the Divisions of astronomical Instmmenls, 281 



the arcs themselves. Lei a denote the real length of the 

 first of these, and ±a\ ±a'^ ±«'", &c., the difference 

 betwixt the first and second, the lirst and third, &c. re- 

 spectively ; let A represent any other arc whose length is 

 known, and which is a multiple of a, as marked upon 

 the instrument, and let this muUiplc be expressed by n. 

 Then will a -f {a + g') + (a + a") + (a + o!'') + &c. ... 



(a 4. a:'' "~') = A, and a = A-a--." - .^-^'-^^ ^^^^^ 



it is evident, that if there is no error committed in t!i» 

 measurement of any oF these arcs, we shall have the value 

 of «, and consequently of a -f a\,a -f af^ y a -\- a'/\ Sec, and 

 of any arc, comprehending any number of these, accurately 

 determined. But if there be an error of e in the measure- 

 ment of the first, of e\ e'\ e^\ &c., in the measurement 

 of the second, third, 8cc., respectively, then we shall have 

 the following equation for determining «, viz a + {a -^ a 



-\-e +e) + (a + a'' + e 4- e'') + &c {a + a'"*"""~ ^ + e 



+ ef'^'" ) = A, and consequently a will appear to be 



equal to , which 



differs from its true value by — 



1 p + f' + e" + .. 



....n-1 



Hence it follows, that the value of the p^^ arc (p being 

 greater than unity), as deduced by this process, will differ 



irom Its true value by -. ■■ 



^e—e^^^'"^^" , and that if we add any number p of these 

 arcs together, in order to determine the value of the arc 

 which is equal to their sum, we shall have an error in this 

 value (and the expression holds when p is unity, or the 



first arc only is taken) equal to p 



,„.... p „,...« — 1 



n — i .e -k- e -^ e + 



p^],e-e'-^e''--,„e"'-'^- 



'•— e— f— ... e ' + V-f ^ + e ^ + ... e , NoW, 



11 



if wc suppose e to be the greatest error to which we are 

 liable in the measurement of any arc, and each of the sue- 

 cccdiner errors to be equal to it, and likewise that e', ^^ 



