366 Mr. J. M. Wilson on the Readings of the Graduated Arc 

 2. Differentiating the four equations given, 

 cos<^=^cos<£'.^+sin</>', 



cos i/r y- =fM cos-^' -£- + sin ->//, 



d£ d f__ _ d ± &t. 

 d/ju dfju dfx, dfju 



w . dd>' dyjr d& 



Eliminating -£*-> -r-> -j-i 



dfju dfjb dfi 



d/j, cos (f> cos yjr 1 — cosyJrco&(j>' 



This result indicates that when ^> = ^|r and o/r' = <£' nearly, the 

 change in (f> is very large for a small change in //,. 



3. Using special values of /* and a, let the angle of the prism 

 be 60°, and let fi x = 1-6801330. (The reason of selecting this 

 value for //, x will appear presently ; and it cannot differ by any 

 appreciable error from the value given by Miiller for/u E .) Then 



D = 2 sin- » . (-8400665) - 60° 

 = 54° 17' 42", 

 and we have therefore the equations 



sin</>=yu,sin</>', (1) 



sin(114 o 17'42"-<£)=/isin(60-<£'). . . (2) 

 Eliminating </>', we obtain 



^~ *V-sin 2 <£=sin ( 114) ° 17'42"-<£) + \ sin 0. 



Now the sine and cosine of the Z 114° 17' 42" differ by -5, 

 and therefore the left-hand side of the equation becomes 



•9114369 (sin <f> + cos <j>), 

 and therefore using logarithms, 



log Ou, 2 - sin 2 <f>) = -3454309 + 2 log sin (45° + <f>) . . . (A) 



4. It will now be easy to calculate the value of //, for any 

 values of (f>. 



When the deviation is a minimum, 



£=2+30°, and = 57° 7' 51". 



I shall proceed to calculate the value of //. for all values of <j> 

 from 57° at intervals of half a degree, as far as is required by 

 the limits of the spectrum, by means of the formula (A). The 

 subjoined Table gives the results: — 



