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Case 3. A beam liko tho last, carrying a weight W at tho distance) a from one end.

In this caeo the function is discontinuous; its forms aro—

Case 5. A boam like that in Case 2, with a straining momentum applied at one end only, as in the exterior tubes of tho Britannia Bridge;

By forming the differential coefficients of F symbolically, L, M, and Q {=y—0) are obtained in a form which admits of numerical calculation for every value of x and y. And from these, B, C, and /3 are computed without difficulty.

In this way the values of B, C, and (3 have been found for every combination of the values x=r x 0-1, x=r x 0-2, x=rx 0-3, &c., with the values y=8xO-1, y=8x0-2, y=sx0-3, &c. In Case 1, 121 points were thus treated: in each of the other cases the computations were made for 231 points.

In the following diagrams are given the curves representing the directions of pressure and tension through tho beam, together with a few numerical values at the most critical points, for each of the cases to which allusion has been made.

Citbv.es BEPKEsrarmra The Strains In Beams, Tinder Dutebent ciBcrtrsrsTANCEs.

The continuous curves indicate the direction of thrust or compression; the interrupted curves or chain lines indicate the direction of pull or tension.

The figures denote the measure of the strain; the sign -f meaning compression, and — meaning tension. The unit of strain is the weight of materia], lamina whose length = depth of beam.

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(Of this case, two instances are given in the curves below.) Case 4. A beam like that in Case 2, with a straining momentum applied at each end, as in tho middle tubes of the Britannia Bridge;

F_ 6xi—12rx+3r' /sjf y\

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No. 5. Beam supported at both ends, and in which a strain (of the nature of a moment or couple) is impressed on each

end, as in the interior tubes of the Britannia Bridge.

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No. 6. Beam supported at both ends, and in which a strain (of the nature of a moment or couple) is impressed on one end,

as in the exterior tubes of the Britannia Bridge. Mac-ar-S o-o -t-as-i Max + S7- 4, + ♦s o ao

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Report on the three Reports of the Liverpool Compass Committee and other recent Publications on the same subject. By Archibald Smith, M.A., F.R.S., and Frederick John Evans, R.N., F.R.S.

The task which we have undertaken, at the request of the British Association, is in some degree lightened by the publication, since the last meeting, of tho 'Admiralty Manual for ascertaining and applying the Deviations of the Compass,' a work which has been compiled under our joint editorship, and published by the direction of the Lords Commissioners of the Admiralty. The publication of this work allows us to treat as known, various methods and formulae which had not previously been published, and to which it will be necessary to refer in the sequel. It, however, makes it necessary that we should give some account of our own work, and this wo think it will be most convenient that we should do at the outset.

The 'Manual' is divided into four parts. Part I. contains the wellknown " Practical Rules " published by the Admiralty, drawn up originally, in 1842, by a committee consisting of the late Admirals Sir F. Beaufort and Sir J. C. Ross, Captain Johnson, R.N., Mr. Christie, and General Sabine. These rules were, and still are, purely practical,—the object being to enable the seaman, by the process of swinging his ship, to obtain a table of the deviations of his compass on oach point, and then to apply tho tabular corrections to the courses steered.

.Part II. is a description of the valuable graphic method known as "Napier's method," in which the deviations of the compass are represented by the ordinates of a curve, of which the "courses " or azimuths of the ship's head which correspond to the deviations are the abscissae. These azimuths may be measured either from tho "correct magnetic north," in which case they are called the "correct magnetic courses," or from the direction of the disturbed needle, in which case they are called "compass courses;" and we should in general obtain one curve if the abscissae represent one set of courses, and a different curve if the abscissae represent the other set. It was, we believe, first observed by Mr. J. R. Napier that, by drawing the two sets of ordinates in proper directions, each may be made to give the same identical curve, and, conversely, that the same curve may be made to give the deviations as well on the correct magnetic courses as on the compass courses, with the additional advantage that the one set of courses may bo at once derived from the other by going from the axis of abscissae to the curve, in a direction parallel to one of the sets of ordinates, and returning to the axis of abscissa) in a direction parallel to the other. The original direction of each set of ordinates is arbitrary, the scale, however, depending on thoso directions. By drawing the ordinates at angles of 60° and 120° from the axis of abscissa), we have the advantage that the scale along each axis of ordinates and also along the axis of abscissae is the same; and these directions are in general the most convenient, although in particular cases, as when tho deviations are very small, it is convenient to take a larger scale for the ordinates than for tho abscissa). The practical advantages of the method aro very great. It enables the navigator, from observations of deviations made on any number of courses, whether equidistant or not, to construct a curve in which the errors of observation are, as far as possible, mutually compensated, and which gives him the deviation as well on the compass courses as on the correct magnetic courses. Various modifications of this method havo been proposed, of which one by Capt. A. P. Ryder, R.N., deserves particular mention from the facility with which it may be used by those to whom tho method is unfamiliar; but for general use there seems to he no form superior to the usual form of Napier's diagram.

Part III. contains the practical application to this subject of mathematical formulae derived from the fundamental equations deduced by Poisson from Coulomb's theory of magnetism. This part was published separately in the year 1851, and afterwards as a Supplement to the " Practical Eules " in 1855. At that time it was considered sufficient to use approximate formula;, going as far only as terms involving the first powers of tho coefficients of deviation. The very large deviations found in iron-plated ships of war rendering it desirable to use in certain cases the exact instead of the approximate formula), this part has been re-written.

It may be desirable to give here some account of these formula?.

Poisson's equations are derived from the hypothesis that the magnetism of tho ship, excopt so far as it is permanent, is transient induced magnetism, the intensity of which is proportional to the intensity of the inducing force, and that the length of the compass-needle is infinitesimal compared to tho distance of the nearest iron.

On this hypothesis the deviation of the compass is represented exactly by one or other of the following formula; :—

sin 2=3 cos sin £'+C cos ('+& sin (2{' + a)+© cos (2£'-M) .. (1) g+g sin Z+C cos £+g sin 2 cos 2( (2) 1+8 cos Z—C sin (+8 Oob 2 f—« sin 2( W in which $ represents the deviation, { tho "correct magnetic course," f the "compass course;" 9, 0, <E are coefficients depending solely on the soft iron of the ship; JS$ and C coefficients each consisting of two parts, one part a coefficient depending on the soft iron and multiplied by the tangent of the dip, the other part a coefficient depending on the hard iron and multiplied by tho

reciprocal of the earth's horizontal force at the place, and by a factor, ^,

A

generally a little greater than unity, and depending on the soft iron. In these equations the sign+indicates an easterly, —a westerly deviation of tho north point of the compass.

If tho coefficients are so small that their squares and products may be

in which it will be observed that the coefficients are now expressed in are, the Roman letters boing nearly the arcs of which the German letters are the sines. When the deviations do not exceed 20°, this equation is sufficiently exact.

As the subject with which we are now dealing cannot bo understood or followed without distinctly apprehending the meaning of the several parts of this expression, we do not apologise for pausing to explain them.

The term A is what is called the " constant part of the deviation." A real value of A can only bo caused by soft iron unsymmetrkally arranged with reference to the compass.

It will easily be seen that such an arrangement of horizontal soft iron rods, such as that in figure 1,

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would give a positive value of A, and no other term in the deviation.

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