84 U. S. COAST AND GEODETIC SURVEY. 



then 



~^=9i(<^ + '^^) 5^2 (<^-'^^) 





30 that from the required condition we have 



The two members of this equation are conjugate complex 

 functions, and the equality can only exist on condition that 

 the members are each equal to a real constant. Let us use 

 i3^ for this constant and, for the sake of abbreviation, let 

 us denote the variable c + iX by z and g^iz) by Z, The 

 differential equation then becomes 



-j y 



Multiply both members by 2-j- and we have 

 dz dz^ dz' 



By integration, 



(iy=^---- 



-7^ being the constant of integration. 



dZ _. 



or 



~JM= = Qdz 



^^2^2 _y ^ • 



