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



function of < + ^. The case of the lower signs is reduced to this by re- 

 versing the sign of ^. Hence the equation (2) contains the complete solution 

 of the problem of orthomorphic projection from one plane to another*. 



The similarity of corresponding infinitely small portions of the planes w 

 and z breaks down at points where the differential coefficient dwjdz is zero or 

 infinite. Since 



dw d(b , .d& .... 



-r = -r- + i>-j L .............................. (")> 



dz dx dx 



the corresponding value of the velocity, in the hydrodynamical application, is 

 zero or infinite. 



A 'uniform' or ' single- valued ' function is one which returns to its 

 original value whenever the representative point completes a closed circuit 

 in the plane xy. All other functions are said to be ' multiform,' or ' many- 

 valued.' A simple case of a multiform function is that of z*. If we put 



z=x+iy = T (cos6-\-i sin $), 

 we have a* =r* (cos 0+1* sin 0). 



Hence when P describes a closed circuit surrounding the origin, 6 increases 

 by 27T, and the function does not return to its former value, the sign being 

 reversed. A repetition of the circuit restores the original value. 



A point (such as the origin in this example), at which two or more values 

 of the function coincide, is called a ' branch -point.' In the hydrodynamical 

 application ' branch-points ' cannot occur in the interior of the space occupied 

 by the fluid. They may however occur on the boundary, since the function 

 will then be uniform throughout the region considered. 



Many-valued functions of another kind, which may conveniently be 

 distinguished as * cyclic,' present themselves, in the Theory of Functions, as 

 integrals with a variable upper limit. It is easily shewn that the value of 

 the interal 



taken round the boundary of any portion of the plane xy throughout which 

 / (z), and its derivative f (2), are finite, is zero. This follows from the two- 

 dimeiisional form of Stokes's Theorem, proved in Art. 33, viz. 



the restrictions as to the values of P, Q being as there stated. If we put 

 P = f ( z \ Q tf( z }i the result follows, since 



Hence the value of the integral (iii), taken from a fixed point A to a variable 

 point P, is the same for all paths which can be reconciled with one another 

 without crossing points for which the above conditions are violated. 



* Lagrange, " Sur la construction des cartes geographiques," Nouv. mem. de 

 VAcad. de Berlin, 1779 ; Oeuvres, t. iv., p. 636. 



