64-65] INVERSE FORMULAE. 81 



65. If w be a function of z, it follows at once from the defini- 

 tion of Art. 62 that z is a function of w. The latter form of 

 assumption is sometimes more convenient analytically than the 

 former. 



The relations (1) of Art. 62 are then replaced by 



dx _ dy dx _ dy 

 ~ ~~ 



A1 dw dd> .d^lr 



Also since -j- ^- -\-i~- = u-\-iv, 



dz dx dx 



, dz 1 1 u .v 



we have 



dw u iv q \q q, 

 where q is the resultant velocity at (x t y). Hence if we write 



and imagine the properties of the function f to be exhibited 

 graphically in the manner already explained, the vector drawn 

 from the origin to any point in the plane of will agree 

 in direction with, and be in magnitude the reciprocal of, the 

 velocity at the corresponding point of the plane of z. 



Again, since \/q is the modulus of dzfdw, i.e. of dxjd<^ + idy/d(f>, 

 we have 



which may, by (1), be put into the equivalent forms 

 1 _ /d*y / ^ y _ tdyv idy 



- " 



^L V _dxdy^_dx_dy 

 / ~ d <l> d ^ d^d$~ 



The last formula, viz. 



( . 



simply expresses the fact that corresponding elementary areas in 

 the planes of z and w are in the ratio of the square of the modulus 

 of dzjdw to unity. 



L. 6 



