From Huygen's Principle we can derive Snell's Law, that the ratio of the sines of the angle of incidence and of refraction is equal to the reciprocal ratio of the speeds of the wave in the respective materials.

Using Snell's Law we can derive a formula for the focal point of a uniform circular cylinder in terms of the index of refraction of the fluid in the cylinder.

If parallel beams of light travel through air then through a uniform, smooth circular cylinder and again out into the air, with all beams traveling in a plane perpendicular to the axis of the cylinder, they will be focused at a point 'behind' the cylinder. The distance of this point 'behind' the cylinder will be (2-n) * (2n - 2) * R, where R is the radius of the cylinder.

The syrup bottle demo can be used to accurately determine the index of refraction of a material. On the appropriate CD, Class Notes #18 (image formation, etc.) about 2/3 of the way down the page is a discussion of this situation and the result the image of a distant source will form at a distance equal to (2 - n) / (2*n - 2) * R behind the cylindrical container (i.e., the syrup bottle), where n is the index of refraction of the material in the cylinder and R is the radius of the cylinder. We do a little algebra to find n from our measurements.

If we let D = (2 - n) / (2 * n - 2) * R, and use just n instead of n, we can divide both sides by R to get D / R = (2 - n) / (2n - 2). The quantity D / R is easy to find, since we can measure D and R (D is the distance at which the image forms behind the container, R is its radius). Let's call this quantity x (i.e, x = D / R). Again, this is just a number you calculate from your observations. Now, x = (2 - n) / (2n - 2). This is easy enough to solve for n and we get n = (2x + 2) / ( 2 x + 1).

So for example if the radius of the container is 3 cm and the image of a distant light source forms at a distance of 2 cm behind the cylinder:

x = D / R = 2 cm / (3 cm) = .6666, so

n = (2x + 2) / (2x + 1) = (2 * .666 + 2) / ( 2 * .666 + 1) = (1.33 + 2) / (1.33 + 1) = 3.33 / 2.33 = 1.44, approx..

If is easy to perform this experiment and find the index of refraction of a clear fluid. Just fill a cylindrical container with water, then honey, then syrup. Use a light source at one end of the room and a wall at the other on which to form the image. Measure the distance D for each. Measure R for the container. Use the formula to find n for each.

The index of refraction is related to the speed of light in a material, which differs form material to material.  The speed of light in a material depends in complicated and not completely understood ways on things like the molecular and crystalline structure of the material, and is therefore quite complicated, explanations being beyond the scope of this course.