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



dwell on the proof of these statements, or to enter more fully into 

 the theory of many-valued integrals of the form (15) ; to do so 

 would be to repeat, with merely verbal alterations, portions of 

 Chapter ill. 



The integral (15) is itself a function of z, according to the 

 definition of Art. 74. For, denoting its value by Z y we have, 



K 1 dZ 



obviously -j = w, 



i. e. Z has a definite differential coefficient. 



78. An important illustration of the above theory is furnished 

 by the integral 



dz 



/T 



The only point at which the function z~ l , or its derivative, is 

 infinite or discontinuous, is the origin. Introducing polar co 

 ordinates, we write 



(18), 



whence = + t00 ........................ (19). 





Hence the value of (17) taken round an infinitely small circle 

 having the origin as centre is 



In the analytical theory above referred to, the logarithm of a 

 complex quantity z is defined by the equation 



J&quot;Z fly 

 - ......................... (20). 

 i z 



Hence log z is a many -valued function, the cyclic constant 

 being 



The properties of the logarithmic function readily follow from 

 the definition (20). Thus if z lt z 2 be any two complex quantities 

 we have 



^ z * ~ *, + *, 



