86 FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. 



Now if we call this Xdx + Ydy it satisfies the condition for a 

 perfect differential 



dZ_9F . &u_&u 

 dy~dx y ] df~dx*' 



r o ^ 



Consequently the line integral \-dx + dy from a given 



point X Q , 2/o to a variable point or,, y is a function only of its upper 

 limit, and represents v. Similarly if v is given, 



fdu j du 7 fdv 7 9w T 

 ^ = ^- dx + ^- d y = ~-dx dy. 

 J dx dy J dy dx y 



Furthermore the first of the equations (A) is the condition 

 that vdx + udy is a perfect differential, and the second that 

 udx vdy is such. 



Accordingly the line integrals 



<> = I vdx + udy, 



^jr = I udx vdy, 



give two new point-functions </>, ty which in virtue of the equations 



d(f> 9"^ d4> 9^ 



dx dy ' dy dx ' 



are conjugate to each other, and give a new analytic function of 2, 

 (f> + ity,or ^ + i<j). From these by new integrations we may obtain 

 any number. 



Examples. The function 



z*=(x + iy)* = x* + 2ixy - y\ 

 gives u = x 2 7/ 2 , v = %xy, 



both harmonic functions. 



The curves u x^ y^ const, and v = 2xy = const, give two 

 sets of equilateral hyperbolas, which intersect everywhere at right 

 angles, Fig. 22. 



The function 



- = - = , 



z x + ^y x 2 + y 2 



x y 



gives u = - . , v = 



* * 



