62] COMPLEX VARIABLE. 75 



depends in general on the ratio &x : 8y. The condition that it should be the 

 same for all values of this ratio is 



7 1" T ~~ V V 7 I L ' 7 I (l)l 



dy dy \dx dx J 



which is equivalent to (1) above. This property may therefore be taken, after 

 Eiemann, as the definition of a function of the complex variable x + iy ; viz. 

 such a function must have, for every assigned value of the variable, not only 

 a definite value or system of values, but also for each of these values a definite 

 differential coefficient. The advantage of this definition is that it is quite 

 independent of the existence of an analytical expression for the function. 



Now, w being any function of 0, we have, corresponding to any point 

 P of the plane xy (which we may call the plane of the variable 2), one or 

 more definite values of w. Let us choose any one of these, and denote it 

 by a point P' of which $, \^ are the rectangular co-ordinates in a second 

 plane (the plane of the function w). If P trace out any curve in the plane 

 of z, P will trace out a corresponding curve in the plane of w. By mapping 

 out the correspondence between the positions of P and P, we may exhibit 

 graphically all the properties of the function w. 



Let now Q be a point infinitely near to P, and let Q' be the corresponding 

 point infinitely near to P. We may denote PQ by dz, PQ' by dw. The 

 vector PQ' may be obtained from the vector PQ by multiplying it by the 

 differential coefficient dw/dz, whose value is by definition dependent only on 

 the position of P, and not on the direction of the element dz or PQ. The effect 

 of this operator dw/dz is to increase the length of PQ in some definite ratio, and 

 to turn it through some definite angle. Hence, in the 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. 

 Any angle in the plane of z is therefore 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 (f> = const., ^ = 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 

 = const., ^ = const., the values of the constants being assigned as before, 

 cut one another at right angles (as has already been proved otherwise) and 

 divide the plane into a series of infinitely small squares. 



Conversely, if <, \J/- be any two functions of x, y such that the curves 

 0=wie, ^ = ?ie, where e is infinitesimal, and m, n are any integers, divide the 

 plane xy into elementary squares, it is evident geometrically that 



dx dy dx _ dy 



If we take the upper signs, these are the conditions that x+iy should be a 



