MEAN INTENSITY OP TRANSMITTED LIGHT. d 



derived from a consideration of any one point of the area, 

 but from a consideration of the whole area; hence the 

 colour we observe is the mean colour. This, however, 

 although very nearly, will not be exactly the same as at 

 the centre. Suppose G to be the centre of a very small 

 area, which we may denote by ds ; then the quantity of 

 light which passes through this small area towards the 

 eye we may denote by ads. Since the area is very small, 

 and ultimately vanishes, we may consider G F as the path 

 of all the rays passing through this small area and reach- 

 ing the eye. Let G F be denoted by x; then the intensity 

 of the light after passing through the liquid will be ak x ds } 

 k being the coefficient of transmission. For simplicity I 

 shall suppose k the same for every species of light, so that 

 we need only consider one term of the above form. The 

 small element ds we may regard as part of an elementary 

 ring of area 2%rdr ) where r denotes G E. The quantity of 

 light passing through this ring towards the eye will be 

 2axrJc c dr. If, then, we integrate this between limits o and 

 R, and divide by 7rR z , we shall obtain the mean intensity. 

 This will be 



j^-l k x rdr (i) 



Let H be the elevation of the eye above the bottom of the 

 cylinder, and h the height of the column of fluid; also let 

 fi be the index of refraction, the angle of incidence, and 

 & the angle of refraction. Then we have the relationship 



sin = fju sin 0', [2) 



■ r=Atan0-j-(H-A)tan0. ... (3) 



We may also write ( 1 ) in the form 



2a rn 



2// i * 



b2 



