Proof of Prentice's rule.

Ray #1 travels through the optical center of the lens, perpendicular to the sides. At this point on the lens the sides are parallel, so the ray is perpendicular to both sides and is not deviated.

Ray # 2 travels parallel to the optical axis of the lens, which is also ray # 1, and enters the lens c cm away from the optical axis of the lens. Because the sides of the lens are angled toward each other at this point the ray is deviated. Because the lens is a plus lens ray # 2 travels toward ray # 1, and at some point crosses ray # 1. The point where ray # 2 crosses ray # 1 is f meters away from the lens.

By definition the amount of prism, P, that is present at the point C on the lens is therefore

   c
P = -------
   f

because the ray # 2 was deviated c cm at a distance of f meters.

But f is the focal length of the lens, and therefore D, the dioptric power of the lens, is equal to 1/f. Substituting 1/D for f in the equation above, we get

   c
P = -----
   1/D
so
P = c D
where c is in cm. Since d, the displacement in mm, is = to c*10, or c = d/10, we get the form of the equation that I prefer,

   d D
P = --------
   10

where
   P = prism present at a point on the lens,
   d = distance from the point to the optical center in mm,
   D = dioptric value of the lens on the meridian through the point.
 


 
 
 

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