of a given Surface on another given Surface. 209 



greatest circles may be calculated with any accuracy that may 

 be necessary by simple formulae. It is therefore possible to 

 calculate the whole system, one side of a triangle having first 

 been duly transferred to the spherical surface, by the angles, 

 entirely as if the whole were on the sphere itself, with the 

 modification just indicated, if necessary ; for all points of the 

 system the values of T and U may be determined, and we 

 may go back from the latter to the corresponding values of eo 

 (in the simplest manner by an auxiliary table of very easy 

 construction). A triangulation never embracing more than 

 a very moderate portion of the surface of the earth, the 

 above-mentioned purpose may be still more perfectly accom- 

 plished if we generalize the preceding solution by putting 

 fv = v + const, instead of fv as v. Clearly nothing would 

 be gained by this supposition, if a real value were assigned to 

 the const, as T and t would then only differ by this constant 

 quantity ; and consequently, the points from which the longi- 

 tudes are counted would only be different. But the result is 

 very different, if we assign to the constant quantity an ima- 

 ginary value. If we put it =s i log k, we have 



T = t, tang J U = k tang \ m . (>£££)* 



In order to decide on the appropriate value of k, it is neces- 

 sary first to determine the scale of the representation. 



In conformity to the notation of articles 5 and 6, we have 



n = a 2 sin u 2 , N = A 2 sin U 2 , $v — 1 



, A sin U A sin U , , _ „ „x 



hence m = -. — = — : . v (1— s cos or) «= 



a . sin u a sin u v 



A ft(l-£3cosfi>3)£-H* 



a cos§«' 2 (l — i cos*.) 8 -f-&2 sin£«2(i_j_ £ cosa/) e 



The scale, therefore, depends on the latitude only. The 

 smallest possible deviation from perfect similarity is obtained 

 by such a determination of k as will give equal values of m 

 for the extreme latitudes, by which the value of m for the 

 mean latitude will be nearly a maximum or a minimum. If 

 we denote the extreme values of m by co° and a/, we obtain in 

 this manner 



COS£»°2(l-£COSft/°) S COS$*/3(l -i COS*/) 6 



, (1 - i» cos fli°Q § + § ' (l-s^cosa/^i + f ' 



~~ sin§o^(l-|-£COS*/) £ sin§6^(l-f£Cos«°) £ 



(l-t^cos*")^* 1 (l-s'cosa^f+i* 



New Series. Vol. 4. No. 2 1 . Sept. 1 828. 2 E In 



