OF ARTS AND SCIENCES. 195 



obtained the values of A, 81/, and dx are found by the method of least 

 squares. 



The contraction of the sun's vertical serai-diaraeter on account of 

 refraction will be J (r' — r ') ; where r' and r" are respectively the 

 refraction of the lower and upper limbs. The contraction of the 

 horizontal semi-diameter on account of refraction, for all zenith dis- 

 tances less than 85°, may be taken as constant and equal to 0".25. 

 Hence, if s represents the sun's semi-diameter, we have 



/s-iir'-r")y 



which is the value to be employed in equation (14). 



As changes of refraction are not strictly proportional to changes of 

 zenith distance, the centre of the sun's image will not coincide rigor- 

 ously with tlie image of the sun's centre. Let the distance between 

 those two points be 8r, and let r'" be the refraction of the sun's centre, 

 r" and r' being respectively the refraction of his upper and lower 

 limbs, as before. Then 



8r = I {r' -\- r") — r'" (16) 



and the co-ordinates of the image of the sun's centre are 81/ -\- 8r, and 

 8x. These rectangular co-ordinates are transformed into polar co- 

 ordinates of our original system by means of the formulae 



li=[(8r/+8ry+{8xy]i] 

 Sy + Sr 



s>n V = ^ 



5.r 

 COS J? = - 



(17) 



where the double sign is to be taken in the same way as in equa 

 tious (10). 



The polar co-ordinates of the image of the centre of the apparent 

 sun have thus been found ; and our original measurements gave the 

 polar co-ordinates of the centres of the image of the apparent Venus 

 and of the photographic plate. Let i?, H', R" be, respectively, the 

 radii vectores, and s, «', e" the angles of these co-ordinates. Passing 

 now to a system of rectangular co-ordinates whose origin is at the 

 centre of the plate, and whose axis of X is parallel to the fixed 



