= (— 



\i+» 



210 Prof. Gauss on the Representation of the Parts 



In order to ascertain for which latitude m has its greatest 

 or smallest value, we have 



dm TT JTT , . & cos u. sin u.du 



= cotang U . aU — cotang co . dco ^ ■ — 



m ° ° 1— «*cos«* 



rf U, d * «* sin « . d u ( 1 — t 2 ) f/ w 



sin U sin « 1 — i^cos*^ ~~ (1 — t'*cos v*) sin a* 



and hence, — r= . ( *"" a ., ■ (cos U — cos co). 



wt sin 4/(1— t 2 cos» i v ' 



It is clear from this that m obtains its greatest or smallest 

 value when U = co; if we denote this value of w by W, we 

 shall have 



l-«cok W\*« - 17 . \-k7 , , . , ~ XT 



rrr ) » Ol', COS W = by which W 



cos vv / * J 



'(i+tO 



may be determined if & is calculated by the above formula. 

 In practice, however, the perfect equality of the values of m 

 for the extreme latitudes will be of little moment, and it will 

 be sufficient to take for 90 — W nearly the middle latitude, 

 and to derive k from it. The general connexion between U 

 and u) is given by this equation 



i TT , i ( (l-«cosW)(l+scos«) )*' 



tang $ U as tang £ m. I j— '), 7 \ 



I ° I (1+* cos W) (1 — i cosu) ) 



For a real numerical calculation it is, however, more advan- 

 tageous to apply series which may receive different forms, but 

 the development of which we shall not here stop to investigate. 

 As it will be easily seen that for co< W, will be U>co, 



d m 



therefore, cos U— cos co, and consequently likewise nega- 

 tive, and that for a>>W we have U<co, and consequently 

 -^- positive, it is evident, that for 00 = U = W the value of 



m is always a minimum and = — ^ (1 — e 2 cos W 2 ). If the 

 radius of the sphere A is therefore assumed 



V^-^cosW')' 



the representation of infinitely small parts of the ellipsoid in 

 latitude 90 — W is not only similar, but also equal to the ori- 

 ginal, but in other latitudes larger. 



The logarithms of m may be developed with advantage in 

 a series of ascending powers of cos U— cos W, the first terms 

 of which will be sufficient for practice, and are as follow : 



Log hyp. m = log j ~ y^l - £ 2 cos W 2 ) \ + £vkl>y ( C0S U 



-cos W) 2 - ggjjk (cos U-cos W) 3 



If 



