of a given Surface on another given Surface. 211 



If, for example, the kingdom of Denmark between the limits 

 of latitude 53° and 58° is transferred in this manner to the 

 surface of a sphere, and W is made = 34° 30', the representa- 

 tion will, for the ellipticity 7 £ 7 , have its linear dimensions near 

 the extreme latitudes increased by only j^Voo* We must 

 content ourselves in this place with having given only a short 

 description of this one method of employing the transfer of 

 figures in higher geodetics, and must reserve a more detailed 

 explanation for another place. 



14. It now remains to take into more close consideration a 

 circumstance which presents itself in our general solution. 

 We have shown in article 5, that there are always two solu- 

 tions ; as P + i Q must either be a function ofp + i q, and P — i Q 

 a function ofp—iq, or P-\-iQ a function ofp — iq 9 and P — iQ 

 a function of p—iq* We will now prove that in the one so- 

 lution the parts of the representation have a similar position 

 as in the original ; whereas in the other they have a reversed 

 position, and we will ,at the same time give a criterion by 

 which this may be ascertained a priori. 



We observe in the first place, that the distinction between 

 a perfect and a reversed similarity can only come into consi- 

 deration if we make a distinction between the two sides of a 

 surface by regarding the one as the upper, and the other as 

 the lower one. As this is something arbitrary in itself, the 

 two solutions are not essentially different, and a reversed simi- 

 larity becomes a perfect one as soon as the side of one surface 

 which was regarded as the lower one is considered as the 

 upper one. This distinction could therefore not present it- 

 self in our solution, as the surfaces were only determined by 

 the coordinates of their points. If this circumstance is to be 

 taken into consideration, the nature of the surfaces must first 

 be established in a manner which shall involve this circum- 

 stance. With this view we will assume that the nature of the 

 first surface is determined by the equation \J/ = 0, where 4> is 

 a given uniform function of x, y 9 z. In all points of the sur- 

 face the value of \J/ will, therefore, vanish ; and in all points of 

 space not belonging to the surface, it will not vanish. In a 

 transition through the surface, vj/ will therefore, at least gene- 

 rally speaking, pass from a positive value to a negative one ; 

 while a transition in a contrary direction will change the ne- 

 gative values of \J/ into positive ones, or on one side of the 

 surface the values of \[/ will be positive, on the other negative. 

 Let us regard the former as the upper, the latter as the lower 

 side. The same may be assumed with regard to the second 

 surface, which is determined by the equation * = 0, where * 



2E2 is 



