Proportions of curvature of the lenses which form an achromatic object-glass.
As some ingenious mechanics may feel a desire to attempt the construction of a compound achromatic object-glass, I shall here state some of the proportions of curvature of the concave and convex lenses, which serve to guide opticians in their construction of achromatic instruments. These proportions are various; and even when demonstrated to be mathematically correct, it is sometimes difficult to reduce them to practice, on account of the different powers of refraction and dispersion possessed by different discs of crown and flint-glass, and of the difficulty of producing by mechanical means, the exact curves which theory requires. The following table shows the radii of curvature of the different surfaces of the lenses necessary to form a double achromatic object-glass—it being supposed that the sine of refraction in the crown-glass is as 1.528 to 1, and in the flint as 1.5735 to 1; the ratio of their dispersive powers being as 1 to 1.524. It is also assumed that the curvatures of the concave lens are as 1 to 2, that is, that the one side of this lens is ground on a tool, the radius of which is double that of the other. The 1st column expresses the compound focus of the object-glass in inches; the 2nd column states the radius of the anterior surface of the crown, and column 3rd, its posterior side. Column 4th expresses the radius of the anterior surface of the concave lens, and column 5th its posterior surface, which, it will be observed, is exactly double that of the other.
| Focus in inches. | Radius of anterior surface, convex. | Radius of posterior surface. | Radius of anterior surface, concave. | Radius of posterior surface. | ||||
|---|---|---|---|---|---|---|---|---|
| Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | |
| 12 | 3 | 4. | 652 | 4. | 171 | 8. | 342 | |
| 24 | 6 | 9. | 304 | 8. | 342 | 16. | 684 | |
| 30 | 7. | 5 | 11. | 063 | 10. | 428 | 20. | 856 |
| 36 | 9 | 13. | 956 | 12. | 513 | 25. | 027 | |
| 48 | 12 | 18. | 608 | 16. | 684 | 33. | 369 | |
| 60 | 15 | 23. | 260 | 20. | 856 | 41. | 712 | |
| 120 | 30 | 46. | 520 | 41. | 712 | 83. | 424 | |
From the above table it will be seen, that to construct, for example, a 30 inch compound object-glass, the radius of the anterior side of the crown must be 7½ inches, and that of the posterior side 11.63 inches; the radius of the anterior surface of the concave 10.428, and that of the posterior 20.856 inches. It may be proper to observe, that in these computations, the radius of the anterior surface of the concave is less than the posterior side of the convex, and consequently admits of its approach, without touching in the centre—a circumstance which always requires to be guarded against in the combination of achromatic glasses. The following table shows the radii of curvature of the lenses of a triple object-glass, calculated from formula deduced by Dr. Robison of Edinburgh.
| Focal length. | Convex lens of crown glass. | Concave lens of flint glass. | Convex lens of crown glass. | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Inches | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. |
| 6 | 4. | 54 | 3. | 03 | 3. | 03 | 6. | 36 | 6. | 36 | 0. | 64 |
| 9 | 6. | 83 | 4. | 56 | 4. | 56 | 9. | 54 | 9. | 54 | 0. | 92 |
| 12 | 9. | 25 | 6. | 17 | 6. | 17 | 12. | 75 | 12. | 75 | 1. | 28 |
| 18 | 13. | 67 | 9. | 12 | 9. | 12 | 19. | 08 | 19. | 08 | 1. | 92 |
| 24 | 18. | 33 | 12. | 25 | 12. | 25 | 25. | 50 | 25. | 50 | 2. | 56 |
| 30 | 22. | 71 | 15. | 16 | 15. | 16 | 31. | 79 | 31. | 79 | 3. | 20 |
| 36 | 27. | 33 | 18. | 25 | 18. | 25 | 38. | 17 | 38. | 17 | 3. | 84 |
| 42 | 31. | 87 | 21. | 28 | 21. | 28 | 44. | 53 | 44. | 53 | 4. | 48 |
| 48 | 36. | 42 | 24. | 33 | 24. | 33 | 50. | 92 | 50. | 92 | 5. | 12 |
| 54 | 40. | 96 | 27. | 36 | 27. | 36 | 57. | 28 | 57. | 28 | 5. | 76 |
| 60 | 45. | 42 | 30. | 33 | 30. | 33 | 63. | 58 | 63. | 58 | 6. | 40 |
The following table contains the proportions of curvature, said to be employed by the London opticians.
| Focal length. | Convex lens of crown glass. | Radius of both the surfaces of the concave of flint glass. | Convex lens of crown glass. | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Inches | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. | Inc. | Dec. |
| 6 | 3. | 77 | 4. | 49 | 3. | 47 | 3. | 77 | 4. | 49 |
| 9 | 5. | 65 | 6. | 74 | 5. | 21 | 5. | 65 | 6. | 74 |
| 12 | 7. | 54 | 8. | 99 | 6. | 95 | 7. | 54 | 8. | 99 |
| 18 | 11. | 30 | 13. | 48 | 10. | 42 | 11. | 30 | 13. | 48 |
| 24 | 15. | 08 | 17. | 98 | 13. | 90 | 15. | 08 | 17. | 98 |
| 36 | 22. | 61 | 26. | 96 | 20. | 84 | 22. | 61 | 26. | 96 |
| 42 | 26. | 38 | 31. | 45 | 24. | 31 | 26. | 38 | 31. | 45 |
| 48 | 30. | 16 | 35. | 96 | 27. | 80 | 30. | 16 | 35. | 96 |
| 54 | 33. | 91 | 40. | 45 | 31. | 27 | 33. | 91 | 40. | 45 |
| 60 | 37. | 68 | 44. | 94 | 34. | 74 | 37. | 68 | 44. | 94 |
From this table it appears, that the two convex lenses, have the same radii of their respective sides and that the concave flint lens has its two surfaces equally concave, so that a triple object-glass formed according to these proportions, would require only three pair of grinding tools. The following are the curves of the lenses of one of the best of Dollond’s achromatic telescopes, the focal length of the compound object-glass being 46 inches. Reckoning from the surface next the object—the radii of the crown-glass were 28 and 40 inches: the concave lens 20.9 inches, and the inner crown-glass lens, 28.4 and 28.4 inches. This telescope carried magnifying powers of from 100 to 200 times.
Although I have inserted the above tables, which might in some measure guide an ingenious artist, yet on the whole, a private amateur has little chance in succeeding in such attempts. The diversity of glasses, and the uncertainty of an unpractised workman’s producing the precise curvatures he intends, is so great, that the object-glass, for the most part, turns out different from his expectations. The great difficulty in the construction is to find the exact proportion of the dispersive powers of the crown and flint glass. The crown is pretty constant, but there are hardly two pots of flint glass which have the same dispersive power. Even if constant, it is difficult to measure it accurately; and an error in this greatly affects the instrument; because the focal distances of the lenses must be nearly as their dispersive powers. In the two preceding tables, the sine of incidence, in the crown glass, is supposed to be to the sine of refraction as 1.526 to 1; and in the flint glass, as 1.604 to 1. Opticians who make great numbers of lenses both of flint and crown glass, acquire, in time, a pretty good guess of the nature of the errors which may remain after they have finished an object-glass; and having many lenses intended to be of the same form, but unavoidably differing a little from it, they try several of the concaves with the two convexes, and finding one better than the rest, they make use of it to complete the set. In this way some of the best achromatic telescopes are frequently formed. I have sometimes found, when supplying a concave flint glass to a telescope where it happened to be wanting, that, of four or five concave lenses which appeared to be the same as to curvature and other properties, only one was found to produce a distinct and colourless image. Should any one, however, wish to attempt the construction of an achromatic lens, the best way for preventing disappointments in the result is, to procure a variety of tables of the respective curvatures founded on different conditions, and which, of course, require the surfaces of the several lenses to be of different curves. Having lenses of different radii at his command, and having glass of different refractive or dispersive powers, when one combination does not exactly suit, he may try another, and ultimately may succeed in constructing a good achromatic telescope; for, in many cases, it has been found that chance, or a happy combination of lenses by trial, has led to the formation of an excellent object-glass.