Part b -- Resultant prism formulas

OK, in the first part of the lesson I asked you how you would mark up or verify glasses with the Rx
        OD pl 1.5BU & 2BI 
        OS pl 1.5BD & 2BI

We are going to do this with a little geometry. Look at the right lens. What is it asking for? The right lens needs a combination of 1.5 up and 2 in. Draw a right lens. Now look at what 1.5 up and 2 in is.

What we actually need is the red line on the diagram to the right: we need the result of moving up 1.5 and in 2.

But up and in are perpendicular to each other, so what we actually have is a right triangle, and basic geometry [Pythagorean Rule] says that the long side in the right triangle is the square root of the sums of the squares of the two shorter sides. So the long side, which we need, and which I will call P for PRISM, is the square root of the sum of the two directions, each squared.

OK, OK. Take out your calculator, and do as I do:

 
• Punch in 1.5.
• Press the "x" key.
• Press "+".
• Punch in 2.
• Press the "x" key.
• Press "=".
• Press the "" key.

Couldn't find the "" key? Look carefully. If it is not on one of the keys then it will be above one of the keys, and colored. If so, you have to press '2nd' or 'shift' or 'inv.' and then that key. [You will have only one of '2nd' or 'shift' or 'inv.'.] If your calculator has the 'SIN' key it has this one also, somewhere. If you cannot find it, get out the instructions for your calculator.

What does your calculator show? Here is what I got, step by step:

• Punch in 1.5.                  1.5
• Press the "x" key.       2.25
• Press "+".
• Punch in 2.                      2
• Press the "x" key.            4
• Press "="                     6.25
• Press the "" key.    2.5

And 2.5 is the amount of prism that results from 1.5 up and 2 in.

We still need to know what meridian the 2.5 will be on. If you remember the trigonometry that we did in the math class(!) you will remember that the tangent of an angle is the opposite side divided by the adjacent side. Look at the angle that we need: it is the angle between the horizontal meridian and the line marked P, which is where the resulting prism will be. The opposite side is the vertical portion of the prism, 1.5 in this case, and the adjacent side is the horizontal part of the prism, 2 in this case.

So, the tangent of the angle that we need is vertical divided by horizontal, or 1.5 / 2.

1.5 / 2 = 0.75, and this is the TANGENT of the angle that we need. So, in your calculator:

 
1. Punch in 1.5.         1.5
2. Press "/".
3. Punch in 2.       2
4. Press "=".       0.75
5. Press 'shift' or 'inv.' or '2nd' or whatever your calculator has.
6. Press 'TAN'. If your calculator does not show 36.869....... press "=" and it should.
7. Round to whole degrees. This becomes 37 degrees.

We now have 2.5 BU&I @ 037. Are we through?

No, we are not. We have to go back to the original problem and ask if the result is in the right quadrant. OD Base U&I is what quadrant? 37 degrees is what quadrant?

If the answers are the same, THEN we have the answer. In this case, both are quadrant I, so yes, the answer is 2.5 BU&I @ 037.

Ready for the left lens?

First we have to find the amount of the prism, P. You will do the same thing here that you did for the right lens: "1.5", "", "+", "2", "", "=", then press the "" key. You will get the same answer as before: 2.5.

Next we identify which of the two original amounts is Vertical and which is Horizontal. Up and down are vertical, the left lens has 1.5 down, so V = 1.5. In and out are horizontal, the lens has 2 in, so H = 2. The tangent of the angle is V / H = 1.5 / 2 = 0.75, and so the angle is 36.789... = 37. Same as for the right eye. [Do not assume, however, that the two will always be the same.]

Now we do the last step: for the left lens we started with prism base down and prism base in. What quadrant is this in? Quadrant III? What should the angle be for Q III? 180-270. Is 37 degrees between 180 and 270? No. So we add 180 to it.

180 + 37 = 217. This is between 180 and 270, so this meridian is in Q III.

The result for the left lens is 2.5 BD&I @ 217, or it can be written 2.5 @ 217, or it can be written 2.5 BD&I @ 037. Any of those three answers would be correct. Do you see why one answer still uses the 37 degrees?

What are you going to look for in your lensmeter when you perform final inspection on these glasses? Next time you have access to the lensmeter get out a and do the follow spherical plus lens, anywhere from +2.00 to +5.00, and do the following:

• Focus the eyepiece. Always. Goes without saying. [If it is someone else's lensmeter and that person has it permanently set where (s)he wants it, note where it is set before you start and put it back there when you are finished. If you do not know how to do this ask, or call your instructor.]


• Look in the lensmeter at the reticle. [That is fancy talk for the black rings and lines that are always there.] First rotate the straight line around. Notice that there are degree markings around the top of the reticle? They usually go from 0 to 180, which is why one of our notations is based on 0-180.
• Rotate the line until it is at approximately 37 degrees. Depending on the manufacturer of your lensmeter you may be setting it at the second of the 4 small lines between 35 and 40, or you may be setting it at the seventh small line between 30 and 40. You are reading this scale just as if it were a ruler.
• Look at the rings. Some nice kind manufacturers scribe little numbers on the rings. The smallest ring may be 0.5, then 1, then 2, 3, 4, . . . , or they may start with 1. There may also be a 1.5 ring between the 1 and the 2. Each manufacturer does the rings their own way. If there are no markings you have to go back to the manual that came with the lensmeter [!] and find out what the rings represent. If there are no markings hopefully someone in your office will know.


• Now you are ready for your lens. Set the power drum on the power of the lens and turn on the light. Center the lens physically -- I mean place it in the lensometer (with the plus side toward you) with the lensmeter looking through about the middle of the lens. You should be getting a bright cross-hatch image.  No?  Rotate the power drum a little until the bright image is nice and clear and sharp.  [I am assuming that you have a spherical power lens, as I requested.  I want you to concentrate on prism here without having to deal with cylinder power as well.]  Move the lens until the cross-hatching is in the center of the rings and dot it.  You now have the thickest part of the lens, the optical center, dotted.  Take the lens out and remove the two 'axis' dots -- just leave the OC.
• Put the lens back in the lensmeter and move it until the center of the bright cross-hatch image is on your straight line and between the 2 and 3 rings.  Dot again.
• Look at the relationship between your new dots (keep the three dots, the axis is now important!) and the center dot.  How far away from the center dot is your new center dot?  That new center dot is where you have 2.5 prism base at 37 degrees.  Does the distance you found in mm meet prentice's rule?  Look at where the OC is with respect to the new center dot.  If this were a right lens, BU&I moves the OC U&I, right?
• Now, find a spherical lens with minus power, in the same range.  Do the same exercise with this lens.  Where is the OC with respect to the 2.5 BU&I dot now?
• Move the lens around so that the base is D&O, at 217 degrees.  Look at where the new dot is with respect to the OC of the lens.

OK, lets do another one. Here is the Rx:
      OD +1.00 3 BD & 1.75 BI
      OS +0.50 2 BU & 3.5 BI

How will you dot this in the lensmeter when it is time to do final inspection?

First you must find the resultant prism. For the OD:

1. Using Pythagorean Rule, find the AMOUNT of the resultant prism.

      "3", ", "+", "1.75", , "=", , ["=" if necessary].
Did you get 3.473....? We will round our prism amount answers to one decimal place unless they are exact quarters. So this gives us 3.5
 
2. What is the vertical amount? V = 3 because down is vertical.

 What is the horizontal amount? H = 1.75 because in is horizontal.
 V / H = 3 / 1.75 = 1.71428....
"inv." "tan" [or "second" "tan" or "shift" "tan", depending on what your calculator has] and = if you need to gives 59.7435.... We round the meridians to whole degrees, so this is equal to 60 degrees.
 
3. For the OD we have D & I. What quadrant is OD D&I? Q IV? We need a meridian that is between 270 and 360. Is 60 degrees in that range? No. Q IV will have the same meridian as Q II in 180 notation, and Q II is between 90 and 180. Is 60 degrees between 90 and 180? No.


The angle that we found is the angle between the horizontal and the prism. What we need is the whole rest of the circle. So we will take 360, which is the whole circle, and subtract the 60 that we got from it. 360 - 60 = 300. Our axis is 300. To get 180 notation we would subtract 180 from that, giving 120. So the answer is either 3.5 BD&I at 120 or it is 3.5 @ 300.

OK, what did we end out with for the right lens? 3.5 BD&I @ 120. Now we need the OS, then we can look at what the final inspection will look like.

1. For the left lens we have a horizontal amount of 3.5 diopters [that is the IN] and 2 diopters for the vertical [that is the up.]
2. For the prism AMOUNT we have:

      "2", ", "+", "3.5", , "=", , ["=" if necessary].
I get 4.03... = 4.0 for the AMOUNT of prism in the left lens.
3. For the MERIDIAN we need V / H = 2 / 3.5 = 0.5714....; "inv." tan = 29.744.... = 30 degrees. No, they do not always come out to such nice round numbers.
4. What meridian is OS U&I? It is in Q II. In Q II we subtract the angle from 180. 180 - 30 = 150 degrees. The meridian is degrees.
5. The answer is 4.0 BU&I @ 150

For the OS we need the moveable line on the 150, and the target centered right on the 4 diopter ring.

STEPS FOR RESULTANT PRISM:

Read through the pages 120-125, 127-131 one more time, carefully following the examples, and this time do the exercises.

OK, here are a few more exercises for you to do.

1. OD 6.5BU & 1.5BI
2. OS 1.5BU & 3.0BI
3. OD 3.0BD & 2.25BI
4. OS 2.25BD & 4.0BI
5. OD 4.0BD & 6.5BO
6. OS 6.5BD & 2.75BO
7. OD2.75BU & 0.5BO
8. OS 0.5BU & 6.5BO
9. OD 3.5BI & 3.5BU
10. OS 3.0BI & 4.0BD
11. OS 4.0BO & 1.0BD
12. OD 1.0BO & 1.5BU
13. OD 1.5BU
14. OS 3.0BI