FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. 



curvilinear rectangles the ratios of whose sides at any point are 

 given by the ratio of the parameters h u and h v . But from the 

 equations (A), we have 



/du\* , fdu\* idv\* fdv 

 h u 2 = ~- + U- = U- + 1 5~ 



so that in this case the plane is divided into small squares. Let 

 us now construct in the second plane, in which u and t; are 



FIG. 21. 



rectangular coordinates, the curves corresponding to u = constant 

 and v = constant. These are of course straight lines dividing their 

 plane into small squares. Moreover the length of any arc da of a 

 curve in their plane, is given by 



da* = du 2 4- dv\ 

 But in virtue of the above relations, this gives 



h = 



dw 



is accordingly the ratio of magnification at the point in 

 question, and varies for different points of the plane. 



Let us now construct, (Fig. 21,) at a point in the x, y plane an 

 infinitesimal triangle made by the intersection of any three curves, 

 and let the lengths of its sides be ds l} ds 2 , ds 3 . Construct the 

 corresponding curves in the u, v plane, intersecting to form an 

 infinitesimal triangle with sides 



da l} d<r 2 , da s .- 



Now we have 



