282 ON A CHART AND DIAGRAM FOR FACILITATING GREAT-CIRCLE SAILING. 



From the ship's place on the chart draw a straight line to touch the latitude circle of 55°, 

 this it does at 0. From Melbourne draw a straight line to touch the same parallel at 0. 

 These two straight lines and the part of the intercepted parallel from to ® constitute the 

 track. The portion from to being due East. 



The lengths of the two parts from the ship to and from ® to Melbourne may be deter- 

 mined as before by the diagram. The portion of the parallel from to © is very simply 

 obtained by the usual rules for parallel sailing. 



The total distance will be found 7003 miles, 



being 400 miles longer than the direct great-circle track, 

 and 1130 miles shorter than the Rhumb track. 



The difference between the first course from the ship to and the course by the Rhumb 

 would be 35". 14', or nearly 3^ points. 



Construction of the Course and Latitude Curves. 



It will not be difficult now to understand how the curves are traced in the diagram. At 

 the highest latitude of a great-circle track, the course is evidently due East or West, and as we 

 move along the track away from the highest latitude, we shall find the course altering continu- 

 ously as the distance alters, the connexion between the course and the distance being determined 

 by the solution of a right-angled spherical triangle. 



If X be the given highest latitude, 



d the distance in nautical miles from the highest latitude, 



6 the course or angle made by the track with the meridian at that distance, 



we have sin— = cot0cot\ . . (1), 



which determines the distance of the point where the course is 9 ; and the curve for the course 

 6 in the diagram must pass through that point where the horizontal line corresponding to the 

 highest latitude X is met by the vertical line at distance d. 



If we thus determine the distance d for the course 9 on each horizontal line we have a series 

 of points through which the curve may be drawn. 



The latitude quadrants are traced in a similar manner : — 

 If X be the highest latitude of a great-circle track, 

 X' ... any other lower latitude on this track, 

 n ... the number of nautical miles between them, 



, . n 



we find sm X = sm X cos — , 



oO 



n . , , . 



.•. cos — =smX .cosecX (2)* 



60 ■ ' 



which determines n, the distance at which the latitude curve X' crosses the horizontal line X. 



H, G. 



