194 



PROCEEDINGS OP THE AMERICAN ACADEMY 



be d ; and if R and £ are the polar co-ordinates of any point of the 

 limb of this ima^e, its rectanofiiUxr co-ordinates, referred to its axes 

 and approximate centre will be 



y' = R sin (e ^ Q) 

 x' = R cos (e =F G) 



(10) 



in which the upper signs are to be taken when the sun is west, and 

 the lower when it is east, of the meridian. 



Owing to atmospheric disturbances, the image of the sun's limb will 

 generally be more or less irregular, and no matter liow carefully it 

 may be centred on the measuring engine, the point from'which the 

 measures are made will seldom coincide quite accurately with the 

 centre of the image. On this account, to obtain the co-ordinates of 

 any point of the limb referred rigorously to the centre and axes, the 

 equations (10) must be corrected by the small quantities 8y and dx, 

 and thus we get 



y = Rsm (£ T 6) + %' 

 X = R cos (e =F ^) + ^^ 



(11) 



The image of the sun's limb being taken as an ellipse, its equation 

 will be 



= — A'B'' + AY + ^x^ (12) 



where A and B are respectively the horizontal and vertical semi- 

 diameters of the sun, affected by refraction and parallax. Putting 

 ^' = nJt^ substituting this value in equation (12), and dividing by 



A% we get 



= — nA'- -{- f -{- nx^ (13) 



Finally, substituting the values of y and x from the equations (11), 

 neglectmg the squares of dy and 8x, and reducing, we obtain 



+ 1 [sin^ (e rp ^) + ^ cos'^ (e ^^ 6)]R' 



= <! _ ^ ^2 + sin (fc- qp 0) 8y 

 -\- n cos (s T 6) dx 



(14) 



Each measurement of the sun's limb, made upon the photograph, 

 gives one equation of the form (14), and from all the equations thus 



