traced upon the Surface of the Sphere. 42 1 



(c /7 ). At all intermediate positions, the values of 6 will be the same as in 

 the other branch (c,) we had for the supplement of these values. Thus if 

 (j>, be the value of $ at any point of (c,), and (ir 0,) be the value of it in 

 the present case ; then we shall have the same value of 6 adapted to both. 



The cases (d tl e tl fi,g^ corresponding to (d t e,f,g^ respectively, are now 

 too obvious to need further remark. 



4. We may now consider its projections. 



(,). In the gnomonic, we have (putting a = rad), 



r =. a tan <p ; 

 .-. r = UK f , 



which is the logarithmic spiral. 



(b,,,) On the equatorial cylinder, referring it to the equator, </> is changed 

 into its complement in the expression, and we have 



cot(f)' = K i , or tan0' = /C~' 



which, when the cylinder is unrolled upon one of its tangent planes, becomes 

 the common or plane logarithmic curve. This logarithmic lies on the con- 

 trary side of the meridian (which we took as origin of co-ordinates) from that 

 upon which the logarithmic spiral lay as it should do. 



(c /7/ ) The gnomonic projection of this spherical logarithmic, then, is 

 identical (except as to its position) with the stereographic projection of the 

 loxodrome. This relation is very beautiful, and leads us, moreover,, to re- 

 mark a considerable number of curious relations between the two curves. 



(d,,,) This relation may perhaps be more elegantly expressed thus : 



From the pole of the logarithmic as a centre, describe another sphere 

 equal to that upon which the logarithmic is traced ; then the gnomonic pro- 

 jection of the logarithmic upon the second sphere is a loxodrome whose 

 rhumb is measured by the subtangent of the logarithmic ; and conversely, 



From the pole of the loxodrome as a centre, describe a second sphere, 

 equal to that upon which the loxodrome is traced ; then the stereographic 

 projection of the loxodrome upon the second sphere will be a logarithmic 

 whose subtangent measures the rhumb of the loxodrome. 



