APPLICATIONS TO THE GEOMETRY OF SPACE 169 
EXERCISES 
1. Show that the total length of a rhumb line on the sphere is 
finite. 
2. The Cartesian coordinates of a point on the surface of a sphere 
are given by the equations : 
x = a cos </> cos 6, y = a cos cf> sin 0, z — a sin <£. 
Deduce (3) from these relations and the equation: 
ds 2 = dx 2 + dy 2 + dz 2 . 
3. Taking as the coordinates of a point on the surface of a cone 
(p, 0), where p is the distance from the vertex and 6 is the longitude, 
show that 
(5) tan w = — d A 
pd6 sin a 
4. Obtain the equation and the length of a rhumb line on the cone. 
5. The preceding two questions for a cylinder. 
fly 
Ans. tan w = , ds 2 = a 2 d6 2 + dz 2 , where r = a 
add 
is the equation of the cylinder in cylindrical coordinates. 
8. Mercator’s Chart. In mapping the earth on a sheet of paper it 
is not possible to preserve the shapes of the countries and the islands, 
the lakes and the peninsulas represented. Some distortion is in 
evitable, and the problem of cartography is to render its disturbing 
effect as slight as possible. This demand will be met satisfactorily 
if we can make the angle at which two curves intersect on the earth’s 
surface go over into the same angle on the map. For then a small 
triangle on the surface of the earth, made by arcs of great circles, 
will appear in the map as a small curvilinear triangle having the 
same angles and almost straight sides, and so it will look very simi 
lar to the original triangle. What is true of triangles is true of 
other small figures, and thus we should get a map in which Cuba 
will look like Cuba and Iceland like Iceland, though the scale for 
Cuba and the scale for Iceland may be quite different. 
A map meeting the above requirement may be made as follows. 
Regarding the earth as a perfect sphere, construct a cylinder tan 
gent to the earth along the equator. Then the meridians shall go 
over into the elements of the cylinder and the parallels of latitude 
into its circular cross-sections as follows: Let P be an arbitrary 
point on the earth, Q, its image on the cylinder.