122 U. S. COAST AND GEODETIC SURVEY. 



By setting aside the condition that the principal meridian 

 shomd be represented by the circumference described 

 upon the line of poles of the map as diameter, we could 

 obtain a series of atractozonic projections instead of a 

 single one, and in this group some would certainly be found 

 the alterations of which would be less than those of the 

 projection that we have just studied. We could still 

 lurther increase the indetermination, and we could intro- 

 duce two parameters in the place of one by not fixing in 

 advance the parallel, the projection of which should be a 

 straight Hne. This remark applies also to the remaining 

 projections in this class. 



In a rectangidar circular projection, in place of deter- 

 mining (p' as a function of (^, so that the projection of each 

 ^one Siould be equivalent to the zone it represents, we 

 can bring about that the ratio of the surfaces should be 

 continually equal to unity along a given meridian or that 

 the lengtns should be preserved upon this meridian. 

 Similarly, we could determine X' as a function of X in such 

 a way that, upon a. given parallel, the same conditions 

 should be fulfilled. By combining each expression of <i?' so 

 obtained with one of the expressions for X we could form 

 several kinds of projections, each of which would possess 

 the t^o properties in question. 



Let us continue to represent the principal meridian by 

 the circumference described upon the line of poles of the 

 map as diameter, the Equator by the diameter perpen- 

 dicular to this line, and let us call It the radius of the cir- 

 cumference. 



The ratio of surfaces at each point, in one of these rectan- 

 gular circular projections, is 



V— Ti2 <^Qs ip' __1 d<p^ dk^ 



"~ cos <p (l + cos X' cos <p'y d<p d\ 



We now propose to bring about that it should remain equal 

 to unjty along the central meridian. For X = we have 

 X' = 0, and the derivative of X' with respect to X assumes a 

 known value n, depending on the nature of the function of 

 A which has been adopted to represent the value of X'. 

 The condition is then 



o, cos <p' dtp' J 



nHr -jir-s T\-> = cos <z> dip 



(l+cos 9? )2 ^ ^ 



or, by integration, 



sm^ = -2-(^l-3tan^2/an2- 



