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



of x + iy there must be one or more definite values of &amp;lt;/&amp;gt; + ity ; but 

 the ratio of the differential of this function to that of x + iy, viz. 



dx + idy 





dx + idy 



depends in general on the ratio dx : dy. The condition that it 

 should be the same for all values of this ratio is 



+ i ..... (13), 

 dy dy \dx dx J 



which is equivalent to (11). 



We may therefore take this property 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. 



The theory of functions of this kind has received considerable 

 development at the hands of Cauchy, Riemann, and others ; and 

 has grown into an important branch of mathematical analysis. 

 We give here only such elementary notions connected with the 

 subject as are of immediate hydrodynamical interest*. 



75. We assume the student to be acquainted with the method 

 of representing the symbol x + iy by a vector drawn from the 

 origin of rectangular co-ordinates to the point (x, y). 



In this method the sum of two vectors is defined to be the 

 vector drawn from the origin to the opposite corner of the par 

 allelogram of which they form adjacent sides. 



The effect of multiplying one vector x + iy by another a + ib 

 is to increase its length in the ratio r : 1, and to turn it in the 



* The reader \vho wishes for an elementary exposition of the analytical theory 

 may consult Durege: Elementc. d&amp;lt;T Tlieorle dcr Functional eincr complexcn rerdn- 

 Gr&ssc. 2nd ed., Leipzig, 1873. 



