SECT. 3.] ANY NUMBER OF OBSERVATIONS. 261 



is not even necessary that the functions V, V, V&quot;, etc. relate to homogeneous 

 quantities, but they may represent heterogeneous quantities also, (for example, 

 seconds of arc and time), provided only that the ratio of the errors, which might 

 have been committed with equal facility in each, can be estimated. 



180. 



The principle explained in the preceding article derives value also from this, 

 that the numerical determination of the unknown quantities is reduced to a very 

 expeditious algorithm, when the functions V, V, V&quot;, etc. are linear. Let us 

 suppose 



V M = v=. m -\- ap -f- bq -(- cr -\- ds -J- etc. 

 V M ^v = m -f a p 4- b q -j- c r -\- d s -f etc. 

 V&quot; M&quot;=v&quot;= m&quot;+ ap + b&quot;q + c&quot;r -f d&quot;s + etc. 

 etc., and let us put 



av-\- a v -f a&quot;v&quot; -f- etc. = P 

 Iv -\- I v -f l&quot;v&quot; + etc. = Q 

 cv -f c v -j- c&quot;v&quot; -\- etc. = R 

 dv -f d v -J- d&quot;v&quot;-\- etc. = 8 



etc. Then the v equations of article 177, from which the values of the unknown 

 quantities must be determined, will, evidently, be the following : 



P = 0, Q= 0, ft = 0, S 0, etc., 



provided we suppose the observations equally good ; to which case we have shown 

 in the preceding article how to reduce the others. We have, therefore, as many 

 linear equations as there are unknown quantities to be determined, from which 

 the values of the latter will be obtained by common elimination. 



Let us see now, whether this elimination is always possible, or whether the 

 solution can become indeterminate, or even impossible. It is known, from the 

 theory of elimination, that the second or third case will occur when one of the 

 equations 



P 0, Q = 0, R = 0, S = 0, etc., 

 being omitted, an equation can be formed from the rest, either identical with the 



