XXIV 



in magnitude and direction, and make it represent the resultant of XH 

 to north and & to east, thus including part of the ship's force, namely 

 (\ 1)H to north and & to east, along with the earth's horizontal force in 

 one circular dygogram, the residue of the horizontal component of the 

 ship's force has also a circular dygogram. The construction thus obtained 

 is fully described and illustrated by a diagram under the heading " Dy- 

 gogram No. II., above. The proof of this is very simple. The follow- 

 ing is the analytical problem of which it is the solution : In the general 

 equations (2) suppose Z to be constant, and put 



X'-(j>,*)Z-P=X", Y'-( ? ,s)Y-Q=Y" ..... (12) 

 We have 



Y"= 



Now imagine two dygogram curves (ellipses or circles) to be constructed 

 as the locus of points (#, y,), (x 1 , y'} given by the equations 



X 2 +Y 2 =H 2 , 



(14) 



and let it be required to find a, /3, y, 8 so that these two curves may be 

 circles ; we have four equations for these four unknown quantities. 

 Then, as 



ar' + ay=X", y'+y=T", 



the resultant of the radius vectors of the two concentric circles thus ob- 

 tained is the resultant of the constituent (X", Y") of the force on the 

 compass ; and by (12) we have only to shift the centre of one of them to 

 the point whose coordinates are (p, z) Z+P, (q, z) Z-fQ, to find two 

 circles such that the resultant of corresponding radius vectors through the 

 centre of one of them shall be the whole horizontal component of the 

 force on the compass. Thus we have Smith's beautiful and most useful 

 Dygogram of two Circles. W. T., January 1874. 



