of a given Surface on another given Surface. 207 



Putting now */{\ — e 2 ) . tang u = tang co (when applied to 

 the terrestrial spheroid 90— co will be the geographical lati- 

 tude and t the longitude), the equation will assume this form : 



= at + idea . - — -r—. — , 



(1 — j^cos*^) sin« ' 



the integration of which gives 



const. = t ± Hog I cotang J co . (i^ii^L) ' \ . 



Consequently f denoting an arbitrary function, we have to 

 take for X the real, and for i Y the imaginary part of 



/(« + ,- kg \ cotang kirlggfii 



If a linear function is chosen for f, putting^u = k v 9 we shall 

 have 



X = kt, Y = h log cotang \ co— \ k s log + tcoSft, ? 



which is a projection analogous to that of Mercator. 



If on the contrary an imaginary exponential function is 

 taken foryj we have 



X = £ tang ) co (j— c — { .cos X*, 



Y = £ tang |co ( -tZTSnr l sm X * 



which putting A = 1 will give a projection analogous to the 

 stereographical, and generally one which is very proper for 

 representing a portion of the earth's surface if the ellipticity 

 is to be taken into consideration. 



The formulae for the other case in which b>a may be im- 

 mediately derived from the preceding ones ; the same notation 



being retained, e will become imaginary, but ( _ * °° s - y will 



again become real. But for the sake of completeness we will 

 separately develope the formulae for this case, and first put 



\f ( — — l)=i). We then determine co by the equation 



s/{\ + >j 2 ) . tang u = tang co, and the differential equation 



= dt + id co. ,, , " - — r-: — has for its integral, the fol- 



( 1 -J- «* cos •>*■*) sin u ° 



lowing : 



Const. = t + i (log cotang \ co + vj. arc tang *j . cos co) so that 

 X will be the real and iY the imaginary part of 



f [t + * log (cotang J co -f *j . arc tang >j . cos co)} 



From 



