82 U. S. COAST AND GEODETIC SURVEY. 



From these we obtain at once 



d<T^~ do- dX~ 5X2 

 b(T^ ' bcr b\ b\^ 

 ^ \b<Tj '^KbaJ \b\J '^\b\J b\ bcr b(T b\ 



Therefore 



W=^f\icr + i\)f,(a-i\). 



1 



If the coorduxates of a plane curve are expressed in 

 terms of an independent variable t in the form 



X^<p(t) 



the expression for the radius of curvature is given in the 

 form 



dxd^y dy d^x 

 1 dt df dt dt^ 



" mnm 



Since in the expressions for x and y in terms of/i andj^, 

 <r is a function oi the latitude and X is merely the longi- 

 tude, <r is constant along a given parallel and X is constant 

 along a given meridian; in other words, <r remaining con- 

 stant, we obtain a parallel by variation of X, and X oeing 

 constant, we get a meridian by variation of <r. Therefore, 

 if we n^lect the sign 



bxb^y byb^x 

 1 b(rb(T^ bcrbtr' 



bx b^y by b^x 

 1 b\b\^ dXdX^ 



'^ [m<mj- 



