would give+A to the starboard compass, combined, however, with+E; and -A, combined with -E, to the port compass. The last arrangement is one sometimes found in the relative positions of the horizontal iron spindle of the wheel and the binnacle compasses placed near it. In compasses placed in the midship line of the ship, such unsymmetrical arrangements of soft iron can seldom have any sensible operation. In such cases A is always small; and when it has a sensible value, it seems more likely to arise from index error of the compass, or from error of observation, and may probably be best dealt with as such, and disregarded in the table of deviations.

The terms B sin '+C cos 'make up together what is called the "semicircular deviation." This is the part of the deviation which it is most difficult to deal with, as well from each coefficient being made up of the two parts which we have described, which cannot be distinguished by observations made in one latitude, as from that part of the ship's magnetism, which we have treated as permanent, being in fact only subpermanent. To this we shall have occasion to revert in the sequel. At present we will only point out that +B indicates an attraction of the north point of the compass to the ship's head, -B to the stern,+C an attraction of the north point to the starboard side, -C to the port side.

The terms D sin 24'+E cos 24' make up what is called the "quadrantal" deviation. This can only be caused by horizontal induction in soft iron. E can only be caused by horizontal induction in soft iron unsymmetrically distributed, and is therefore, except in such cases as those represented in fig. 2, very small. +D may be caused by the following arrangements of symmetrically arranged soft iron, in which the ship's head is supposed to be directed towards the top or bottom of the page. -D may be caused by the same arrangements, the ship's head being now supposed to be directed to the right or left of the page.

Fig. 3.

N:1.

N:2.

N: 3.

N:4

N:5.

FFF

Between these various arrangements there is this most important difference, that in No. 1 and No. 4 the directive force of the needle would be increased, while in No. 2 and No. 5 it would be diminished. In No. 3 it

might be either increased, or diminished, or left unaltered, according as the effect of the longitudinal and the transverse iron preponderated. We may, therefore, by observing the effect on the directive force, as well as on the quadrantal deviation, ascertain how much of the latter is caused by fore-andaft iron, how much by transverse iron.

This explanation of the coefficients will probably be sufficient for the purposes of this Report, and we now revert to Part III. of the Manual.' The principal object of this part is to find the means of computing A, B, C, D, E, from the deviations observed or derived by Napier's curve for a certain number (8, 16, or 32) equidistant points. This is easily done by formulæ founded on the method of least squares; and the method is made of ready application by tabular forms and tables given in this part.

The direct computation of the exact coefficients A, B, C, D, E by the method of least squares would be a matter of very great labour; but they are easily derived to terms of the 3rd order inclusive from the approximate coefficients A, B, C, D, E by formule which are given for the first time in this part.

There are two other coefficients, the knowledge of which is of great importance, but which can only be derived from observations of force, viz. X, or the ratio of the mean force to north at the place of the compass to the earth's horizontal force, and μ, the ratio of the mean vertical force at the same place to the earth's vertical force.

One of the most important errors in the modern iron-built and ironplated vessels is the heeling error. The deviations obtained by the usual process of swinging are for a vessel on an even keel. It is found by experience that as the vessel heels to one or other side, the north point of the compass is drawn either to the weather or lee side, generally the former; and the deviation so produced, when the ship's course is near north or south, often exceeds the angle of heel. This not only produces a deviation which may cause a serious error in the ship's course, but if the ship is rolling, and particularly if the period of each roll approximates to the period of oscillation of the compass, produces a swinging of the compass-needle, which may amount to many times the angle of heel, and make the compass for the time useless for steering.

This is a part of the deviation which has been involved in some obscurity. Mr. Airy, in a paper in the Transactions of the Institution of Naval Architects,' vol. i. p. 107 (1860), says that the disturbance produced by heeling has not been well observed, and its correction has not yet been reduced to easy laws; and that the effect of heeling is the only part of the magnetic disturbance in regard to which the practical correction of the compass is really at fault; and the Reports of the Liverpool Compass Committee refer to it as one of the most perplexing parts of the subject. It therefore appeared to us desirable to deduce from Poisson's formulæ, expressions for the alteration of the coefficients introduced by the inclination of the ship. This has been done in the 'Manual,' and the result is, we think, to remove entirely the obscurity which rested on the subject. The effect of the heeling error is, as might have been anticipated, to leave unaltered the coefficients which depend on fore-and-aft action, viz. B and D, to alter C, and to give a value to A and E. The latter appear to be, except when the compass is near either extremity of the vessel, of small amount. The alteration of C is the only one which is important. The formulæ show that it consists of two parts, which are caused by arrangements of iron, such as that in the following figure, in which the vertical line represents iron permanently magnetized,

or vertical iron magnetized by induction, drawing the north end of the needle downwards in the northern hemisphere; the horizontal line a rod, such as that in fig. 3, No. 2, which would give +D, and which, when the ship's head is

Fig. 4.

north or south, will have no effect till the ship heels, when its upper (weather) end will attract the north point of the compass. Each rod in the figure will therefore cause a deviation of the north point of the needle to the weather side. In order to correct this, the vertical magnetism must either act upwards, or the transverse magnetism must be such as would be caused by a horizontal transverse rod on each side of the compass, the formula indicating the relation which must exist between the vertical and the transverse horizontal magnetism in order that the heeling error may be zero.

6

The 4th Part of the Manual' contains charts of the lines of equal variation, equal dip, and equal horizontal force over the globe; the first for the purpose of enabling the navigator at sea to determine the deviation by astronomical observations; the two latter to throw light on the changes which the deviations undergo on a lengthened voyage, and to enable the navigator to anticipate the changes which will take place on a change of geographical position.

Of the Appendices, one (No. 2) contains a short account of the method proposed by Mr. Airy for the mechanical correction of the semicircular and quadrantal deviation, and a notice of a method lately proposed by Mr. Evans for the correction of the quadrantal deviation when excessive. No. 3 is on the mathematical theory of the deviations of the compass, being the deduction from Poisson's equations of such formulæ as may be most conveniently applied to the analysis of the tables of deviations derived from actual obser

vation.

There is a graphical method of representing the magnetic state of a ship as regards deviation, described in pp. 106 and 107, which we may shortly describe.

If

If from the centre of a compass, in any part of the ship, we draw a horizontal line, representing in amount and direction the ship's disturbing force on the north end of the needle of that compass, the ends of all the lines so drawn will, as is shown in this appendix, trace out an ellipse. the soft iron of the ship be symmetrically distributed, so that A and E are zero, the construction of this ellipse is simplified, as its axes are then parallel and perpendicular to the fore-and-aft lines of the ship. The position of the centre of the ellipse gives the amount of the force to head, and force to side, which cause the semicircular deviation. The fore-and-aft and transverse

axes of the ellipse give the amount of the fore-and-aft transverse inductive forces which give rise to the quadrantal deviation. An ellipse so drawn, therefore, gives to the eye, at a glance, the whole magnetic character of the ship as regards deviation on an even keel.

If the mean directive force of the needle is not altered, the ellipse becomes a circle, the coordinates of the centre of which are B and C, and the radius, on the scale in which the mean force to north represents unity. If we have no observations of horizontal force, the circle is all we can draw; it gives all the information to be derived from the ellipse, except the diminution of the directive force. For the complete representation of the deviation and force, it is convenient to have both the circle and the ellipse drawn.

In the diagrams the direction and force of the earth's magnetism as the ship is on different azimuths are represented by the radius of a circle, of which the compass is centre, and which is divided in the reverse order of the compass-card. A line drawn from a point in the circle to the corresponding point in the ellipse or small circle represents, on the common principle of the parallelogram of forces, the direction and amount of the force on the needle*. A modification of this diagram is described at p. 96 of the 'Manual' under the name of "dygogram" (dynamo-gonio-gram), applied to it from its showing the force as well as the angle of deviation of the needle.

The principle of its construction is the following. If we draw a vertical line representing the magnetic meridian, and from a given point in it draw lines representing in length and direction the directive force and direction of the needle for each azimuth of the ship's head, the extremities of such lines will trace out an epicycloid which is very easily constructed by points when the coefficients A, B, C, D, E are determined. The method is applied in plate 2 to the deviations of the standard compass of the Warrior,' and has been applied by us to many other ships, and has been found a most efficient aid in discussing the observed deviationst.

We now come to what we consider the proper subject of this Report, viz., the practical results as to the deviations of the compass which have been deduced from actual observation on board ship; and the works to which we shall principally confine our attention are the following:"Account of Experiments on Iron-built Ships, instituted for the purpose of

* A practical application of the diagram to the correction of the compass was suggested by its being accidentally held to the light and looked at from behind. When this is done, it will be seen that the large circle is divided in the same way as the compass-card. If, then, the radius of the large circle represent the direction of the disturbed compass-needle, the line joining the corresponding points in the large circle and on the ellipse or small circle will represent the direction of the magnetic meridian.

By therefore drawing on an ordinary compass-card a circle of which the coordinates of the centre are -B and +C, and the additional coordinates of the north point -, and dividing the small circle in the reverse order, we get the following rule for the correction of the compass:

"Take the given course on the card, and also on the small circle, and suppose a straight line drawn through these. Then keep the ship's head in the direction of the line, disregarding, of course, the lubber-line."

+ If X be the force to north in terms of the mean force to north, Y the force to east, then X and Y representing rectangular coordinates,

2+2 from the centre of

which is the equation to an epicycloid traced out by a point

a circle whose radius is √B2+C2, and which rolls on a circle of equal size, and the coordinates of the centre of which are X=1, Y=A.

discovering a Correction for the Deviation of the Compass produced by the Iron of the Ship, by G. B. Airy, Esq., Astronomer Royal" (Phil. Trans. 1839, p. 167).

"Discussion of the Observed Deviations of the Compass in several Ships, woodbuilt and iron-built, by G. B. Airy, Esq." (Phil. Trans. 1856, p. 53). "Practical Illustrations of the Necessity for Ascertaining the Deviations of the Compass, &c., by Capt. Edward J. Johnson, R.N., F.R.S., Superintendent of the Compass Department of the Royal Navy." 1st edition, 1848; 2nd edition, 1852.

"Magnetical Investigations by the Rev. W. Scoresby, D.D." 2 vols. 1844-1852. "Journal of a Voyage to Australia and round the World, for Magnetical Research, by the Rev. W. Scoresby, D.D." Lond. 1859.

"First and Second Reports of the Liverpool Compass Committee to the Board of Trade, 1857."

"Third do.,

1861."

"Reduction and Discussion of the Deviation of the Compass observed on board of all the Iron-built Ships, and a Selection of the Wood-built Steam-ships in Her Majesty's Navy, and the Iron Steam-ship Great Eastern,' by F.J. Evans, Master R.N., Superintendent of the Compass Department of H. M. Navy" (Phil. Trans. 1860, p. 337).

The first and most important general result which is derived from all the observations recorded in these works, and from many more which have not been published, is, that the observed deviations are represented by the formula derived from Poisson's theory with a correctness which is within the limits of error of observation.

In saying this, we are in some degree differing from a conclusion which the Reports of the Liverpool Compass Committee draw from observed deviations, viz. that there is a difference in the amount of the quadrantal deviation in different quadrants, depending either on some quality of the iron as regards its capacity for induction in different directions, or on the greater or less time occupied in moving the ship's head over one or other of the quadrants. That some difference may, under certain circumstances, be caused by the latter cause we do not dispute, but we are not satisfied that it is appreciable in the ordinary process of swinging. On the contrary, we believe that, within very small limits of error, Poisson's theory may be considered as exact for the ordinary process of swinging a ship. As regards more lengthened periods, particularly when the ship has been exposed to mechanical violence, the hypothesis no doubt ceases to be exact; but even then the most convenient mode of treating the subject is analogous to that which is familiar in physical astronomy and other mixed sciences, viz. to consider the theory as exact, but the coefficients derived from that theory as being themselves subject to changes to be derived from observations, and reduced or not, as the case may be, to law.

Mr. Airy, in the first paper to which we have referred, describes very careful observations made by him on board of two iron ships, the 'Rainbow' iron-built steamer, and the Ironsides' iron-built sailing-ship. In the first, observations were made at four stations: station 1, near the binnacle, 13 feet 2 in. from the stern; station 2, at a part in which a standard compass would probably be placed, being 31 feet 9 in. from the stern; station 3, 48 feet 3 in. from the stern; station 4, 47 feet from the knight-heads, or 151 feet from the stern. Each compass was raised 4 feet from the deck. In the 'Ironsides' the compass was placed in the position of the binnacle compass.