64-65] INVERSE FORMULA. 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 

 dxdy dx dy 



d&amp;lt;f&amp;gt; dr } d^ 



dw dd&amp;gt; .dty 



Also since -= - = ?~ + i I - 



dz dx dx 



, dz \ \ in .v 



we have 





dw u iv q \q q. 

 where q is the resultant velocity at (x, 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 f 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 l/q is the modulus of dzjdw, i.e. of dx/d(f&amp;gt; + idyjd&amp;lt;}&amp;gt;, 

 we have 



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



f 



W. 



\ttyy wcp u^ n^r uy&amp;gt; 



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 dz/dw to unity. 



I, 6 



