208 Prof. Gauss on the Representation of the Parts 



From this expression may be immediately derived the for- 

 mulae for tliis case corresponding to those above given for the 

 particular suppositions made for the function f. 



In the first supposition we shall have to take 

 X = kt 9 Y=z k log cotang £ # -f- »j k arc tang y . cos » 

 In the second case 



X = k tang i **«jr "$« ten S ■ • cos » . cos A t 



Y = k tang ico X .e- vXaTCt&net, ' :coS6 ' .sin \t. 



13. Lastly, we will consider the general representation of 

 the surface of an ellipsoid of revolution on the surface of a 

 sphere. For the latter we will retain the solution of the pre- 

 ceding article, and put the radius of the sphere = A, and 

 X = A cos T . sin U, Y = A sin T . sin U, Z = A . cos U. 



Applying the general solution of art. 5. we shall find, f de- 

 noting an arbitrary function, that T must be made equal to 

 the real, and i log cotang \ U to the imaginary part of 



/(^noglcotang^^^i^fl)*. 



The supposition fv = v will give the simplest solution, by 

 which will be 



T = Man g 4U = tangJ a) .(i±if^) i ' 



This presents a transformation exceedingly useful in higher 

 geodetics ; on the application of which we can, however, give 

 in this place only a few short hints. If we regard as corre- 

 sponding points on the surface of the ellipsoid and the sphere 

 those which have the same longitude, and whose latitudes 

 90°— U, 90°— on respectively are connected together by the 

 above-given equation, we shall have for a system of compara- 

 tively small triangles (as those which can serve for real measure- 

 ment must always be) which are formed by shortest lines on 

 the surface of a spheroid, a corresponding system of triangles 

 on the surface of the sphere whose angles are exactly equal to 

 the corresponding ones on the spheroid, and the sides of which 

 deviate so little from arcs of greatest circles, that in most cases 

 where the very extreme of accuracy is not required, they may 

 be supposed to coincide with them ; and even where such 

 extreme accuracy is required, the deviation from parts of 



* We pass over the second solution of art. 5. which is distinguished from 

 the above by a substitution of — T for -f-T only, and which would give a 

 reversed representation ; and we pass likewise over the case of an oblong 

 spheroid , which, agreeably to the treatment of the analogous case in the 

 preceding article, results immediately from that of the oblate spheroid. 



greatest 



