SECT. 4.] PLACES IN SPACE. 159 



114. 



By multiplying equation [1] by sin a&quot; tan B&quot; sin L&quot; tan /?&quot;, equation [2] 

 by cos L&quot; tan /3&quot; cos a&quot; tan B&quot;, equation [3] by sin (L&quot; a&quot;), and adding the 

 products, we get, 



[4] = n ((0. 2. II.) d 4- (0. 2. II.) D) ri ((1. 2. II.) &amp;lt;T + (I. 2. II.) Z&amp;gt; ) ; 

 and in the same manner, or more conveniently by an interchange of the places, 

 simply 



[5] = n ((0. 1. 1.) d 4- (0. 1. 1.) D) -f n&quot; ((2. 1. 1.) d&quot; -f (II. 1. 1.) ZX ) 

 [6] = ((1. 0. 0.)&amp;lt;T + (I. 0. 0.)Z&amp;gt; ) n&quot; ((2. 0. 0.)&amp;lt;T + (II. 0. 0.) D&quot;). 



If, therefore, the ratio of the quantities n, n , is given, with the aid of equation 4, 

 we can determine d f from d, or d from d ; and so likewise of the equations 5, 6. 

 From the combination of the equations 4, 5, 6, arises the following, 



m (o.2.ii.)a-f(o.2.n.).p (i.o.o.)y+(i.o.o.)zy (2. ij.) 

 L -l (o. i.i.) 



by means of which, from two distances of a heavenly body from the earth, the 

 third can be determined. But it can be shown that this equation, 7, becomes 

 identical, and therefore unfit for the determination of one distance from the other 



two, when 



B=B =B&quot;=Q, 

 and 



tan F tan &quot; sin (L a) sin (L&quot; L } + tan 0&quot; tan sin (I! a ) sin (L L&quot;} 



-\- tan p tan sin (L&quot; a&quot;) sin (I/ L) = 0. 



The following formula, obtained easily from equations 1, 2, 3, is free from this 

 inconvenience : 



[8] (0. 1. 2.) 8W + (0. 1. 2) Z&amp;gt;cT&amp;lt;r -}- (0. 1. 2) ZWd&quot; -f (0. 1. TLj ff dd 



-}- (o. i. n.) pTTtf -f (o. i. n.) Djyy 4- (0. i. 2) z&amp;gt; w 4- (o. i. n.) DW = o. 



By multiplying equation 1 by sin a tan /?&quot; sin a&quot; tan /T, equation 2 by 

 cos a&quot; tan /5 cos tan 0&quot;, equation 3 by sin (a&quot; a ), and adding the products, 

 we get 



[9] = n ((0. 1. 2) d 4- (0. 1. 2) D) n (L 1.2)I/ + n&quot; (H. 1. 2) 2/ 



