and explosion twenty miles above it, upon the meridian of Land's End. The flight of 360 miles appears to have occupied seven or eight seconds of time.

Meteor, 1861, November 15th, 10h 14m P.M.

The meteor described by Mr. Nash at Greenwich, and Mr. Herschel at Shooter's Hill, although identical, do not admit of useful comparison with one another, nor with that observed by Mr. Greg at Styall, near Manchester,-the base-line in the former case being too small for such a purpose, and the third meteor being apparently distinct from the former two.

Meteor, 1861, November 19th, 9h 38m 30s P.M.

The Ipswich and Norwich accounts place the audible explosion of this brilliant meteor at no great height between the two towns; thirty miles of height must be allowed to it for the altitude as seen from Exeter, although such a height is at variance with the view obtained from Greenwich and North Foreland. It is not impossible that explosion, audible at Norwich and Ipswich, and perhaps also at North Foreland, may have depressed the last portion of the flight, for this was hidden from view at Exeter. The near verticality at North Foreland, the passage over the moon (whose altitude was 38° E. by S.) in the eastern parts of Kent, and the low southern position of the nucleus as first perceived by Messrs. Hill at Woodford, Mitchell and Harmer at Tunbridge, and James Rock, jun. at Guestling, show this meteor to have taken a nearly meridian and nearly horizontal course. A flight of 260 miles in 10 or 12 seconds, from fifty-five miles above Paris to thirty miles above Beccles (between Suffolk and Norfolk), is found to satisfy the whole of the accounts with considerable accuracy.

Meteors, 1861, November 24th, 8h 0m P.M.

The resemblance of these meteors is casual,-the lines of sight of commencement lying widely upon opposite sides of the base-line between the stations, while those of termination approach no nearer than twenty-six miles upon the southern side of the base-line.

Meteors, 1861, December 1st, 9h 15m P.M.

The resemblance of these meteors is not borne out by the uranographical positions assigned to them at the two distant stations,-the point of commencement having little or no parallax with considerable deviation of the lines of sight, while the lines of sight of termination lie upon opposite sides of the base-line.

Meteor, 1861, December 8th, 8h 16m P.M.

At Dungannon in Ireland this meteor appeared to fall vertically, while at Wakefield (Yorkshire) it passed overhead. The observation of Dr. Walker at Birkenhead (Seacombe), assigns Strangford, on the Irish coast, as the spot between these two towns where the body would have struck the earth. By Mr. Redford's account, from Silloth near Carlisle, the height at disappearance is found to be fifty miles above the sea, halfway from Lancaster to the Isle of Man; the height above Wakefield eighty-five miles, and at Hull 110 to 115 miles. Modified by the remaining accounts, a course of 160 miles from 110 miles above Hull to forty-five miles above the Irish Sea, twenty miles E. of Douglas Town, performed in six or eight seconds of time, appears to be a near approximation to the truth. It is possible that an explosion loudly heard at Lancaster and Southport, but not heard at Douglas, may have caused the deflection by which the meteor in the latter portion of its flight appeared sta

tionary at Castletown some seconds. On the 3rd of the same month, a similar detonating meteor appeared in Germany, bursting sixty miles over Dessan, and directed almost from the Pole (see the Calculation of Professor Heis). Mr. Greg at this time observed the radiant point of shooting-stars to lie between Gemini and Auriga. On the 24th of December it was in Taurus.

Meteors, 1861, December 9th, 5h 30m P.M.

The resemblance is casual. The uranographical position at Hawkhurst places this meteor at a great height towards Edinburgh, upon the latitude of Glasgow.

Shooting-star (F), 1862, January 28th, 11h 4m P.M.

The base-line of forty miles between the stations of London and Stone affords a good determination of this shooting-star. The lines of sight for the commencement are only three miles apart at their nearest approach, namely, at 44 miles above Melton Mowbray in Leicestershire, while those of termination are only 2 miles asunder at 471⁄2 miles above Macclesfield in Cheshire. The horizontal fiight of sixty miles was performed in 1 to 1 second, by careful estimation at the time of the observation. Direction from 32° S. of E. At 880 yards it would have equalled the full moon.

Meteor, 1862, February 2nd, 8h 20m P.M.

The astronomical accounts of Mr. E. J. Lowe and Mr. Alcock at Beeston Observatory and Newark, together with similar details from Tarporley in Cheshire, appear to fix the disappearance of this meteor with precision at fourteen or fifteen miles above Cheadle, on the borders of Derbyshire, where the meteor arrived after a flight in the air of 236 miles from 190 miles above Lyme Regis, occupying six seconds of time and directed to earth in the valley of the Dove, or at the foot of the Peak of Derbyshire. The point of first appearance in Orion or the Pleiades, as seen at Liverpool and Tarporley, places this meteor among the few whose true courses are observed to lie from W. to E. of the meridian.

Meteor, 1862, February 23rd, 9h 25m P.M.

This meteor, which passed nearly over Liverpool towards S.W., appeared to Mr. W. H. Wood, at Weston-super-Mare, to move 30° horizontally in the N. at 20° from the horizon. It appears to have sought the earth at Pembroke, and had its flight from forty miles above Stockport, near Manchester, to twenty miles above Aberystwith, in Wales.

The following comparison of the brightness of these meteors is offered as leading to an estimation of their probable dimensions.

The photometric tables of the light of certain stars compared with that of the full moon, published by Sir John Herschel, enable us to compare the light of ordinary shooting-stars with a standard generally familiar; and the same may be done when fireballs are compared in their illuminating power to different phases of the moon; but the class of meteors intermediate between these in the scale of brilliancy are usually compared with the planets of whose light at different phases no tables are prepared. Among the preceding known meteors, one only of the latter class (shooting-star e) is found. The following deductions aim at no greater accuracy than is commensurate to the character of the observations themselves.

(A) I. Meteor, 1861, July 16th, 10h 15m P.M.: shone apparently as half of a moon two days old, at Furness, 150 miles from the meteor's termination. At 25 miles it would have equalled the full moon.

(B) II. Meteor, 1861, July 16th: shone as one-fourth of moon two days old, at Flimwell, distant 220 miles from bursting. At 37 miles it would have equalled full moon.

(C) Meteor, 1861, August 6th, 11h 21m P.M.: shone one-tenth of moon two days old, at London, 150 miles from brightest point. At eight miles it would have equalled full moon.

Shooting-stars, August 8th, 10th, 11th, would have equalled full moon at distance of 352, 398, 692, 274, 1484 yards.

(D) Meteor, 1861, November 12th, 5h 49m P.M.: lighted the turnpike-road at Hay fully as much as the moon itself shining upon it, and ten days old. Meteor overhead, seventy-five miles from Hay. At sixty-three miles it would have equalled full moon.

(E) Meteor, 1861, November 19th, 9h 38m P.M.: threw shadows half as deep as the moon, then full, at Tunbridge, seventy-seven miles from the first burst of light. At fifty-four miles it would have equalled full moon.

(F) Meteor, 1861, December 8th, 8h 16m P.M.: exceeded the light of the moon then shining clear and six days old, at Hull, 130 miles from the flash over Walney Isle. At eighty-eight miles it would have equalled full moon.

(G) Meteor, 1862, February 2nd, 8h 20m P.M.: shone as brightly as the moon unclouded and ten days old, at Beeston, forty miles from the explosion. At thirty miles it would have equalled full moon.

(H) Meteor, 1862, February 23rd, 9h 25m P.M.: threw a bright light from the sky which filled the streets at Liverpool and Bromborough, distance forty miles; perhaps equal to a moon four days old. At 16 miles it would have equalled full moon.

Assuming an ordinary flame of street gas to measure a cubic inch of incandescent matter, and at 15 yards to throw a light equal to the direct light of full moon, we have 13,690 gas flames at a mile equivalent to full moon; and the following are the globes of burning coal-gas which would shed the light produced by the separate meteors and shooting-stars of the foregoing list.

Meteors.

I.

II.

July 16. July16. Aug. 6. Nov. 12. Nov. 19. Dec. 8. Feb. 2. Feb. 23.

It is possible that these results afford a juster idea of the real sizes of the luminous bodies than those derived from angular measurements of their apparent discs.

[For Errata of the Catalogue, &c., see Appendix I. at the end of the Reports in this volume.]

1862.

G

On the Strains in the Interior of Beams.

By GEORGE BIDDELL AIRY, F.R.S., Astronomer Royal.

[A communication ordered to be printed among the Reports.]

THE author states that he had long desired to possess a theory which should enable him to compute numerically the strains on every point in the interior of a beam or girder, but that no memoir or treatises had given him the least assistance*. He had therefore constructed a theory which solved completely the problems for which he wanted it, and which appears to admit of application at least to all ordinary cases.

The theory contemplates forces acting in one plane. A beam therefore is considered as a lamina in a vertical plane, the same considerations applying to every vertical lamina of which a beam may be conceived to be composed.

The author remarks that it is unnecessary to recognize every possible strain in a beam. Metallic masses are usually in a state of strain, from circumstances occurring in their formation; but such strains are not the subject of the present investigation, which is intended to ascertain only those strains which are created by the weight of the beam and its loads. The algebraical interpretation of this remark is, that it is not necessary to retain general solutions of the equations which will result from the investigation, but only such solutions as will satisfy the equations.

After defining the unit of force as the weight of a square unit of the lamina, and the measure of compression-thrust or extension-pull as the length of the ribbon of lamina, whose breadth is the length of the line which is subject to the transverse action of the compression or tension, and whose weight is equal to that compression or tension, the author considers the effect of tension, &c. estimated in a direction inclined to the real direction of the tension, and shows that it is proportional to the square of the cosine of inclination. He then considers the effect of compounding any number of strains of compression or tension which may act simultaneously on the same part of a lamina, and shows that their compound effect may in every case be replaced by the compound effect of two forces at right angles to each other, the two forces being both compressions or both tensions, or one compression and one tension. Succeeding investigations are therefore limited to two such forces.

Proceeding then to the general theory of beams, it is remarked that if a curve be imagined, dividing a beam into any two parts, the further part of the beam (as estimated from the origin of coordinates) may be considered to be sustained by the forces which act in various directions across that curve, taken in combination with the weight of the further part of the beam, the load upon that part, the reaction of supports, &c. Expressing the forces in conformity with the principles already explained, and supposing that there is one compression-force B making an angle 6 with y (in the direction of y diminishing for increase of a), and another compression-force C making an angle 90°+ẞ with y, it is easily seen that the element ds of the curve, supposed to make the angle 0 with y, sustains the forces

In x,
B.dsx sin (ẞ+0)x sin ẞ+C.dsx sin (B+90°+0)× sin (3+90°).
In y, B. sx sin (3+0) x cos B-C.sx sin (B+90°+0) x cos (6+90°).
The weight of lamina bounded by y and y+dy, and estimated as acting

Subsequently to the communication of this Report, the author learned that one instance (the second) of those given here had been treated by Professor Rankine, by methods peculiar to that instance.

upwards, is yox. And the reaction R of a support may act upwards at distance h.

Expanding the sines and cosines, putting dx for sin e. ds, and dy for cos 0.ès; putting also

L=B.sin' 6+C.cos2 ß,

and forming the equations of equilibrium in the usual way, they will be found to be

Equation for forces in x, fdx. (Lp+M)=0.

Equation for forces in y, fdx(Mp+0)-R=0.

Equation of momenta, fdx(Lyp+My+Mxp+0x)—Rh=0.

Now these equations, applying to any curve, will apply to any two curves very close together; and therefore their variation, taken by the rules of the Calculus of Variations, will be 0. The proper equation (in the usual nota

tion) is N-d(P)=0. Applying this, the results are

dx

From this it follows that (omitting some arbitrary functions which represent original strains in the formation of the beam) L, M, O, are partial differential coefficients of the same function of x and y, which we may call F: so that

dF

J.a(d)=0, S.a(dF)—R=0, S.d(y+21

-F) –

-Rh=0.

dy Considerations, of a somewhat detailed character, depending partly on the relation assumed to exist between tension-force and material extension, are necessary to show the form which must be assumed for F in the various cases to be examined. The conditions to be secured are— -that the horizontal part of the thrust, &c. shall be the same as that given by ordinary theories, on the relation just mentioned; and that the equations above shall be satisfied. After due application of these in the following five cases, these forms are found for F..

Case 1. A beam of length r and depth s projecting from a wall;

Case 2. A beam of length 2r and depth s supported at both ends;