86 U. S. COAST AND GEODETIC SURVEY. j 



Since /i(o- + iX) is equal to x — iy, the constant C tends 

 only to translate the origin. Let us suppose that C is st. 

 complex quantity in the form of a + ih. If we transpose 

 C to the left-hand member, we have 



x-a-i(y + h)== j/g2^(.+ix) + j^' 



a and h may be either positive or negative and either or 

 both may be zero. No generality is lost if we set them 

 both equal to zero, since they may be accounted for by a 

 mere translation of axes. 



Now, let M= - Ai and N= —Bi and we get 



By multiplying both terms of the fraction by Ae^^'^~^^^ + 

 ^g-^(cr-ix)^ we get 



_. ^^g-2effX_^^^g-2ff. 



^ ^^ ^2g2^a _f_ 2AB COS 2^X + ^2g-2^.r 



A sin 2^\ + i(A cos 2^\ + Be-^^'') 



^^^•^ + 2^^ cos 2/3X+^2g-2^. 



By equating the real parts and the imaginary parts, we 

 obtain 



^ A sin 2^\ 



^~A'e^^'^ + 2AB cos 2^\ + B^e-^^'' 



^ A cos2^\ + Be-^^'' 



y j^2^pa _|. 2AB cos 2i3X + B^e-^^'' 



On the sphere 



ana on the elhpsoid 



That the meridians and parallels are both circles, we 

 already know, since the function j^j was determined on 

 this condition; but in order to obtain their equations, we 

 must proceed in the usual way. If we eliminate or, we 



