116 U. S. COAST AND GEODETIC SURVEY. 



By substituting these values we obtain 



p sin (9 /p du ^ dp\V'{\) , 



2u r(x) 



but 



aj^ cos (f 



sm 6 = 



cos 6 = 





Hence 



2p2 Ju it2-P(X) 2pu 1 (Zp 



a 



-^cosi^ d<p [u'-^T^Wf a'cos^p u' + T^X) d<p r'(X) 



This is an equation that must be identically satisfied by the 

 values of u (a function of <p) and r(X) (a function of X). 

 The right-hand member is independent of <p ; hence the left- 

 hand member must also be independent of <p. The condi- 

 tion will be identically satisfied if u equals a constant and 



if -^ — J- is equal to a constant. 



a^cos<p dip ^ 



If It is a constant, s is also a constant, and the projection 

 would pass into one of the limiting cases of the poly conic 

 projections. 



The integration of the equation 



2p dp = a^c cos <p dip 

 gives 



P^ = Po^-\-a?c sin <p. 



By assigning particular values to the constants po and c, 

 we may obtain Lambert's central equal area projection, 

 Lambert's isospherical stenoteric projection (sometimes 

 called Lambert's fifth), or, finally, Albers' projection. 

 None of these are polyconic projections in the accepted 

 sense, and hence no investigation of their properties will 

 be given at this time. 



No one of the strictly polyconic equivalent projections 

 has ever become of practical importance, because they 

 would generally be complicated both for computation and 

 construction. 



