1 12 Prof. Gauss on the Representation of the Parts, fyc. 



This agrees evidently when applied to the earth with Mer- 

 cator's projection, if we make t the geographical longitude, 

 and 90°— a the latitude. For the scale of linear dimensions 



k 



the formulae of article 7 give m = — : — . 



° a sin u 



If we assume for f an imaginary exponential function, and 

 in the first place the simplest of all, fv = k e lv , we have 



/(/ + l log cotang * u) = £e logtang * M + i < = 

 £ tang J z* (cos £ + i sin 2) and X= k tang ^ tf. cos t, Y= & tang 

 ^ u . sin £ which is, as will be easily seen, the stereographical 

 projection. 



If we put more generally fv = ke tXv , we have 



X = k tang \ u . cos Kt , Y= & tang J w . sin A. U 

 For the scale of linear dimensions in the representation, we 

 obtain here n = a 2 sin u 9 , N = 1, $£p = i\ke xXv , and hence 



, A A- tang * m 

 a sin w 



It is evident that the representation of all points for which u 

 is the same, will form a circle, and the representation of those 

 points for which t is constant, a straight line, as also that the 

 different circles corresponding to the different values of u are 

 concentric. This affords a very useful projection for maps, if 

 a part only of a sphere is to be represented. It will then be 

 best to choose A in such a manner as to make the scale the 

 same for the extreme values of u which will make it smallest 

 towards the middle. If we suppose the extreme values of u 

 to be u° and u' 9 we must put 



log sin u'— log sin u° 

 log tang f u'— log tang £ u \ 



The sheets Nos. 19—26 of Prof. Harding's Celestial Maps, 

 are drawn agreeably to this projection. 



11. The general solution of the example given in the pre- 

 ceding article, may be exhibited in another form, which de- 

 serves to be mentioned on account of its neatness. 



In conformity to what has been proved in article 6, we have 



["tang \ u (cos t + i sin t) being a function of t -f i log cotang \ u 



, A T , , . • • .\ sinw .cos t-\- i sin m .sin/ x-\-iy~i 



and tang \ u (cos t + z sm t) = — - =— -— I 



& z v * i + cos u a+z J 



for the general solution likewise these formulas : 



X+iY =/ ^L, X-t Y =/'^ that is, X must be 



u a+z u a-\-z 



made equal to the real, and i Y to the imaginary part of/ 7 ~^- 



■ f f denoting 



