of a given Surface on another given Surface* 109 



The differential equation cg = dP + du 9 = gives these two 

 integrals 



t + iu — const, t — iu = const. 



and in like manner the two integrals of the equation ft = 

 dT+rfU 2 = 0, are the following T+fU = const., T-/U = 

 const. The two general solutions of the problem are ac- 

 cordingly 



I. T + iU=f(t + iu), T-i\J=f'(t-zu) 



II. T + *U =f(t-iu\ T-*U =/'(*+*?/). 



This result may be thus expressed : f signifying an arbitrary 

 function, the real part off(x + iy) is to be taken for x, and the 

 imaginary part divided by i for y or for — y. 



If the functional characteristics <p, <p' are taken in the same 

 signification which they have in article 7, and if we put 



$(x+iy) =£ + *>j, tf{x— iy) = $-1* 



where £ and >j will be clearly real functions of x and y, we 



have by the first solution 



dX+idY = (g + iyi)(dx+idy) 

 dX — idY = (£—/»}) (dx—idy) 



and consequently, dX = £dx—r\dy 

 dY = i\dx-\-%dy 

 If we now put £ = <r . cos,/ , 19 = <r . sin,/ 

 dx = ds. cos g, dij=:ds.smg 

 dX=zdS.cosG, dY =zdS.sinG, 

 so that ds is a linear element in the first plane, g its inclina- 

 tion to the line of abscissae, d S the corresponding linear ele- 

 ment in the second plane, and G its inclination to the line of 

 abscissae, the above equations give 



d S . cos G = o- . d s . cos (g -\-j) 

 d S . sin G = <r ds sin {g+j) 9 and consequently, 

 if we consider cr as positive, as we may do 

 dS = a-.ds , G = g+j. 

 We see, therefore (in conformity to article 7), that <r is the 

 index of the ratio of increase of the element ds in the repre- 

 sentation d s, and is, as it ought to be, independent of g ; and 

 in the same way the angle,/ being independent of g, proves 

 that all linear elements of the first plane proceeding from one 

 point are represented by elements in the second plane which 

 form to each other, and, as we may add, in the same direction, 

 the same angles. 



If we now choose for/ a linear function, so that/p = A + 

 B v where the constant coefficients are of the form A=a + b. i y 



B = 



