GEODESIC INVESTIGATIONS. 175 



And as cot- h I,, must be positive, it is evident that the radical in eqiiatiou fjj 

 must have the sign -f- before it, and that U must also be positive. More- 

 over, from this and the fact that cot^ i I, is greater than cot- ^ l^^ (or from 

 the relation, cot- i I, • cot^ i I,, = M) it is evident that the radical in 

 equation (i) must have the sign -j- before it. 



Having found l„ l,„ from the equations d, e, f, g, h, i,j, we have 



V = 90° — I 



I" = 90° — I,, 



The various other entities can be found as in problem 1 . 



^^ If we were to assume !>,, = A,„ so as to find l„ from the triangle 

 S,PD", the result would be too small (&.•. the latitude I" too great) by the arc 

 S^, , which amounts to about 6 seconds in the case worked out in my last paper. 

 (See the last note to problem 5.) 



The equations — 



cos i {I, — Q ^ cot i (A' + A" ) 

 coa ^ (I, -}- Ij,) tan | « 



cot^^.+ p 



sin2 A" 



can be put under the forms : — 



, j^ cot i I, cot i Z/, -f" 1 cot i A" cot ^ A' — 1 



'^^ ^ '^ ■ cot I I, cot \l„ — l ~ cot i A" 4- cot h A! 



ctM ^„ { MH ;,-!)■ + 4 g.cofiZ,} ^ „„,,,^,„^„„^,^^). 

 cot'i !, ((cotH!„-lf + 4p.cotH!") ""tH -1" (1 + ootH ^r 



It is evident that by knowing any three of the iive entities V , I", A', A", 

 w, we can, by solving these equations, find the remaining two.* 



1^^ In determining entities, all correct methods available should be 

 employed, and the means of the results taken in order to ameliorate the 

 discrepancies which inevitably occur, owing to imperfections in the data. Want 

 of space has compelled me to omit many useful improvements and additions 

 of this character in the solutions of the preceding problems. 



For the same reason I have omitted expressions for the magnitudes of the 

 various straight lines of the figure, and the angles they make with each other ; 

 and also the relations or formulae derivable from the anharmonic divisions 

 and pencils. 



To establish the lengths and positions of great Base-Lines more accurately 



than by the usual method of triangulating from a short Base measured 



by means of rods and bars : — 



Select three mutually visible stations Si, S^, <?.■,, no two of which shall difi'er 



in longitude by less than half of a degree, or be of a less distance asixnder than 



50 miles. (The greater the difference in longitude, the better as respects the 



two more distant stations Si and S3). 



* If I', A', u ^® ^^® given entities, we can easily find X,„ z„ D„ from the triangle 

 D,P S,. And by equating the expressions for cos CFl) in the two triangles D„PI, SyPI, 

 ■we can find the value of tan A 2'- Then by means of formula (25) we can find A" ; and 

 by means of formula (4) or (30) we can find I". &o., &e. 



