200 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



105. Example. Conformal Representation of Sphere 

 on Plane. Let the surface q 3 be a sphere of radius R, and take 

 for the coordinates q 1 and q z , the co-latitude 6 and the longitude </>. 

 Then by 17, we have 



hl= R> hz 



ds* = R 2 (dd* + sin 2 6d<p) = (du? + dv z ) M, 

 and the differential equation satisfied by u and v is 

 8 ( . du\ 9(1 du _ 



= 



If we take 



jMdu = Rdd, JM dv = R sin 

 and if we choose JM = R sin 6, then 



dv = dd>, du = -; ^ . 

 sin 



Integrating we obtain 





 v = $, u = log tan ^ . 



If now we take u and v for rectangular coordinates in a plane, 

 the surface of the sphere is conformally represented upon the 

 plane by means of the above transformation. This particular 

 representation is known as Mercator's Projection. The meridians 

 <p = const, correspond to the straight lines v = const., and the 

 parallels 6 = const, correspond to the lines u = const.* 



Since the whole sphere is covered by a variation of </> between the 

 limits 0, 2-7T, the projection on the plane has the finite width 2?r, 

 but the length of the projection is infinite, the poles 9 = 0, 0= TT 

 corresponding to u = oo , u = oo . If we make a conformal trans- 

 formation of the 7F-plane by means of the function 



u + iv = log (x + iy\ 

 we obtain the formulae, 



u = log r = log Jx 2 + y*, v = tan" 1 - , 



CO 



= r = tan ^ , $ = tan~ x - 



-j X 



* For an example see Fig. 71, 177. 



