Theory of Transit Corrections. 2G3 



be three equations of the error of the clock, the transit axis being 

 assumed perfectly horizontal. Then if c is sought, the coef- 

 ficient of one part of the effect of the errors of observation on 

 its value is 



1 



{n—n') — (n'—n''). {m-m') 



{rnf — m'') 



But n—n' is always a proper fraction in our latitude for instru- 

 ments having only a southern exposure, and is in fact as far as 



the latitude whose tangent is ^37 The same is true of »'—*", 



TV — tlf* 



m-m', and ?n'-m". But —. - is also a fraction, since the 



ml - m' J 



numerator is a less fraction than the denominator. 



For m' - **'**= sin (<p —8') . sec 8' - sin (<p - $") . sec 8" 



(tan 8' -tan 8") 

 n - n" = sec 8 9 — sec 8 f/ . 

 Differentiating cos q> . tan 8" we have cos q> . sec 2 *'' . d8'\ 

 and sec S" we have sin <5" . sec 2 8" . d8". 

 Hence we have cos g>.sec 2 *".€&"> sin <J".sec 2 <J".d*" when 

 '"< 90° - 9. From this we have 



cos <p . (tan 8' — tan d")>(sec 5' — sec 8''). 



1 he same is true when one of the declinations is negative, since 

 observations will not be made near the horizon. Then 

 m - m'' > n' — n" when the greatest declination is less than the 

 implement of the latitude of the place of observation. Hence 



« follows that the reciprocal of (n -n')- ,_ „ m (™-m'), 



which is the coefficient of one part of the effect of the errors of 



observation, is integral when the transit has only a south ex- 

 posure. 



If the instrument commands the whole meridian, we may 

 s how that the coefficient of one part of the effect of the errors of 

 °bservation, cannot he largely fractional without making the co- 

 integral. The denominators of the two 



— *weiu oi tne other part 

 Efficients are 



m—m 



n" -n' n-ri x 



(m' — m") 



m' f — m' m-m' 

 n" -ri n — n' , 



m" -m' m-m' 



l he first expression is largely integral, m-m' must be so also, 



can 



n f ._ n t S ec 8" — cf 



m"—m'~~tan #".cos y-c 



u 



