Theory of Transit Corrections. 259 



is probably less than - - For if the numerator of the first ex- 



IV 



pression is not probably greater than that of the second while its 

 denominator is greater than the denominator of the other, then 

 the first expression is probably the smallest. But there is no 

 probability that the numerator of the first is greater than that of 

 the second, since there is as much chance that e will be negative 

 as positive. 



As the probable reduction of the error of observation is in pro- 

 portion to the number of wires, it becomes important to connect 

 as many with the instrument as possible. All the advantage, 

 however, of an additional wire to the transit, in the use of the 

 common method of taking a mean, may be gained by adding the 

 times at all the wires to the time at the middle wire and dividing 

 by the number of wires increased by one. This method gives 

 a result probably as accurate as that which would be given by 

 the other with a new wire added to the instrument. Let 

 error at the middle wire and e'= sum of errors at all the wires. 



Then according to the method just stated, ZTTi is the mean er- 

 ror : a result just the equivalent of that given by the common 

 method with an additional wire, inasmuch as there is no proba- 

 bility that the error at an additional wire will be more favorable 

 m respect to amount and sign to the diminution of the numera- 

 tor of the fraction expressing the mean, than the error at the mid- 

 dle wire. Indeed, the error at the middle wire is probably less 

 than at an extreme one. 



When the intervals between the wires of the transit are une- 

 qual, a correction must be applied for this inequality. It is given 

 by eminent authority, that this correction is most perfectly applied 

 by reducing the time at each of the wires separately to the mid- 

 d| e wire and then taking a mean, thus requiring as many separate 

 eductions as there are wires less one. But this process is tedious, 

 jnd may be avoided with advantage in the following manner. 

 * >nd the place of the mean of the wires. The product of the 

 equatorial distance of this from the middle wire into the secant 

 ot the declination, applied with its proper sign to the time of 

 be transit over the mean of the wires, gives the time of the 

 transit over the middle wire. The time of the passage over the 

 ^eaii is found by dividing the sum of the times of the transits 

 °y the number of wires, and found with as much probable accu- 

 ^y as would be connected with the time over the middle wire, 

 obtained in the usual manner, when the intervals are equal. 

 et '< ~ true time at the mean of the wires. 



d > d x,d 3 , etc. = equatorial distances of the wires from the mean. 



e > e i j e 3 , etc. = errors of observation at the wires. 

 = declination of the star. 



