SECT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 239 





With the aid of these two equations x and x&quot; can be determined from a , 5 , c , tf, 

 Q . a&quot;, b&quot;, c&quot;, d&quot;, Q&quot;. If, indeed, x or x&quot; should be eliminated from them, we should 

 obtain an equation of a very high order : but still the values of the unknown 

 quantities x , x&quot;, will be deduced quickly enough from these equations by indi 

 rect methods without any change of form. Generally approximate values of 

 the unknown quantities result if, at first, Q and Q&quot; are neglected ; thus : 



j _&amp;lt;! + d&quot; (V -f- Q -f- d d&quot;V 

 1d d&quot; 



_ c 4- d (V + c&quot;) -f d d V 

 x ~ l d d&quot; 



But as soon as the approximate value of either unknown quantity is obtained, 

 values exactly satisfying the equations will be very easily found. Let, for ex 

 ample, be an approximate value of x , which being substituted in equation I., 

 there results x&quot; = &quot; ; in the same manner from x&quot; = &quot; being substituted in 

 equation II., we may have x = X. ; the same processes may be repeated by sub 

 stituting for x in I., another value -\- v , which may give x&quot; = &quot; -(- v&quot; ; this 

 value being substituted in H, may give x = X -\- N . Thereupon the corrected 

 value of x will be 



t [ IS * ) r S &quot; -&quot; y 



and the corrected value of a/ , 



&quot; + jy-v 



If it is thought worth while, the same processes will be repeated with the cor 

 rected value of x and another one slightly changed, until values of x , x&quot; satisfy 

 ing the equations I., II. exactly, shall have been found. Besides, means will not 

 be wanting even to the moderately versed analyst of abridging the calculation. 



In these operations the irrational quantities (x x -\- a a } 1 , (x&quot;x&quot; -\-a&quot;a&quot;} , are 

 conveniently calculated by introducing the arcs z , / , of which the tangents are 



