METHOD OF EXAMINING ASTRONOMICAL INSTRUMENTS. 365 



prehending any number of these, accurately determined. Method of ex- 



But if there be an errour of e in the measurement of the *m»»»gthe di- 



visions by ir. 

 first, e 9 e'\ e"', &c, in the measurement of the second, 



third, &c, respectively, then we shall have the following 



equation for determining 0, viz. a -f- (a -f- a' -}- £ + e) -f- (« 



+ a "+e+e') + &c. ... (a + a'"v w ~ *-f e + <?"•••"- A) 

 = A, and consequently a .will appear to be equal to 



A - g'- ^-...q""*- 1 -w-l g-e'-g"-...c /// - * - 1 • 

 which differs from its true value b^-le + e' + e" -f ...» 



„v/ .... w „ i 



• Hence it follows, that the value of the pth arc 



(p being greater than unity,) as deduced by this process, will 



A-tr e -, , i i «£l*e *¥*+*' + ...*"•••/>-* 



diner from its true value by , 1 ' — 



n 



±1 T-g _ g e ///...p- 1 and that if we 



n 

 add any number p of these arcs together, in order to deter- 

 mine the value of the arc which is equal to their sum, we 

 shall have an errour in this value (and the expression holds 

 when p is unity, or the first arc only is taken) equal to p 



JT^T . e + e' -f e" + ... e'-P ~ l +e'"- p + ... e'"- n ~ l 



n 

 Now, if we suppose e to be the greatest errour to which 

 we are liable in the measurement of any arc, and each 

 of the succeeding errours to be equal to it, and likewise 



v 1 



that <?', g",. ..£"'•••* are all negative; then it will appear, 



n — p 

 that — — x 2pe will be the greatest errour that can be 



committed in determining the value of any arc, by adding 

 together the values of the (p) smaller arcs of which it is 

 compounded. For instaace ; if the interval betwixt the 



micrometer 



