Translated from the Portuguese by Commander L. H. Chandler, U. S. Navy.
[TRANSLATOR'S NOTE.I have ventured to make this translation and to present it to the members of the Naval Institute, because it is the only full description I have been able to find of this method of compensation, a method which certainly possesses sufficient advantages to repay the study necessary to enable it to be used when circumstances prevent a reliance upon our ordinary methods. In reading it must always be remembered that the author is describing the methods employed in the use of Lord Kelvin's compass, binnacle and accessories, which methods would necessarily be somewhat modified with our apparatus, although our binnacle, with its magnet tray or racks, affords excellent facilities for the use of the author's special methods. This translation is published with the permission of the author, Lieutenant Radler de Aquino, Brazilian Navy, who also kindly consented to revise the manuscript before it was sent to the printer.]
I. Why Employ Special Deflectors?
"The deflector is an instrument by means of which the deviation of the compass may be determined and the compass accurately compensated at any time, independently of the sun and regardless of overcast skies, fog or darkness. It makes use of the principle that, by means of a magnet which attracts or repels the north end of the compass needle, the exact strength of the proper correctors may be determined." (Gareis: "Uber Deflectoren," Mittheilungen aus dem Gebiete des Seewesens, Pola, Vol. XXV, No. 1, 1897, p. 10.)
In the year 1846, an English general, Sir Edward Sabine, in a little pamphlet explained the use of his "deflecting apparatus," showing the possibility of determining the directive force of the compass needle on any heading of the ship, and also how to find, independently of observed bearings, the values of the two terms which when combined represent the semicircular deviation.
In 1862, Lieutenant Raphael, of the French Navy, again took up the problem, but without practical results.
Still later, the French admiral, Ernest Fournier, then a lieutenant, constructed an apparatus similar to that of Sabine, which he called an "alidade of deviation," and demonstrated the possibility of obtaining by its use the separate values of B' and C', the exact coefficients of the terms involving semicircular deviation, and also the exact coefficient, D', of the term involving the major part of the quadrantal deviation.
In 1878, Lord Kelvin, then Sir William Thomson, presented an explanation of the principle and use of his "adjustable deflector," by means of which it is possible to effect the horizontal compensation of the compass, and, if desired, to determine the deviation.
Since that date, various instruments called "deflectors" have appeared, all based on the same theoretical principles. The inventors of these were Lieutenant Collongue, of the Russian Navy; Lieutenant Clausen, of the Danish Navy; A. Gareis, the wellknown director of the hydrographic department of Austria; and Heinrich Florian, the eminent nautical expert of the Adria Society of Steam Navigation.
In foggy weather, at sea or in port, it was formerly impossible to regulate the compass, until Poisson's theory, as modified by Archibald Smith, demonstrated that the deviation on each heading depended upon five coefficients, which coefficients could not only be obtained from the values of observed deviations, but also directly from the directive forces which control the compass on the different headings.
In a previous paper (by the author of this one), Poisson's theory was presented, as aptly modified by Smith, and his formulae were expressed as functions of certain parameters expressing the effect upon the compass of iron on board ship, which gave for the deviation formula:
tan? = A’+B’sinz+C’cosz+D’sin2z+E’cos2z / 1+B’coszC’cosz+D’cos2zE’sin2z
and also
sin? = A’cos? + Bsinz’ + C’cosz’ + D’sin(2z’+?) + E’cos(2z’+?)
in which
?A’ = db/2
?B’H = P+cZ
?C’H = Q+fZ
?D’ = ae/2
?E’ = d+b/2
? = 1 + a+e/2
The above formula show that the deviation of a compass is a function of the exact coefficients A', B', C', D' and E' (in themselves functions of ?), and that it is extinguished when they disappear from the equation.
Before passing on from this final result, let us turn back to the expressions for the forces which deflect the compass from the plane of the magnetic meridian, which are
H’cosz’ = ?H(cosz+A’sinz+B’+D’coszE’sinz)
H’sinz’ = ?H(sinz+A’cosz+C’+D’sinz+E’cosz)
in accordance with the accepted notation.
H' cos z' represents that component of the total directive force acting on the needle, H', which acts to draw the north point of the needle to the bow, while H' sin z' represents the component of the same total force which acts to starboard.
As z=z'+? (that is, the magnetic heading, z, differs from the compass heading, z', by the deviation, ?), we have
H’cosz’ = ?H[cos(z’+?)+A’sin(z’+?)+B’+D’cos(z’+?)E’sin(z’+?)]
H’sinz’ = ?H[sin(z’+?)+A’cos(z’+?)+C’+D’sin(z’+?)+E’cos(z’+?)]
As we know, the coefficients A' and E' in the above formula are the result of unsymmetrical soft iron represented by the paramaters b and d. In actual practice, as the standard compass is placed in the plane of symmetry, these two coefficients can be considered as nonexistent.
The formula then reduce to
H’cosz’ = ?H(1+D’)cos(z’+?) + ?HB’ to head
H’sinz’ = ?H(1D’)sin(z'+?) – ?HC’ to starboard
The form in which D' and B' appear in the above equations proves that the parameters a, e, c and P act to deflect the needle to head; and in which D and C appear, that a, e and Q (for f=0, from symmetry) act to deflect it to starboard. The general formula for the deviation also reduces to
tan? = (B’sinz + C’cosz + D’sin2z) / (1 + B’cosz – C’sinz + D’cos2z)
and
sin? = B’sinz’ + C’cosz’ + D’sin(2z’+?)
The relation between the magnetic and compass headings, as
taken from the above formula, is
tanz’ = (1D’)sinz–C’/(1+D’)cosz+B’
THEOREM.—When the horizontal magnetic forces on four equidistant headings are equal, the coefficients B', C' and D' will be zero, and there will be no deviation.'
For greater simplicity, in practice we take as the equidistant headings the four cardinal points, N., E., S. and W., for which we have z'=0°, 90°, 180° and 270° respectively.
When the heading is N. or S. by compass, we have from formula (13)
H_{n}’ = ?H(1+D’)cos?_{n} + ?HB’ to head
H_{s}’ = ?H(1+D’)cos?_{s} – ?HB’ to head
When the heading is E. or W. by compass, we have from formula (14)
H_{e}’ = ?H(1D’)cos?_{e }– ?HC’ to starboard
H_{w}’ = ?H(1D’)cos?_{w} – ?HC’ to starboard
Owing to the symmetrical disposition of the iron on board ship:
?_{n} = ?_{s}
?_{e} = ?_{w}
If we suppose ? to be not greater than one point (11° 15’) we may consider cos?=1 (actually cos11° 15’ = 0.98), and can add and get
H_{n}’+H_{s}’+H_{e}’+H_{w}’ = 4?H
From the same formulae we have
(H_{n}’H_{s}’) / ½(H_{n}’+H_{s}’+H_{e}’+H_{w}’) = 2?HB’/2?H = B’
(H_{w}’H_{e}’) / ½(H_{n}’+H_{s}’+H_{e}’+H_{w}’) = 2?HC’/2?H = C’
(H_{n}’+H_{s}’) – (H_{e}’+H_{w}’) / (H_{n}’+H_{s}’+H_{e}’+H_{w}’) = 4?HD’/4?H = D’
The approximate formula for the deviation is
? = Bsinz’ + Ccosz’ + Dsin2z’
in which
B=B’(1D’/2)
C=C’(1+D’/2)
D=D’
Admitting the hypothesis that
H_{n}’ = H_{s}’ = H_{e}’ = H_{w}’
we see clearly that
B’ = C’ = D’ = 0
which is what we sought to prove.
The above demonstration is all that is necessary to enable us to clearly understand the methods of compensation and for determining the exact and approximate coefficients (and from them the deviations) by the use of the instrument for measuring the horizontal forces, that is to say, by the employment of the "deflector."
The introduction of the new practical methods explained later renders unnecessary the use of the special deflectors with the Thomson compass and others, and facilitates greatly all of the operations necessary for the solution of this important problem.
II. Deflectors in General.
As says Gareis: "The deflector, in principle, is a magnet, or magnets, by the use of which the compass may be deflected while the ship is on the various headings, from the resulting observations of which process we can measure the directive force on each heading," which is the desired result.
Diagrammatically, from O, the center of the compass rose (Fig. 1), the line ON is drawn in the direction of the directive force of the needle, the distance ON representing, on some convenient scale, the horizontal force of the earth, H. From O, draw OA in the direction of the magnetic axis of the deflector, and lay off the distance OM=K to scale, equal to its magnetic moment.
Under the influence of the deflector the compass needle will be deflected from the meridian, ON, to a new direction, say ON', forming an angle ? with its original direction ON.
Let us express the angle N'OA by the character a. Then we see from the figure that
Ksina = Hsin?
When the magnetic axis of the deflector is perpendicular to the direction of the deflected needle, ON', then the angle a=90°, and sin a=1; and in this case from (33) the magnetic force of the deflector will be proportional to the sine of the deflection which it causes. Therefore such a procedure is called the method of sines, which method was first elucidated by the eminent French physicist Pouillet, as a means for measuring the force of electric currents by means of the sine galvanometer.
If a=?=90°, we have K=H, in which case the horizontal magnetic force of the earth is equal to the force of the deflector.
When the magnetic axis of the deflector is perpendicular to the original direction of the needle, ON, we will have a+?=90°, whence K=H tan ?, in which case the force of the deflector will be proportional to the tangent of the deflection which it causes. This method is called the tangent method, and is the basic principle of the tangent galvanometer employed in measuring electric currents.
If ?=45°, we will have K=H, which is to say that the magnetic force of the deflector is equal to the horizontal force which attracts the needle.
Lord Kelvin's deflector, the description of which and the rules for the use of which follow, and that of Clausen (his first instrument) employ the first principle, that of sines. On the contrary, those of Gareis, Clausen (his later instrument) and Florian use a special method which is a modification of the tangent method, and which is due to Lieutenant Clausen, which principle, because of its superiority over the method of sines, also serves as the basis of the new practical method to be described later in this paper.
Any deflector can be used with any of the methods explained, but it is thought that Clausen's method is the best.
III. Lord Kelvin's Deflector.
Compensation and Determination of the Coefficients.
Description and Manipulation.As clearly shown in the plate,' the new model of Lord Kelvin's deflector (which differs only slightly from the old one) is composed of a frame to which are attached two pairs of magnets, a, a' b and b' (b' not visible in the plate). Each pair stands in the form of an inverted V, and is so arranged that it is possible to increase or decrease the angular opening between the two legs of the pair by turning the screwed spindle dcc', which is also secured to the frame. The horizontal axis, by swinging which we rotate the frame, is mortised into the vertical columns.
The angular opening between the two pairs of magnets (which are crossconnected), is recorded on a scale, e, graduated in millimeters. The decimals of a millimeter are recorded on the head of the screw shaft which opens and closes the magnets. When the angle of opening between the magnets is zero, the scale also reads zero.
The whole apparatus is mounted upon a horizontal metal frame similar to the alidade of a prismatic azimuth instrument, and is placed in a similar manner upon the top of the compass bowl, being centered by two points on the under side at the righthand end of the spring which is shown on the plate at the lefthand end of the instrument. The pointer, f, indicates the direction of the magnetic axis of the deflector.
When the reading of the scale is zero, the red ends of a and a' are very close to the blue ends of b and b', respectively, and the magnetic moment of the system will be very small, as it is a function of the distance between these opposite poles. With deflector No. 270, aboard the cruiser Barroso, the deflector, when the scale reads zero, produces only 4.2° deflection of the standard compass (a Kelvin compass).
As the screw is turned, the poles of the magnets separate, and the magnetic moment of the system is increased in proportion to the increase in the reading of the scale.
The same deflector, No. 270, with the same compass, aboard the Barroso, at Rio de Janeiro, on October. 2, 1900, for a scale reading of 16.5, produced a force equal to the horizontal force of the earth at that place; viz., 0.250 C. G. S.
The method of using Lord Kelvin's deflector is simple. Suppose that we are to do the work both on share and afloat, in the same locality, and that the horizontal force on board which attracts the needle is H'. By the method of sines we see that it is possible to measure this force by means of the deflector by creating with it a deflection, called the "normal deflection," in which the angle ? created is 90°, when the "angle of position," or a, is also 90°. In this case K=H'. Then the deflector will give immediately the value of H', by setting the deflector to the reading of the scale corresponding to the magnetic moment of the instrument necessary to produce a force equal and opposite to H', by which the equilibrium of the needle is established at 90° from its original position.
In practice it is advisable to make ?=85° and a=78° 45', in view of the fact that the directive force will be zero when a=?=90°, on the assumption that the original heading is N., or 0°, and that we desire to know the reading of the deflector corresponding to the force H_{n}', which attracts the north end of the compass needle. Turn the screw of the deflector to separate or bring together the magnets, and swing the deflector in azimuth, to the right, let us suppose, until we deflect the compass 85° (that is, until it shows a heading of the ship of N. 85° W.), with the pointer of the deflector over the reading of N. 78° 45' E. of the deflected card. Proceed in a similar manner on each of the other headings. If the reading of the deflector scale be 10.5, we know at once that:
K=some function of H' proportional to 10.5. (34)
To quickly establish the equilibrium between the horizontal magnetic force of the deflector and the horizontal terrestrial force, it is necessary that the compass card should not swing rapidly when the deflector is moved. It is best to start with the pointer of the deflector over the north point of the compass, in which position no deflection is produced. Then swing it quickly to the position over N. 78° 45' E. of the rose. Practice is necessary in order to enable the operator to establish the equilibrium of the card without loss of time. To place the deflector on top of or remove it from the compass bowl, it should be raised or lowered vertically, with its magnetic axis parallel to the compass needle, in order not to disturb the card and set up oscillations.
To effect the horizontal compensation by means of this instrument and to determine the approximate coefficients of the equation of deviation, it is not necessary to know the absolute values of the horizontal magnetic forces corresponding to the readings of the deflector scale, as the work may be readily done as follows:
Graduation of Deflector (Lord Kelvin's) No. 270. Compass No. 9556.
Carry the deflector and the compass ashore to some place free from local magnetic forces, and where the horizontal magnetic force of the earth is known.
Set up the bowl horizontally, with the lubber's point coincident with the north point of the card. Set the scale of the deflector at zero, and place the instrument in its place on top of the bowl. Then observe the deflection, ?, when the angle of position is 90°+ ?; that is, when the pointer is at E. or W. by the deflected compass. Make the same observations for different readings of the scale (generally for differences of reading of two millimeters will be enough), and also observe accurately the reading of the scale for a deflection of 90°, under which conditions the terrestrial horizontal force is equal to that of the deflector. For readings for which the deflection is greater than 90°, it is necessary to observe also the point of the compass rose over which the pointer rests. For accuracy, these observations should be carried to the end of the scale. Then we may calculate the forces in the customary way by the use of the preceding formulæ.
As an example, we have observations made on October 2, 1900, on Fiscal Island, Rio de Janeiro, as set forth in the following table. If the curve for such a set of observations be plotted, it will point out any errors in one or more individual observations.
Sensibility of the compass 0.1°
Temperature 23°C
Plane of needle below point of suspension of card 0.027m
Reading of the scale  Observed deflections  Force of the deflector 
0  4.2  0.0727 
2  10.9  0.1894 
4  17.8  0.3062 
6  24.9  0.4200 
8  32.6  0.5398 
10  41.1  0.6583 
12  50.8  0.7753 
14  62.2  0.8843 
16  81.1  0.9881 
16.5  90.0  1.0000 
Pointer at  
18  66.5  1.0900 
20  57.0  1.1920 
22  51.0  1.2870 
Proceed to compensate the Thomson compass with this deflector, by the use of the compensating magnets and the softiron spheres, in the following manner:
Put the ship on compass heading north, by the compass that is to be compensated, and make a comparison with another compass, to enable the ship to be steered by the latter during the operation. By means of the deflector, as already explained, deflect the compass normally, to either the right or left hand, to the known graduation of the scale which corresponds to a force equivalent to that which attracts the needle when heading north. This will be t, which we will express for convenience
H_{n}’=t_{n}
Thus the deflector has turned the rose to the primitiveposition of equilibrium, which is north. Remove the deflector vertically, and proceed as follows:
Place the ship on headings E. and S. by the compass that is to be compensated, and take a comparison with the other compass on each to enable these courses also to be steered during the operation. Deflect the compass normally, and write similarly to the above:
H_{e}’=t_{e}
H_{s}’=t_{s}
If H_{s}'=H_{n}', there is no necessity for fore and aft compensation, for, by the formulæ, in this case B'=0. If they be different, set the scale of the deflector to the mean reading, t_{n}+t_{s}/2, and by means of the fore and aft magnets bring the deflection to normal. Place the blue ends of the magnets in the direction towards which it is necessary to draw the north end of the compass needle.
Repeat this operation with the ship heading E. by compass, and then on heading W. and we will have
H_{w}’ = t_{w}
If the readings corresponding to H_{e}' and H_{w}' are the same, there is no need for thwartships compensation, for the formulæ show that in that case C'=0. If they differ, set the scale of the deflector to the mean reading, t_{e}+t_{w}/2, and place the thwartships magnets to obtain the normal deflection. Place the blue ends of the magnets in the direction towards which it is necessary to draw the north end of the compass needle.
Finally, if H_{n}’+H_{s}’=H_{e}’+H_{w}’ (that is, if t_{n}+t_{s}=t_{e}+t_{w}), from the formulae it follows that D'=0, and the quadrantal deviation is already compensated. If this be not true, set the deflector to the scale reading of t_{n}+t_{s}+t_{e}+t_{w}/4 , and, with the ship heading either E. or W., set the spheres to produce the normal deflection when t_{n}+t_{s}>t_{e}+t_{w}, or to remove it when t_{n}+t_{s}<t_{e}+t_{w}.
When the compensation is carefully made by this method, it rarely leaves a residual deviation of over
We now present two examples to show the practical compensation and the determination of the approximate coefficients by the use of Lord Kelvin's deflector.
1. For the four headings indicated in the preceding work, H_{n}’=36.5, H_{s}’=15.2, H_{e}'=32.7, H_{w}'=16.9:
H_{n}’ = 36.5  H_{e}’ = 32.7 
H_{s}’ = 15.2  H_{w}’ = 16.9 
Sum = 51.7  Sum = 49.6 
Mean = 25.8  Mean = 24.8 
Head south and set the deflector to 25.8, and insert fore and aft magnets until the normal deflection is obtained.
Head west and set the deflector to 24.8, and insert thwartships magnets until the normal deflection is obtained.
As the difference of the readings on headings N. and S. and on E. and W. exceeded 10 divisions of the scale, which, by virtue of the method of construction of the scale, is beyond the limits admissible under these conditions, take a new set of observations, as follows:
H_{n}’ = 25.2  H_{e}’ = 27.0 
H_{s}’ = 26.6  H_{w}’ = 25.6 
Sum = 51.8  Sum = 51.6 
Mean = 25.9  Mean = 25.8 
The magnets are then arranged to suit these new conditions, in the same manner as before.
As the mean of 25.9 and 25.8 comes within the practical limits of accuracy (0.5 being the greatest admissible difference), we conclude that the quadrantal deviation is already compensated and that the spheres are therefore well placed.
2. The exact coefficients may be easily determined by the formulæ given, or by the process indicated in the English Admiralty Manual of 1893, page 101.
Example:
H_{n}’ = 26.5  H_{e}’ = 16.0 

H_{s}’ = 19.6  H_{w}’ = 20.3  
H_{n}’ + H_{s}’ = 46.1  H_{e}’ + H_{w}’ = 36.3  4?H = 82.4 
H_{n}’ – H_{s}’ = 6.9  H_{w}’ – H_{e}’ = 4.3  2?H = 41.2 
B’ = 6.9/41.2 = 0.168  C’ = 4.3/41.2 = 0.104  D’ = 46.136.3/82.4 = 0.119 
B’, C’ and D’ are expressed in circular measure; to express them in degrees (that is, as the approximate coefficients), we have
B = 9.6°, C = 6.0°, D = 6.8°
Eliminating the exact coefficients from the formula given, we will have, for all headings, that H'=?H. There may possibly be a value for A' or A, there may be errors inherent in the instrument, or accidental errors of observation, but they will be removed by the first operation, and will thus disappear from the second.
We have thus eliminated all the forces which produce deviation, and the compass will indicate the correct magnetic course on all headings.
In conclusion, the method of using Lord Kelvin's deflector is based upon the balancing of various forces by means of the scale readings corresponding to the angle of normal deflection; that is, the angle remains constant and the forces vary.
The method of tangents, as modified by Lieutenant Clausen, which will now be explained, is preferable on account of the advantages which it possesses and because of the great ease with which experience shows that it may be used. It is based on the use of a constant magnetic force in the deflector, and by observations of the different angles which result, which give the values of the approximate coefficients at once.
IV. Modified Tangent Method.
New Compensation of Lord Kelvin's Compass.
We have ON in the direction of the magnetic meridian (Fig. 2), where ON=?H. We have OA, the vector which represents the magnitude of the magnetic force of the deflector, which is equal to K, or √2?H, inclined to the magnetic meridian, NOS, at an angle equal to 135° (angle NOA), with the red pole 45° to the left of north. The deflector will thus give two components, —?K, or OS, opposite to ON, and +?H, or OB. Thus a force of ?H' will establish equilibrium with the force ?H which controls the needle. Now, under the single action of the force OB, the needle swings to the direction OB, perpendicular to its original direction; that is, to the magnetic meridian. In other words, it will be deflected through 90° under the influence of OB.
A fundamental advantage of this method is that it keeps the needle from swinging, contrary to what we have seen in the method of sines. The formula will lose its accuracy when we place the deflector at 78° 45' instead of 90°, because then the force of the deflector will not establish equilibrium with the magnetic force which attracts the needle, and will set up oscillations because of the difference between the forces which we use to establish the equilibrium.' If the angle of position be made 78° 45', we will have a small component of the force left which will disturb the needle, instead of exactly nullifying the directive force, whereas with a normal deflection of 90° this would not be the case.
If now we suppose that the directive force is ON'=H', instead of ?H, the deflection will be equal to the angle NOB'=?=90°—a.
From this we conclude immediately that the greater the force the greater the deflection, the difference H'—?H (small in practice) is proportional to tan a, because the direction of this force is perpendicular to the original direction of the compass needle, OB’.
Let us now consider the magnitude of the deflection 90° —a, or the same a as before, to estimate the value of H'. As aforesaid, ?_{n}, ?_{s}, ?_{e}, and ?_{w} are the deflections corresponding to the same headings as before, whence,
tan a_{n} = +?HD’ + ?HB’ = cot ?_{n}
tan a_{s} = +?HD’ – ?HB’ = cot ?_{s}
tan a_{e} = ?HD’ – ?HC’ = cot ?_{e}
tan a_{w} = ?HD’ + ?HC’ = cot ?_{w}
We may practically consider the arcs instead of the tangents, and B, C and D instead of their sines, B', C' and D', and the formula then become:
B = a_{n}a_{s}/2
C = a_{w}a_{e}/2
D = ½[(a_{n}+a_{s}/2) – (a_{e}a_{w}/2)]
in which we use the observed deviations; or
B = ?_{s}?_{n}/2
C = ?_{e}?_{w}/2
D = ½[(?_{e}+?_{w}/2) – (?_{n}+?_{s}/2)]
in which we use the observed deflections and consider ?H equal to unity.
The deflection, a, is positive, or greatest, when the north point of the needle is deflected to the left of the lubber's point of observation; and negative, or least, when it is deflected to the right of the same line.
V. The New Method.
With Lord Kelvin's binnacle, a new method may be employed to compensate the compass and to determine the values of the coefficients, and to determine the deviations from the latter. This method is very simple, easy of manipulation, and involves little labor.
1. Draw, on the interior of the bowl of the compass, lubber's points corresponding to the thwartships plane through the center of the compass. These lines will be 90° distant from those already existing in the fore and aft plane. These can be located by any of the methods of elementary geometry.
2. Place two boards inside the binnacle on which to place two magnets at right angles to each other, for use instead of the deflector, by means of which we may deflect the compass according to the second of the two methods already described, by moving them toward or away from the compass. As before, suppose the ship to be heading north by compass'. Place the magnets on the board under the compass needle (in the horizontal plane), in the NW.SE., or the NE.SW. plane, as indicated in the figure (Fig. 3, Pos. 1), selecting, say, NW.SE. by the compass when in its position of equilibrium under the action of earth, ship and the compensators already in place in the binnacle.
The magnetic force of these magnets, which serve instead of a deflector, will be practically constant during a sufficient length of time to enable the magnets to be so placed as to exert a force equivalent to √2?H, but not to make the compensation or to verify the accuracy of their place of lodgement. It is enough to be able to replace them in the same place. The position is found by means of observations ashore, about as already shown, and the rest will be merely the ordinary observations of deviation and the simple calculations for determining the coefficients B, C and D necessary for compensation.
Beyond the practical advantage above assigned, it is important to complete the mechanical stability of the compass. Almost all the other deflectors require the manipulation of the deflector on top of the compass bowl, and therefore demand the exercise of great care not to disturb the needle mechanically. By this new method the bowl is unencumbered, having nothing on top of it.
Finally, it is practicable to so vary the positions of the two magnets that we employ as deflectors in relation to the binnacle, that we can cause any variation in the position of their magnetic axis that we require.
Certainly the advantages of this new method will be appreciated by those who know from practical experience how much labor is involved and how difficult it is to acquire the manual dexterity necessary for a successful use of the other deflectors, and especially that of Lord Kelvin with his compass.
This new method is easily capable of application to our type of binnacle (Brazilian Navy, Lord Kelvin's patent compass).
VI. Compensation of the Compass by the New Method.
Three operations are necessary to effect the compensation and to verify its exactness.
1. The determination of the normal installation of the magnets which are to be used in place of the deflector.
2. The calculation of the coefficients B, C and D, and the compensation of the forces which they represent.
3. New observations of the deviations, and new calculations of the coefficients to verify the indications of the compass.
VII. The Normal Installation of the Magnets.
The final force which acts to attract the north end of the needle on all headings is ?H, called the mean force to north.
If we place the compass ashore in a place free from local magnetic forces, and, by the method described for the Kelvin deflector, determine the position of the deflecting magnets necessary to create a deflecting force equal to ?H, we shall have found the position for them corresponding to the normal installation on board, from which position we may determine ?.
Replace the binnacle aboard ship, and, admitting the hypothesis that we do not have deviation resulting from the deflector magnets when in their normal installation, there would be in the compass a normal deflection of 90°.
If when we consider the normal installation corresponding to the force ?H we also consider the corresponding horizontal terrestrial force in the place of H, we can determine the parameters a and e, which enter into the formation of the most important coefficients of the theory and practice of the magnetism of ships; that is to say,
? = 1 + a+e/2 , ?D’ = ae/2
1+a = ?(1+D’), 1+e = ?(1D’)
These reduce to
H_{n}’ = H?(1+D’) + ?HB’
H_{8}’ = H?(1+D’) – ?HB’
And
H_{e}’ = H?(1D’) – ?HC’
H_{w}’ = H?(1D’) + ?HC’
By virtue of the above expressions for ? and D’, we have
H_{n}’ = H(1+a) + ?HB’
H_{8} = H(1+a) – ?HB’
And
H_{e}’ = H(1+e) – ?HC’
H_{w}’ = H(1+e) + ?HC’
From which
H_{n}’ – H = tan a_{n}
H_{8}’ – H = tan a_{8}
H_{e}’ – H = tan a_{e}
Which become
tan a_{n} = aH + ?HB’
tan a_{s} = aH – ?HB’
tan a_{e} = eH – ?HC’
tan a_{w} = eH + ?HC’
whence, supposing H=1
?B’ = ½(tan a_{n} – tan a_{s})
a = ½(tan a_{n} + tan a_{s})
?C’ = ½(tan a_{w} – tan a_{e})
e = ½(tan a_{w} + tan a_{e})
As before, the arcs may be substituted for the tangents of a_{n}, etc. Then ?B', a, ?C', and e will be expressed in degrees. However, the value of ? will be better determined by the foregoing formula.
Example:
The normal installation of the deflector corresponding to the horizontal terrestrial force H, gives the following deflections:
a_{n} = 6.7°  a_{e} = 18.2° 
a_{s} = 9.9°  a_{w} = 8.4° 
tan a_{n} = +0.117  tan a_{e} = 0.329 
tan a_{s} = 0.175  tan a_{w} = 0.148 
?B’ = +0.146  ?C’ = +0.091 
a = 0.029  e = 0.239 
? = 0.866
B’ = 0.165, C’ = 0.103, D’ = 0.119
B = 9.5°, C = 5.9°, D = 6.8°
As this case is almost never used, except to determine A and D', proceed in the following manner:
If the ship be under way (which is the practical and most convenient case), head north by compass, compare the course with that shown by another compass, and steer by the latter. Place the deflector magnet NW.SE., with the blue end to starboard (Fig. 3, Pos. 1), and when the deflection produced is 90° note its location.
Withdraw the deflector, by so doing restoring the rose to its former position (that is, with the ship heading north by it), placing the magnet at such a distance that it exerts no influence upon the compass.
Then head south. Place the deflector again in the same place (Fig. 3, Pos. 3), with the blue end to port. Observe the deflection.
If it be 90°, then the force B' and the coefficient B will be zero, and there will be no need of altering the compensation as already made with the fore and aft magnets. If there be a difference, take the half, and raise or lower the deflector magnet to obtain a deflection equal to that half. The magnetic force of the deflector is then practically equal to √2?H. If D'=0, it will be equal. Note the number of the place, and that will be the place in which the magnet was normally installed during the operation.
VIII. Calculation of the Coefficients and their Immediate Compensation.
With the deflector normally installed, and with the ship heading south by compass, observe the port lubber's point for the deflector. In the same manner observe the deflection on headings west and north.
If the deflection on N. is the same as on S., the force B' and the coefficient B will be zero, and there is no need of altering the position of the fore and aft magnets, which are already in place, or of placing new ones in the binnacle. If they be not equal, take their algebraic mean, and remove half of it by the fore and aft magnets, until the needle shows a deflection equal to that half. Then the force and the coefficient aforesaid will be eliminated.
Then observe the deflection on E., and proceed in the same manner on E. and W. to compensate for C', using the thwartships magnets.
The halfdifference of the deflections observed on N. and S. represents the coefficient B.
The halfdifference of the deflections observed on E. and W. represents the coefficient C.
When the halfsums of these deflections are equal, the force D' and the coefficient D will be zero.
When the halfsums of these deflections differ, half their difference will be the halfsum of the deviations due to D. Correct half of this by the spheres, as when compensating by means of the Kelvin deflector, while on heading E., or place the sphere according to the tables for the correction of the quadrantal deviation; that is, at a distance necessary to correct a quadrantal deviation equal to D.
As is known, the force B'=cZ+P, which varies with the geographical position of the ship. Compensate this by means of the fore and aft magnets. The separate parts, c and P, are the result: (1) c, of the vertical, symmetrical soft iron; and (2) P, of the horizontal hard iron, which, under certain conditions, makes compensation by the fore and aft magnets alone possible.
When such compensation is not possible, install a Flinder's bar (compensation of c) in such a manner as to compensate half of B (effect of B'). The other half should then be compensated by the fore and aft magnets. By this the spheres and the side corrector magnets may be accurately installed. This is the most probable compensation.
IX. New Observations of the Deviations, and New Calculations of the Coefficients for Verification.
Owing to the inevitable errors of observation, prudence always demands that all results shall be verified. To do this repeat the preceding operations, chiefly using the approximate methods. In practice, however, if we have confidence in our own work, the residuals will be slight and may be neglected.
Practical example:
1. Normal installation of the deflector magnet.—Place the ship on heading N. (Fig. 3, Pos. I), and place the magnetic axis of the deflector magnet in the NW.SE. line, with the blue end to starboard, in the place corresponding to a deflection of 90°. Withdraw the magnet, and bring the rose to its original position.
2. Swing the ship to either starboard or port, and bring her to head S. (Fig. 3, Pos. 3). Place the deflector magnet in the same plane, with the blue end to port, in a position which always makes an angle of 45° to the left of the N.S. line of the needle. Observe the deflection. Suppose it to be 20.8° (to the right of the port lubber's point) ; that is, with the heading by the compass N. 20.8° W. Half of this is 10.4°; so move the deflector magnet as described before until the rose indicates a heading of N. 10.4° W. This will be the normal installation.
Repeat the operation on heading W. (Fig. 3, Pos. 7). Place the deflector magnet with the blue end to port, in the position of normal installation (this corresponds to an equal deflection of the rose). Observe the deflection from the compass heading. Suppose this to be o.°; that is, the heading by compass being N. 0.4° E. Then head N., and install the deflector magnet normally as before. Observe the deflection from the starboard lubber's point. Suppose it to be 11.0°; that is, with the rose showing a heading of N. 11° E.:
a_{n}+a_{s}/2 = 1110.4/2 = +0.3°
B = a_{n}a_{s}/2 = 11+10.4/2 = +10.7°
Compensate B by the means of the fore and aft magnets, it being the mean that the rose indicated, N. 0.3° E., from the starboard lubber's point.
Head finally on E. (Fig. 3, Pos. 5), and observe the deviation. Then place the deflector magnet in the position of normal installation, with the blue end to starboard.
It will be —10.4° deflection; that is, the rose will indicate a heading of N. 10.4° W. by the forward lubber's point.
a_{w}+a_{e}/2 = 0.410.4/2 = 5°
C = a_{w}a_{e}/2 = 0.4+10.4/2 = +5.4°
By means of the thwartships magnets compensate for this value of C, in such a way that the rose indicates N. 5° W. by the forward lubber's point.
As the difference between a_{n}—a_{s} and a_{w}—a_{e} is greater than 10°, proceed with a new compensation. This will rarely be necessary in practice.
For it we will have:
A_{n}=+0.7° and a_{s}=+0.6°
and
a_{e} = 2.4°
a_{w} = 14.8°
½(a_{w}+a_{e}) = 8.6°
½(a_{w}+a_{e}) = 6.2°
Readjust the thwartships magnets on heading E., until the rose indicates N. 8.6° by the forward lubber's point.
The compensation of the quadrantal deviation is made as follows:
½[(a_{n}+a_{s}/2) + (a_{e}+a_{w}/2)] = ½(+0.658.6) = 3.98°
D = ½[(a_{n}+a_{s}/2) – (a_{e}+a_{w}/2) = ½(+0.65+8.6) = 4.63°
While on the same heading, E., move the spheres until the rose indicates N. 3.98° W., by the forward lubber's point.
Finally determine the residual deviations on the cardinal and semicardinal points, which are:
?_{n} = 0.8°  ?_{s} = 0.5° 
?_{ne} = 1.4°  ?_{sw} = 0.3° 
?_{e} = 1.5°  ?_{w} = 0.2° 
?_{se} = 0.0°  ?_{nw} = 0.0° 
It is significant that the results of compensation by this method, without the knowledge given by azimuth observations, are comparable with, and at times will be better than, those dependent upon such observations and calculations based on observed azimuths and subject to errors in the calculations upon which they depend.
The advantages of this method of compensation are obvious, for it makes compensation possible in an hour, before leaving port, and gives results actually not more erroneous than those taken by other methods under doubtful conditions.
In many cases circumstances will not permit the loss of time involved in compensation by the method of observed azimuths and the attendant preliminary operations, and in such cases it is of great advantage to know that this simple method is available.
X. Compensation and Regulation of the Compass without Azimuths in Accordance with the Ideas Set Forth.
In Lord Kelvin's compass bowl there are drawn the lubber's points referred to above as necessary (that is, four, at 90° apart: bow, stern, starboard and port). There are also places provided for placing the deflector magnets near the compass, parallel to those provided for the compensating magnets.
In addition to the magnets necessary for compensation, two more are provided," one called the "deflector magnet," and the other the "director magnet." Each of these is marked to show its purpose, by a label pasted on it reading "deflector " or "director."
If both of these magnets have the same magnetic moment, either may be used for either purpose indiscriminately. Ordinarily this is impossible of accomplishment, however, and it will be necessary to distinguish between them.
The deflector magnet produces a force of —?H (see Fig. 2), and the director magnet a force which, when parallel to the line N.S., causes a deflection of the needle to the position OB.
As it is only necessary to use the magnet which enters by the starboard door, the port door should be shut to guard against mistakes.
Proceed as before, placing the ship on headings N. and S., to determine the normal installation of the deflector magnet.
On heading N. (Fig. 3, Pos. 1), the director is placed in athwartships position (the closer to the needle the greater the effect), with the blue end to starboard. Place the deflector fore and aft with the blue end aft, and at such a distance that the rose will have the normal deflection of 90°.
Now head S. (Fig. 3, Pos. 4); reverse the positions of the ends of the director and deflector, and observe the deflection. Take the half, and install the deflector normally.
Now head W. (Fig. 3, Pos. 8), placing the director fore and aft, and the deflector thwartships, corresponding to the normal installation, with the blue end to port. Observe the deflection.
Now head N., replacing the director in the original position and the deflector normally. Observe the deflection. Determine the value of the coefficient B and compensate as before.
Head finally on E. (Plate 3, Pos. 6), placing the director fore and aft, with the blue end forward; and the deflector normally, with the blue end to starboard.
Observe the deflection. Determine the value of C and D and compensate as before.
The simplicity of this method is manifest, for the process is very easy. The director magnet is always parallel to the line E.W. of the needle (before deflection), with its blue end to E. The deflector magnet is always parallel to the line N.S. of the needle (before deflection), with the blue end to S. The binnacle is so arranged as to permit the proper placing of the two magnets above the regular positions of the compensating magnets, thus avoiding confusion.
XI. Conclusion.
We here submit to competent criticism these ideas intended to throw light on this important question of the compensation of the compass without the use of azimuths. The indifference of the nautical world to this truly ideal solution of the problem is absolutely unjustifiable.
The two new practical methods described in this paper will, I believe, introduce new possibilities to those who will study them. The first method makes use of a single magnet to produce the two indispensable magnetic forces; the second employs the two perpendicular magnets to produce the same forces. It is possible to employ indifferently the whole process or any of its component parts. Necessarily, with the second method there is more manipulation of the magnets necessary than with the first.
We have in our (Brazilian) navy a large number of Lord Kelvin's deflectors, and it is possible to use these in conjunction with one or two magnets serving as "directors."
It is, however, advisable to employ exclusively the method of tangents as modified by Lieutenant Clausen, in view of the superior facilities which it offers.
The writer's personal experience is authority for this advice, and confirms all the theories that have been advanced.
How great is the contrast between the simplicity of the solution and the complexity of the problem! Honor to them to whom we owe the final solution!
XII. Compensation of the Heeling Error.
We will now show how to accomplish the theoretical compensation of the vertical force which produces deviation and mechanical disturbance of the compass prejudicial to its indications when the ship rolls. The principal part arises from the coefficient J, and is a maximum on headings N. and S. The spheres and the Flinder's bar compensate this deviation in part. There remains the correction for the parameter c and for that part of e which enters into the expression for the heeling deviation. J represents then the permanent magnetism of the parameter R and of the induced magnetism represented by k. We will say that this will be compensated by means of a permanent magnet producing a force —R'=J.
When i=o° (ship horizontal), the vertical forces do not disturb the indications of the needle. When the ship rolls, if the vertical forces are not zero, the rose oscillates from one side to the other.
Place the ship on heading N. or S., and one or more vertical magnets ill suspension, with the red or blue ends upward, according to the circumstances. Raise or lower the magnets until the oscillations are one degree or less (allow for movement of ship's head in azimuth due to the sea). In virtue of the continuity of the roll it is easy to proceed to the verification.
The magnets, finally, will be in the right position when, while steering some determined course, the rose indicates the same heading in spite of the roll.
With a change in magnetic latitude the compensation must be corrected.