Astronomical and Nautical Collections. 419- 



■ We may therefore consider, in the first place, the state of 

 a plane surface rolled round a cone, and inquire into the 

 form assumed by a straight line on that surface. Calling 

 the line a tangent to a circle, of which the centre is applied 

 to the apex of the cone, it is obvious that the secants will be 

 made to correspond with the superficial radii of the cone, 

 the point of contact becoming the vertex of the geodetic line, 

 and the diameter of the circle constituting an asymptote 

 which the curve can never reach at any finite distance 5 as 

 will be obvious on inspection, if we wind a sheet of paper 

 round a funnel, or twist it into a cornet. The angle of the 

 secant with the tangent is obviously equal to the angular 

 distance from the diameter which becomes the asymptote, 

 to which it becomes the alternate angle. Hence as we roll 

 the surface equably round the cone, it is obvious that the 

 inclination of the geodetic line to the superficial radius must 

 also vary equably, the change amounting for instance to a 

 quadrant between the vertex and the asymptote ; a distance 

 which will correspond to the same curvilinear length, on the 

 base of the cone, whether referred to the apex of the cone 

 or to the centre of the base, and will subtend an angle in the 

 base as much greater than from the apex as the oblique side 

 of the cone is longer than the axis, or as the sine of the 

 angle formed by the side with the axis is less than the radius. 

 Hence it follows that the angular change of azimuth of this 

 geodetical line is always as much smaller than the angular 

 increment of the base, as the sine of the inclination of the 

 side to the axis is less than the radius. 



We have, therefore, the general equation d/x = sdr, tx being 

 the azimuth of the geodetical line, z the longitude, and s 

 the sine of the inclination of the surface of the cone touching 

 the given solid to the axis ; or, in other words, the sine of 

 the latitude. 



Hence for any short distance, the difference of longitudes 

 may readily be deduced from the difference of azimuths, by 

 taking the sine of the mean latitude, or that of the latitude 

 of the middle of the arc, for a divisor. 



It may, however, often be easier, and it will be perfectly 

 accurate, to consider only the plane triangle formed by the 



