w, and the rate of increase of temperature in the direction perpendicular to them, W, are known by the formulas tan = -^, W=J(U*+ F) 2 . The values of u and W for the several months are given in the Table. ON THE METEOROLOGY OF IRELAND. 333 ELEMEMTS OF MONTHLY ISOTHERMAL LINES. tf T U V W u 1851. Mean. January, . . . M 41-7 + -0080 - -0102 0130 52 February, . . 44-2 42-3 + -0093 -0119 0151 52 March, . . . 44-6 44-1 + -0131 -0064 0146 26 AprU, . . . . 46-9 47-1 + 0043 -0070 0082 59 May, .... 52-0 52-9 + -0012 - -0139 0140 85 June, .... 56-8 56-7 - -0031 - -0109 0114 106 July, . . . . j 58-9 58-8 -0049 - -0202 0208 104 August, . . .1 60-6 58-3 + -0029 - -0121 0124 77 September, . . i 57-4 October, . . . 52-7 November, . . I 44-1 57-8 50-2 48-4 + -0101 + 0059 + 0304 -0090 -0070 + -0077 0135 0092 0313 42 50 -14 December, . . 1 45-7 45-4 + -0103 -0017 0104 9 Year, . . . . 50-7 50-3 + 0073 -0085 0112 49' We see then that, on the mean of the whole year, the isothermal lines are inclined to the meridian by the angle N. 49 W. ; and that the temperature increases in a direction perpendicular to these lines, by -0112 of a degree for each geographical mile, or at the rate of 1 degree for 89 miles. The increase of temperature, in proceeding from north to south, is V= '0085, or 1 in 118 geogra- phical miles ; the corresponding increase, in proceeding from east to west, is U = -0073, or 1 in 137 geographical miles. We learn further, that the mean annual isothermal lines furnish a very inadequate representation of the progression of temperature ; and that when we follow the course of these lines from month to month, we find them to vary within very wide limits. The extreme positions of -these lines, as given in the preceding Table, are those for the months of June and November. But the result obtained for the latter month must, I think, be regarded as anomalous, on account of the irregularity in the distribution of temperature already 334 ON THE METEOROLOGY OF IRELAND. noticed ; and, rejecting it, the extreme positions correspond to the two solstitial months. They are the following : June, . . December, . = N. 106 W., = N. 9 W., w= 0114, 0104; so that the direction of the isothermal lines varies through an angle of 97 in the course of the year, being nearly parallel to the meri- dian in December, and nearly perpendicular to it in June. (See Plate i.) We may now employ the formula = T+ Vy, to deduce the probable temperature at any place, and compare it with that actually observed ; we shall thus find the effect [due to local causes. Making this calculation for the four inland stations, we obtain the results given in the following Table : EXCESS OF CALCULATED TEMPERATURES AT INLAND STATIONS. Month. Armagh. Markree. Portarlington. Athy. January, . . r-8 4-4 3-6 3'5 February, . . 1 -5 2-7 3-7 3-4 March, . . . 1-3 2-4 4-1 2-7 April, ... 1-0 1-4 3-9 1 -6 May, . . . 0-9 1 -3 3-1 1 -8 June, . . . 0-1 0-5 2-2 0-3 July, . . . 0-9 1 -2 2-0 1-4 August, . . 1-3 1-4 3-1 0-1 September, . . 1-3 2-2 3-7 3-8 October, . . 1-7 2-7 2-6 1 -8 November, . . 2-4 3-5 4-1 3-4 December, . . 2-1 3-7 3-3 4-2 Year, . . . 1-4 2-3 3-4 2-3 ON THE METEOROLOGY OF IRELAND. 335 We learn that the defect of temperature due to inland position is, as might have been expected, least in summer and greatest in winter. A small part of this defect is due to elevation ; but it is easily eliminated. The mean height of the instruments at the coast stations above the level of the sea is 30 feet. We have, therefore, only to subduct' this from the known heights at the inland stations, and to correct for the difference of level at the rate of 1 Fahr. for 276 feet, which is the mean of the determinations made by Mr. Welsh in his balloon ascents, for the lower portion of the atmo- sphere lying beneath the great vapour plane. The mean yearly results at the four inland stations, thus corrected, are as follows : Observed Defect. Height above Sea. Correction. Reduced Defect. Armagh, . . . 1M 211 -0^-7 0-7 Markree, . . . 2 -3 132 -0-4 1-9 Portarlington, . 3 -4 230 -0-7 2-7 Athy,. . . . 2 -3 200 -0-6 1-7 Mean = 1-S DIURNAL EANGES OF TEMPERATURE. Climatology depends upon the ranges of temperature (whether diurnal, monthly, or annual), no less than upon mean values; and their investigation is accordingly a necessary part of the present inquiry. In the present series of observations, the diurnal ranges of temperature are given by means of the results obtained with self- registering thermometers. These results are the least satisfactory portion of the whole series. It is well known that the ordinary self -registering thermometers are extremely apt to get out of order, the maximum, by the index becoming entangled in the mercury, and the minimum, by the distillation of the spirit into the upper part of the tube ; and although the observers were carefully in- structed in the mode of remedying these derangements, no one (I believe) who has handled such instruments will wonder that men previously unaccustomed to them should have sometimes failed, in what is in all cases a somewhat delicate operation. 336 OF THE METEOROLOGY OF IRELAND. But there is another source of error affecting the maximum thermometer, which it is still more difficult to avoid. If the in- strument be exposed to the influence of radiation for any portion of the day, however short, it will, from its construction, retain the impression made upon it ; and, consequently, if the abnormal tem- perature to which it has been thus subjected exceeds the greatest temperature of the air in the day, an erroneous result will be re- corded. The difficulty of guarding thermometers completely from such influences is well known; and although some trouble was taken to insure this protection, the observations themselves show that it was not effective at all the stations. I have, accordingly, been compelled to reject a portion of the results obtained with the maximum thermometer, as defective from this cause. In the following Table are given the diurnal ranges deduced from the maximum and minimum temperatures, combined in yearly and half-yearly periods, retaining only those stations at which one or other of the two half-years is complete : DIURNAL BANGES (HALF-YEARLY AND YEARLY MEANS). Station. Summer. Winter. Year. Portrush, . . 12'2 lQ^-3 no.o Donaghadee, .... 10-9 8-3 9-6 Armagh, .... 13 -3 10 -4 11 - 8 KillmipV . 9 *9 Markree, 14-3 11 -0 12-6 Dublin, . . . 11 '5 8*K 1ft .1 Portarlington, . . , . 17-1 12-6 14-9 Athy, 15-0 11-0 13-0 Kilrush, . . 9.7 Cahirciveen, .... 7-4 Castletownsend, . . . 11-8 8-2 10-0 Coast Stations, . . . ll-6 8-9 10-3 Inland do., ... 14-9 11-3 13-1 Differences, . . . 3-3 2-4 2-8 ON THE METEOROLOGY OP IRELAND. 337 From the mean results of the preceding Table, we learn that the diurnal range is greater at the inland than at the coast stations, the mean excess being 2-8 degrees. The excess is greater in summer than in winter, being 3'3 in the former, and 2'4 in the latter season. "We are now in a position to refer to one, at least, of the practical inferences which may be deduced from the preceding results. The climatological conditions connected with temperature, which favour the prevention or cure of pulmonary diseases, are, firstly, a high winter temperature ; and secondly, a small amount of diurnal range. It has been already stated that Ireland is well circumstanced as to these conditions ; let us now inquire which is its most favourable region as respects them. The months of lowest temperature in Ireland, and which are on that account the most trying to the sufferers from the diseases referred to, are those of December, January, February, and March. During these months the mean temperature varies very little in Ireland, the mean range at Dublin varying from 41'7 in January, to 45 0< 4 in March, or only 3'7 degrees. Now the mean direction of the isothermal lines for these four months is N. 37 W. ; so that the highest mean temperature for these months is to be found on the south-western coast, not far from Valentia. The second condition above mentioned, although not frequently taken into account, is, perhaps, still more important. In proof of this it may be mentioned that in Norway, which is remarkable for the small amount of the diurnal range of temperature, consump- tion is uncommon, even in the highest latitudes ; while in parts of Sweden, where this condition does not hold, it is prevalent. Now, we learn from the preceding Table that, among the stations at which observations were made in 1851, the winter diurnal range of temperature is least at Cahirciveen. Both conditions, therefore, point to the sputh- western coast of Kerry as the region in Ireland most favourable to patients affected with these formidable maladies. I am not in possession of any statistical data bearing upon this question, and am therefore unable to say how far the conclusion thus drawn is borne out by facts. 338 0& THE METEOROLOGY OF IRELAND. TEMPERATURE OF THE SEA. Provision was made that the temperature of the sea should be observed at all the places at which tidal observations were taken. For this purpose each station was furnished with a thermometer, having its bulb inclosed within a small reservoir of copper, for the double purpose of guarding it from accident, and of protecting it (by means of the contained water) from rapid changes of temperature, when it was lifted into the 'air for obser- vation. The observer was instructed to note its indications twice in the day, at intervals of about twelve hours, the thermometer being attached to a pole, and plunged to the depth of about one foot in deep water. The diurnal change of the temperature of the sea being very small, it is completely eliminated by two such observations. At many of the stations the instrument was lost, or broken, in the attempt to use it during boisterous weather. We are, therefore, only in possession of the results from six stations, which are contained in the following Table. In the last two lines of the Table are given the mean results of the six stations, and the differences between them and the mean of the entire year. These numbers accordingly exhibit the law of the annual variation of sea temperature, around the coasts of Ireland ; and the remarkable regularity in their progression shows that, even from the results of a single year, we obtain a close approximation to the actual law. ON THE METEOROLOGY OF IRELAND. :339 00 -tl CO i-H 00 CO CO OS 8 p I OS OS 00 CD OO 1 l : W CO ,H CO 9 CM 3 3 3 5 J> CO * 7 . OS CO 1-- ,-H ,-H 00 rtn >O "^t >O CO "^ o >o *o *o *o *o CM O "o eo 3 , t- eo 5 o os co 5 CM i C 1C *O CO CO CO 1 - I r^ O <* OS >0 CO kO kO ^O CO CO CO ? 9 > * oo o o O -" O 10 >0 3 CO CO CO & CO 1 ep i us ap * cp Tt< CM CM r- 00 CO >o >o c o >o >o CM O E? tj m 9 ,-. cp TH CM p o os o TH sc- eo kO Tfl >O O O >O 9 "? H 1 1 OS (M ^H * CO l~- t^ I 00 i-H OO OS 00 *O OO CO j 1^- CO CO OS 00 00 *- CO I >O O *O O *O CO 1 "? 1 9 r 9 ^ -r 9 co 9 1 g. 9 9 ? T 9 T 1 cb co eo oo co i- o Station. s 1 1 g 6 | 6 S 3 1 i ^ ii II 1 z2 340 ON THE METEOROLOGY OF IRELAND. We learn from these numbers that the annual variation of the sea-temperature, at the surface, differs considerably from that of the air above it, the difference consisting chiefly in a retardation of the epochs of maximum and minimum. Thus the minimum temperature occurs in the middle of February, and the maximum in the middle of August, or about a month after the corre- sponding epochs of the temperature of the air. The annual range is also, as might have been expected, considerably less than that of the air. These results accord sufficiently well with the conclu- sions drawn by Ksenitz, from a comparison of the results of many voyagers. But the most interesting result is that concerning the relation between the temperature of the sea at the surface, and that of the superincumbent air. Upon this subject the greatest discordance exists in the statements of different observers. According to Humboldt, the mean temperature of the Atlantic Ocean, at the surface, is in all cases higher than that of the atmosphere above it. This conclusion is confirmed by the observations of Peron and Fitzroy, and is contradicted by those of Irving, Forster, and Kotzebue. From an elaborate discussion of the observations of many voyagers, Ksemtz infers that the temperature of the sea is less than that of the air over the land in the lower latitudes, while in the higher latitudes it is greater. The original conclusion of Humboldt, however, seems to be placed beyond all doubt by the recent obser- vations of Captain Duperrey, which appear to be more numerous, and taken with more precautions to insure accuracy, than any preceding. It seems now to be generally admitted that, in the temperate and polar regions, the temperature of the sea is higher than that of the air ; and the only question that remained was as to the tropics. Now the observations of Duperrey were made all round the globe, between 10 N. and 10 S. latitude ; and they were taken at intervals of four hours, so as completely to eliminate the effects of the diurnal change. From these observations it appears that the temperature of the sea is higher than that of the air within the zone already mentioned, the mean excess in the Atlantic being 0'83 Fahr., and in the Great Ocean about half that amount. The present observations possess much interest in connexion with these questions. In order to perceive their bearing, I have, in the Table which follows, given the half-yearly and yearly ON THE METEOROLOGY or IKEEAND. 341 means of the sea-temperature at the several stations, together with the differences between them and the corresponding means of the temperature of the air. At Cushendall and Bunown no observa- tions of the temperature of the air were actually made ; and for these stations^, consequently, the latter means are calculated from the isothermaf lines. TEMPERATURE or THE SEA (YEARLY AND HALF-YEARLY MEANS). Station. Summer. Winter. Year. Temp. Excess. Temp. Excess. Temp. Excess. Portrush, . . 54-6 + r-o 48M + 3-8 51-5 + 2M Cuahendall, 53-2 - 1-0 49-1 + 4-4 51-1 + 1 -7 Donaghadee, . 53-6 -0-6 48-6 + 3-6 51 -1 + 1 -5 Bunown, . . 58-0 + 2-4 50-2 + 3-2 54-1 42-8 Courtown, . . 57-7 + 2-2 47-9 + 3-0 52-8 + 2-6 Castle to wnsend, 56-9 + 0-1 48-9 + 1 -5 52-9 + 0-8 ! Meandiff., . 40-7 + 3-3 + 2-0 It appears from the last line of this Table, that the temperature of the sea is, upon the mean of the entire year, 2'0 higher than that of the air above the coast. The excess is 3'3 in winter, and 0-7 in summer. There appears also to be considerable diversity in the amount of the excess at the different stations; it is greatest, on the mean of the entire year, at Bunown, and least at Castletownsend. This excess of the temperature of the sea above that of the air furnishes the explanation of the fact already noticed, namely, the diminution of the temperature of the air in proceeding from the coasts inland ; for it is obvious that the air in the vicinity of the sea must have its temperature raised by contact with the water. It follows also, that the absolute excess of sea-temperature considerably exceeds that above stated. Thus, w Imv M6O, th. 342 ON THE METEOROLOGY OF IRELAND. temperature of the sea, on the average of the entire year, exceeds that of the air over the coasts by2'0 ; while the latter temperature exceeds that of the air inland (for the same latitude and longitude) by l-8. The total excess of the sea-temperature above that of the air amounts, therefore, to 3'8 Fahrenheit. This excess, which appears to be much greater than has been observed elsewhere, is to be ascribed, mainly, to the influence of the gulf-stream upon the temperature of that part of the ocean which bathes our shores. But there is likewise another cause which undoubtedly contributes to the effect. It has been shown by Mayer and Joule, that heat is generated by the friction of fluids in motion, and the latter experimentalist has established the important physical law, that there is a definite relation between the heat so produced, and the mechanical power ex- pended by the moving mass. Mr. Eankine has applied this principle to explain the fact, observed by M. Eenou, namely, that the temperature of the river Loire at Vendome is higher than that of the air above it ; and it is obvious that a similar explanation is applicable to the phenomenon under consideration. There is no doubt as to the reality of the cause ; the only question is as to the magnitude of the effect to be ascribed to it. That such effect is, at all events, sensible, I infer from two circumstances. The first of these is, that the phenomenon of the excess of sea- temperature appears to be general, and must, therefore, be the effect of some general cause ; the second is, that on the coasts of Ireland there is no sensible difference between the amount of the excess on the eastern and on the ivestern shores. Should the effect of this cause be found to be sensible, and its amount be determined, our views of the cycle of meteorological phenomena would be much enlarged. The elevation of tempera- ture rarefies the air ; the denser air flows in to supply the partial vacuum, and wind is produced ; and finally, this wind, both by its own motion, and by that of the ocean which is so subject to its power, restores again the heat which had been converted. Thus the normal condition of temperature is preserved, not only throughout the changes which render it latent and sensible, in the generation and condensation of vapour, but also in its conversion into mechanical power, and its reproduction, in the phenomena of the tempest and of the billowy sea. ON THE METEOROLOGY OF IRELAND. 343 BAROMETRIC PRESSURE. An attempt* was made to correct the barometric results, by a careful comparison of the several instruments with the Dublin standard, by means of portable barometers, and by the reduction of the results to the sea-level, calculated at the rate of '0011 of an inch for each foot of altitude. These results are, however, incom- plete, no comparisons having been made of the barometers at the four inland stations. For this reason, and also because of the uncertainty attending the comparison of barometers by means of portable instruments, I have thought it necessary to seek th corrections by a comparison of the observed results themselves. I comparisons of this kind, where the stations are widely separate; it seems necessary to employ the means of a somewhat extendt series of observed results, during which the fluctuations of bare metric pressure are small. I have accordingly selected for the purpose the monthly means of May, July, and September, in which months there was but little variation of barometric equilibrium. The defects of the means at each station, compared with those at Dublin, for these months, are given in the following Table, and the last column contains the inferred corrections, which are equal to the mean differences + *021, the added number being the reduc- tion of the Dublin results to the sea-level. ON THE METEOROLOGY OF IRELAND. BAROMETRIC CORRECTIONS. Station. Defect from Dublin. Correction by Comparison. May. July. Sept. Portrush, .... 058 070 056 + 082 Buncrana, . . . 120 100 088 + 124 Donaghadee, . . 052 062 054 + -077 Killybegs, . . . -056 032 080 + 077 Armagh, .... 252 257 248 + 273 Killough,. . . . 058 054 068 + -081 Markree, .... 134 130 155 + 161 Westport, . . . 046 058 114 + -094 Dublin, .... 000 000 000 + 021 Portarlington, . . Athv, . 254 325 022 268 304 008 309 317 014 + -298 + 336 + -036 Courtown, . . . Kilrush, .... 090 062 089 + 101 Dunmore, . . . '078 056 076 + -091 Cahirciveen, . , . 062 034 085 + 081 Castletownsend, 016 006 054 + 046 Applying the foregoing corrections, we obtain the numbers of the following Table. In order to perceive more clearly the simultaneous variations in the distribution of pressure, I have, in the last four lines of the Table, combined the stations, and their results, into four groups, as hereafter described in treating of the observations of wind-force : ON THE METEOROLOGY OF IRELAND. 345 - ^ " _r 346 ON THE METEOEOLOGY OF IRELAND. The phenomena of the distribution of pressure are very clearly shown in the foregoing Table. It will be seen from it that, on the average of the entire year, there is an excess of pressure in the south of the island, and a defect in the north, the minimum being at the north-western extremity. This excess of pressure in the south is shown in the means for the several seasons of summer, autumn, and winter, respectively ; and the cause of it will, I think, here- after appear upon the discussion of the phenomena of storms. In the separate months, the points of greatest and least pressure vary somewhat irregularly ; but they are, in nearly every month, at opposite extremities of the island. Thus, in January, the maxi- mum pressure is in the south-east, and the minimum in the north- west ; and so for the others. This circumstance is what should have been expected d priori ; and it affords satisfactory evidence of the general accuracy of the results themselves. DIRECTION AND FORCE OF THE WIND. Direction of Wind. The direction of the wind was observed, at most of the stations, by means of the ordinary wind-vane. Much care was taken, not only in placing these instruments truly in azimuth, but also in selecting positions for them which seemed least exposed to eddies or other local irregularities. At Armagh and Dublin the direction of the wind was recorded continuously, by means of self-registering anemometers. The following Tables give the number of times, out of 100, in which the wind blew from each of the eight points at the several stations, for the summer and winter half-years respectively, and for the entire year. The winds from the intermediate points, when observed, were divided equally between the two adjacent jmncipal points : ON THE METEOROLOGY OF IRELAND. 347 FREQUENCY OF THE SEVERAL WINDS (SUMMER). Station. - * N. N.E. E. S.E. S. S.W. w. N.W. Portrush, . . 20 5 6 6 24 15 11 13 Buncrana, . . 13 8 6 12 11 19 12 19 Donaghad.ee, . 23 6 5 13 11 14 16 11 Killybegs, . . 15 9 14 7 8 13 21 14 Armagh, . . 12 9 7 6 16 19 18 15 Killough, . . 11 7 14 12 16 18 8 14 Markree, . . 14 5 4 17 14 15 10 21 Westport, . . 10 10 14 10 3 3 32 19 Dublin, . . . 2 10 12 13 8 23 11 19 Portarlington, 5 28 2 11 7 14 13 21 Athy, . . . 13 1 2 12 16 12 25 19 Courtown, . . 13 17 5 7 8 23 15 13 Kilrush . . 12 10 14 8 8 13 19 17 Dunmore, . . 15 8 14 5 7 18 8 16 Cahirciveen, . 11 9 12 9 12 18 16 14 Castletownsend, 8 9 11 12 2 37 15 6 FREQUENCY OF THE SEVERAL WINDS (WINTER). Station. N. N.E. E. S.E. S. S.W. W. N.W. Portrush, . . 15 3 4 7 35 23 9 4 Buncrana, . . 10 5 5 10 15 27 15 13 Donaghadee, . 9 7 4 7 14 25 26 9 Killybegs, . . 11 6 8 9 13 19 18 15 Armagh, . . 6 5 2 5 26 35 12 10 Killough, .' . 9 4 4 5 18 25 12 24 Markree, . . 12 4 4 19 19 23 9 10 Westport, . . 13 4 12 10 6 8 26 21 Dublin, . . . 2 1 2 14 14 38 14 13 Portarlington, 5 11 1 4 10 19 20 30 Athy, . . . 7 2 1 15 28 11 23 13 Courtown, . . 7 5 4 4 18 23 -'! in Kilrush, . . 10 9 10 6 16 23 13 11 Dunmore, . . 16 4 3 5 1 1 'Jo 22 16 Cahirciveen, . 8 8 13 12 1-J 20 17 10 Castlrtownsoml, i:j 5 :$ 6 :;.-, 11 13 348 ON THE METEOROLOGY OF IRELAND. FREQUENCY OF THE SEVERAL WINDS (YEAR). Station. N. N.E. E. S.E. S. S.W. W. N.W. Portrush, . . 17 4 1 5 6 30 19 10 8 Buncrana, . . 12 7 6 11 13 23 14 16 Donaghadee, . 16 7 4 10 13 19 21 10 Killybegs, . . 13 8 11 8 10 16 20 15 Armagh, . . 9 7 4 5 21 27 15 12 Killough, . . 10 5 9 8 17 21 10 19 Markree, . . 13 4 4 18 17 19 10 16 "Westport, . . 11 7 13 10 4 5 29 20 Dublin, . . . 2 6 7 14 11 31 13 16 Portarlington, 5 19 1 7 9 16 16 25 Athy, . . . 10 2 j 1 13 22 12 24 16 Courtown, . . 10 11 1 4 6 13 23 19 14 Kilrush, . . 11 9 12 7 12 18 16 16 Dunmore, . . 15 6 9 5 11 19 20 16 Cahirciveen, . 9 8 13 10 12 19 17 12 Castletownsend 10 7 7 9 3 36 18 9 The following are the mean results for the whole island : N. N. E. E. S. E. s. s. w. W. N. W. Summer, . 12 9 9 10 11 17 16 16. Winter, . 10 5 5 9 16 23 18 14. Year, . 11 7 7 9 14 20 17 15. We learn from them that, in the year 1851, the wind blew, on the average of the entire year, most frequently from between S. W. and W., and least frequently from between N. E. and E. The same thing holds also for the summer half-year, the point of maxi- mum frequency being, very nearly, W. 8. W., and that of mininmm frequency E. N. E. In the winter half-year the point of maximum frequency is more nearly S.W., that of the minimum being as before. The ratio of the numbers representing the greatest and least frequency is greater in winter than in summer. ON THE METEOROLOGY OF IRELAND. 349 It is not necessary to enter more minutely into the discussion of the numbers of the preceding Tables, as it is probable that the results of a single year, as to the frequency of the several winds, will deviate considerably from the means of several. I may observe, however, that they afford some indications of a law of distribution, depending upon the aspect of the coast. Thus, on comparing the numbers denoting the frequency of any particular wind at the several stations, with their mean for the whole island, it would seem that easterly winds are slightly in excess on the western coast, and westerly winds on the eastern. In other words, there appears to be a preponderating tendency of the wind to blow front the tand, at each place, as compared with the mean of all. It will remain for future inquiry to ascertain whether this holds good in other years, and is, therefore, to be referred to a general law. If so, it is probably the effect of the land and sea breezes, the former preponderating in the average of the winds at 9 A. M. and 9 p. M. Pressure of the Wind. For the measurement of the pressure of the wind, a Lind's anemometer was furnished to each station. The difficulty of obtaining accurate results with this little instrument arises, partly, from the smallness of its indications, and, partly, from the oscillations of the fluid in the tube ; the latter are so considerable as to render the instrument of little value, except in the hands of a patient and somewhat practised observer. After some trial, accordingly, it was deemed advisable that the force of the wind should be in all cases estimated, and that the use of Lind's anemometer should be limited to that of furnishing a check upon this estimation in the case of the stronger winds. The first thing to be determined, then, was the choice of a scale of force. The scales in use are various : in one of them there are four degrees of wind- force ; in another six ; and in a third (the Admiralty scale) there are ticeke. The last of these appears to be too minute for the ordinary powers of unaided esti- mation, and the first not sufficiently so. The intermediate scale (from to 6) was accordingly adopted ; and it appears to be further recommended by the circumstances, 1, that it is the subdivision most generally used on the Continent; and 2, that, as its numbers represent the same degrees of wind-force with the alternate numbers of the Admiralty scale, the latter are readily 350 ON THE METEOROLOGY OF IRELAND. convertible into the former. The six degrees of wind-force were de- signated as follow : 1. Light breeze; 2. Moderate breeze; 3. Strong breeze ; 4. Moderate gale ; 5. Strong gale; 6. Storm. In order to know the amount of confidence which may be placed in such observations, it is necessary to determine how far, in respect of accuracy, six degrees of wind- force can be estimated, the observations being supposed to be made by practised observers. And to be able to apply the observations, we must further know what are the pressures and velocities of the wind corresponding to the several terms of the scale. For these purposes I made a somewhat extended series of observations, estimating the force of the wind according to the prescribed scale, and, at the same time, measuring its velocity by means of Robinson's anemometer. The following Table gives the mean results of these observations. The numbers in the first column are the terms of the scale ; those in the second are the corresponding times of 100 revolutions of the instrument, expressed in seconds ;* the third column contains the corresponding velocities of the wind, in feet per second; and the fourth the calculated velocities, deduced as hereafter described. * Dr. Robinson has shown (Trans. S. I. A., vol. xxii., p. 167) that the velocity of the wind is to that of the centres of the hemispherical cups, as 3 to 1. But r being the length of the horizontal arms of the instrument, measured to these centres, the circumference of the circle described by them is 2irr ; and if r be expressed in feet, and n be the number of revolutions performed in a second, their velocity is 2*r x n. The corresponding velocity of the wind therefore is V= 6- x . In the instrument in my possession the radius is 5-5 inches. Hence 2r = , and substituting for IT its nu- merical value, V= 8-64 x n. Instead of noting the number of revolutions, and parts of a revolution, performed in a given time, I have found it convenient to observe the time of performing 100 revo- lutions. I have had the instrument accordingly provided with a little hammer, which is pressed against the registering wheel by a spring, and which, being raised by a pro- jecting pin at one point of its circumference, falls again with a sharp noise when this has passed. The interval between two such strokes of the hammer, therefore, is the time of one whole revolution of the registering wheel, or of 100 revolutions of the arms. Accordingly, a chronometer being held close to the ear, the whole observation effected by the help of that organ. The velocity of the wind in this case is given by the formula V = ~, T being the observed time of 100 revolutions. ON THE METEOROLOGY OF IRELAND. 351 VELOCITIES OF THE WIND CORRESPONDING TO THE TERMS OF THE f SCALE (0-6). n T V (observed). F (calculated). I. 71 s 12 12 II. 35 25 23 III. 25 35 35 IV. 20 43 46 V. 16-8 51 58 YI. 11-6 75 70 We see that the terms of the estimated scale correspond, nearly, to an arithmetical progression of velocities, and not of pressures. This fact has been already noticed by Dr. Eobinson. The common difference in this series, which is equal to its first term, is obtained from the numbers of the third column by means of the formula V = n Fi. The following are the deduced values : I. V, = 12-0. IY. V, = 10-8. II. 12-5. Y. 10-2. III. 117. VI. 12-5. The mean of these values is Fi = 11*6. The calculated values of F, contained in the last column of the foregoing Table, are, accord- ingly, obtained from the formula r- 11*6x11; their agreement with the observed values is sufficient to establish the assumed law. As a verification of the preceding result, I took also a tolerably extended series of measurements of the pressures of the wind, corre- sponding to the highest term of the scale, with Lind's anemometer. Their mean gave 2'06 inches for the reading of the instrument corresponding to that term ; and the corresponding pressure on one square foot of surface, computed in the proportion of 5 -20 pounds to the inch, is 10*7 pounds. Hence, the pressure belonging to the unit of the scale is Pi = 0'30. The corresponding velocity is in- ferred from the formula F 2 = 437 P. Its value is F, = 11 '5; a result which agrees very closely with that already deduced from Robinson's anemometer. The results hitherto given rest only on my own estimations ; it remains to POP bow far tlvv accord with those <>f otln-r obsr-r. 352 ON THE METEOROLOGY OF IRELAND. I have selected for this purpose the results of the observations with Lind's anemometer, made at Portrush and Donaghadee by two of the best of the Coast-guard observers, and have placed my own beside them, for comparison. The results, converted into pressures (expressed in pounds on the square foot) are contained in the fol- lowing Table. The numbers in the last column are the calculated pressures, deduced from the formula P = Ptf, n being the number of the term of the scale, and P 1 ( = 0'30) the pressure corresponding to the first term. PRESSURES or THE WIND CORRESPONDING TO THE TERMS or THE SCALE (0-6). Term. Dublin. Portrush. Donaghadee. Calculated. I. 0-5 0-4 0-5 0-3 11. 1-3 1-3 1-1 1-2 III. 3-0 2-9 2-9 2-7 IV. 4-2 5-3 5-3 4-8 V. 7-0 7-3 7-9 7-5 VI. 10-7 10-8 It will be seen, that the differences of the corresponding num- bers at the three stations are small, and that their means agree very well with the calculated pressures. It seems, therefore, to be fully proved that the velocity of the wind may be estimated to six degrees, by practised observers, with sufficient accuracy. In the following Table are given the results of the observed wind-force for the entire year, and for its two principal divisions. The excess of the force in winter appears at all the stations, ex- cepting Dublin, Portarlington, andAthy.* At these three stations, also, the force of the wind is below the average. The mean force for the entire year is 1'76, corresponding to a velocity of 20'4 feet per second. The force is, of course, greater in winter than in summer, the mean force for the winter half-year being 1*87, and that for the summer half-year 1'65. * Buncrana is likewise an exceptional case ; hut the exception is there probably due to inaccuracy of observation. ON THE METEOROLOGY OF IRELAND. MEAN FORCE OF THE WIND FOR THE SUMMER AND WINTER HALF-YEARS, AND FOR THE WHOLE YEAR. Station. Summer. "Winter. Year. Portrush .... 1'68 1'85 1 *77 Buncrana, Donaghadee, .... Killybegs, Armagh, Killough, 2-15 1-50 1-40 1-23 1-60 2-15 1-82 1-75 1-78 1-92 2-15 1-66 1-58 1-51 1-76 Markree, Westport, Dublin 1-68 2-20 1-33 1-82 2-78 1-32 1-75 2-49 1*33 Portarlington, .... Athy 1-42 1-42 1-27 1-20 1-34 1*31 Courtown, 1-40 1-70 1-55 Kilrush, 1-72 2-02 1-87 Dunmore, 1-88 2-03 1-96 Cahirciveen, .... Castletownsend, . . . 1-80 1-93 2-17 2-38 1-98 2-16 If, to eliminate local irregularities, we combine the preceding results in groups, according to the arrangement hereafter described, we find the following values for the mean forces of the entire year: North-east, . . 1'64. North-west, . . 1-94. South-east, . . 1-61. South-west, . . 2-00. From this it appears that the mean force of the wind is consider- ably greater in the icest of the island than in the east, the ratio being somewhat greater than that of 1 '2 to 1. There is but little difference between the forces in the northern and southern portions of the island. CYCLONIC MOVEMENTS. In analyzing the phenomena of rotation, the first step was to note those cases in which the mean directions of the wind, in any two districts, differed by 90, or upwards. It was soon perceived, that no conclusion could be drawn as to a general movement of the atmosphere, when the wind was very moderate, the direction being 2 A 354 ON THE METEOROLOGY OF IRELAND. then greatly influenced by local causes. Accordingly, excluding those cases in which the wind did not exceed a light breeze at most of the stations, the remainder were examined in detail, by laying down the simultaneous directions of the wind upon a series of skeleton charts prepared for the purpose ; and there was no diffi- culty in ascertaining, by the inspection of these charts, the exist- ence or non-existence of rotatory movement. The Fame means sufficed to determine, very nearly, the position of the centre of the vortex at each epoch ; and the places of the centre being thus found, for epochs distant by intervals of twelve hours, the direction and velocity of its progressive movement are ascertained. The position of the centre of the vortex at any instant may be determined, more accurately, by calculation. Thus, if y and x denote the distances (in geographical miles) of the place of obser- vation from any assumed central point, measured on the meridian, and on the perpendicular to the meridian, respectively ; ?/ and ^ the corresponding co-ordinates of the centre of the vortex ; and the angle which the direction of the wind at the point (y, x) makes with the meridian, measuring from north to east : y - y + (x - a*,) tan = 0, the direction of the wind being perpendicular to the line connect- ing the points (y, x) and (y , x ). Now, all the quantities in this equation are given, excepting y n and as a ; so that, if the direction of the wind be accurately known at two stations, the co-ordinates of the centre of the vortex may be completely determined. The irregularities due to local causes, and the errors of observation themselves, forbid this ; and, in order to lessen their influence, it is necessary to know the direction of the wind at several stations. There will then be as many equations of the preceding form as there are places of observation ; and the unknown quantities, ?/ and ar , are to be determined by combining these equations by the method of least squares. It is found, that the centre of the vortex is also the point of least barometric pressure, and that the pressure increases regularly with the distance from it. Hence the position of the centre may be inferred from the barometric observations alone. The positions thus determined have been found to coincide in all cases, very nearly, with those deduced from the observed directions of the wind. ON THE METEOROLOGY OF IRELAND. 3;V> The following are the well-marked instances of aerial rotation whichhave -occurred in Ireland in the course of these observations. No case has been included in the enumeration, in which the simul- taneous directions of the wind did not differ, at two points, by at least 90 ; and thus, probably, many cases of cyclonic movement are passed over, in which the centre of the vortex was remote. 1850. Oct. 6, 7. Cyclone and storm, moving from S.W. to N. E., with a velocity of about 290 geographical miles per diem. (Plate II., figs. 1, 2, 3.) Oct. 6, 9 A.M. Centre of the vortex on the south-western coast of Ireland, west of Kilrush. Least pressure at Cahirciveen. Mean velocity of the wind = 25 feet per second ; greatest do. (on the west coast) = 45 feet. The atmosphere at the northern stations unaffected by the vortex at this epoch. Oct. 6, 9 P.M. Centre of the vortex over the north of Ireland, a few miles north of Killybegs. Absolute barometric minimum (= 28-836) at Killybegs ; increase of pressure in 100 miles = 0'30 inch. Mean velocity of wind = 35 feet per second ; greatest do. (Markree) = 70. Southern stations unaffected by the vortex. Oct. 7, 9 A.M. Centre on south-western coast of Scotland. Least pressure at Donaghadee. Mean velocity of wind = 45 feet per second ; greatest do. (north coast) = 60 feet. Hail fell at Markree ; wind amounting to a gale in the north, in the evening of the same day. The diameter of the vortex may be estimated with tolerable precision in this case, by measuring from the centre to the limits of the region affected by the movement ; it was about 280 geogra- phical miles. Oct. 22, 23. An interesting and instructive case of conflicting currents generating a rotatory movement. The velocity of the wind was uniform throughout the island, and was from 30 to 35 feet per second. (Plate II., figs. 4, 5, 6.) Oct. 22, 9 P. M. Wind from N. W. in the north of Ireland, and from S. W. in the south-east. The central point of junction of these currents was over the channel, to the north-east of Dublin. Least pressure at Donaghadee. Oct. 23, 9 A.M. A distinct rotatory movement, whose centre was a little to the north-east of the point of junction above referred 2A2 356 ON THE METEOROLOGY OF IRELAND. to, not far from Donaghadee. Least pressure at Donaghadee, as before. Oct. 23, 9 P. M. Rotatory movement continued. Centre of vortex had moved from S.W. to N.E., at the rate of about 100 miles per diem. Absolute minimum of pressure (=29 -3 60) at Donagh- adee; increase of pressure in 100 miles = O'lO inch. Nov. 18, 19. A cyclone, with violent storm, crossing the island from W. S.W. to E.N.E. (Plate III., figs. 1,2,3.) The movement of the centre of the vortex appears to have been curvilinear, and to have varied considerably in velocity. Between 9 p. M. of the 18th, and 9 A.M. of the following day, its path was from S. W. to N. E., and its velocity about 320 miles per diem ; in the succeeding twelve hours its course was nearly from W. to E., with a greatly diminished velocity. The mean velocity of the wind, throughout the storm, was from 45 to 50 feet per second. Nov. 18, 9 P. M. Centre of the vortex on the south-western coast, about 30 miles to the north of Cahirciveen. Least pressure at Kilrush. Maximum velocity of wind (in south of island) = 60 feet per second. Nov. 19, 9 A.M. At this epoch the wind was blowing from N. at Killybegs, and from S. at Donaghadee ; it was blowing from S. E. at Portrush, and from N. W. at Castletownsend ; from S. S. E. at Armagh, and from N. N. W. at Markree. The centre of the vortex was therefore over Ireland at that time, and between the stations above mentioned, its exact position being about 15 miles to the west of Armagh. Absolute minimum of pressure (= 28*248) at Armagh ; increase of pressure = 0'31 inch. Maximum velocity of wind (in south) = 65 feet per second. Nov. 19, 9 p. M. Centre over the Channel, to the south-east of Donaghadee. Absolute minimum of pressure (= 28*410) at Donagh- adee ; increase of pressure = 0-28 inch. Maximum velocity of wind (in south) = 55 feet per second. We have seen that the centre of the vortex was between Armagh and Markree at 9 A.M. of the 19th ; and, as the direction of its progressive movement was not far from the line connecting these places, it must have passed nearly centrally over both. Hence we should expect there the peculiar phenomena the Ml of the wind, and the sudden reversal of its direction which are observed to occur at places in the path of the centre of a cyclone. I shall, ON THE METEOROLOGY OF IRELAND. 857 therefore, briefly describe the series of changes at these two stations. The observations at Armagh are taken from the records of the self- registering anemometer, which were, of course, continuous; those at Markree were made at short intervals. At Armagh the wind began to blow at 7 p. M. of the 18th, with a velocity of 32 feet per second. The maximum velocity, with the exception of a short squall* at 5 A. M., occurred at 7 A. M. of the 19th, and amounted to 43 feet per second. From this time the wind abated rapidly almost to a calm, its velocity at noon amount- ing only to 6 feet per second ; but at 3 p. M. it rose again, with a velocity of 22 feet. The initial direction of the gale was from the E. S. E. From 9 p. M. on the 18th, to 1 A. M. on the 19th, it veered to S., at which point it continued for several hours, including the period of greatest force of the gale. At 11 A.M. its direction had returned to S. E., and it then suddenly shifted to W. N. W., alter- ing through 160 in 24 minutes. The minimum of pressure took place at ll h 30 m , at the close of this movement; its amount was 27-930 inches.f At Markree the gale commenced at 4 h 30 m P.M. of the 18th, with a rapidly falling barometer. At 7 P. M. the wind abated to a breeze, the barometer still falling. It recommenced at 10 p. M. During the squall, which lasted only three minutes, the velocity reached 90 feet per second. t The following are the anemometric observations above referred to. The direction is measured from S. through W. to N. ; the velocity is expressed in miles per hour. On the 19th, from 4 A. M. to 8 A. M., the direction-registering pencil was thrown out of gear, but there appears to have been no change of any magnitude in the interval: Nov. 18, A.M. Nov. 18, P.M. Nov. 19, A.M. Nov. 19, P.M. Hour. Vel. Dir. Vel. Dir. Vel. Dir. Vel. Dir. 12-2 28-8 4-8 212-3 23-4 335-2 4-1 124-0 1 8-3 28 -2 5-3 243 -6 24-4 354 -3 10-5 104 4 2 5-9 47 '8 8-7 276 -2 22-2 353 -4 12-0 93 9 3 7-2 49 -8 7-8 274 -5 23-6 345 -6 15-4 00 2 4 8-1 48 -1 14-7 278 -8 23-2 16-9 108 8 5 3-8 41 -3 Hf-1 281 -7 25-3 13-8 136 7 6 2-9 76 -0 16-6 281 -9 27-4 16-1 i:;s 3 7 8 9 19 11 3-7 1-5 2-3 4-9 5- 162 -3 162 -0 212 -4 206 -4 209 -2 22-0 19-2 19-3 17-6 20-1 283 -5 285 -5 282 -1 321 -3 330 -6 29-4 29-0 17-6 14-6 7-6 339 -8 320 -6 324 -5 16-2 16-6 15-8 9-2 11-8 161 1 163 6 161 -2 147 '6 159 -5 358 ON THE METEOKOLOGY OF IRELAND. from the S. E. ; and at 3 A. M. on the 19th it appears to have attained its maximum. At 6 A. M. the wind again abated ; and at 7 A. M. there was a calm. The minimum pressure took place at this time, and amounted to 28170 inches. At 9 A. M. the wind rose again from the N. N. W., but not with such force as before ; and in the afternoon there was a strong gale again.* From these facts it is evident, that the centre of the vortex passed nearly over Markree at 7 A. M., and over Armagh at ll h 30 m A. M. At Donaghadee, which is nearly in the prolongation of the line connecting the two former places, the wind ceased at 1 P. M., and recommenced at 5 p. M. ; so that the vortex passed nearly cen- trally over this station at about 3 p. M. From these data we learn that the cyclone moved from W. S. W. to E. N. E. ; and that the velocity of the progressive movement was then about 12 miles per hour. The dimensions of the vortex may likewise be collected from the same data. The interval between the commencement of the storm, and the passage of the centre, at Armagh, was 16| hours ; and, the velocity being 12 miles an hour, the radius of the vortex * The following are the extra observations at Markree above referred to. The numbers in the column headed " Barometer," are the excesses above 28 inches : Date. Nov. 18. Nov. 20, Hour. Bar. Ther. Remarks. 4" 30" 6 1-124 1-002 46 -8 48 -8 Blowing a gale ; storm and rain began about noon. Ditto; rain. 7 8 0-926 0-835 49 -5 50 -5 Strong breeze ; heavy rain. Ditto ; ditto. 10 0-652 53 -7 Gale. 11 12 0-577 0-538 54 -0 52 -5 Wind rising to gale ; mizzling rain. Strong breeze ; ditto. 3A.M. 5 6 0-315 0-120 0-027 50 -5 46 -5 45 -5 Wind higher than at any previous time. Gale ; mizzling rain. Strong breeze. 7 0-009 47 -3 Calm. 8 0-023 48 -2 Light breeze ; mizzling rain. 9 0-067 45 -8 Gale. 10 0-124 46 -0 Strong breeze ; wind N. N. W. 3 P.M. 0-303 48 -8 Strong breeze. 7 0-483 49 -5 Strong gale. 10 11 30 0-533 0-578 48 -0 49 -0 Gale from N. W. ; light showers. Strong gale. 10 A.M. 1-084 48 -4 Ditto ; showers. 11 1-128 48 -3 Ditto. 1 30 1-227 47 -5 Ditto. 3 1-282 46 -5 Ditto ; heavy rain. 5 1-376 44 -8 Moderate gale. ON THE METEOROLOGY OF IRELAND. 359 was about 200 miles. The magnitude of the nearly quiescent por- tion of air in the centre of the vortex is better defined. At Armagh the lull lasted from three to four hours ; at Markree three hours ; and at Donaghadee four hours. The diameter of the quiescent central portion was, therefore, about 40 miles. We may now refer to some particulars connected with this gale, which appear to merit attention although probably, in the present state of our knowledge on this subject, we should not be justified in offering any suggestions in explanation. Among the first of these are the abnormal variations in the rotatory movement, especially along the track of the centre. The most curious of these irregularities is that of the direction. At Armagh this began to change rapidly at 9 p. M. of the 18th. At 9 p. M. it was E. S. E. ; at 10 p. M., 8. E. ; at midnight, S. S. E. ; and at 1 A. M. on the 19th, 8. At this latter point it remained for several hours ; and the direction then retrograded through an arc of about 45. At 9 A. M. on the 19th it was S. S. E. ; and at 1 1 A. M. it came back to S. E., after which the sudden shift to W.N.W., already noticed, took place. The next point which seems to merit notice is the fact, that the force of the gale was considerably greater to the south of the line of passage of its centre, than on that line itself, or to the north of it. Thus, at Killiney, where I made frequent observations during the gale, I found the maximum velocity to be 80 feet per second ; at Armagh it was little more than half that amount. It has been already mentioned that the greatest force of the storm occurred at Armagh and Markree, before the epoch of mini- mum pressure, the interval at both places being about four hours and a half. A similar interval took place at Killiney, but in the opposite direction, the epoch of greatest intensity following that of least pressure by four hours and a half. The last point which appears to demand notice is the fact, thai there was a considerable interval between the epochs of the ^-i intensity of the storm at Dublin and at Killiney, places only ten miles apart. The greatest force of the gale, at Dublin, took place between 1 p. M. and 2 p. M. ; at Killiney it occurred between 5 p. M. and 6 p. M. There is a similar interval between the times of mini- mum pressure at the two places, the least height of the barometer occurring later at Killiuey than at Dublin by two or three hours. 360 ON THE METEOKOLOGY OF IRELAND. These differences are probably connected with the difference of altitude of the places of observation. 1851. Jan. 15, 16. A remarkable case of a double cyclone with storm, and a double minimum of pressure. (Plate III., figs. 4, 5, 6.) The first of the two vortices crossed the island from S. to N. on the 15th, and the second traversed the north-western portion of it, from S. W. to N. E., on the following day. The velocity of the former is not well determined ; that of the latter was about 270 miles per diem. The mean velocity of the wind was from 30 to 35 feet per second on the former day, and from 55 to 60 on the latter.* Jan. 15, A. M. Centre of vortex about 20 or 30 miles south of Dunmore. Absolute minimum of pressure (= 28'718) at Dunmore ; increase of pressure = 0-15 inch. Maximum velocity of wind (west coast) = 60 feet per second. Jan. 15, 9 p. M. Centre of vortex appears to have been at this time a few miles north of Buncrana ; the cyclonic movement was, * The following extra observations were taken at Markree : Date. Hour. Bar. Ther. Wind. Remarks. Jan. 15, 2 30 0-786 0-832 42-l 41 -5 N. W. 4 N. W. 4 Mizzling rain. Mizzling rain. 4 0-921 40 -8 W.N.W. 4 Rain. 5 0-986 41 -5 W.N.W. Rain. 6 10 12 5 12 17 1-040 1-204 1-242 1-240 42 -5 37-9 34 -4 34 -1 W.N.W. S.W. S.W. S.W. Clouds breaking all round- Began to clear at 6 h 30 m . Clouds rising from south-west. -Jan. 16, 1 A.M. 1-225 34 -3 S.W. A large halo round moon. 8 30 9 0-776 0-751 42 -0 43-7 S.E. 5 S. E. 5 Mizzling rain. 9 30 0-726 45 -5 S.S.E. 5 10 0-718 47-2 S. S. E. 5 Lind's anemometer = 0*85 inch. 11 0-668 49-7 S. 5 Wind not as strong as at 10 A.M. 12 1 P.M. 0-614 0-566 50 -0 51 -5 S. 4 S. 4 Rain ; rough, with heavy rain. 2 9 0-534 0-620 50 -5 46 -8 S. 4 S.W. 5 Clouds began to break at 1 A.M. 9 30 10 0-661 0-715 46 -7 46 -1 S.W. 5 S.W. 5 Gusts very high occasionally* Lind's anemometer = 0-80 inch. 11 0-794 45 -2 S.W. 4 11 30 0-824 44 -8 S.W. 5 Rain. 12 0-835 43 -0 S.W. 4 Jan. 17, 1 A.M. 0-890 43 -5 S.W. 4 2 0-920 42 -7 S.W. ON THE METEOROLOGY OF IRELAND. 361 however, not distinctly marked, probably owing to the influence of the second cyclone. Least pressure at Buncrana. Velocity of wind uniform throughout the island. Jan. 16, 9 A. M. Centre of second vortex to the south-west of Westport. Least pressure at Westport. Jan. 16, 9 P.M. Centre about 20 miles west of Buncrana. Absolute minimum of pressure (= 28'671) at Buncrana ; increase of pressure = 0'20 inch. Jan. 30, 31. A very interesting cyclone traversing the western portion of the island, in direction from N. to S. nearly, at the rate of about 150 miles per diem. The wind light, the mean velocity being about 20 feet per second. Jan. 30, 9 p. M. Centre of vortex over north-western portion of the island, a little to the north of Killybegs. Least pressure at Killybegs. Maximum velocity of wind (in south-west) = 40 feet per second. Jan. 31, 9 A. M. Centre a little to the eastward of Westport. Absolute minimum of pressure (=29*032) at Westport ; increase of pressure = 0*10 inch. Maximum velocity of wind (in south-west) = 25 feet per second. Lightning observed in north in evening of this day and day preceding. March 18. A cyclone, with storm, traversing the island from S. to N., at the rate of about 200 miles per diem. March 18, 9 A. M. Centre of vortex near Markree. Absolute minimum of pressure (= 29 '328) at Armagh ; increase of pressure = 0-10. Mean velocity of the wind = 45 feet per second ; greatest do. (west coast) = 50 feet. March 18, 9 p. M. Centre of vortex north of the island. Ab- solute minimum of pressure (= 29-371) at Portrush ; increase of pressure = 0'13 inch. Mean velocity of the wind = 35 feet per second ; greatest do. (north-west) = 50 feet. March 19, 9 A. M. Rotatory movement broken up, and wind lessened. Barometer fell, and wind rose again to a gale in the evening ; greatest velocity (north-west) = 65 feet per second. March 25. A distinct rotatory movement at 9 A. M. of this dav, the centre of which was ;i liHl<> 1<> tin- north "f 362 ON THE METEOROLOGY OF IRELAND. Absolute minimum of pressure (= 29*408) at Westport; increase of pressure = 0'13 inch. The velocity of wind uniform, and about 30 feet per second. The wind was very light at the preceding and subsequent observations, so that the progressive movement of the vortex cannot be traced. June 11, 12. Cyclone crossing the island from S. W. to N. E., with a velocity of about 260 miles per diem. June 11, 9 p. M. Centre of the vortex a little to the west of Cahirciveen. Least pressure at Cahirciveen. Mean velocity of wind = 40 feet per second. June 12, 9 A. M. Centre over the island, between Kilrush and Westport. Absolute minimum of pressure (= 29 - 347) at Kilrush ; increase of pressure = 0'04 inch. Mean velocity of wind = 25 feet per second. June 12, 9 p. M. Centre over the channel, to the east of Kil- lough. Least pressure at Dublin. Mean velocity of wind = 20 feet per second. July 27, 28. Cyclone traversing the western coast, in direc- tion from S. S. W. to N. N. E. Velocity of wind = 30 feet per second. July 27, 9 A. M. Centre of vortex west of Cahirciveen. Least pressure at Cahirciveen. Greatest velocity of wind in south-west. The wind at the north-eastern stations uninfluenced by the vortex. July 27, 9 P. M. Centre south of Westport. Absolute mini- mum of pressure (=29'559) atMarkree ; increase of pressure = (HO inch. Velocity of wind uniform. July 28, 9 A. M. General current from S. W. ; mean velocity = 30 feet per second. August 23, 24. Well-defined cyclone advancing in a curvili- near path, the movement of the centre being at first from N. W. to S.E., and afterwards from S. W. to N.E. Aug. 23, 9 p. M. Centre of the vortex north-west of the island. Least pressure at Buncrana. Mean velocity of wind = 25 feet per second. Lightning along the whole of the eastern coast during the day. Aug. 24, 9 A. M. Centre near Armagh. Absolute minimum of pressure (= 29 -439) at Armagh ; increase of pressure = 0-13 inch. ON THE METEOROLOGY OF IRELAND. 363 Mean velocity of wind = 35 feet per second ; greatest do. (south) = 55 feet. The centre of the vortex appears to have passed over Donaghadee about noon. At 9 A. M. the direction of the wind at that place was E. S. E. ; at 12 (noon) W. S. W. ; and at l h 30 m p. M. W. N. W., the shift being accompanied by strong gales and heavy rain. Aug. 24, 9 P. M. Centre north-east of the island. Least pressure at Donaghadee. Mean velocity of wind = 25 feet per second. September 29, 30. Interesting cyclone and storm, crossing the island from S. S. W. to N. N. E., with a velocity of about 270 miles per diem. (Plate IV., figs. 1, 2, 3.) Mean velocity of wind on the 29th = 45 feet per second.* Sept. 29, 9 A. M. Centre of vortex off the south-western coast, to the west of Cahirciveen. Force of wind greatest at the same station at 3 A. M. ; but the barometer continued to fall until noon, when the pressure was 28'970. Increase of pressure = 0*22 inch. Greatest velocity of wind (north-west) = 60 feet per second. Sept. 29, 9 P. M. Centre over the island, about midway between Kilrush and Dublin. Absolute minimum of pressure (= 29' 030) at Markree ; increase of pressure = 0'12 inch. Least pressure in * The following extra observations were taken at Markree : Date. Hour. Bar. Ther. Wind. Remarks. Sept. 29, 7A.M. 197 50 -3 S.E. 5 Rain; 100 rev. anem. in 29". 8 169 51 -1 S. E. 5 Ditto. 9 145 51 -1 S.E. 6 Ditto. 10 125 51 -4 S.E. 5 Ditto ; 100 rev. anem. in 29 1 . 11 '111 52-7 S.E. 6 Ditto. 12 070 64 -2 S.E. 5 Ditto. IP.M. 059 55 -0 S.E. 5 Ditto. 3 0-998 66-2 S.E. Ditto. 5 0-955 55 -3 E. Ditto. 7 0-932 55 -0 E. Ditto. 10 0-869 55 -0 N.E. 100 rev. anem. in 164' ; aurora. Sept. 30, 7 A.M. 0-952 55 -0 W. A little rain. 8 0-979 54 -4 W.N.W. Mizzling rain. 10 0-948 64 -0 W. 4 11 1-062 54 -2 W.S.W. 4 11 P.M. 0-833 50 -2 S.E. 5 12 0-801 50 -2 S.E. 5 Oct. 1, 1 A.M. 0-752 51 -0 S.E. 5 i 0-700 61 -0 S.E. 4 364 ON THE METEOROLOGY OF IRELAND. south-east at 6 p. M. Greatest velocity of wind (north-east) = 55 feet per second. Sept. 30, 9 A. M. Centre near Malin Head, at northern ex- tremity of the island. Absolute minimum of pressure (= 29*020) at Portrush ; increase of pressure = 015 inch. Mean velocity of wind = 35 feet per second ; greatest do. (north-west) = 55 feet. Sept. 30, Oct. 1. Cyclone moving apparently in curvilinear path, its course being at first from W. to B., until it reached the centre of the island, and afterwards from S. S. W. to N. N. E. Mean velocity of wind between 25 and 30 feet per second. Sept. 30, 9 p. M. General southerly current. Centre of vortex to the west of the island ; least pressure on west coast. Greatest velocity of wind (on west coast) = 45 feet per second. Oct. 1, 9 A. M. Centre of vortex over the island, between Kil- rush and Courtown. Absolute minimum of pressure (= 28'838) equally distant from Dublin, Courtown, and Dunmore. Northern stations beginning to be affected by vortex. Greatest velocity of wind (north-east) = 50 feet per second. Oct. 1, 9 p. M. Centre north of Portrush. Absolute minimum of pressure (= 28'853) at Portrush ; increase of pressure = 0'09 inch. At Donaghadee a sudden shift of the wind from S. S. E. to "W.took place at 4 h 30 m P.M. Oct. 4, 5. Distinct cyclone moving fromW.S.W. to E.N.E., and passing over (or near) the northern extremity of the island. (Plate IV., figs. 4, 5, 6.) Mean velocity of wind = 35 feet per second. General electrical disturbance. Oct. 4, 9 A. M. General current from S. W. ; centre of vortex north-west of the island. Greatest velocity of wind (on west coast) = 45 feet per second. Oct 4, 9 p. M. Centre close to northern extremity of the island. Absolute minimum of pressure (= 29-182) at Portrush ; increase of pressure = Oil inch. Greatest velocity of wind (north-west) = 55 feet per second. Oct. 5, 9 A. M. Centre north of the island ; least pressure at Portrush. Greatest velocity of wind (in the north) = 60 feet per second. ON THE METEOROLOGY OF IRELAND. 365 From the facts above stated, we may draw the following general conclusions : 1. The occurrence of cyclonic movements in the atmosphere is not infrequent in Ireland, and may be traced even in the case of moderate winds. 2. The rotatory movement is invariably in the same direction, namely, that opposite to the diurnal movement of the sun in azimuth. 3. This rotation is always accompanied by a considerable dis- turbance of barometric equilibrium, which is greater in proportion to the velocity of the rotatory movement, the pressure being a mi- nimum at the centre of the vortex, and increasing regularly with the distance from that point. 4. The place of greatest velocity appears to have no very defi- nite relation to that of the centre of the vortex, sometimes nearly coinciding with it, and at others being situated in front, or in the rear, on the right hand, or on the left, of the centre.* 5. The vortex itself has a progressive movement, at the rate of from 100 to 300 miles per diem, the average velocity of those observed being 220 miles per diem. The direction of this movement in Ireland is generally from S. W. to N. E. 6. If a line be drawn through the centre of Ireland, in the direction from S. W. to N. E., the track of the centres of by far the greater number of the cyclones, passing over or near Ireland, lies to the north of that line. 7. There is reason to conclude, that these rotatory movements are caused by the conflict of two rectilinear currents moving in dif- ferent directions. STORMS. For the purpose of eliminating local irregularities, and (to a certain extent also) inequalities of estimation, I have, in examining the distribution of the higher winds, combined the stations into four groups, omitting Portrush and Buncrana, which lie somewhat apart. These groups are as follow : I. NORTH-EASTERN. Donaghadee, Killough, Armagh. Mean latitude = 54 24'; mean longitude = 5 57'. In the remarkable cyclone of November 18, 19, 1850, the wind raged with greatest violence on the right hand of the centre (looking in the direction of the pro- gressive movement) ; and this appears to be the case of most frequent occurrence. 366 ON THE METEOROLOGY OF IRELAND. II. NORTH-WESTERN. Killybegs, Markree, Westport. Mean latitude = 54 13'; mean longitude = 8 51'. III. SOUTH-EASTERN. Dublin, Courtown, Dunmore. Mean latitude = 52 43' ; mean longitude = 6 29'. IV. SOUTH-WESTERN. Kilrush, Cahirciveen, Castletownsend. Mean latitude = 52 2', mean longitude = 9 37'. The line joining groups I. and IV. lies, almost exactly, N. E. and S. W. ; and that joining groups II. and III., N. W. and S. E. The following are the numbers of times in which the average force of the wind, in each of these groups, amounted to a strong breeze : or the average velocity to 35 feet per second, and upwards : NUMBER OF TIMES IN WHICH THE VELOCITY OF THE WIND WAS 35 FEET PER SECOND AND UPWARDS. Month. North-East North-West, South-East South- West ! January, .... 18 19 13 29 February, . . . 7 15 10 13 March, .... 5 15 4 14 April 3 11 3 12 May 2 10 j June, 2 15 5 10 July 3 9 4 August, .... 9 3 7 September, . . . 2 11 3 6 October, .... 3 21 4 17 November, . . . 14 1 6 December, . . . 3 9 9 14 Spring, .... 10 36 8 30 Summer, .... 5 33 12 26 Autumn, .... -. 5 46 8 29 Winter, .... 28 43 32 56 Year, 48 Io8 60 141 ON THE METEOROLOGY OF IRELAM). f 'J07 From the foregoing numbers it appears, that high winds are much more frequent on the western than on the eastern coast, the numbers denoting the relative frequency, on the average of the entire year, being nearly as 3 to 1. This preponderance of high winds on the western coast holds at all seasons of the year, the maximum occurring at the north-western extremity in autumn, and at the south-western in winter. The greatest frequency is in the north-west, on the average of the entire year. The following are the cases in which the mean force of the wind, over the whole island, amounted to a gale ; or in which the mean velocity was 45 feet per second and upwards : Nov. 23, 24, 1850. Storm along the western coast, blowing at first from S. S. W., and veering through S.W. to W. Least pressure in north-west throughout. Nov. 23, 9 p. M. Storm began at south-western extremity of the island ; velocity = 45 feet per second. Nov. 24, 9 A. M. Wind continued to blow in same district ; velocity increased to 60 feet per second. Absolute barometric mi- nimum (north-west) = 28*644. Nov. 24, 9 P. M. Storm extended over whole of western coast ; velocity of wind = 55 feet per second. Dec. 14. Storm affecting the whole island, but chiefly the western coast. Wind at first from S. S. W., but veering to W. S.W. at 9 p. M. Least pressure in north-west throughout. Electrical disturbance over the whole island. Dec. 14, 9 A. M. Velocity on western coast = 65 feet per second. Absolute barometric minimum (north-west) = 28*952. Dec. 14, 9 P. M. Velocity on western coast = 50 feet per second. Dec. 31, Jan. 1, 1851. Storm from S. W. and S., beginning on western coast, and extending over the whole island. Dec. 31, 9 A.M. Velocity on western coast = 50 feet per second. Direction S. S. W. and S. W. Dec. 31, 9 P.M. Gale affecting the whole island, except north- eastern extremity. Greatest in south-west ; velocity = 60 feet per second. Direction as before. Absolute barometric minimum (north) = 29177. 308 ON THE METEOROLOGY OF IRELAND. Jan. 1, 9 A. M. Wind abated. Jan. 1, 9 P.M. Gale from S. W. and S. over the whole island, except north-western extremity. Velocity (south-east) = 55 feet per second. Absolute barometric minimum (north) = 28*975. In this case, therefore, there were two storms succeeding each other on consecutive days, with a double fall of the barometer. The direction of the wind on Jan. 1, 9 P.M. was remarkable. The prevailing current was from S. W., and extended over the central parts of the island ; while there appears to have been an indraught towards it, from the north-western and south-eastern quarters. Jan. 12, 13. Storm from S. andS. W., beginning in the north* west, and advancing in the direction from N. W. to S. E. Velocity of wind = 60 feet per second. Least pressure in north-west throughout. Jan. 12, 9 P.M. Gale in north-west. Jan. 13, 9 A.M. Storm advanced to line joining north-east and south-west centres. Absolute barometric minimum (north-west) = 29' 174 ; pressure least at Markree at noon. Jan. 27. Storm from S. and S. W. in the afternoon of this day, chiefly along the western coast. Velocity of wind = 55 feet per second. Absolute barometric minimum (north-west) = 29'309. June 15, 16. Gale from S. W. and W., on the western coast. June 15, 9 A. M. Wind from S. W. Velocity on western coast = 50 feet per second. Least pressure in north-west. June 15, 9 P.M. Velocity = 45 feet per second. Absolute barometric minimum (north) = 29'575. June 16, 9 A. M. Wind from W. Velocity = 50 feet per second. July 13, 14. Storm chiefly in north-west, blowing at first from S. S.W. and veering through S. W. to W. This appears to have been a cyclonic gale, the centre of the cyclone passing to the north of the island ; it is not included in the former series on account of this circumstance. The velocity of the wind was greatest in the north-west throughout ; the barometric pressure was least ON THE METEOROLOGY OF IRELAND. 869 in the north-.west on the 13th, and in the north-east on the fol- lowing day. July 13, 9 A. M. Storm from S. S. W., in the north-west of the island. Velocity = 60 feet per second. July 13, 9 p. M. Gale veered to S. W., and affected a large portion of the island. Velocity of wind = 60 feet per second, as before. Absolute barometric minimum = 29-052. July 14, 9 A. M. Wind veered to W. Velocity in north-west increased to 65 feet per second. Dec. 7. Storm began in south-western extremity of the island, and extended thence over the whole. Direction of wind between S. and S. W. Dec. 7, 9 A. M. Gale from S. S. W. in the south-west. Dec. 7, 9 p. M. Storm over the whole island. Greatest velocity and least pressure in north-west. Velocity = 70 feet per second. Absolute barometric minimum = 29*267. At Cahirciveen the baro- meter fell until 7 p. M. ; and the wind shifted from S. to W. at the same time. Dec. 9. Storm from S. W. along the western coast. Least pressure in north-west throughout. Dec. 9, 9 A. M. Velocity of wind in west = 50 feet per se- cond. Dec. 9, 9 P. M. Velocity = 60 feet per second. Absolute baro- metric minimum = 29-632. Dec. 20. Gale blowing from S. S. "W., beginning on western coast, and advancing to eastern. Least pressure in north and north- west. Dec. 20, 9 A. M. Gale on west coast. Velocity = 55 feet per second. Dec. 20, 9 P.M. Gale transferred to east-coast. Velocity = 50 feet per second. Absolute barometric minimum (north) = 29--l~>7. At Markree there was a sudden shift of the wind from S. S. W. to N. W. at 7 h 35 m p. M. From the foregoing facts we may draw the following general conclusions : 1. The greater gales are much more frequent on the western, 370 ON THE METEOROLOGY OF IRELAND. than on the eastern coast, the numbers denoting the relative fre- quency being nearly as 5 to 1. The frequency of storms is nearly the same in the northern and southern portions of the island. 2. The direction of the wind, in all the cases enumerated, was between S. and W. In about half of these cases the wind blew, throughout, from the same point ; in half it veered from 4 to G points of the compass, the veering being in the direction produced by a cyclone moving from S. W. to N. E., and having the path of its centre to the north of the island. 3. The axis of the gale is in some cases transferred parallel to itself, to the eastward. Remarkable instances of this movement occur in the gales of January 12, 13, and December 20. 4. The least barometric pressure occurs, in almost every instance, in the north-western quarter of the island.* 5. The locality of the highest wind sometimes coincides with that of least pressure, and sometimes does not. In the latter case, the axis of least pressure is generally to the westward of the axis of the storm. 6. On either side of the axis of a storm, the wind appears to blow towards that line. A remarkable instance of this phenomenon occurred in the storm of January l.f We are now in a position to consider the question, whether all storms are cyclonic ? and if not, what proportion do rotatory storms bear to the whole ? Of the greater storms which have occurred since the commencement of these observations, the rotatory charac- ter of five those of October 6, 1850, November 18, January 15, 1851, March 18, and September 29 has been completely established. We have seen in this section, that the same character may be pre- dicated, with great probability, of five more ; while there remain five in which the wind has blown, throughout, in the same direction. In fifteen months, accordingly, there have occurred fifteen storms, of which two-thirds were cyclonic. As respects the remaining one- third, the phenomena are characterized, not only by the absence of any veering of the wind, but also by the fact, that the pressures * In one case only, the locality of least pressure shifted from the north-western to the north-eastern extremity of the island. This is consistent with the supposition, that, the storm in question was a cyclone, whose centre had a progressive motion eastward. t The conclusions numbered 3, 5, 6, have already been drawn by Mr. Espy, from an examination of the storms in the United States in the early months of the votr 18-13. Fir at Report on Metioro'.ogy. ON THE METEOROLOGY OF IRELAND. ,371 appear to increase with the distance from a line or axis of minimum pressure, rather than from a point ; or, in other words, that the iwbaric lines are parallel right lines, instead of concentric circles. It is true that these facts are by no means decisive in disproving rotatory movement ; for they are consistent with a rotation of the wind in a plane perpendicular, or highly inclined, to the horizon. Still we are perhaps not justified in assuming the existence of a rotation of this kind, without further evidence ; and it seems more reasonable, in the present state of our knowledge, to admit two dif- ferent kinds of winds, than to endeavour to reduce all to one by the help of a gratuitous hypothesis. Hourly Observations. It has been already stated, that hourly observations were appointed to be made during twenty- four conse- cutive hours, at the equinoxes and solstices, in the hope that their results might throw light upon the simultaneous atmospheric changes occurring over the island, and especially upon the direc- tion and rate of progress of atmospheric waves. The results of the observations on the first two of these term-days March 21, and June 21 are here given. During the two latter September 22 jmd December 22 no atmospheric change of a marked kind occurred. March 21. A gale occurred on this day, accompanied by a marked barometric depression. The minimum of pressure took place during the observations, the time of its occurrence varying considerably at the different stations. At Cahirciveen, there was a sudden fall of the barometer between 9 A. M. and 10 A. M., fol- lowed by a sudden rise between 12 and 1 P. M., the mercury being nearly stationary from 10 A. M. to 12. A similar change took place at Dunmore East, and at the same hours. For these two stations, accordingly, the epoch of minimum pressure may be taken to be 11 A.M. ; the subsequent changes were small and irregular. At Courtown, the barometer descended very slowly and gradually until 5 p. M. ; it then ascended until 10 p. M., after which it de- scended again. All the changes were, however, very small. At the northern stations the fall of the barometer was more considerable, and more regular. At Markree, where it was most rapid, it amounted to 0*210 in 6 hours. The minimum at Markree occurred between 3 P. M. and 4 P. M. ; at Armagh, the minimum took place at 6 p. M. ; and at Portrush, at 8 p. M. 2 B2 372 ON THE METEOROLOGY OF IRELAND. From these results it would appear that the trough of the wave travelled from south to north, nearly, with a velocity of about 22 miles per hour. The barometric depression was greatest at Markree, where the barometer stood at 28'689, when lowest. The lowest pressure increased from that point in the south-easterly direction, being 28-972 at Dunmore. At Markree the wind shifted from S. S. E. to S. S. "W. at the time of greatest depression. The same phenomenon took place at Armagh and Portrush, although not with such precision; the change of direction at the former station being from S. S. E. to S., and at the latter from S. E. to S. No similar change occurred at the southern stations. The foregoing phenomena are not necessarily to be ascribed to the transit of a rectilinear wave. They are all consistent with the effects of a cyclone, coming from the S. or S. W., the track of its centre lying to the west of the island. June 21, 22. The changes of the direction and of the pressure of the wind, on these days, are manifestly the effects of a cyclonic movement, the centre of the vortex sweeping round the north coast of Ireland, in a somewhat curvilinear path, from west to east. It has not been included in the former series, the force of the wind having been below the limit there adopted. At 9 A. M. of the 21st, the centre of the vortex was off the north-west coast, to the west of Killybegs. At 9 p. M. of the same day it had arrived to the north of Portrush ; and at 9 A. M. of the 22nd it was to the north- east of Donaghadee. The veering of the wind was, on the average, about 90; its duration was very different at the different stations, being shortest for those near the path of the centre of the vortex, and longest for- t-hose remote. The wind, which was very light throughout, fell about the time of veering at most of the stations. The descent and subsequent rise of the barometer were regular, and the minimum well-defined. The time of least pressure coin- cided at all the stations, very nearly, with the middle of the time of veering of the wind ; it was earliest on the western coast, and latest on the eastern, the epoch of its occurrence being between 12 and 1 P. M. at Markree and Cahirciveen, and between 5 p. M. and 6 p. M. at Dublin and Courtown. The barometric depression was small, the mean pressure at the epoch of minimum being 29 '74. ON THE METEOROLOGY OF IRELAND. 373 HUMIDITY OF THE AIR. The first two of the Tables which follow give the results of the psychrometrical observations. The first contains the monthly means of the tension of vapour at the several stations, calculated by iiegnault's Table ; and the second the corresponding values of the relative humidity, the state of saturation being represented by 100. Very few results of a general nature can be drawn from these observations, the distribution of vapour being governed by the proximity of the station to the sea, or by other local circumstances. It will be seen, from the last column of the first Table, that the yearly mean tension of vapour increases, although not in any regular progression, in proceeding from the north to the south of the island. Its mean value for the entire island is 0*326 of an inch ; its greatest value (at Westport) is 0'374. The distribution of humidity is still more under the influence of local circumstances, and therefore still less regular. Thus, Portrush and Castletownsend the one at the northern, the other at the southern extremity of the island have nearly the same mean humidity ; while Portarlington and Athy places near each other, and both inland are almost at the opposite extremities of the .scale. The driest station is Portarlington; the most humid. Westport. At the latter place, in fact, the air is nearly saturated with moisture, the place of observation being entirely surrounded by water, and but a few feet above the sea. The mean humidity for the entire island, for the year 1851, is 87. EAIN. Before proceeding to the observations of rain-fall throughout Ireland in the year 1851, it is important that we should know its normal amount at one or more stations, as deduced from the mean of several years. We have, for this purpose, two series of obser- vations, one at Dublin, and the other at Armagh, extending uninterruptedly over eleven and twelve years respectively. The results of these two series are contained in the third and fourth of the following Tables. The lowest line in each gives the mean monthly fall of rain. It will be seen, from an inspection of the numbers, that there is ;J74 ON THE METEOROLOGY OF IRELAND. 110 regular progression in the amount of rain-fall throughout the year, such as is observed in the phenomena of temperature or humidity. In Dublin the greatest rain-fall, in the mean of the eleven years, occurs in October, and the least in February ; their amounts are 3 '34 and 1'74 inches respectively. At Armagh the maximum is in January, and the minimum in May; and they amount to 375 and 215 inches. The fifth and last Table gives the monthly fall of rain in the year 1851, at all the meteorological stations. It will be seen that the greatest diversity exists in the amount of rain-fall in different localities. To render this more apparent, ;md to facilitate the examination of the causes which influence the distribution, I have, in the following Table, given the yearly rain- fall at the several stations arranged in the order of magnitude, beginning with the smallest : TOTAL RAIN-FALL IN THE YEAR 1851, AT THE SEVERAL METEOROLOGICAL STATIONS. 20-25 inches ( Portarlington, .... 21-23 inches. (KiUough, ..... 23-19 /Dublin, ...... 26-40 25-30 j Donaghadee, .... 27-93 ,, \ Courtown, ..... 29-64 /Kilrush, ...... 32-58 30-35 \ Armagh, ...... 33-05 j Killybegs, ..... 33-20 \Dumnore, ..... 33-54 35-^0 . j Por tsh, ..... 37-24 ( Buncrana, ..... 39-28 ,, 40-45 jMarkree, ...... 40-31 ,.. ( Castletownsend, . . . 42-53 4550 ,, .... Westport, ..... 45-86 50 60 .... Cahirciveeu, .... 59-37 Thus, the greatest rain (at Cahirciveen) is nearly treble of the least (at Portarlington). The mean rain-fall throughout Ireland, in the year 1851, was 34'50 inches. If we assume the proportion of rain at the different stations to be constant, or nearly so, the numbers of the preceding Table may ON THE METEOROLOGY OF IRELAND. 375 all be reduced to their mean values, by multiplying by the factor which expresses the relation of the rain of 1851 to the mean at any one station. We already possess two such mean values : viz., at Armagh and Dublin. They are 29'14 and 34'68 inches re- spectively ; and the factors thence deduced are T10 and 1'05. When we examine the results of the preceding Table, taken in connexion with the geographical position and physical circum- stances of the stations, we arrive at the following conclusions : 1. The places of least rain are either inland, or on the eastern coast ; while those of greatest rain are at, or near, the western coast. Thus the stations at which the yearly fall of rain exceeds 40 inches are all on the western and south-western coasts ; while those at which it is below 30 inches are either inland or on the eastern. 2. The amount of rain is greatly dependent on the proximity of a mountain chain or group, being always considerable in such neighbourhood, unless the station be to the east or north-east of the same. Thus, of the places of least rain, Portarlington lies to the north-east of Slieve-bloom ; Killough, to the north-east of the Mourne range ; Dublin, to the north-east of the Dublin and Wicklow range ; while, on the other hand, the places of greatest rain Cahirciveen, Westport, and Castletownsend are in the vicinity of high mountains, but on a different side. These facts are easily explained. The prevailing wind blows from the S. W., and reaches this island loaded with the vapour of the Grulf-Stream. This vapour is condensed and precipitated in rain, when it first meets the colder air over the land, namely, on the western and south-western shores. But the principal con- densing centres are the mountains, in the neighbourhood of which, consequently, the precipitation is more abundant, especially on their western and south-western sides. And the same circumstance which causes the greater precipitation at these points must al> protect the region over which the wind next passes (the north-east), the air being thus deprived of a large portion of its vapour before arriving there. NOTE A few of the latter Tables of this Paper, aa originally printed, hare been omitted here as unnecessary. 376 ON THE METEOROLOGY OF IRELAND. ^ suusOoor j~3 c't vii en ^u ^ii 5 ^ScocpcNcocococpcpepcococococo K^ J i^ 5 o od * i~- t- CN 10 >o co o K. CX1!NCNCS(N(N J I wMWCpcpcpepcpcpcpcpcpcpepcpS f ^ I *....... |> g opptr90?QOtr-*^o>rHcpr-iopp PM O I | sssssssKiggssslls H % I si^5s^i2^^??"^ < I I I *! f.< : < i 'g ^ i s ^ S ^ cS Poo ON THE METEOROLOGY OF IRELAND. 377 3 8 S S S 8 3 8 S 8 8 8 8 >oo*e5'OOC& OCiOOOCiaO OOCOt--OWO5C5t--COt~00Or-irHI O OOQOOOQOQOOOQOC300r-QOQOOiaOOOO5 'H(M'*QOOO-*OOC.'3 CJOQOQOr-QOQOOSt r-QOOOCSt^-0000 aoocsi-t~o>-i t--QOOOOOQOC5QOl--QOOOOOOOQOaO or-t--r~t-t o eo I-H CM t^ eo 3 "^ ^ t~* Oi CO 00 CNCMi-Hi-iCOeboCMCMCMcb ^HCNCN^H^rH^ebOTH Oi-HCMCMOCMCMT-i CO S 00 05 ^ 2 O eb eb o o cb cb CM >O * CM >O CM O l>" O CO J- CO CM O ^1 CM ^ CM rH CM -< 8 S S i-ICNOr-ICNCOO'-' 55 12 co eo .-i eo CM ON THE METEOROLOGY OF IRELAND. 379 g CM CO CO CO $ CM 5 S S3 g 2 S 3 O rH T* HH 6 O CO CO CO CO CO T? 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CO rH CO rH rH CM o t- eo i^ * co t~ CM t- CM CM CO CO t 3 rH ^1 CO co O5 r- o o co * CM CM CM rH CO i-H rH rH CO CM CM CM 9 -H O O5 >f5 * O5 05 I- CM CO 05 oo co ^ o oo co JO rJH CM CM CM CM Tj< ^ co rH eo * o CO i 3 S 1 i 2 1 S S 1 S 1 380 ON THE METEOROLOGY OF IRELAND. 1 (N /ear ; but the influence of the Gulf-Stream is much greater upon the temperature of winter. Thus the winter temperature of Dublin is considerably higher than that of Milan ; and Stromness, in the Orkneys, has a winter temperature greater than that of Paris. In fact, the isothermal line for January, which passes through the Shetland Islands, runs almost exactly from north to xouth ; and thus the winter temperature is nearly the same along the whole eastern coast of Great Britain, while it increases as we proceed from that coast to the westward. In Norway the effect is even more remarkable. Owing to the interposition of the British Islands, the Gulf-Stream is intercepted from the southern portion of the Norwegian coast, while it reaches the northern. The southern limit of the stream thus falls on the town of Bergen ; and accordingly, in winter, the temperature actually increases in proceeding northicard from that point. But the disturbing effect of local causes on the temperature is measured exactly by the method employed by Professor Dove. In this method the mean temperature, corresponding to any parallel of latitude, is deduced from the observed temperatures at 36 equidistant points on the parallel, and is regarded as th normal temperature of all places on that parallel ; and the difference between this, and the actual temperature of the place, is obviously u measure of the local influence. This difference is called by Professor Dove the thermic anomaly. We thus find that the measure of the influence of the Gulf-Stream, or the thermic anomaly, at the Orkneys is, on the mean of the entire year, 15 ; while, in the month of January, it amounts to 34. The excess of the actual above the normal temperature is greatest between Jan Meyen and the Lofoden Islands, a little above the Arctic Circle. The mean thermic anomaly is there 22'5 ; while, in the month of January, it amounts to 45. 388 THE CLIMATE OF IRELAND, AND The mean thermic anomaly for Ireland, or the excess of the mean temperature of the year (50'3) above that due to the latitude (37'7), is 12 J. The corresponding excess for the month of January is 27. Our winter temperature is thus raised, by the Gulf-Stream, as much as if our place on the globe had been 15 nearer to the Equator ! The Gulf-Stream changes its boundaries from time to time ; and these changes are sometimes indicated in a singular manner. The Medusa?, or " sea-nettles," as they are commonly called, are often found in vast numbers in the stream ; and it seems to be owing to the attraction which these creatures have for the whale, who feeds upon them, that the leviathan is so often found hovering on its skirts for its temperature is too high to permit him to enter it. Now this fact furnishes some curious evidence respecting the boundary of the stream, and its changes. Thus M. Babinet tells us, that in the autumn of 1846 he learned that the whalers had been obliged to run up to a higher latitude in search of their prey. From this fact he concluded that the stream had advanced further northward than usual ; and he predicted, in consequence, that the following winter would be one of unusual mildness in Europe. The prediction was fulfilled. I now proceed to consider, somewhat more particularly, the influence of these and other causes on the climate of Ireland. I have already adverted to the system of meteorological obser- vations adopted in 1851, under the direction of the Boyal Irish Academy. In addition to the observatories of Armagh, Markree, and Dublin, where such observations are made continuously, the observations were taken at eight coast-guard stations, by boatmen belonging to the service, and at three of the lighthouses, by the light-keepers. The instruments were furnished by the Academy. They were constructed on a common plan, and carefully compared with standard instruments by members of the Council of that body, by whom also the men were instructed in their use. The instruments were recorded daily at 9 A. M. and 9 p. M. ; and obser- vations were also taken, on the same system, at two stations in the interior of Ireland, where the task was voluntarily undertaken by private individuals. The mean temperatures of the several months having been calculated, and reduced to their normal values by means of the Dublin observations, it was found that the temperatures of the THE CURRENTS OF THE ATLANTIC. 380 inland stations were in defect, as compared with the coast stations. The cause of this is obviously their greater distance from the warm waters of the surrounding sea; and it follows evidently from the fact, that the actual isothermal lines in Ireland are inflected in passing from the sea to the land, and must even, in part of the island as Mr. Hennessy has pointed out take the form of closed curves, dependent on the position of the places traversed with respect to the coast-line, as well as upon their longitude and latitude. To determine such curves with exactness would require a far greater number of stations than those em- ployed, and would not seem likely to yield results of corresponding value. The most feasible course seems to be to determine, in the first instance, the law of distribution of temperature depending on geographical position alone ; the disturbing influence of the land can afterwards be computed approximately, and allowed for. Dealt with in this manner, the observations show that, on the mean of the whole year, the isothermal lines are inclined to the meridian by the angle N. 49 W., and that the temperature increases, in a direction perpendicular to these lines, at the rate of 1 Fahr. for 89 miles. "We learn further, that the mean isothermal lines for the entire year furnish a very inadequate representation of the progression of temperature, and that, when the course of these lines is traced from month to month, they vary within very wide limits. The extreme positions correspond to the months of June and December. In the former, the inclination of the isothermal lines to the meridian is N. 106 W. ; in the latter, it is N. 9 W. Thus these lines vary in direction through an angle of 97 in the course of the year, being nearly parallel to the meridian in December, and nearly perpendicular to it in June. The rate of increase of temperature changes little. The mean yearly temperature for the central station, whoso latitude and longitude are the arithmetical means of those of the stations of observation, is 50-3. It is thus distributed throughout the year : 390 THE CLIMATE OF IRELAND, AND Month. Mean Temperature. Month. Mean Temperature. January, . . 41"-7 July, . . . 58-8 February, . . 42 -3 August, . . 58-3 March, . . . 44 -1 September, . . ! 57-8 April, . . . 47 -1 October, . . j 50-2 May, . . . 52 -9 November, . . 48 -4 June, . . . 56-7 December, . . 45-4 These facts are in entire accordance with the general course of the isothermal lines, as laid down by Dove. After what has been said, they are readily accounted for. The heating effect of the Gulf-Stream is greatest about the time of the winter solstice, or soon after. It then overpowers altogether the effect of solar radiation, which is then weakest ; and the temperature increases as we approach the waters of the stream i. e., as we proceed to the westward. On the contrary, the effect of the stream is least, and that of the sun greatest, about the time of the summer solstice; and the heat increases as we proceed southward. The maxima of the curves in Europe, at this period of the year, show that the burning soil of the African Continent then influences most the distribution of temperature to the north of it.* * The following are the mean temperatures of the several places of observation in Ireland, reduced to their normal values, for the four seasons of the year, and for the entire year : Place. Spring. Summer. Autumn. Winter. Year. Portrush,. . . . 4G"-5 56'0 5l'l 41'5 48-8 Buncrana, . . . 46 -6 56 -5 oO -6 41 -0 48 -7 Donaghadee, . . 47 -1 56 -4 51 '4 42 -2 49 -3 Killybegs, . . . 48 -1 57 -6 52 -8 4:$ -1 50 -4 Armagh, . . . Killough, . . . 46-3 47 -4 56 -5 56 -6 49 -6 01 '9 40 -6 43 -5 48 -3 49 -8 Markree, .... Westport, . . 46 -1 48-7 56 -2 57 '7 49 '6 53 -8 39 '4 44 -9 47 "8 51 -3 Dublin, .... 48 -0 59 -5 50 -4 42 -0 50 -0 Portarlingtoii. . . 44 -4 55 -8 48 -5 39 -0 46 -9 Athy, ... 46 -1 57 -8 48 -8 39 -4 48 -0 Courtown, . 48 -0 58 -3 50 -7 42 -5 49 -9 Kilrush, . . 48 -2 57 "9 52 -o 43 -4 50 -5 Dunmore, Cahircivcen, . Castletownscn il , 48-7 49 -8 49 -4 59 -9 59 -3 59 -0 52 -6 o:; -8 r>:5 -9 43 -9 44 -7 44 -6 51 -3 51 -9 51 -7 * THE CURRENTS OF THE ATLANTIC. 391 But the climate of a place depends upon the ranges of tempera- ture, whether diurnal or annual, no less than upon the mean rallies. It is a well known meteorological fact, that the ranges of temperature on the globe vary within very wide limits, being least of all at sea, and increasing with the distance of the place from the ocean, until in the interior of the continents they become very great. Thus, the difference between the mean temperatures of summer and winter, or the annual range, is, on the average, 19 in Great Britain ; 27 in France ; 36 in the eastern parts of Germany ; 40 in European Russia ; 60 in Siberia until at Jakoutsk, in that inhospital clime, it reaches the enormous amount of 101 ! And these differences exist even among places having the same mean temperatures. Thus Dublin, Prague, and Astrakhan on the shores of the Caspian, have all nearly the same mean annual temperature. But the difference between their summer and winter temperatures are respectively, 17'5 for Dublin, 37 for Prague, and 57 for Astrakhan. These differences are due to the different effects of solar radiation on water and on dry so/7, already explained ; and climates have been distinguished into marine and continental, according to the amount of this variation. The annual range in Ireland is the smallest in Europe ; on the coasts it falls as low as 14. The difference between the mean temperature of day and night, or the diurnal range, follows the same law, being least on the coasts, and greatest in the interior of the continents. I have already adverted to the principal cause of this. A great portion of the solar heat which falls on the water during the day is employed in changing its state into that of vapour, and does not affect its temperature ; and, on the other hand, the heat so employed is, in part, restored and rendered sensible, when the vapours are condensed during the night. Thus the two extremes approach one another. But another reason is, that the vapour itself stops, or absorbs, a large portion of the sun's heat ; and the effect, of course, is greatest when the air is most humid. Dr. Livingstone found a difference of temperature, amounting to 48, between sun-rise and mid-day, on the eastern side of South Africa ; while in the valley of the Zambesi, on the opposite side of the Continent, where the air is loaded with moisture, the difference amounted only to 12. In accordance with this, the diurnal rmigo of temperature 392 THE CLIMATE OF IRELAND, AND in Ireland is small, the mean difference between the greatest temperature of the day, and the least temperature of the night, being only 117. The range is greater inland than on the coast, the mean range of the inland stations being 13'l, and that of the coast stations 10'3. Among the places of observation, it is greatest at Portarlington, and least at Cahirciveen at the south- western extremity of Ireland. Having spoken of the temperature of the air, I must now say a few words of its movement and, first, of the direction of that movement. During the period of the simultaneous observations already referred to, the wind blew, on the average of the entire year, most frequently from S.W. and W., and least frequently from N.E. and E. Thus, taking the mean of all the stations, the number of times in which the wind blew from the N.E. is only 7 per cent, of the whole, while the number of S.W. winds is 20 per cent. The ratio of the numbers is greater in winter than in summer. The following Table gives the number of times out of 1000, in which the wind blew from each of the eight principal points of the compass, for the whole island. Direction of Wind. Summer. Winter. Year. N. 122 96 108 N.E. 94 52 73 E. 89 50 69 S.E. 100 86 92 8. 107 164 136 S.W. 170 234 202 w. 163 176 170 N.W. 156 144 150 The velocity of the wind varies from to 70 feet per second, and upwards this last being the speed corresponding to a storm. If we confine our attention for the present to winds whose average velocity is not less than 35 feet per second, which is the velocity of a strong breeze, we find that the high winds are much more frequent on the western than on the eastern coast, the numbers denoting their relative frequency being, on the average of the "THE CUERENTS OF THE ATLANTIC. 393 entire year, nearly as 3 to 1. This preponderance of high winds on the western coast holds at all seasons, the maximum occurring at the north-western extremity in autumn, and at the south- western in winter. But of these winds, there are two distinct kinds. In one of these, the wind blows steadily in the same direction for a con- siderable time, and the axis of the gale is a straight line, or a line but slightly curved. In the other, the wind rotates round a centre, while the centre itself has a progressive motion just as we see, on a miniature scale, in the vortices of dust which sweep along the road in gusty and unsettled weather. These rotating winds are called cyclones. They are from 200 to 300 miles, and upwards, in diameter, and their centres move with a speed of from 100 to 300 miles per diem. The direction of the rotation is invariably the same. In the northern hemisphere the wind revolves in the direction opposite to that of the sun in its daily course, i.e., in the direction N. W. S. E. In the southern hemisphere, the direction of rotation is with the sun, or N. E. S. W. When a cyclone passes over any point on the earth's surface, the wind must veer. If the centre of the cyclone passes directly over the place of observation, the veering is through 180 degrees, or the wind changes to the opposite. The amount of veering is less, the smaller the chord of the circle which passes over the place; until, for places at the circumference, there is no veering whatever. But there is another remarkable distinction between these two classes of winds. In the case of the cyclone, the barometric pressure diminishes to a point, which is the centre of the vortex. In the non-rotatory gales, the pressures diminish to a line, which is the axis of the storm. The most remarkable of the cyclonic movements traced in the course of the simultaneous observations was that which occurred on the 18th and 19th of November, 1851. The centre of the vortex passed over two of the principal stations, Markree and Armagh, at both of which all the changes were accurately ob- served ; and the velocity of the wind reached a maximum of Go feet per second. The lull oi the wind during the passage of the central portion of the vortex, and the reversal of its direction, were observed at Markree, Armagh, and Donaghadee. The diameter of the vortex was about 400 miles ; that of the quiescent central portion 40 miles. 394 THE CLIMATE OF IRELAND, AND The facts collected relating to these winds lead to the following conclusions : 1. The occurrence of cyclonic movements in the atmosphere is not infrequent in Ireland, and may be traced even in the case of moderate winds. 2. The rotatory movement is invariably in the same direction namely, that opposite to the diurnal movement of the sun in azimuth. 3. This rotation is always accompanied by a considerable disturbance of barometric equilibrium, which is greater in pro- portion to the velocity of the movement, the pressure being a minimum at the centre of the vortex, and increasing regularly with the distance from that point. 4. The vortex itself has a progressive movement, at the rate of from 100 to 300 miles per diem, the average velocity of those observed being 220 miles per diem. The direction of this move- ment is generally from S. W. to N. E. 5. If a line be drawn through the centre of Ireland, in the direction from S. W. to N. E., the track of the centres of the greater number of the cyclones, passing over or near Ireland, lies to the north of that line. With reference to the first of these conclusions viz., the pre- valence of cyclonic movements in the atmosphere over, or near, this island I may here state, that of all the greater storms which occurred during the period of observation, no fewer than two-thirds were cyclonic ; while several cyclonic movements of smaller velo- city were distinctly traced. The area of observation may be thought, perhaps, to be too small to furnish conclusive evidence relating to these great aerial movements. But I have the satis- faction of knowing, that of the cases recorded in the Memoir on the subject published in the Transactions of the Royal Irish Aca- demy, the cyclonic character of most (if not all) has been since confirmed by the late Admiral Fitzroy, from the records of the logs of ships. The prevalence of such movements is just what we should be led, a priori, to expect, from the vicinity of the GruH'-Stream. The waters of the stream in mid-Atlantic have a temperature of 80 Fahr., while the air on either side of it is, in winter, at the freezing temperature. We have here all the conditions required for the production of cyclonic storms. Aqueous vapour is much lighter than air at the same temperature, and conse- THE CURRENTS OF THE ATLANTIC. 395 quently ascends in it ; and, in its ascent, it drags the heated air along with it, and produces an upward current. There is thus an upward rush of heated air and vapour above the heated waters, and an inward rush of cold air from either side to supply its place ; and such a combination of movements will produce a great rotat- ing eddy, much in the same manner as the downward flow of water through a hole in the bottom of a vessel will give rise to a vorticose movement of the whole liquid mass. Accordingly, the most terrific hurricanes of which we have any record have been generated on the borders of the Gulf-Stream. In the hurricane of 1780, in the Bermudas, houses were levelled by the gale ; forts were washed away by the waves ; heavy pieces of ordnance were lifted into the air ; and the bodies of animals were carried aloft, and dashed to pieces in the fall. The loss of human life was terrible. It is computed that not fewer than 20,000 lives were lost on shore; and on the water no vessel could stand the gale. And the Gulf-Stream appears also to exert a marked influence on the course of storms which are engendered in other parts of the Atlantic. Erom the examination of the logs of ships, Captain Maury has ascertained that the gales, which are produced in the Atlantic to the south-east of the current, usually travel to the north-west until they meet it, after which they turn along with it, and follow its course. And, according to Mr. Espy, a similar effect is produced in the case of storms which have their birth- place in the valley of the Mississippi. There are two facts which have been brought to light by the Irish meteorological observations, which confirm in a remarkable manner this explanation. The main branch of the Gulf-Stream approaches the coasts of Ireland most nearly at the north-western extremity of the island ; and it is, accordingly, there, if this ac- count be the true one, that the evidences of cyclonic movement .should be most marked. And such is the fact : the track of the centres of most of the cyclones whose course has been investigated lies to the north-west of the island. And in exact accordance with this fact is another of a different kind. I have said that, in the area covered by a cyclone at any moment, the minimum of baro- metric prcaaure is at the centre of the vortex. Hence the frequent passage of cyclones in any particular direction must affect the mean distribution of atmospheric pressure, the prepsuro being most ;}96 THE CLIMATE OF IRELAND, AND diminished at those places which are nearest to the track of their centres. It has been found, accordingly, that there is an ine- quality in the distribution of the atmospheric pressure in Ireland, and that the minimum occurs in the north-western quarter of the island. I now proceed, lastly, to the effects of the Grulf-Stream on the humidity, and on the rain-fall, in this island. The amount of evaporation from the surface of water increases rapidly with its temperature. Consequently the air above the Gulf-Stream is loaded with vapour to a much greater degree than that which rests on other portions of the Atlantic. This vapour is borne to the British Islands by the south-west winds, which are the predominating winds in this portion of the globe ; and, in consequence, these islands have more than their average share of humidity. The effect is, of course, greatest at those places which the wind first reaches. It is, accordingly, greatest in Ireland, less in England, and still less on the Continent of Europe, diminish- ing with the distance of the place from the Atlantic. Thus, in Paris (and probably in France generally) the mean amount of vapour in the air is about 77 per cent, of the maximum which it is capable of- holding. In London, it is 84 per cent. ; and in Dublin it amounts to 88 per cent., which is about the average for Ireland. The humidity is greatest, as we should expect a priori, on the western coast of Ireland, and least on the eastern the former being about 90 per cent., and the latter 85. The following Table gives the mean humidity of the several months of the year at Dublin. It is greatest in December, and least in June : Month. Humidity. Month. Humidity. January, . . 92 il July, . . . 84 February, . . 88 : August,. . . 86 March, . . 88 ! September, . . 90 April, . . . 87 October, . . 91 May. . . . 85 Xovember, . . 92 June, . . 83 !{ December, . . 92 THE CURRENTS OF THE ATLANTIC. 397 The distribution of rain is very unequal. In the east of Eng- land, the annual rain-fall is between 20 and 25 inches only ; in the west of Ireland, it is between 40 and 50 inches. In Ireland, in the year 1851, the greatest amount of rain fell at Cahirciveen, and the least at Portarlington, the amounts being 59*4 inches, and 21*2 inches, respectively. The mean rain-fall for the whole island, in that year, was 34'5 inches. The causes already adverted to explain, in part, the distribu- tion of rain. The vapour which is borne to us by the south- westerly winds is partly condensed into rain when the air is cooled by contact with the land. The chief condensers are the mountains, because they interpose the greatest obstacle to the movement of the vapour-laden air. Accordingly the high moun- tains of Kerry, which offer the first barrier to the progress of these winds, bring down on their flanks the largest amount of rain ; and, generally, the rain-fall is greatest on the western and south- western coasts, while the places of least rain are either inland or on the eastern coast. The disparity is greatest in winter, when the clouds are low ; in summer, when they are high, they escape the condensing effect of the land over which they pass, and the rain is more evenly distributed. The amount of rain is always considerable in the neighbour- hood of a mountain chain or group, except at places to the east- ward of it, where it is small. Thus, the places of least rain in Ireland Portarlington, Killough, and Dublin all lie to the north-east of a mountain range ; while the places of greatest rain. Cahirciveen, Westport, and Castletownsend are all in the vicinity of high mountains, but on a different side. All this is readily understood from what has been stated. The prevailing wind is the south-west, which comes to us laden with the vapour of the Gulf-Stream, and this vapour is precipitated chiefly on the flanks of the mountains. And the same cause protects the region over which the same wind next passes the east and north-east the air having been deprived of its excess of vapour before arriv- ing there. The distribution of rain throughout the year in Ireland will be understood from the following numbers, which give the mean rain- fall in each month at Dublin, as deduced from the observations of the eleven years (1841-1851). The mean yearly rain-fall is 29'1 inches. 398 THE CLIMATE OF IRELAND, AND Month. Rain-fall. Month. Rain-fall. January, . . 2-9 inches. July, . . . 2-4 inches. February, . . 1-7 August, . . . 2-7 March, . . . 1-9 September, . . 2-3 April, . . . 2-5 October, . . | 3-3 May, . . . 2-0 November, . . j 2-9 ,, June, . . . 2-2 December, . . 2-2 ,, In dealing with the subject of Climate, even in the cursory manner which alone is suited to a discourse such as this, it is impossible to overlook its effects upon animal and vegetable life ; and therefore, although no physiologist myself, I will offer no apology for laying before you some broad facts connected with the sciences of life, in relation to Meteorology. To begin with the vegetable kingdom it is well known that plants are dependent, in a remarkable degree, upon every change of heat and moisture ; and that, accordingly, the genera and species of plants are distributed on the earth's surface so that each shall receive its proper supply. The distribution of temperature being much more regular than that of moisture, the general features of the geography of plants are determined by the former ; and thus the several divisions of the vegetable kingdom are found disposed in broad bands on the earth's surface, which are closely connected with the isothermal lines. And the succession of these bands is similar but of course more rapid, and therefore more readily seen on the flanks of high mountains. Thus we meet, near the base of the Andes, first the palms ; and then the tree-ferns. At greater altitudes there, or at lower heights in the temperate re- gions, we come to the deciduous trees among which the oak, the beech, and the birch, are the principal. Then, as we ascend, we reach the pines ; next the rhododendrons, and other dwarf shrubs ; after these, at still higher elevations, the creeping herbaceous plants ; and, lastly, at the very limit of vegetation, the lichens. But in studying the effects of temperature upon vegetable life, it is important to observe, there are two pairs of limiting tempera- tures to be considered, one wider, and the other narrower. The former are the limits beyond which the existence and continued THE CURRENTS OF THE ATLANTIC. 399 life of the plant is impossible ; the latter are the limits which de- termine its flowering, and the ripening of its seed or fruit. It is hardly needful to remind you, that the very existence of plants is dependent upon two extremes of temperature, one higher and the other lower. In the temperate regions of the earth, in which we ourselves have the happiness to live, the effect of the higher extreme is seldom felt ; for there are few plants which will' not bear our highest summer heat. Accordingly, with us it is the winter temperature, or rather the lowest temperature of the year, which determines the existence of most perennials. We are all fami- liar with this fact in the case of our ornamental greenhouse plants, which are for the most part natives of warmer climates, and will not bear the rigours of our own without protection ; and the same thing is true of many native and acclimatized plants in seasons of unusual severity. On the other hand, in the south-west of Ireland, which has the highest winter temperature of any in the British islands, there are twelve species of wild plants which are natives of Spain. But within the limits beyond which the plant cannot live, there are two other limits which determine its fruitfulness. It is now well established that, in order to blossom, and to ripen its seed or fruit, each species of plant requires a certain definite amount of summer heat, above that which is necessary to the continuance of its existence as a vegetable ; and that, provided it receives this fixed amount, the time in which it is imparted is not essential. This determines the lower limit to the successful culture of the fruit-bearing plants. But there is also a higher limit, which is less obvious. Thus in the case of the annuals, which include the plants most useful to man, there is a certain temperature at which the plant becomes perennial, and is propagated by lateral shoots, and not by seed. When the temperature reaches this limit for any plant, its culti- vation for seed becomes impossible, and we can use it only for its leaves or root. On the other hand, tuberous plants become seed- bearing, by the lowering of the temperature. We have a familiar instance of the latter transformation in our fields. Most of our green crops, such as turnips and mangolds, are cultivated for their roots, and under ordinary circumstances produce neither flower nor seed. But it sometimes happens that certain plants in a field have received a smaller amount of heat than is required for this mode 400 THE CLIMATE OF IRELAND, AND of growth ; and these plants throw tip a lofty stem, and flower, while the tuberous growth ceases altogether. This is one of those many beautiful adaptations which compel us to look up from nature to nature's Grod. When the plant is under such conditions that it must perish, it sends up its spikelet of flowers as it were by an instinct and matures the seed, which is destined to continue the life of the species, while the individual fades and dies ! And here I cannot refrain from throwing out a suggestion which, if found to be true, may be of some practical importance. The plant which we chiefly cultivate for its root, and whose life and health is of such importance to this part of the kingdom, pro- duces (as you know) flower and seed, as well as tuberous root. I venture to suggest to physiologists the inquiry, whether this cir- cumstance may not be closely connected with the uncertainty of the crop ? and whether, if means could be found to repress the flower, the root might not acquire vigour and strength to resist disease ? But however this may be, the transition from the con- dition of perennial to that of annual is a subject of much import- ance in connexion with the question of the acclimatization of plants. It is probable that there are many tropical plants, unknown to this climate, which are perennial in their native place, but which would become annuals, with seed and fruit, when transferred to a colder clime. When we consider the acquisitions which have been already made by such transfers, we can hardly over-estimate the import- ance of future gains, or the benefits which may be thus acquired for the human family. But the lower limit of summer temperature is by much the most important to us ; for it is that which in our climates determines the question of success in the culture of the cereals. I will there- fore refer to it a little more particularly. There is no doubt that the integral of solar heat, taken between the limits of time when the vegetation becomes active in Spring, and when the seed is ripened in Autumn, is the element which determines one of the geographical limits which Nature has im- posed to the culture of the plant. To find this, for each important species of plant, would require an amount of observation which has not yet been bestowed. In the meantime, however, a tolerable approximation to its value may be made, by taking simply the mean summer temperature i.e., the average temperature of the THE CUREENTS OF THE ATLANTIC. 401 months of June, July, and August. Now, if we look to the dis- tribution of the cereals in Europe, as given by Berghaus or John- stone in their physical Atlases, we observe that they are arranged in belts, or zones, whose limits correspond nearly with the lines of equal summer temperature. Thus we have, first, a belt of barley alone, which extends from the North Cape, in latitude 70 N. to about midway in Norway and Sweden. This is followed, as we proceed southward, by a belt containing oats and rye, as well as barley. Then comes a belt of rye and wheat, in the south of Scan- dinavia, and in the north of central Europe. In the southern half of Europe, we have wheat alone ; and, beyond this, wheat and maize on the coasts of the Mediterranean. And the boundary lines of these several districts are all related, more or less closely, to the isothermal lines of mean summer temperature. It is of the first importance to the agriculturist to know these limits in his own district ; for on their position will depend the chances of successful cultivation of the particular crop. I shall therefore ask your attention for a few moments longer, while I endeavour to ascertain the limit of wheat, the most important of the cereals, in these islands. The result of the inquiry is some- what startling, and it deserves to be more fully known. There is some doubt still as to the native place of the cereal grasses. Most of them have been found growing wild in Persia, on the banks of the Euphrates, and in Tartary. Whatever their original habitat may have been, it was a warmer clime than ours. The mean summer temperature of the British Islands is under 60, while that of the plains of Lombardy, where wheat is grown in perfection, is 73 ; and that of Sicily " the granary of ancient Rome " is 77. "We are, therefore, in these countries, probably near the lower limit of the wheat crop, beyond which its successful culture is impossible. It is important that we should know how near. We have some of the data requisite for the determination of this question, in the long series of observations of temperature made at the apartments of the Eoyal Society, in London, as com- pared with the prices of wheat, given for an equally long series of years in " Tooke's History of Prices." From the former we learn how far the summer temperature of each year deviated from the mean ; while the latter furnish us with a measure of the abundance 2 D 402 THE CLIMATE OF IRELAND, AND or scarcity of the years in question.* The comparison has been made by Mr. Whitley, of Truro, in Cornwall, and the result which he has obtained from it is interesting and important. The observations of temperature to which I refer were begun in the year 1774, and were continued (with only an interruption of five years) to the year 1842, inclusive ; they thus extend through sixty-four years. They have been carefully reduced by Mr. Grlaisher, and the means of each month and season computed. One of the results which may be inferred from them and it is a result of some interest in connexion with the effects of season is that, for the most part, the spring and summer of the same year are of the same character, either both above, or both below the mean. In the course of the sixty-four years, the character of the two seasons was the same forty-six times, and opposite only eighteen times ; so that the chances that a warm summer will follow a warm spring, or vice versa, are as two and a-half to one. A like fact appears also from Dr. Butty's observations of the weather in Dublin, which extended through forty-one years in the last century (1725-1765). In these observations the character of each season is defined, with reference to the rain-fall, as dry, wet, or variable. The observations have been discussed by Kirwan ; and it appears from that discussion, that wet springs are followed by wet summers five times out of six. But to return to the London observations the greatest devia- tion of any particular year, from the mean of all, was 4'8 in defect ; it occurred in the year 1816, which was a year of famine. In the course of the sixty-four years, a deviation of 2, and upwards, occurred twenty-three times, the summer temperature being thirteen times above the mean to that amount, and ten times below it. IS ow it is deserving of notice, that there is no appearance of a regular cycle in these good or bad years, such as Luke Howard and many other meteorologists have imagined. On the contrary, both the warm nnd the cold seasons usually occur in groups, comprising three or four years in succession ; and the groups themselves do not recur * It is not, of course, meant that prices furnish an accurate measure of the abund- ance or scarcity of the crop ; for they are dependent, we know, on other causes also, ;imong which legislative enactments regulating the import of foreign grain are the principal. It is enough for the argument in the text, that prices varied with abundant or scanty harvests, as with their principal cause ; and this we know to have been the fact, as long as any restriction on the import of com remained. N THE CURRENTS OF THE ATLANTIC. 403 in any order or regularity. The most remarkable of these groups is that of the years 1809-1817, comprising no fewer than nine years in which the temperature was below the mean. The average price of wheat in these years was 95.$. the quarter, the ordinary average being 70s. ; while in the years 1811 and 1812, it reached 122s. The total deficiency of summer temperature in these nine years amounted to twenty-three degrees. I need not remind you of the incidental confirmation of this law of groups, in the account which we have in Grenesis of the seven years of famine, following seven years of abundance, in Egypt and Syria. Our own recent experience at home has instructed us, that years of deficient harvest come in succession. But I wish particularly to point out, that this recurrence of bad years by no means justifies the conclusion which some have drawn from it namely, that our climate has changed, or was changing, for the worse. On the contrary, we learn from observation that such unpropitious years are not likely to occur more frequently than three or four in succession ; and that they will probably be suc- ceeded, at some future time, by three or four years of an opposite character. Now when these years of low summer temperature are ex- amined, with reference to the price of wheat, it is found that they are, with few exceptions, years of scarcity and high prices. A de- ficiency of temperature, amounting only to 2, is most injurious to the wheat harvest in England ; while a deficiency of 3 is almost destructive. The reverse is the case when the summer tempera- ture exceeds the average by the same amounts. This result is of considerable importance. The lowest summer temperature at which wheat can be successfully cultivated in Eng- land, is only 2 below the mean ; and, as the mean summer tem- perature of England is 60, it follows that the minimum for wheat is 58. This minimum, however, is not absolute. It varies a little with the soil, and other circumstances ; and thus it is, that we find the culture of wheat in Scotland extended as far as Inver- ness, where the mean summer temperature is only 57. Now taking this lower limit as the most favourable, let us see what we are to learn from it in Ireland. In this island, the mean temperature of the three Bummer months is 58 ; and, accordingly, for places about the centre of Ireland, a deficiency of a */;///' tl^ir,-.- of summer temperature brings us to the very limit of wheat culti- 40-1: THE CLIMATE OF IRELAND, AND vation ; while a greater deficiency is fatal to the crop. The case is somewhat better in the southern half of the island, and some- what worse in the northern ; and we are therefore justified in concluding, that it is contrary to the rules of all sound experience to attempt the culture of this cereal in Ireland, except in the most favoured localities. In the preceding discussion I have considered only the relation of temperature to the culture of wheat. I have not adverted to the frequently destructive en 3 ect of the autumn rains, which set in with the fall of temperature, and often before the harvest is gathered. Time will not permit me to discuss the corresponding questions for the other cereals ; nor to advert to the fact, which is now be- ginning to gain general acceptance, that the climate of this island, while it is unfavourable to the higher cereals, is adapted in a pecu- liar manner to the cultivation of root-crops, and of fodder.* I hasten to say a few words, before I conclude, of the effects of climate in general, and especially of our own climate, upon the health of man. We are all familiar with the command of the physician to the patient, " to change the air ;" and most of us know something, although probably less than the importance of the subject demands, of the salubrity or insalubrity of different districts. Of all the meteorological elements, that which exerts the most direct effect upon human health is the temperature of the air ; and in this respect the two forms of organized life are subject to one law. We all know the effects upon the bodily frame of an ex- treme cold in winter, and of harsh winds in spring ; and some of us are obliged to leave our homes and daily occupations, and to take refuge from these dangers in the more genial climates of the south, f M. Quetelet, in his important work " On Man," has brought together many curious facts connected with the influence of climate upon human life. Dividing Europe into three zones which we * The turnip depends chiefly on a proper supply of moisture during the summer months ; and it is injured by much heat. Accordingly, moist and cool summers such as are frequent in Ireland are the fittest for this vegetable. The same rule applies to grass. t The connexion of the mortality with the cold of winter is very distinctly marked in the weekly returns of the Registrar-general. In the severe frost of Christmas, 1860, the increase in the number of deaths in London was forty daily, being greater than the increase caused by cholera. ^ THE CURRENTS OF THE ATLANTIC. 405 may call northern, central, and southern Europe, respectively he finds that the mortality in the last is by much the greatest, while it is nearly equal in the two other divisions. In this comparison M. Quetelet has considered merely the effect of latitude upon mortality. But as we have already shown that places in the same parallel of latitude often have widely different temperatures, it seems plain that the comparison should be made differently, and that we should group together for the purpose places under the same, or nearly the same, isothermal line. When we compare in this manner the death-rates of the several countries of Europe with their mean temperatures, we find that the mortality is greatest in Italy and Greece, where the temperature is highest ; and that it decreases as the temperature decreases, down to a certain limit, after which it appears to increase again. But this is obviously an inadequate mode of considering the problem. It seems plain that, for animals as well as for plants, the salubrity of a climate will depend on the extremes of tempera- ture, much more than on the mean ; and that, of these extremes, the most influential on life will be that most removed from the normal temperature which is best adapted to the species. Accord- ingly, we should expect the mortality to be influenced chiefly by the summer temperature in hot climates, and by the uinter tempera- ture in cold ones. Arid this I find to be the case in Europe, where alone we have the data requisite for the comparison. The following are the principal conclusions which we seem to be warranted in drawing : 1. In the southern half of Europe the mortality depends upon the temperature of summer, being greatest where that temperature is greatest, and diminishing with it down to a certain limit. 2. In the northern half of Europe, on the contrary, the mor- tality depends on the temperature of winter, being greatest when that is least. 3. The boundary line between these two regions is not far from the mean yearly isothermal of 50, which is accordingly the line of least relative mortality. 4. The mortality attains an absolute minimum in the British islands, at the western extremity of this line, the annual range of temperature being there least. These conclusions will be evident on the inspection of the fol- lowing Table, in which the mortality is compared with the 406 THE CLIMATE OF IRELAND, AND of the temperature of the hottest month above 50 Fahr., or with the defect of temperature of the coldest "below the same.* Countries. Excess of Summer Temperature. Defect of Winter Temperature. Death rate i per 1000. Italy; Turkey, . . . + 25 33 France; Austria, . . + 18 25 Central Germany, . . + 15 22 British Islands, . . . + 12 -13 21 Belgium, -16 23 Holland . . -18 26 Prussia -22 28 -36 37 I believe that this decrease of the mortality, in proceeding from the hotter to the colder parts of Europe, and its subsequent increase, result from fas joint operation of different diseases having opposite relations to temperature. Epidemic and endemic diseases appear to diminish indefinitely, as the temperature diminishes ; while consumption, and other diseases of the respiratory organs, increase. These are the maladies most destructive of human life ; and it will be readily understood in what manner their contrary tendencies may produce a minimum mortality at some temperature far removed from either of the two extremes.! I have already stated that the absolute minimum of mortality is to be sought for at the western extremity of the yearly isother- mal of 50 i. e., in our own island ; and the facts bear out this conclusion. The average mortality of Ireland, so far as it can be * There is, of course, some uncertainty in comparing numbers, one set of which are dependent on political boundaries, and the other upon physical features. In Russia, the mean winter temperature adopted is that of the January isothermal passing nearly through three of the great centres of population Petersburgh, Moscow, and Astrakhan. No attempt has been made to include Sweden and Norway in the comparison, the extraordinary flexures of the January isothermals in these countries rendering the deduction of a mean winter temperature uncertain. t This law appears to be exhibited, even within the narrow limits of our own island. It appears from the valuable statistics of disease collected during the Census, and discussed by Sir W. Wilde, that deaths by consumption are least numerous in the south-west of Ireland, while deaths from epidemic and contagious diseases are most numerous in the same district. THE CURRENTS OF THE ATLANTIC. 407 determined by the imperfect method of a decennial census, is 21 per 1000, that of England being 22. And here I cannot avoid remarking, that the connexion of these facts with temperature has been strangely mis-stated by the eminent French statist, M. Moreau de Jonnes, by whom these vital statistics were first collected. After observing that, of all the states of Europe, the British Islands are the most favoured by a low mortality, he adds with evident sur- prise " It is on the frozen rocks of Ireland (les rochers glaoes de 1'Irlande) and in the midst of the eternal mists of Scotland, that man reaches the most advanced age !" But the season of the year, and probably even the hour of the day, exert their influence upon mortality. The following numbers give the relative mortality of the several months of the year in Bel- gium, that of the whole year being taken as unity : Month. Mortality. Month. Mortality. January, . . 1-21 July, . . . 0-81 February, . . 1-20 August, . . 0-82 March, . . . 1-19 September, . . 0-89 April, . . . 1-12 October, . . 0-93 May, . . . 0-98 November, . . 0-94 . June, . . . 0-88 December, . . 1-03 The regularity of these numbers is very remarkable. The mortality is greatest in January, the coldest month of the year, and least in July, which is the warmest ; and the proportion of the former to the latter is that of three to two. These facts are in accordance with the laws of mortality stated above ; for Belgium is to the north of the neutral line, and in it, therefore, the mortality should depend chiefly upon the cold of winter. To the south of that line we should expect an opposite result namely, that the mortality should be greatest in the hottest months of the year. We do not possess the data necessary to verify, or to disprove, this conjecture. It is worthy of remark, however, that in Ireland, which 'is traversed by the mean isothermal of 50, the periods of greatest and least mortality are neither the coldest nor the hottest months, but the intermediate periods of spring and autumn. It appears from the Census of 1851, that the number of 408 THE CLIMATE OF IRELAND, AND burials in Dublin was greatest in March, and least in October and November; and in Ireland generally the mortality is greatest in spring, and least in autumn. The following numbers represent the relative mortality at each of the four seasons, that of the entire year being taken, as before, as unity : Season. Relative Mortality. Spring, , 1-31 1-11 Autumn, Winter 0-57 1-01 Even for places under the same isothermal line, and at the same season of the year, the difference between the temperatures of day and night, or the diurnal range, exerts an effect upon health. When it is great, the extremes are, of course, more divergent. But, in- dependently of this circumstance, the suddenness of the change of temperature very probably adds to its influence, especially in the case of individuals predisposed to lung-disease. I have already pointed out the fact that, in islands this variation is small, com- pared with the interior of continents. Throughout Ireland, the diurnal range of temperature is small. It is less on the coast than in the interior of the island ; and it is least of all on the coast of Kerry, in the neighbourhood of Valentia, Now it is remarkable that this region is also that most favoured by a high winter tem- perature. The mean direction of the isothermal lines in Ireland, in the winter months, is N. 40 W. ; so that the highest winter temperature is at the south-western extremity of the island, and the lowest at the north-eastern. The former, therefore, seems to be the region most favourable to the patients I have referred to, as well on account of its high winter temperature, as because of its small daily range ; while the latter is the least so. This conclusion (which I ventured to advance some years ago, before the facts were known) has since been fully verified by Sir William Wilde, in his analysis of the Census of 1851. Neglecting the inland districts, for which the meteorological data are insuffi- cient, he finds that the deaths from consumption, at the several THE CURRENTS OF THE ATLANTIC. 409 parts of the coast-line of Ireland, are as follow, the deaths from all causes being 1000 : North-east, from Glenarm to Dublin, . . South-east, from Dublin to Youghal, . . South, from Youghal to Bantry Bay, . . South-west, from Bantry Bay to Galway, North-west, from Galway to Donegal Bay, North, from Donegal Bay to Glenarm, . 172 161 106 76 83 127 The proportion of the highest of these numbers to the lowest is nearly that of 2 to 1. I have hitherto spoken only of the relation of the temperature of the air to health : it remains to say a few words of its moisture. We all know that the outer covering of the human body is porous, and that these pores are incessantly engaged in the per- formance of an office, upon the due discharge of which the health is closely dependent ; but we are perhaps not all aware of the magnitude of the operation. The experiments of Lavoisier and Seguin show that the quantity of water, which transudes through the skin of an average-sized man, in 24 hours, is 30 ounces ; and this, in the ordinary state of things, is carried off by evaporation as fast as it is secreted. Now, it is a well-known meteorological law, that evaporation is less, the greater the amount of moisture actually present in the air ; and that it ceases altogether when the air is saturated with moisture. It follows, therefore, that the due performance of one of the most important functions of the animal system is dependent on the dryness of the air. It is not to be in- ferred from this, however, that the transpiration, as Blumenbach denominates the process, ever wholly ceases, even when the air is saturated with moisture. For the air in contact with the body rises in temperature, and is thus enabled to take up a fresh supply of moisture from the skin. But besides the effect of humidity in checking the process of the insensible perspiration, it also operates in reducing the animal heat, when the temperature of the air is low. This seems to re- quire some explanation. One of the immediate consequences of evaporation is, we know, the abstraction of heat ; and as the eva- poration from the surface of the body is less in moist air than in 410 THE CLIMATE OF IRELAND, AND dry, it would seem at first sight that the chilling effect should be less also. But this is more than compensated by the increased con- ducting power of air when charged with moisture. When such air is much below the temperature of the body itself, the heat of the latter will escape through it, much more rapidly than through air that is dry. The chilling effect of the damp fogs of winter, with all their train of attendant evils, is due to this cause. On the other hand, extreme dryness of the air produces an in- jurious effect of a different kind. When the air in contact with the lungs is excessively dry, inflammation of the organ is apt to ensue ; and thus, to those who have a constitutional tendency to such disease, a very dry air is often fatal. To this cause are to be ascribed the painful, and even fatal effects of the simoom, the dry wind of the African desert. And it is on this account also that, in rooms heated by iron stoves, which are so common in Germany, a vessel containing water is usually placed at the top of the stove, to give the suitable degree of moisture to the air. The mortality in towns is much greater than in the rural districts adjoining. Thus in Belgium, where vital statistics have been collected with much care, the death-rate in the towns is 27 per 1000, while in the country it is only 21. This is in part accounted for by the deficiency of pure air produced by over-crowd- ing. Every adult inhales about 13 cubic feet of air in an hour, from which his lungs extract the vital element, and replace it by a noxious gas ; and in populous cities and especially in the dwell- ings of the poor this waste is not adequately supplied.* The effects of overcrowding have been clearly shown by Dr. Farr, in his useful analysis of the Eeports of the Eegistrar-general on the mortality of London. The following are the results of his com- parison of the mortality with the density of the population, in the three principal districts : Square yards to each person. Death-rate per 1000. I. Whitecnapel district, . 57 33 II. St. Saviour do. 78 28 III. Kensington do. 217 22 It has only lately been brought to light, by the philanthropic exertions of the THE CURRENTS OF THE ATLANTIC. 411 But the supply of air in towns is not merely deficient in quan- tity it is also greatly deteriorated in quality. This is mainly owing to the impurity caused by decaying animal matter. The fluids of the body have a tendency to decompose, which is resisted by the vital power ; and when that power is finally withdrawn, we all know how soon the work of destruction is completed. Life is thus a continued struggle between the chemical and the vital powers ; and the effect of decaying animal matter is to stimulate the chemical forces, and to a degree which the vital power may be unable to resist. The actual contact of such matter with the blood will destroy life speedily ; and every surgeon knows the danger of a wound from the dissecting-knife. But the poison is usually administered more slowly. The effluvium pollutes the air we breathe, and reaches the blood through the lungs ; and thus the process of destruction goes on more slowly, indeed, but as surely. These agents of disease and death may be, and have been, suc- cessfully controlled by means within our own power. The death- rate of London was formerly 57 per 1000 as great as it is in Constantinople now. At present it is only 23 ; while, in some of the London districts, the death-rate has been reduced to 17, a proportion as small as that of the healthiest of the rural districts. This is very encouraging to the sanitary reformer. The first attempt to legislate for the sanitary improvement of towns was made as recently as the year 1848. Since the passing of the " Public Health Act " of that year, the death-rate in the towns to which it has been applied has diminished by 6 per 1000 ; and, in Liverpool, the death-rate is said to have been reduced from '38, which was its amount in 1846, to 24. Much useful work has also been effected by voluntary associations, and especially by the " Metropolitan Association for Improving the Dwellings of the Industrial Classes." In one of the great lodging-houses of that Society the " Metropolitan Buildings," in Pancras-road the death-rate has been reduced to 13| per 1000, which is little more, than one half the average death-rate of London. The effects of these improvements are even more marked in the case of children, who are far more susceptible than adults to the deleterious effects of impure air.* The death-rate of children under five years of Lite lamented Lord Herbert, that one half of the barracks of the United Kingdom furnished little more than 400 cubic feet of space to each soldier. * It appears from the Census of Ireland, that the number of deaths of children, in 412 THE CLIMATE OF IRELAND, AND age, in the buildings of the Association, is said to be only 5 per 1000, the corresponding death-rate for the whole Metropolis being 46 ! The annual death-rate in Dublin, as deduced from the returns of the Census of 1851, was 29 per 1000. The Act for the registra- tion of births and deaths in Ireland came into operation only at the beginning of last year ; and the death-rate in Dublin, deduced from the returns made under that Act, was 27 per 1000. Thus there has been some amelioration in our sanitary state, although much still remains to be done. We have every reason to hope that the sanitary measures now in progress in this city, under the direction of a gentleman so competent to the task as the present Medical Officer of Health, will before long effect the desired im- provements. I have ventured to touch upon these painful topics, on account of their urgent importance at the present time. A fearful epidemic has, in its stern undeviating march, revisited Great Britain, and, if the arm of the Almighty be not stretched forth to arrest its progress, will shortly reach our homes. Whatever be the original source of this formidable disease, it seems to be now generally believed that its proximate cause is the poisoning of the blood ; and we cannot doubt that this poison is communicated to others, whose vital powers are unable to resist it, through the air for we can in no other way explain the fact, that the disease in its progress fol- lows the great highways of human intercourse, while at the same time it is little, if at all, contagious. It is thus refemBle to the general cause already adverted to namely, the power of disorgan- ized matter to generate decay although the effect bears in this, as in other familiar cases, the impress of the original type. It is true, we have not yet succeeded in tracing the organic element in the air, to which it owes its destructive power. But there are many facts which prove that the air does contain elements which evade all the resources of chemistry, while, nevertheless, their presence is proved by their physiological effects. No chemist has detected the substance which imparts the perfume of the rose to the air ; and yet there are individuals so sensitive to the odour of flowers, as to be rendered ill by inhabiting the room in which they lie. It is even stated I quote from a high French authority the first year of their age, is 9 per cent, of the total number in the rural districts, while it is 16 per cent, in towns. THE CURRENTS OF THE ATLANTIC. 413 that the air taken in the great sewer of Montmartre, and that collected in the Place de la Concorde (one of the most open spaces in Paris), have been carefully compared, and found to be undistin- guishable by the most delicate chemical tests. But, whatever be the source of the virus, its action is, at all events, powerfully stimulated by the co-operation of decaying organic matter in its ordinary forms ; and as this ally of disease is within our power, we shall be without excuse if we neglect the known precautions for defence.* I regret to have to close this brief sketch of our atmospheric relations with a disagreeable topic. I have endeavoured to bring before you the more salient features of our climate, so far as they can be exhibited in a popular form ; and, on a review of all, we have, I think, no reason to repine. If the humidity of the air by which we are enveloped is at times depressing to our energies of body, or of mind, we must remember that we are indebted to it for the luxuriant crops, which furnish food to our cattle, and through them to ourselves. Our summer temperature may be too low to bring some of our seeds and fruits to perfection ; but, on the other hand, it is to its moderated heat that we owe our exemp- tion from some of the maladies which afflict the inhabitants of more southerly climes ; while our mild winters press gently on the springs of life, and suffer man to reach an advanced age. Let us be thankful to-ihe Giver of all good gifts for these His blessings ! and if as all earthly blessings are they are at times checquered by afflictions, let us remember that these too come from a Father's hand. Let them serve to remind us how far we may have strayed from Him, and bring us back, as sorrowing and re- pentant children, to His outstretched arms ! * This seems to be conclusively shown by the effect of elevation on the number of deaths by cholera in London, in the year 1849. The Table contained in the Reports of the Registrar-general gives the heights above the Thames, and the corresponding number of deaths by cholera, out of a population of 10,000. The numbers decrease at first rapidly, and afterwards slowly, until, at heights above 100 feet, the variation in the number of deaths is very small. The facts are justly ascribed to the imperfect drainage of places situated at low levels. XIX. ON THE RISE AND PROGRESS OF MECHANICAL PHILOSOPHY. Introductory Lecture delivered in the Philosophy School of Trinity Colkge, in Hilary Term, 1834. GENTLEMEN, I have lately endeavoured to lay before you an outline of the methods which have been pursued in the culti- vation of Physical Science; and I dwelt in particular on the principles of the inductive philosophy as laid down by Bacon. At present I propose that we should confine our attention to that branch of natural philosophy which is to form the subject of our consideration during the ensuing Term ; and review briefly the rise and subsequent advances of Mechanical Science. The fallacious methods adopted by the ancients in physical science, generally, did not exclude a certain progress in the science of force. The doctrine of equilibrium, as you will soon learn, is altogether independent of experience ; and the laws which deter- mine the relations of balancing forces may be deduced wholly by a priori reasoning. In this department, accordingly, some steps were early made. From the metaphysical principle of sufficient, reason, Archimedes derived the properties of the lever; and showed that two weights attached to it are in equilibria, when they are to one another inversely as their distances from the fulcrum. This important and fundamental principle contained the germ of Statical Science : Archimedes himself was able to deduce from it the rules for the composition of parallel forces ; and to show that there existed in every body, or system of bodies, a point now known by the name of the Centre of Gravity in which its weight might be supposed to be concentred. KISE AND PEOGKESS OF MECHANICAL PHILOSOPHY. 415 The foundations of Hydrostatical Science were also laid by the same great geometer. His work de humido imidentibiis is derived from the principle of the equality of fluid pressure a principle still adopted as the basis of this science. In this work he has shown that a heavy body, when immersed in a liquid, loses a portion of its weight equal to that of the liquid displaced. It is by the aid of this theorem a theorem on which the present mode of determining the specific gravities of bodies is made to depend that the philosopher of Syracuse is supposed to have solved the famous problem of the crown, proposed by King Hiero, and to have detected the fraud of his workman. A long period of darkness followed the discoveries of Archi- medes. The philosophers of Alexandria, Ctesibius, and Hero, Pappus Alexandrinus and others, pursued the inquiries which he had begun, and by their inventions added much to practical mechanics. To the two first of these authors we owe the analysis of the various classes of machines, and their reduction to five simple ones ; and the name of $vvajj.ttc, or powers, which they affixed to these elementary machines, is still retained. But yet the theory remained nearly as it had been left by Archimedes, and the doctrine of equilibrium was destined to receive no addition until the close of the 16th century. Hitherto the theory of the Mechanic Powers was understood only so far as it could be derived from the principle of the lever ; and no method was known of determining the conditions of equili- brium of forces whose directions were inclined. It was at this point that the progress of the ancients in Mechanical Philosophy was arrested, and it is here that we are to look for the first im- portant extension of this science. The problem was attempted, but unsuccessfully, by Ghiido "Ubaldi, a mathematician of Italy, in the 16th century ; and was finally solved by Stevinus, a cele- brated engineer of the Low Countries, who demonstrated the principle of equilibrium on the Inclined Plane. In this remarkable demonstration, he supposes a chain of uniform thickness to encom- pass the plane entirely round : part of this will rest on the plane, along its hypothenusal side, part will hang vertically beside its altitude, and the remainder will form a loop below the base. Now the whole is in equilibria, for if not, a perpetual motion must ensue in the direction of the greater force ; and as the part which hangs below the plane draws equally in both directions, its effect 416 RISE AND PEOGEESS OF may be disregarded. There is equilibrium, therefore, between the other two portions that which is equal to the length of the plane, and that which is equal to its altitude ; and it follows that two weights, which balance by means of the inclined plane, one hanging freely, and the other resting on the plane, must be to one another in the ratio of the height of the plane to its length. This singular solution of the problem was given in the year 1585 ; and from it the author has derived the general conditions of equilibrium among any three forces meeting at a point. Thus, though he does not seem to have been aware of the full value of the prin- ciple, he touched the very corner-stone of statical science. The theory of the Equilibrium of Fluids is also largely indebted to Stevinus ; and he was the first to determine the pressure which the bottom of a vessel sustains from the contained liquid. The second grand division of mechanical philosophy the Doctrine of Motion is of much later birth. We are here beyond the circle of abstract truth, and experience alone can furnish the principles of our reasonings. We are not to wonder, therefore, that no progress was made in this department of science, when the true principles of experimental inquiry were themselves unknown. The school of Aristotle taught that motions are either natural or unnatural The natural tendency of all terrestrial bodies was said to be, either to fall directly to the ground, or to ascend from it until they reached their place ; while bodies that were impelled obliquely were believed to pursue a violent or unnatural course, and the motions thus generated were supposed to tend continually to decay. In the heavens all this was different : the natural motions of the heavenly bodies were pronounced to be circular ; and, as their matter was incorruptible, so their motions were eternal and immutable. Amid these reveries, we can still trace throughout the workings of genius ; and in some parts of the mechanical speculations of Aristotle, we find very just views respecting the nature of gravity, and of force in general. The principle of the Composition of Motion that fertile principle which served as the basis of the reasonings of Gralileo, and was more fully developed by Newton is to be found distinctly stated, and reasoned on in Aristotle's writings. For more than two thousand years the dogmas of Aristotle continued to extort the assent of mankind, and though here and there a bold inquirer might be found, who ventured to doubt the MECHANICAL PHILOSOPHY. 417 authority of the " great master," and, to appeal to nature herself, yet the science of motion was not destined to profit by these glimmering lights, and its very foundations remained unlaid until the close of the 16th century. Galileo was born at Pisa, in the year 1564, and at an early age devoted the energies of an ardent and powerful mind to the mathematical and physical sciences ; and here was first formed that union of abstract and experimental reasoning, whose force and value in explaining the phenomena of nature is now so well under- stood. While yet pursuing his studies at the university, Galileo had begun to make experiments on the laws of falling bodies; and he then discovered the fact, that all bodies, whether light or heavy (the effect of the air's resistance being abstracted), fall to the ground from the same height in the same time. This im- portant law of gravity was in direct contradiction to the principles of the Aristotelian philosophy, then in full repute ; and, unac- countable as it now seems, it was believed that two unequal weights, if let fall at the same instant, and from the same height, would reach the ground in times inversely proportional to the weights that a weight of 100 Ibs. (for example) would descend in the y^oth part of the time occupied by a weight of 1 Ib. Galileo tried this very case before many witnesses, from the summit of the famous leaning tower of Pisa. The two weights were found to reach the ground in precisely the same time ; and the decisive result of the experiment did more, perhaps, to shake the empire of the Aristotelian philosophy, than any single effort of opposition hitherto made. Setting out from the principle that bodies, in their free descent towards the earth, receive equal increments of velocity in equal times, Galileo deduced, by the application of mathematical reasoning, the known laws of falling bodies; and having proved that these laws must hold good also in the case of bodies de- scending on inclined planes, where a portion of the weight is sustained, and where consequently the acceleration is not so rapid, he was enabled to bring the whole theory to the test of experi- ment. He applied himself, in the next place, to the consideration of the motion of a heavy body projected obliquely ; and by the aid of the principle of the composition of motion a principle tacitly assumed in his reasoning he proved that the path traced by a projectile was the well known curve of Appolonius the parabola, 2K 418 RISE AND PROGRESS OF All these results were confirmed by observation, and thus the nature and laws of terrestrial gravity were fully established. Galileo seems to have turned his attention at an early period to the motions of pendulous bodies ; and he is said to have been conducted to the knowledge of the fact that all the vibrations of the same pendulum, whether great or small, are performed in the same time by observing the swinging of the lamps in the Cathedral of Pisa. The tautochronism of the pendulum, however, seems to have been noticed, and even applied, at a much earlier period. The astronomers of Arabia employed the instrument in some of their observations, patiently counting the number of its oscillations during the period of an eclipse, and renewing the motion with a slight push of the finger, when the arcs of vibration became too small. The same mode of observation was practised in more recent times by Gassendi, Biccioli, and other astronomers of Europe, when the attention of the scientific world was again drawn to the principle of tautochronism. By these important discoveries, Galileo has justly earned the title of " father of dynamical science." But to estimate duly the merits of this great man, we must consider the age in which he lived, and the darkness with which he was encompassed. Alone and unsupported, he assailed the Aristotelian philosophy with all the weapons of sound argument and caustic ridicule ; and the inveterate spirit of hostility which he thus excited, the rancour of rivals, and the intolerant persecution of the Church, ended only with his life. Notwithstanding all this opposition, however, truth was progressive. The proofs which the astronomical dis- coveries of Galileo brought in support of the Copernican system roused the attention of philosophers; and it may be fairly ques- tioned whether the discoveries of the Florentine philosopher have not had as great an influence on the progress of physical science, as the works of his immortal contemporary Lord Bacon. While to Galileo we owe the foundation of the doctrine of motion, to his pupil Torricelli we are indebted for the first step which was made in that branch of the doctrine which relates to fluids. The fundamental problem of hydraulics that which de- termines the velocity of efflux of a fluid through an aperture in the bottom or side of a vessel was first solved by this writer, in his work de motu gravium. As the terrestrial mechanics of Aristotle were overthrown by MECHANICAL PHILOSOPHY. 419 Galileo, so, about the same time, his theory of the heavenly motions was shown to be baseless. According to Aristotle, it has been already mentioned, the motions of the celestial bodies are circular and uniform because the circle is a perfect figure. Many of his followers, however, attempted to support the doctrine on physical grounds, and asserted with Eudoxus, that the planetary bodies are confined by crystalline spheres, the movements of which gave rise to the motions we perceive. It is somewhat remarkable that the very same bodies, whose motions now tend to establish the existence of a resisting medium in the heavens, should be the first to give evidence against the theory of the crystalline spheres. It was ascertained by Tycho Brahe, from astronomical observations, that the comets were not, as Aristotle supposed, meteors in our atmosphere, but moved throughout the planetary spaces ; and it was plain that, as they crossed the orbits of the planets in every direction, they could have encountered no obstacles such as the spheres of Eudoxus. Yet the doctrine of circular motion remained still undisputed, until Kepler, the successor of Tycho Brahe, announced to the world his great discoveries the elliptic motion of the planets, the equable description of areas, and the harmonic law. It is impossible to read the account given by Kepler of these discoveries, without feeling animated with a portion of his en- thusiasm. " I have stolen," says he, " the golden vases of the Egyptians, to build up a tabernacle for my God, far from the confines of Egypt." But when from these laws of observation he attempts to ascend to their physical causes, his reasonings are deeply tinged with the mysticism which seems to have gained so powerful an ascendancy over his great mind. Of the existence of gravitation, as the ruling principle of the universe, he enter- tained very strong and clear views ; but the attraction itself he conceived to be of the nature of animal force. "If the moon and the earth," he wrote, " were not retained in their orbits by their animal force, or some other equivalent, the earth would mount to the moon by a fifty-fourth part of their distance, and the moon fall towards the earth by the other fifty-three parts, and they would thus meet, the substance of both being assumed to be of the same density. If the earth should cease to attract its waters to itself, all the waters of the sea would be raised, and would flow to the body of the moon." The earth itself he supposed to be an enormous living animal, of which the tides 420 RISE AND PROGRESS OF constituted the act of respiration ; and as this vital function was supposed to be produced by the influence of the moon, so its other living energies were dependant on the configurations of the planets. In the Harmonies of the celestial motions, he assigns to Jupiter and Saturn the bass, to Mars the tenor, the counter-tenor to the Earth and Venus, and the treble to Mercury. The cosmical system of Des Cartes is too important in the history of science to be passed over without notice, although it seems to have been but a refinement on the doctrine of Eudoxus, and was as remote from real existence. Kepler, we have seen, had endowed the great bodies of the universe with vital powers, and chained them together, in their wanderings through space, by the ties of animal sympathy. The astrologers of the day borrowed from the region of spirit itself, and, by a beautiful fiction, assigned to each planet its guardian angel, whose office it was to guide its career through the trackless void. Des Cartes was the first who, with any show of plausibility, attempted to reduce the whole to physical principles, and to derive the phenomena of the material world from the fundamental properties of body. Matter, accord- ing to this ingenious philosopher, fills all space ; and its parts are endued with motion in every possible direction. By the combina- tion of these infinitely varied motions, matter is supposed to be continually deflected from its rectilinear course ; and thus at length to form itself into vortices, in which the denser bodies of the universe floated, and of whose motion they partook. In this manner the earth and planets were supposed to be borne round the sun, in the vortex of the solar system : and each planet itself was the centre of a lesser vortex, which carried its secondaries. Such was the system which preceded the theory of universal gravitation. Mankind looked up with wonder on the symmetrical fabric which the enchanter's wand had raised, and they examined not too narrowly into its foundations ; but a mightier wizard soon after came upon the scene, and the rod of Aaron swallowed the rod of the Egyptian. The actual contributions of Des Cartes to mechanical science are inconsiderable. He was an a priori philosopher, in the strictest sense of the word; and in the barrenness of that philosophy aided as it was by powerful talents, and by the force of these talents gaining an ascendancy, which for a time closed men's eyes to the truth we have, perhaps, as striking a proof as could be MECHANICAL PHILOSOPHY. 421 given of the soundness of Bacon's views. " We wish," says Des Cartes, "to deduce effects from their causes, and not, con- versely, causes from their effects. We appeal to experience only, that out of innumerable effects which may he produced from the same cause, we may direct our attention to one rather than to another." The axioms, to which he endeavoured to reduce all his theories, were grounded on metaphysical notions of the attributes of the Deity ; and we have a remarkable instance of the presump- tion and error, which characterized most of his reasonings, in the dogma, that the quantity of motion in the universe must remain always the same, because the Divine nature is immutable. To Des Cartes, however, we are indebted for the first distinct enunciation of the laics of motion; although his own conception of these laws was far from accurate, and he seems, in particular, to have regarded the inertia of matter as a kind of active force. The first actual contributions of importance which dynamical science received subsequently to the discoveries of Galileo are due to Huygens. By the aid of the most refined geometry, of which he was so perfect a master, this mathematician discovered the relation which subsists between the length of a pendulum and the time of its vibration ; and having developed the theory of the instrument, he realized the conception of Galileo, and applied it to the regulation of the clock. On pursuing his inquiries, however, Huygens ascertained that the property of tautochronism belonged to the circular pendulum only when the arcs of vibration are indefinitely small; and he was thus led to inquire what curve possessed the property universally. Finding this curve to be the cycloid, he conceived the idea of making a pendulous body move in it'; and his mathematical skill soon after enabled him to point out the means of accomplishing this object. His account of this construction, and of the principles on which it depends, was pub- lished in the year 1670, in a work entitled Horologimn Oscillatorinm, many years after the date of the discovery. But interesting as the cycloidal pendulum is in theory, it has been long since abandoned, as useless, in practice ; and the clocks to which the circular pendulum was adapted were found far to excel the cycloidal clocks of Huygens. To Huygens we are indebted also for the next great step which was made in dynamical science the theory of enrufur )uoti