195, 196, 197] IRROTATIONAL MOTION. 521 



so that if we put z/qp = tf , 6 is the amount of liquid produced per 

 unit volume per unit of time. The total amount 



dt i i i Gdr 



is called the strength of the source. If (3 is given as a function of the 

 point we have 



Accordingly the velocity potential has the properties of a force 

 potential, the density of attracting matter being represented by 



L times the strength of source per unit volume. The negative sign 



occurs here from the different convention employed, it being customary 

 to define the force as the negative parameter, the velocity as the 

 positive parameter of its potential. In particular a point source of 



strength m produces a radial velocity of magnitude -^^ This 

 system is called by Clifford a squirt. 



197. Uniplanar Motion. A simple and interesting case is that 

 of uniplanar flow as defined above. We then have all quantities 

 independent of s, so that Laplace's equation reduces to 



nn 



A powerful method of treatment of such problems is furnished by 

 the method of functions of a complex variable. The complex number 

 a 4- ib, where a and & are real numbers and i is a unit defined by 

 the equation 



=-i, 



(the same root being always taken) is subject to all the laws of 

 algebra, and vanishes only when a and ~b both vanish separately. 

 Any function of the complex number obtained by algebraic operations, 

 after substituting for every factor i 2 its value 1, becomes the sum 

 of a real number plus a pure imaginary, that is a real number 

 multiplied by i. Any equation between complex numbers is equivalent 

 to two equations between real numbers, being satisfied only when 

 the real parts in both numbers are equal as well as the real coeffi- 

 cients of i in both members. If z denote the complex variable x + iy, 

 any function of & may be written 



w = f(z) == u + iVj 



