522 XI. HYDRODYNAMICS. 



where u and v are real functions of the two real variables x and y. 

 For instance 



z* = C + ty) 8 = x* - i 



w = # 2 ?/ 2 , v 



1 _ 1 x iy 



z 



U = 



Let us examine the relation between an infinitesimal change in z 

 and the corresponding change in /"(#) We have, x and y being real 

 variables capable of independent variation, 



114) dz = dx + i dy, 



115) 



Consequently by division, 



du , d* . /# , dv \ 



7 i . -, o dx + -^ dy -+- 1 1 75 dx -+- -^ a w I 



^.,,N die du -\-idv dx dy \o x cy } 



dyl dx 



The ratio of the differentials of w and z accordingly depends in 

 general on the ratio of dy to dx, that is, if x and y represent the 

 coordinates of a point in a plane, on the direction of leaving the 

 point. If the ratio of dw to dz is to be independent of this direc- 

 tion and to depend only on the position of the point x, y, the 

 numerator must be a multiple of the denominator, so that the expression 



containing ~ divides out. In order that this may be true we must have 



du . . dv 



that is 

 117) 



\dx n dx/ dy dy 



Putting real and imaginary parts on both sides equal, 

 -,^ Q ^ du cv dv du 



HO) O = ^~ > 7T~ = a" > 



dx cy ex cy 



and 



11Q . dw du . dv dv .du 



= + * = -' 



