SECT. 4.] PLACES IN SPACE. 157 



by means of Z, B, D, a, /9, d, and the coordinates relating to the second and third 

 places in a similar manner, the preceding equations will assume the following 

 form : 



[1] = n (8 cos a -\- D cos Z) it (8 cos a -f V cos Z ) 

 + n&quot; (8&quot; cos a&quot; -f D&quot; cos Z&quot;), 



[2] = w (&amp;lt;? sin a + D sin Z) ri ($ sin a -f- V sin Z ) 



+ n&quot;(T sin &quot; + /?&quot; sin Z&quot;), 



[3] = n (d tan /} + Z&amp;gt; tan ,5) w (d tan + Z&amp;gt; tan Z*) 

 + n&quot; (d&quot; tan 0&quot; -f D&quot; tan Z&quot; ). 



If , /?, Z 1 , Z, Z 1 , and the analogous quantities for the two remaining places, are 

 here regarded as known, and the equations are divided by n , or by n&quot;, five un 

 known quantities remain, of which, therefore, it is possible to eliminate two, or to 

 determine, in terms of any two, the remaining three. In this manner these three 

 equations pave the way to several most important conclusions, of which we will 

 proceed to develop those that are especially important. 



113. 



That we may not be too much oppressed with the length of the formulas, we 

 will use the following abbreviations. In the first place we denote the quantity 



tan /? sin (a&quot; a } -\- tan |3 sin (a a&quot;) -\- tan ft&quot; sin ( ) 



by (0. 1. 2): if, ha this expression, the longitude and latitude corresponding to 

 any one of the three heliocentric places of the earth are substituted for the longi 

 tude and latitude corresponding to any geocentric place, we change the number 

 answering to the latter in the symbol (0. 1. 2.) for the Koman numeral which 

 corresponds to the former. Thus, for example, the symbol (0. 1. 1.) expresses the 

 quantity 



tan fi sin (Z a ) -)- tan /? sin (a Z ) -|- tan B sin ( a) , 

 also the symbol (0. 0. 2), the following, 



tan (3 sin (a&quot; Z) -f- tan B sin (a a&quot;) -f- tan $&quot; sin (Z a) . 

 We change the symbol in the same way, if in the first expression any two helio- 



