110 Prof. Gauss on the Representation of the Parts 



Bsc+ei, we shall have ^» = Bac+e/, therefore tr =c 



V{c*+e*)> j = arc tang -^-. 



The ratio of increase or the scale is consequently constant 

 throughout, and the whole representation similar to the surface 

 represented. For every other function f 9 it may be easily 

 proved that the scale cannot be constant, and that the simila- 

 rity can only take place in the smallest part. If the places 

 are given wnich are to correspond in the representation to a 

 determinate number of given points of the first plane, we may 

 easily determine by the common method of interpolation the 

 simplest algebraical function f, which will fulfill those condi- 

 tions. If we denote the values of x + iy for the given points 

 by a, b 9 c, &C. and the corresponding values of X + i Y by A, 

 B, C, &c* then it will be necessary to put 



* (a— b)(a— c)... (6— o)(6— c)... ' (c— a){c— b)... 



which is an algebraical function of v of a degree one unity 

 lower than the number of given points. For two points, where 

 the function becomes linear, a perfect resemblance will conse- 

 quently take place. 



An useful application may be made of this in geodetics, for 

 converting a map founded on moderately good measurements, 

 which in its minute detail is good, but on the whole somewhat 

 distorted, into a better one, if the correct position of a number 

 of points is known. 



Going through the second solution in the same manner, it 

 will be found that the only difference is, that the similarity is a 

 reversed one ; that all elements form indeed with each other 

 the same angles as in the original, but in a contrary direction, 

 so that that which is to the right in the one, is to the left in the 

 other. But this difference is not an essential one, and vanishes 

 if the side of the plane which was first considered as the upper 

 one is made the lower one. This latter remark may be always 

 applied whenever one of the surfaces is a plane ; and we shall 

 confine ourselves in the following examples of this kind to the 

 first solution. 



9. Let us now consider (as a second example) the repre- 

 sentation of the surface of a perpendicular cone in a plane. As 

 the equation of the former, we take 



where we put x = K t . cos u, y = K t . sin #, z = t, and as 

 before, X = T, Y = U, Z = 0. 



The 



