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



* + *>=/(* + *?) ..................... (2), 



we have (f +tf)-V(+40-<(4+tf) ...... (3), 



whence, equating separately the real and the imaginary parts, we 

 obtain (1). 



Hence any assumption of the form (2) gives a possible case of 

 irrotational motion. The curves </> = const, are the curves of equal 

 velocity-potential, and the curves ^ = const, are the stream-lines. 

 Since, by (1), 



d<f> d-^r d<f> dty _ . 

 dx dx dy dy 



we see that these two systems of curves cut one another at right 

 angles, as already proved. Since the relations (1) are unaltered 

 when we write ty for </>, and < for i/r, we may, if we choose, look 

 upon the curves ty = const, as the equipotentiai curves, and the 

 curves </> = const, as the stream-lines ; so that every assumption of 

 the kind indicated gives us two possible cases of irrotational motion. 



For shortness, we shall through the rest of this Chapter follow 

 the usual notation of the Theory of Functions, and write 



(4), 

 ........................... (5). 



At the present date the reader may be assumed to be in possession of at 

 all events the elements of the theory referred to*. We may, however, briefly 

 recall a few fundamental points which are of special importance in the hydro- 

 dynamical applications of the subject. 



The complex variable x-\-iy may be represented, after Argand and Gauss, 

 by a vector drawn from the origin to the point (.v, y). The result of adding 

 two complex expressions is represented by the geometric sum of the corre- 

 sponding vectors. Regarded as a multiplying operator, a complex expression 

 a+ib has the effect of increasing the length of a vector in the ratio r : 1, and 

 of simultaneously turning it through an angle 0, where r = (a* + 6 2 )*, and 



The fundamental property of & function of a complex variable is that it 

 has a definite differential coefficient with respect to that variable. If $, v//- 

 denote any functions whatever of x and y, then corresponding to every value 

 of x + iy there must be one or more definite values of <t> + tyi hut the ratio 

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



*?+.%' 



* See, for example, Forsyth, Theory of Functions, Cambridge, 1893, cc. i., ii. 



