THEORY OF POLYCONIC PROJECTIONS. 117 



Let us investigate the case in which the scale should be 

 held constant along the parallels. We should then have 



1cp=l and Icra cos ^ = 1, 

 or 



1 /^ Q_dp\_. 

 a \d(p d(p/ 



ds cos d — dp — a dip 



or 



ds cos d=^dp-\-a d(p. 



On any given parallel the right-hand member of this eq^ua- 

 tion is a constant, since dp is a function of <p; but is a 

 function of <p and X, for we have 



or, by integration, 



^ a cos (p dX 



^ ^ g cos (p ^ 



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



It follow^ that the left-hand member of the above equa- 

 tion must vanish identically; that is to say, ds = 0. The 

 circles of parallels are, therefore, concentric and 



dp= —a dip J 

 or, by integration, 



p = Po + a(<^o-^)- 



This is Bonne's projection; but, of course, it is not a poly- 

 conic projection, since s is constant; that is, the parallel 

 arcs are concentric. It appears, however, in the attempt 

 to attain certain things by means of the equal-area poly- 

 conic projection and can be looked upon as a limiting case 

 of the same. 



If we assume 



p = a cot ip 



s = a{ip+ cot <p)y 



then 



dp „ 



j-=—a cosec^ v> 



ds 

 T-=a{\ — cosec^ <p)= —a cot^ tp. 



