309-312] 



Conjugate Functions 



259 



There is a geometrical interpretation of multiplication. 

 In fig. 84, let OA = 1, OP = z, OP' = 2:' and OQ = zz'. 



Then the angles QOA, P'OA being equal to + 6' and 0' respectively, 

 the angle QOP' must be equal to 0, and therefore to POA. 



Moreover Q 



OQ _OP 

 OP 7 ~ OA ' 



each ratio being equal to r, so that the triangles 

 QOP' and POA are similar. Thus to multiply 

 the vector OP' by the vector OP, we simply 

 construct on OP' a triangle similar to AOP. 



The same result can be more shortly ex- 

 pressed by saying that to multiply z' (= OP') by 

 z( = OP), we multiply the length OP' by | z and 

 turn it through an angle arg z. 



So also to divide by z, we divide the length of 

 the line representing the dividend by z and 



turn through an angle arg z. In either case an angle is positive when the 

 turning is in the direction which brings us from the axis of x to that of y 

 after an angle Tr/2. 



FIG. 84. 



Conformal Representation. 



312. We can now consider more fully the meaning of the relation 



U + iV = $ (as + iy). 



Let us write z x + iy, and W = U + iV, z and W being complex imagin- 

 aries, which we must now suppose in accordance with equation (230) to be 

 connected by the relation 



W = $(z) (232). 



We can represent values of z in one Argand diagram, and values of W in 

 another. The plane in which values of z are represented will be called the 

 -plane, the other will be called the TF-plane. Any point P in the ^-plane- 

 corresponds to a definite value of z and this, by equation (232), may give one 

 or more values of W, according as </> is or is not a single-valued function. 

 If Q is a point in the TF-plane which represents one of these values of W, 

 the points P and Q are said to correspond. 



As P describes any curve S in the 3-plane, the point Q in the TF-plane 

 which corresponds to P will describe some curve T in the TT-plane, and the 

 curve T is said to correspond to the curve 8. In particular, corresponding 

 to any infinitesimal linear path PP' in the ^-plane, there will correspond 



172 



