XX111 



is, according to (2) above, that m is the position to which M is brought 

 translation (-P, Q, -K) and strain* with rotation, represented by 

 the matrix 



(q, #), (q, ?/), (<, 2), (11) 



(r, oo), (r, y), (r, z). 



Take any plane section (large or small circle) of the spherical surface. 

 The corresponding line on the ellipsoid is also a plane section, but gene- 

 rally in a different plane from the other. For example, let the ship 

 revolve round a vertical axis OZ ; in other words, relatively to the ship 

 let ON revolve round OZ in a cone whose semi- vertical angle is 0, the 

 dip. The locus of N is a horizontal circle whose radius is H, the hori- 

 zontal component of the earth's magnetic force. The corresponding 

 locus of m is an ellipse, not generally in the plane perpendicular to OZ 

 that is to say, not generally horizontal. This ellipse and that circle are 

 Smith's " Ellipse and Circle " (Admiralty Manual, 3rd edition, 1869, 

 App. 2, page 168). The projection of the ellipse on the plane of the 

 circle is the dygogram of what is wanted for the practical problem, namely 

 the horizontal component of the ship's force. 



By a curious and interesting construction (Admiralty Manual, page 175) 

 Smith showed that, when 21 and < are zero, the ellipse and circle are 

 susceptible of a remarkable modification, by which, instead of them, an 

 altered circle and another circle (generally smaller) are found, with a 

 perfectly simple law of corresponding points, to give, in accordance with 

 the general rule stated above, the magnitude and direction of the resul- 

 tant of horizontal force on the ship's compass. But in point of fact the 

 comparison with Dygogram No. I., by which (pages 168, 169) Smith in- 

 troduced Dygogram No. II., taken along with his previous mechanical 

 construction of Dygogram No. I. (pages 166, 167), proves that Dygogram 

 No. II., simplified to two circles, is not confined to cases in which & and 

 ( vanish, and so gives to this beautiful construction a greatly enhanced 

 theoretical interest. It is to be also remarked that, although the necessity 

 for supposing <E and ( zero has been hitherto of little practical moment, 

 as their values are very small for ordinary positions of the compass in all 

 >or nearly all ships at present in existence, the greatly increased quantity 

 of iron in the new turret ships, and its unsymmetrical disposition in the 

 newest projected type (the ' Inflexible '), may be expected to give unpre- 

 cedentedly great values to ( and 21. The happy artifice by which Smith 

 found two circles to serve for the " ellipse and circle " consisted in alter- 

 ing the radius of the first circle from H to XH. If, further, we alter it 



* This strain must include reflexion in a plane mirror so as not to exclude negative 

 values exceeding certain limits in the constituents of the matrix. It is to be borne in 

 mind that, imaginary values of the elements being excluded, strain and reflexion can 

 only alter spheres or ellipsoids to spheres or ellipsoids, riot to hyperboloids. 



