46 Wadsworth — Simple and Accurate Cathetometer. 



A = t sin cp . I 1 — 



cos q> 



,] 



\Zn 2 — sin'cp- 



^> being the angle of rotation measured from the position in 

 which the plate is normal to the ray. This expression may be 

 written : 



A = 



n . 



t tan 



g( p. |(i jcos<? W C0S <p i 



n — 1 



rc 



£ tang (?./(» 



The quantity in brackets or f(<p) may be shown to be very 

 nearly unity for all values of <p between and 30°. In order 

 to determine its exact value we may develop it into a series as 

 Poynting does, but since this series is only rapidly convergent 

 for low values of <p, it is on the whole better to compute it 

 directly from (1), which is in a form well adapted to logarithmic 

 computation. I have calculated the values of f{<p) for values 

 of <p from 5° to 30° and for two values of n, viz : n = 1*5 and 

 n = 1*55, about the mean indices of the glass most likely to be 

 used for this purpose. These values are given in the following 

 tables : 



15 



Table I. 

 n — l /n =i= 0-3333. 



<p 



tang (p 



n 



/(*)= 



1 + rf* 



\rs = tang a tang 



rs—o 



5° 



•02916 



1-00044 = 



1 + -00044 



+ •0011 



+ •00066 



10° 



•05878 



1*00162 = 



+ -00162 



+ •0021 



+ •00049 



15° 



•08932 



1-00333 



+ -00333 



+ •0033 



•ooooo 



20° 



•12132 



1-00528 



+ •00528 



+ •0045 



— •00078 



25° 



•15544 



1-00681 



+ •00681 



+ •0057 



— •00111 



30° 



•19245 



1-00708 



+ •00708 



+00708 



+ •0000 



n = 1-55 



Table II. 

 n— 1 



= -35484 







tang (p 



n 



/(*) 



1 + 6 



5° 



•03104 



1-00024 



1 + -00024 



10° 



•06257 



1-00085 



+ •00085 



15° 



•09508 



1-00162 



+ •00162 



20° 



•12915 



1-00216 



+ •00216 



25° 



•16547 



1-00188 



+ •00188 



30° 



•20487 



1-00002 



+ -00002 









~-F A ,"c n .*.s™/<l» 



* [In a similar table given by Poynting, the sign of 6 is erroneously written 

 negative (probably a typographical error), and there is also a small error in the 

 value of 6 for (p — 10°, which is however unimportant since 6 itself is so very 

 small for this angle.] 



