THEORY OF POLYCOXIC PROJECTIONS. 115 



Integrating partially with respect to X and B with (p re- 

 maining constant, we obtain 



a^cos 



(p\a<p dip J 



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

 In this expression s and p are any function of <^ that we 

 may choose. Q would then be determined by the above 

 equation. Inversely, if we give the relation which should 

 obtain between ^, <^, and X subject to the condition that 

 X should be a linear fimction of B and sin ^, there would be 

 an infinity of equal- area poly conic projections which 

 would satisfy this relation. In fact, u and v being given 

 functions of «^, the assigned relation would be 



u sin 6/ — v ^ = X, 

 iu wliich 



p ds 



u = 



a^cos (p dip 



«, P dp 



a^cos (p dip 



or 



p^ = Po^ + 2a^ I V cos <p d<p. 



a^ I - cos ip dip 

 J o p 



= So + 



Po and So denoting the two constants of integration. 



There is no equivalent poly conic projection that is at the 

 same time rectangular. In a rectangular polyconic pro- 

 jection we have 



^_P du 



dip u dip 

 and 



. d r(x) 



tan — 



2 u 



be r'(x) 



dx r(x) 



sm^. 



