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familiarity which all acquire with any appearance which they have many times reproduced, will only serve to mislead, if at the beginning spurious lines have been confounded with the true, for then the deceptive spurious appearance will be sought for as eagerly as though it were the true one.

A realization of this circumstance has led several eminent observers to propose a criterion of resolution which appears at first sight to meet the case, but which I must really think is more difficult and less accurate than the third method. This plan is very well described in the letter of President Barnard quoted in the article to which I have referred.* “ When, for instance, I found that the value by micrometer of twenty spaces on the nineteenth band as counted, was exactly equal to the value by the same micrometer of ten spaces on the ninth band, I could not doubt that the nineteenth band was resolved." This method presupposes of course that the lines are ruled at exactly the distances Nobert intended, viz. those of the ninth band goooth of a Paris line from centre to centre, and those of the nineteenth to booth from centre to centre. It also presupposes absolute accuracy in comparing the portion of the two bands selected.

Now, the supposition that Nobert's estimate of the distances between the rulings is mathematically correct, appears to me highly improbable from many considerations; but when the attempt is made to demonstrate the precise degree of success attained, many difficulties are encountered. For example, as the ninth band contains twenty-seven lines and the nineteenth fifty-seven, if the distance of the lines of the nineteenth band from centre to centre is exactly half that given to those of the ninth, it follows of course that the nineteenth band ought to be broader than the ninth by the space of four of its lines. I have, however, been quite unable to satisfy myself that this is the case ; repeated measurements of the nineteenth band of my plate inclining me to think rather that it is somewhat narrower than the ninth. It is indeed extremely difficult to make such comparison with the requisite precision, so difficult that I do not believe anyone could tell merely by the count and measurement of twenty lines whether he was examining the seventeenth, the eighteenth, or the nineteenth band. In the most careful and experienced hands, therefore, this plan offers at the best greater difficulties than a simple count across the band, and except in such hands it is only liable to mislead.

On the other hand, the third method is more positive, for the number of lines in each band being now known, a complete count gives results which cannot reasonably be questioned. But two objections have been made to this plan. First, that an objective of

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high angle may have exquisite definition combined with such curvature of field that a part only of any given band may be resolved at a time; and secondly, that in the case of the higher bands at least, a count of the whole band from edge to edge is so difficult as to be almost impracticable unless special costly and troublesome apparatus is employed.

The first of these objections falls to the ground if the actual width of the bands is considered in connection with the aperture of the objectives employed. It is asked, “If Nobert had covered a whole inch with the 112,000 and some odd lines, would anyone claim that all must be seen at once ?” Now, the fact is that each of the bands on the plate is really only about the zoooth of an inch in width; and the question is not whether an imaginary band of greater width could all be resolved at once, but simply whether the modern objectives as actually made have a field sufficiently flat to resolve from edge to edge a series of lines occupying a space the two thousandth part of an inch wide in breadth. I have already expressed my opinion on this matter, but desire here to offer a few considerations in its support.

My Powell and Lealand's immersion sixteenth, with the short eye-piece I generally employ on the plate, gives a field •004 of an inch in diameter, or eight times the width of one of the bands. The Tolles' fith, belonging to the Museum, with the same tube and same eye-piece, gives a field •008 of an inch in diameter, or sixteen times the width of a band. The Tolles' th , belonging to the Museum, under the same circumstances, gives a field .017 of an inch in diameter, or thirty-four times the width of a band.

With such an eye-piece only the central portion of the actual aperture of the objective is utilized ; and I find that with the fifteenth band sharply in focus at one side of the field I get at the same time complete resolution of the fourteenth and thirteenth bands with both Powell and Lealand's immersion oth and Tolles' sth. I cannot, therefore, admit that the actual curvature of field is such as to prevent any given band from being resolved from edge to edge by an objective capable of resolving any part of it, and pass on to consider next the question of the difficulty of a count. .

I have published elsewhere * what appeared to me to be a very easy and simple method of counting the lines. The circumstance that no one appears to have adopted it, on account probably of its requiring some special apparatus, induces me to mention some still simpler methods, which I have frequently employed with success. If after resolution is attained, a cobweb micrometer be substituted for the ordinary eye-piece, the well-known difficulties resulting from tremor will be encountered if any attempt be made to turn the screw and move the cobweb from line to line, as is ordinarily

*"Quarterly Journal of Microscopical Science,' October, 1868.

done. But if instead, the blackened brass teeth which serve to record the movements of the cobweb be used simply as reference points to enable the eye to keep its place in counting across the band, a little practice will soon enable the observer to count the whole band with certainty and precision. The micrometer not being touched, there will be no tremor. It will of course be understood that a little strip of brass furnished with fine teeth, gummed to the diaphragm of an ordinary eye-piece, will answer the same purpose. A glass eye-piece micrometer will be found to impair definition too much. After a little experience, however, even the specks on the ordinary eye-piece and on the plate itself can be employed as reference points, and will render a successful count feasible, though not so easy as with the help of the economical contrivance I have described. Of course it is best in any case to begin with practice on the lower bands.

Having acquired the degree of skill necessary to enable him with such help to count a number of fine lines without losing his · place, the conscientious student of the Nobert's plate has not, however, disposed of all the difficulties in his path. With such oblique light as is necessary for the resolution of all the higher bands with our present objectives, he will find a certain number of spurious lines on both sides of the band, and he will have to learn some method of determining where to begin and where to end his count. Nothing gave me greater difficulty in my earlier investigations of the plate, and I hope, therefore, that a short account of the practical results at which I arrived may prove of service to other microscopists.

If one of the higher bands of the plate be examined by an objective capable of resolving it, while it is illuminated by a pencil of insufficient obliquity, a mere tint or shade of the width of the band will be observed, with more oblique illumination a wavy irregular appearance comes into view; increasing the obliquity of the pencil still further, a series of lines make their appearance, occupying about the width of the band, but fewer in number than the real lines; finally, when the degree of obliquity necessary for actual resolution is attained, the true lines start into view, accompanied, however, by certain spurious lines on the margins of the bands. Four appearances of the bands are thus indicated, viz. a mere tint or shade, a wavy irregular appearance, spurious lines occupying the place of the band, and true lines with spurious lines on the margins.

The last two of these appearances require consideration.

A. Spurious Lines occupying the Place of the Band.—These are generally clean smooth lines, quite sharp, and much narrower than the apparent interspaces. They are well calculated to deceive the unwary, as they have often done. On a count, however, they will be found too few; for instance, twenty to forty in the nineteenth band, instead of fifty-seven. The laws governing the production of these false lines and the relation of their number to the aperture of the objective and the obliquity of the illuminating pencil, have perplexed the most eminent students of optics, and are well worthy of future investigation. It is enough for my present purpose to state that while false lines of this character may be seen in the place of the true lines of the higher bands with objectives perfectly capable of resolving them with more oblique illuminating pencils, a similar appearance is often the best that can be produced with the most oblique pencils, if the resolving power of the objective is inadequate. These thin spurious lines differ in several particulars from the real ones of the higher bands. The latter are not smooth, they are irregular, they are thicker than the interspaces, they are wavy. The cutting tool has moved with a certain tremor; some of the lines are ploughed deeper than others, the distances from centre to centre are not always equal. Such inequalities might be expected in ruling such fine lines, and with adequate defining power they will readily be recognized

B. Spurious Lines seen on the Margins of the Resolved Bands. - These require careful consideration by those who attempt to count the higher bands; but by selecting for study at first a moderately fine band, say the eleventh or twelfth, their characters can be learned, and the method of distinguishing them from the true ones mastered.

Suppose, for example, the twelfth band is under observation ; when the pencil is sufficiently oblique to show the true lines, a series of fine spurious lines, closely resembling true ones, are seen adjoining the band, which, when the objective is in the lowest focal position compatible with definition, appear on the side from which the oblique light comes (as the microscope reverses, they are of course really on the opposite side). Of these spurious lines those next adjoining the real ones measure from centre to centre the same as the real ones, but the more distant ones grow gradually fainter and more separated till they disappear from view. On the opposite side of the band are a few coarser spurious lines, easily distinguished from the real ones. Such is the condition of things shown in my photograph of the nineteenth band. I do not think anyone could tell by mere inspection either of the photograph or of the image in the microscope, if the focal adjustment remains unaltered, which was the last true line, and which the first spurious one, on the side of the fine spurious lines ; on the other side it is easy enough. But if now the direction of the light is reversed, the fine and coarse spurious lines change places, and what is still more important, a similar change can be effected by a mere change of focus. In fact, when the band is seen as above, if the fine spurious lines are on the

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right and the coarse ones on the left, it will be found that on very slightly withdrawing the objective from the plate the fine and coarse false lines change place almost exactly as though the direction of the light was changed. This change is not a sudden one; fine spurious lines begin to be seen on the second side before they have all disappeared from the first, and there is no intermediate position such as may be attained with the lower bands, in which there are absolutely no spurious lines to be seen on the edges.

These results of a change of focus enable the observer to distinguish in the microscope the position of the last real line on either side, and thus to attain accuracy in his count; and if a successful photograph be compared with the object as seen in the microscope, there will be no great difficulty in determining, first on one side and then on the other, the portions of the picture which correspond to the true limits of the band. It was in this way that I determined the limits to be given to the enlargement of my photograph of the nineteenth band, which was distributed in every case pasted on the same card with a print from the unmodified original negative. The object of the limitation given to the enlargement was to make it serve as a representation of conclusions, readily obtained in the microscope by change of focus, but which the possessors of the photograph could not arrive at without repeating my observations, since the photograph represented of course but a single focal position.

If the first and last real lines are fixed in such a photograph by comparing it with the appearances in the microscope, the photograph will answer a useful purpose in verifying the count. Even if the observer is not sure as to this, if he can find, as he generally can, any reference points in the microscope and in the photograph by which fixed points in the band can be identified, a count of the intermediate lines will serve as a useful check. I have made use of all such aids in my study of the Nobert's plate, but I desire to say expressly that my statement of the number of lines in each band rests essentially on my counts in the microscope, and that I have repeatedly counted in the microscope the lines of each of the bands, from edge to edge. I may also make a single remark in this place with regard to our knowledge of the number of lines in the bands. I believe I was the first to state the actual number for the higher bands of the new nineteen-band plate. Nobert gave the distance of the lines apart in fractions of a Paris line, the actual width of the bands and the number of lines in each was not stated. From his figures the number of lines per millimètre and per inch has been computed. But the practical difficulty of measuring the width of a band with sufficient precision to deduce the number of lines from these figures has caused all the writers on the subject, whose papers I have seen, to observe a judicious reticence as to the actual number


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