322 ELECTROSTATICS. [PT. II. CH. VII. 



161. Logarithmic Transformation of last case. If we 



transform the last figure by means of the function w' log w, we 

 obtain the whole of the 7F-plane represented on a strip of the 

 U' F'-plane of width 2?r, so that the transformation 



(1) w = log cosh z 



transforms the right-hand half of a horizontal strip of width 2?r in 

 the XF-plane into the whole of the same strip in the U' F'-plane. 

 Taking the antilogarithms we have 



(2) e w ' = cosh z = cosh x cos y + i sinh x sin y, 

 that is, 



(3) &u> ( cos v ' ~t~ * s ^ n v ') ~ cosn x cos y + * sinh x sin y, 

 so that 



(4) e u ' cos v = cosh x cos y, e u> sin v' sinh x sin y, 

 from which, in the same manner as above, 



cos 2 v' sin 2 v 1 cos 2 v' sin 2 v 1 

 cosh 2 x + sinh 2 x~~^" cos 2 y ~ sin 2 y = ~^' ' 



and taking logarithms, 



, 1 , f cos 2 ?/ sin 2 v' 

 2 (cosh 2 a? sinh 2 a 



, 1 , f cos 2 v' sin 2 v' 



nt' _ __ IfiCT < 



2 5 (cos 2 y sin 2 y 



From these equations the curves corresponding to x = const, 

 and y = const, may be immediately plotted by the aid of tables of 

 logarithms and hyperbolic functions. They are shown in Fig. 67. 

 It is at once seen that u' is a periodic function of v', the period being 

 TT. The figure is the same for negative x and y as for positive. In 

 order to represent the whole of the 7F-plane corresponding to the 

 half strip in the X F-plane, we must however let v' vary from to 

 2-7T. The curves x = const, are sinuous curves, u' having maxima for 

 v = 0, TT, 2-7T, . . . and minima for v = Tr/2, 3?r/2, .... The maxima 

 u = log cosh x and minima u' = log sinh x differ but little for large 

 values of x, since then approximately cosh x = sinh x = e?/2 so that 

 we may then take out this factor from u', obtaining u x log 2 

 for all values of v, so that the curves x = const, are nearly straight 

 lines. 



As x diminishes the maxima and minima both diminish, but 

 get farther apart, the maxima being always positive, while the 



