of using tJte Theodolite, 179 



sible way ; let i be the number of readers, a- p* will occur 



(n — 1) (n — 2) (n — J3) (m — i + 1) times, Vfh\\e2a^pq 



will occur 2 (n — 2) (n — 3) (n — i + 1) times; hence the 



entire sum is 



(»— 2)(n— 3) (n—i+l) {(«— 1)2««. 2^«4- 4 2«/5.2/)9} 



and thus 



VS^/ n(»-.l) n(»— 1) (2«)« 



In our case a = 5, /3 = 3, 7=:2, 6 = 2, 4 = 4; hence 



^^^ - 24». 7n-7 ^^ 



24 7» — 7 ^ ^^ 

 that is, since w is a very large number, the measure of inaccu- 

 racy by this method is ^ of that by one reading. Now had 



the four readers been arrang^ at 90° apart, and their simple 

 average taken, the measure of inaccuracy would have been 



^ or 24, so that g part is added to the chance of error by* this 



improper arrangement ; while the labour of averaging is much 

 increased. 



It is well known, however, that an observation with the 

 theodolite is only complete when the telescope has been reversed 

 in its V*s, and the readings again taken. Let us then seek for 

 the measures of inaccuracy of the different arrangements of 

 readers, supposing the reversion to be used. If the readers be 

 so arranged that, on reversion, they come opposite new portions 

 of the limb, the chance of error will be diminished one-half; 

 but if their number be even, the chance of correcting the errors 

 of graduation is only improved if they be unsymmetrically ar- 

 ranged. It is then highly improper to have an even number of 

 readers uniformly distributed ; for example, if we have four, 



the error on reversion is | (e^y ; while if three verniers only had 



been used, the chance of error would have been diminished to 



6 (^i)* ' ^^^ errors of collimation and verticality being corrected 



