XI 



(, 19, (, is that which was introduced by Smith when he first substi- 

 tuted the rigorous formulae for the approximate harmonic formulae 

 which had previously sufficed : it is explained in the Appendix to this 

 notice. 



Dygogram Number II. may be deduced from dygogram Number I. by 

 attaching a piece of paper to the half-speed revolving arm, and letting 

 the tracing-point of the limagon leave its trace also on this paper, which 

 will be a circle, while at the same time the fixed point from which the 

 resultant radius-vector is drawn will trace another circle on the moving 

 paper. The fresh diagram thus obtained consists of two circles. Mark 

 one of these circles (the second in the order of the preceding descrip- 

 tion) with the points of the compass*, like a compass card ; or (better) 

 mark simply degrees all round from North taken as zero ; and mark 

 with degrees counted in reverse direction the other circle, which, for 

 brevity, will be called the auxiliary circle. Mark the ship's compass 

 course on the circumference of the ideal compass card. From this 

 point to the corresponding point on the auxiliary circle draw a straight 

 line. The direction of this line shows by the parallel to it, through the 

 centre of the ideal compass card, the compass course corresponding to 

 any chosen magnetic course. The length of the line, drawn in the 

 manner described, represents the horizontal resultant force of the earth 

 and ship, at the point occupied by the compass needle, in terms of the 

 radius of the ideal compass card, as unity. The writer of the present 

 article believes that this construction will yet prove of very great practical 

 utility, although hitherto it has not come into general usef. Its geo- 

 metrical beauty attracted the notice even of Cayley, who has contributed 

 to the Admiralty Compass Manual a second method of solving, by 

 means of it, one of Smith's compass problems. 



Construction from ship's and earth's magnetic elements. With O as 

 centre and OH equal to "mean force on compass to north " (AH) describe 

 a circle. Make 



NL = &, OB = -B; BC=-C; CD=-$; Dft = -e ; 



with C as centre describe a circle through h. 



The following diagram shows (for an ideal case, as possibly a turret 

 ship of the future, with very large values of the usually small magnetic 

 elements & and () the Dygogram of two circles, modified to suit the 

 Chinese compass (or needle unloaded with compass card, which is un- 

 doubtedly the compass of the future). This modification is also conve- 



* The ancient system of marking 32 points on the compass card, and specifying 

 courses in terms of them, has always been very inconvenient, and is now beginning to 

 be generally perceived to be so. 



f A short demonstration of it, deduced directly from Smith's fundamental formula?, 

 is appended to the present article for the sake of mathematical readers who may not 

 have the Admiralty Compass Manual at hand. 



