OF ARTS AND SCIENCES. 199 



To test the amount of inaccuracy introduced in equation (79), by 

 writing for 0^ in (75), and omitting ^ in (78), the rigorous expres- 

 sion for 8r, involving 6, d^, and jW, was formed ; and, upon substituting in 

 it the numerical values of these quantities for the extreme case where 

 the sun has a zenith distance of 80°, the numerical coefficient in (80) 

 became 0.374. This differs so little from the value before found as to 

 lead to the conclusion, that equation (79) is probably as accurate as 

 the refraction tables themselves. 



We have thus found the values of 8i/, dx, and dr, and it only remains 

 to deduce from them the expressions for H sin e and jR cos e ; i? and e 

 being respectively the radius vector and angle of the image of the true 

 sun's centre, referred to the original system of polar co-ordinates given 

 by the measuring engine. An inspection of Fig. 4 shows that 8r is 

 measured along the line S'^^v', while 8i/ and dx are measured along 

 lines respectively parallel to, and perpendicular to, S' ^Z. Hence, as 

 the angle e is measured from a line at right angles to S'^Z, and 0^ is 

 the angle ZS'^S', 



H sin e = 8y -\~ 8r cos 0^ ) 



H cos s =z 8x ^^ 8r sin d^ ) 



in which 8r must be regarded as essentially negative, d^ must always 

 be taken less than ninety degrees, and the double sign must be under- 

 stood in the same way as in the equations (10). It is, perhaps, 

 scarcely necessary to add, that these are the values of li sin s and H 

 cos s, which must be employed in the equations (18). 



To facilitate the computation of 8r from equation (80), it is desira- 

 ble to have a table giving the numerical values of the second deriva- 

 tive of the refraction. To form such a table, we assume Bessel's 

 expression for the refraction ; namely, 



r= atan C (82) 



where r is the refraction, expressed in seconds of arc, and a is a quan- 

 tity which varies slowly with the zenith distance, (^. The first deriv- 

 ative of this expression is 



which becomes 



dr 1 



d( cos- C ^ ■' 



