traced upon the Surface of the Sphere. 359 



NOTE D. 



IN HYMEES'S Analytical Geometry of Three Dimensions, p. 186, the orthogra- 

 phic projection of the rhumb-line on the plane of the equator is stated to be the hyper- 

 bolic spiral. A moment's consideration will shew that this is a mistake. It is this : 

 The projection of the spherical surface itself upon the plane of the equator is con- 

 Jined to the limits of the circle of the equator ; and hence the orthographic projection 

 of any curve traced upon that spherical surface must also be confined to those same 

 limits : but the hyperbolic spiral has a rectilinear asymptote, and hence this curve is 

 not confined within the limits assigned above. The projection, therefore, whatever it 

 might be, is not the hyperbolic spiral. The mistake was most probably an oversight 

 in writing the name of the spiral, the projection being really one of Cotes's spirals, 

 or at least a case of one of them. 



Dr LAEDNEE also, in his valuable treatise on Algebraic Geometry, has made an 

 oversight concerning the projections of this curve. See p. 478. He says, " If a 

 logarithmic spiral be described upon the plane of a great circle of the sphere, with 

 the centre as a pole, and perpendiculars be drawn from every point in it to meet the 

 surface of the sphere, the extremities of those perpendiculars will trace out a loxo- 

 drome curve, or, in other words, the projection of the loxodrome curve on the plane of 

 the equator is the logarithmic spiral."" 



We have seen that the equation of the stereographic projection of the Rhumb- 

 line is the logarithmic spiral (XXXIII. b, c, &c.) ; and we have seen (XXXIII. d) 

 that the orthographic is defined by a different equation, and therefore is a different 

 curve. The equation of the hyperbolic spiral, too, is v = a 6~ l , which is a very 

 different expression from that we have found in (d). Hence the equation of the 

 hyperbolic spiral does not, as Mr HYMEES inadvertently supposed, coincide with 

 that of the orthographic projection of the loxodrome. These curves are not there- 

 fore identical. It remains to ascertain the equation of the spherical curve formed 

 by the perpendiculars raised upon the points of the logarithmic spiral for compari- 

 son with Dr LAEDNEE'S other statement -that it is the loxodrome. We shall also, 

 en passant, show the converse of (XXXIII. b, c); viz. that the stereographic pro- 

 jection of the logarithmic spiral upon a concentric sphere, from one of the poles of 

 the plane of the spiral, is the loxodrome. 



1st, The stereographic projection of the logarithmic spiral upon the surface of a 

 centric sphere is the loxodrome. 



Let AG = CB = CL = , be the unit of radii-vectores, and CL the origin 

 of polar angles. Join DB cutting CF in V. Denote the log.-spiral by 



00 



z z 2 



