OF ARTS AND SCIENCES. 235 



The function 



is constant along the circumference drawn about the origin with 

 radius r, and is evidently the flow function due to a simple 

 2" + ^ -plot at the centre of a circular disk. 



15. The rational algebraic function 0, , , where the numerator 



■^ (z) 



and denominator are algebraic polynomials involving only integral 

 powers of 2, can be put by simple division into the form of a rational 



algebraic polynomial in z plus the proper fraction, -LA^ , where the 



F(z) 



numerator is of degree at least one lower than the denominator. 

 This proper fraction can be represented in the usual way as the sum 

 of a series of fractions the numeratois of which are in general com- 

 plex constants and the denominators of the form [z — {a -\- b i)]". 



The real part of -^ >-^ is therefore the velocity potential function 



of a collection of motors such that each has for its own velocity po- 

 tential function the real part of a function of the form 



a -{- (3 t ^ fi (cos 8 + ^ sin 8) .oo\ 



(z — {a+b i)y (z — (a + b i)y 





Fig. 11. 



Let us take as the axis of a simple multiplet a line which, drawn as 

 a diameter through the symbol which represents the motor, bisects 

 the black portion nearest the axis of x. If this line goes through 

 two black portions, the sense in which the direction of the axis is 

 taken is of no consequence. If it goes through one black sector and 

 one unshaded sector of the symbol, the direction shall be from the 

 unshaded part to the black part. The simple motor, for instance, 

 which has for its flow function the real factor of the imaginary part of 



— , has its axis coincident with the axis of x. 



z" 



