XX11 



2. Dyyograms of Class II. 



Take lengths numerically equal to X, T, Z and X', T', Z' for the co- 

 ordinates of two points. The axes of coordinates being fixed relatively 

 to the ship, conceive the ship to be turned into all positions round a 

 fixed point taken as the origin of coordinates ; or for simplicity imagine 

 the ship to be fixed and the direction of the earth's resultant force to take 

 all positions, its magnitude remaining constant : the point (X, T, Z) will 

 always lie on a spherical surface, [(9) below) ; and the point (X', T', Z') 

 will always lie on an ellipsoid fixed relatively to the ship. Por we have 



X 2 + Y 2 + Z 2 = F, (9) 



where I denotes the earth's resultant force. Now by (2) solved for 

 X, Y, Z, we express these quantities as linear functions of 



X'-P, Y'-Q, Z'-K. 



Substituting these expressions for X, Y, Z, in (9) we obtain a homoge- 

 neous quadratic function of X' P, Y' Q, Z' B,, equated to I 2 , which 

 is the equation of an ellipsoid having P, Q, R, for the coordinates of its 

 centre. 



It is noteworthy that the point (X', Y', Z') is the position into which 

 the point (XYZ) of an elastic solid is brought by a translation (P, Q, E-), 

 compounded with a homogeneous strain and rotation represented by the 

 matrix 



), O, z), ") 



), (<?,*), f (10) 



(?,*), (r,y), l + (r, z). \ 



Instead of drawing at once the dygogram surface for the resultant of 

 the force of earth and ship (X', Y', Z'), draw according to precisely the 

 same rule, the dygogram surfaces for (X, Y, Z), the earth's force, and 

 (X' - X, Y' Y, Z' -Z), the force of the ship. The first of these will be 

 a sphere of radius I. The second will be an ellipsoid having its centre 

 at the point (P, Q, B). Let ON and OM be corresponding radius vectors 

 of these two surfaces. On OM, ON describe a parallelogram MONK. 

 OK is the resultant force of earth and ship at the point occupied by the 

 ship's compass. Vary the construction by taking a " triangle of forces " 

 instead of the parallelogram, thus : Produce MO through O to m, making 

 Om equal to MO ; in other words, draw the dygogram surface represent- 

 ing (X X', Y Y', Z Z') ; and of it let Om be the radius vector cor- 

 responding to OM of the spherical-surface dygogram of the earth's force. 

 Join Nm ; through O draw OK equal and parallel to Nm. OK (the 

 same line as before) is the radius vector of the resultant dygogram sur- 

 face, corresponding to ON of the spherical dygogram. The law of cor- 

 respondence between N on the spherical surface and m on the ellipsoid 



