12 U. S. COAST AND GEODETIC SURVEY. 



Therefore 



or, at the limit 



M'N= p (y)d<p + ds sin ^. 



MN=SM-SN=SM-S'N-SP, 



since at the limit 



S'N=PN, 

 But 



By substituting this value and the value of SPj we obtain 



MN= -dp + ds cos e. 



If we denote Z -M*' i/iV by xj/, we have at the limit 



dd ds . ' 



^an^=^^=-3^ 7-37* 



J- cos d—j- 

 dcp (lip 



If we denote the change in scale or the magnification 

 along the meridian by ^m and that along the parallel by 

 ^Tp, we shall obtain the following expressions for these 

 quantities: 



M' M= MN sec \l/=(ds cos 6 — dp) sec \p. 



The arc of the meridian on the earth that is represented 

 by M' M is given by 



-, , a(l-e^)d(p 



^^=^-^^=^(l-e^sinV)''^' 



Hence we have 



d-e^sm'^pY'^ /ds , dp 



JCm — 



ad-e') 



/ds - dp\ , 



