THEORY OF POLYCOXIC PROJECTIONS. 123 



No constant of integration is added, since <^ and ip' vanish 

 at one and the same time. If each pole is to be a single 



point this equation must be valid for 2 or —2- This gives 



7ii?2 = 3. If we wish that the ratio of surfaces should be 

 equal to imity along the Equator, it would be necessary to 

 have 



(l+cosX)2 ' 



n' being the value of the derivative of tp' with respect to 

 <p for «p = 0. We deduce from this equation, by integra- 

 tion, the relation 





(l+itan»^)tan|, 



no constant being added, since X and X' vanish together. 

 Since the meridian of 90° of longitude is to be represented 

 by the circumference described upon the line of poles of 

 the map as diameter, it is necessary that this equation 

 should be satisfied when we make in it at the same time 



X=2 ^^^ ^'°°°2' ^® tave then 



4 



We can unite the two conditions; then the mode of pro- 

 jection will be defined by the two relations which we have 

 just obtained, the first between tp' and (p, the second be- 

 tween X' and X; in addition, ti' will be found joined to n 

 by the relation nn^B?' = 4, which we obtain either by making 



^ = and ^- = 7i' in the first differential equation or by 



making X = and ~jr = '^ in the second. From this we 

 conclude that 



