74 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



transition from the plane of z to that of .w, all the infinitesimal 

 vectors drawn from the point P have their lengths altered in the 

 same ratio, and are turned through the same angle. Hence any 

 angle in the plane of z is equal to the corresponding angle in the 

 plane of w, and any infinitely small figure in the one plane is 

 similar to the corresponding figure in the other. In other words, 

 corresponding figures in the planes of z and w are similar in their 

 infinitely small parts. 



For instance, in the plane of w the straight lines &amp;lt; = const., 

 -fy = const., where the constants have assigned to them a series of 

 values in arithmetical progression, the common difference being 

 infinitesimal and the same in each case, form two systems of 

 straight lines at right angles dividing the plane into infinitely 

 small squares. Hence in the plane xy the corresponding curves 

 &amp;lt; = const., T/T = const., the values of the constants being assigned 

 to them as before cut one another at right angles (as has already 

 been proved otherwise) and divide the plane into a series of in 

 finitely small squares. 



The similarity of corresponding infinitely small portions of the 

 planes w and z breaks down at points where the differential 



/v. , dw . . ., . dw dd&amp;gt; .d^lr . 



coefficient -=- is zero or infinite. Since -=- = - + i -^ , the 

 dz dz dx dx 



corresponding value of the velocity, in the hydrodynamical ap 

 plication, is zero or infinity. 



77. The processes of differentiation and integration, as applied 

 to functions of a complex variable, claim a little notice. 



The conditions (13) that w should be a function of z may be 

 written 



dtv . dw . . 



T- = I-J- (14), 



dy dx 



a form which the student should interpret. It is obvious that (14) 



is satisfied when -j- = -= is written for w, and therefore that 

 dz L dx\ 



-j- is itself a function of z. Hence all the derivatives of a function 

 dz 



of a complex variable are themselves functions of that variable. 



