Theory of Transit Corrections. 261 



It is well known that the time occupied by a star in passing 

 over a given part of the field of view, is not accurately expressed 

 by the product of its equatorial value into the secant of the de- 

 clination. This arises from the inequality of the arcs of differ- 

 ent diurnal circles, intercepted between the same limits. The 

 formula which is strictly accurate, is, in its common form, 



1/ 

 i 



x 



arc . sin (15 . yy . arc V . sec <?) 



15 



Where pe s± time of passage from one wire to another, 



iy— equatorial interval between them. 

 But there is another known form, which, while it indicates the 

 operations to be performed more clearly than the last, is some- 

 times otherwise more convenient for use. It may be thus de- 

 monstrated. 



Let x = arc of a diurnal circle intercepted between two wires. 

 y = arc of equator intercepted between the same. 

 x'~ a part of x equal to y. 

 Then yx = yy . sec S. 



yx 1 ; yx\\x' : x 



yx 



yx 1 . x yy . sec 8 . x 



x ~ y 



But y=z sin#= sinf — M ) -cos S= sin (m.x.seo S). cos S 



™ . cos S 



v 



*, sin 3 . (m . x . sec 8) 

 AnQ - arc (m . x . sec 8) = sin (m.x. sec <J+ ^-^ +etc. 



y 3 . sec 3 8 

 y . sec <?-f- — ^ + etc. 



But arc (m . # sec 8)=x . sec 8 



y 3 . sec 2 8 

 .'.x=y-\- — ~— o — + etc - 



/ y 2 . sec 2 8 

 And yx=yy . sec 8 ( 1-f — g-g — 



=yy . sec <*+37 . 5 . arc 3 1" . yy 3 . sec 3 8. 



The two corrections of the time for error of collimation and 



Nation of the transit axis from the plane of the prime vertical, 



jra given respectively by the expressions c . sec . 8 and a . sin 



Vg~-*).sec 8 ; so that the equated error of the clock becomes 



It is 



», / , ov mat u^ v,^.*^- ~~~- 



*r - 7 - a . sin (<*>-<*). sec 8— c.sec 8. In 



l «e latitude of the place of observation, M tne ngm as™ 



as far, and a, c, the deviation and error of collimation 



P'ain that the values of a and c might be determined fiw, «., 



fnree equations of the error of the clock, but this method is much 



le ss accurate, on account of errors of observation, than another 



w hich is snm t™ 01 , «r«T^,vi7^ This latter method may be ex- 



