THEORY OF POLYCONIC PROJECTIONS. 



73 



But 



2 tan 



sin = 



1 + tan 

 l-tan2 



cos 6 = 



2 ^ 2uT(\) 

 2 



e_ 



2 u'-TW) 



l+tan2 



e u' + VKX) 



Substituting these values and the value of 



ds _p du 

 dip ud(p 



we obtain 



r'(x) = 



>i^ + rHX)]cos<p /_p du u'-T^jX) dp\ 

 2pu \u dipu'+T^CK) dip) 



, ./ cos ip dp cos tp dv\ y cos <p du cos <p dp\ 

 ^ ^^\~2^dip'^ 2u^ dipj'^^y 2u' dip 2pu dip J 



(■ 



_{dp, duX cos ip 

 dip ^dipj 2pu^ 



u 



T^(\)+u^ 



dp 

 dip 



du 

 ^d^p 



dp du 

 dip dip 



Since r(X) is independent of ip, r'(X) is also independent 

 of ip; consequently the two expressions dependent upon ip 

 must reduce to constants. We can set one of them equal 

 to unity, because u can be multiplied by any constant 

 without changing the value of either s or p; and if so, 

 r(X) would be multiplied by the same constant, so that 

 d would not be changed thereby. 



