4 
to the circumference. For colorimetric measurements we 
take the vertical length of the column of liquid. In what 
follows I propose to consider if this would lead to any 
noticeable error. Our impression of colour will not be 
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 GF as the path of all the rays passing 
through this small area and reaching the eye. Let GF 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? rrdv, where 
r denotes GE. The quantity of light passing through this 
ring towards the eye will be 2 y (&rk x dr. If then we inte- 
grate this between limits 0 and ft, and divide by ttE 2 , we 
shall obtain the mean intensity. This will be 
~ J k x rdr - (1) 
o 
Let H be the elevation of the eye above the bottom of the 
cylinder, and li the height of the column of fluid ; also let 
ju be the index of refraction, d the angle of incidence, and d' 
the angle of refraction. Then we have the relationship 
sin0 = /xsin0'... (2) 
r = ht an0 + (H - h) tan0' (3) 
