THfi TfiUESGOPE-MlRROR-SCALE METRO ADJUSTMENTS AND TESTS HOLMAN - THE TELESCOPE-MIRROR-SCALE METHOD ADJUSTMENTS AND TESTS SILAS W. HOLMAN PROFESSOR OF PHYSICS (EMERITUS) MASSACHUSETTS INSTITUTE OF TECHNOLOGY Reprinted from The Technology Quarterly, September, 1898 NEW YORK JOHN WILEY & SONS LONDON : CHAPMAN & HALL, LIMITED 1898 THE TELESCOPE-MIRROR-SCALE METHOD; ADJUSTMENTS AND TESTS. 1 BY SILAS W. HOLMAN. Received May, 1898. IN the telescope-mirror-scale method for measuring small angles, an illusory estimate of the accuracy attained may easily arise through inattention to the requisite adjustments, tests, and corrections. Yet an adequate and well-ordered presentation of them is not to be found. Partial statements are given in text-books on manipulation, but Czer- mak 2 alone has given a discussion approaching completeness. Even this omits some essential points, and moreover is not arranged in a manner to lend itself readily to the practice of the method. The following presentation of the subject is designed for the observer who would put the method into direct service for obtaining measurements of a specified or of a determinate accuracy. It there- fore not only discusses the several sources of error, but describes the various adjustments and tests in the sequence in which they should ordinarily be made, and gives a numerical measure of the closeness with which each must be carried out to secure a specified precision in the result. Some remarks on the selection of instruments are appended. The general plan adopted in treating each adjustment or test is as follows : To deduce a general, though usually approximate, expres- sion for the error attending its omission, or preferably for the correc- tion therefor. To deduce therefrom a numerical measure of the close- ness with which the adjustment must be made or the test fulfilled, in order to insure a designated precision in the use of the method, so far 1 Copyrighted, 1898, by S. \V. Holman. Published separately by John Wiley & Sons, 53 East loth Street, New York. Cloth, 75 cents. 2 CZERMAK. " Reduction Tables for Readings by the Gauss-Poggendorff Mirror Method." Besides a discussion of many of the errors of the method, this book gives extended tables of corrections and reductions, of much service in long series of observations. The text is in German, French, and English, in three parallel columns. 271629 2 The Telescope-Mirror-Scale Method. as that source of error is concerned ; and to make certain comments based thereon. To render the problem definite, the tangent of the angle of deflec- tion of the mirror is assumed to be the desired result of an observa- tion. The deductions are easily adaptable to other less frequently employed functions. The expressions for the errors or corrections are brought into the form of fractional errors or corrections ; that is, they are expressed as a fraction of the value of the tangent of the deflection. The correction is of course equal to the error, but has the opposite algebraic sign, a positive error requiring a negative correction, and vice versa. It is more convenient in the present work to deal directly with corrections than with errors. The numerical solutions are computed on the basis of a desired precision of one part in one thou- sand, or one-tenth of one per cent., in the resulting value of the tan- gent. The precision discussion based on methods elsewhere stated 1 is sufficiently obvious. As there are some fifteen sources of error, the average effect to be assigned to each consistently with the prescribed limit of o.ooi in the result is o.ooi -f- */ 15 = o.oo 030 nearly enough, as several of the errors are usually rendered insignificant. The solu- tions are also made for a scale of millimeters at a distance of one meter from the mirror, and for a maximum deflection of 500 mm. It will be found by inspection of the results, especially under I, III, XII, XIII, XIV, XV, that o. i per cent, is about the limit of accuracy attainable under these conditions. In the employment of the apparatus absolute values of the tangent of the deflection are sometimes sought, in which case the measure- ments may be called primary. More often, however, only relative values are required, or rather we are concerned as to the accuracy of only relative values of the tangents. The measurements may then be called secondary. An example of the secondary use is where the tele- scope and scale are employed with a reflecting galvanometer to meas- ure currents, and where the "constant" of the apparatus is found by sending a known current and reading the deflection. In such cases any constant fractional errors in the telescopic method enter into the "constant," as well as into subsequent observations, but with opposite signs, so that they are eliminated from the results. It is therefore needful to discuss the errors with respect to both primary and second- 1 " Precision of Measurements." John Wiley & Sons, New York. Silas W. Holman. 3 ary use of the method, and relief is thus found possible in the latter from some of the exactions of the former. When reversals are taken, that is, when the mirror is deflected so that a reading can be made first on one and then on the opposite half of the scale, certain sources of error are reduced by averaging the two deflections. This is usually precluded in practice, however, by such conditions as continual change in the reading, avoidance of delay, etc. Both cases must therefore be discussed. The desired tangent or other function of the angle of deflection is computed from the observed scale-reading d and scale-distance r as stated under XVI. We will assume that it is found, with due allow- ance for the approximation involved (cf. XVI), from the expression tan y = . (i -\- u -\- v -\- w -\- . . .) 2 r where u, v, w, etc., are the fractional corrections to the observed tangent to allow for the various sources of error to be pointed out. As d enters as a direct factor in this expression for the tangent, the fractional corrections may be applied directly to d. In fact it is more convenient in deducing the formulae to find an expression for either Sd, the numerical correction to d, or for 8d / d, the fractional correction to d. Thus Sd is such a quantity, expressed in scale divisions, that when added to the observed scale-reading d it will give the correct scale reading, as far as the designated source of error is concerned. And &d I d = u or v, etc., is this correction expressed as a fraction of the observed reading d. The algebraic sign of any correction may be either determinate or indeterminate. ' Also it may be noted that as r enters as a factor in the denominator, the corrections u, v, etc., may be applied to or deduced for r instead of d, if desired, but with the difference that the algebraic sign would be reversed. Any corrections which may prove to have a constant value may be included once for all in the numerical constant i / 2r. Constant fractional corrections disappear when merely relative deflections are used, or when the con- stant is determined by calibration, as above stated. The assumption is made that none of the errors with which we have to deal are in excess of i or 2 per cent. Greater ones must be reduced by instru- mental rearrangement before they can be determined with sufficient closeness. The Telescope-Mirror-Scale Method. The algebraic expressions which will be deduced for u, v, w, etc., are useful in two ways : First, they enable us to compute the correc- tion to be applied to any observed reading ; second, they are used in the precision discussion, which shows how closely each correction must be worked out to secure a prescribed accuracy in the value of tan, cp or how close an adjustment is needed in order that the correction may be omitted without introducing more than its due share of error into the result. These will be called negligible corrections. Any error of this magnitude produces an effect not more than is admissible on the result, and the effect of one as small as one-third of this amount will be inappreciable. The results deduced for the telescope-mirror-scale method are in general directly applicable to the mirror method where a beam of light replaces the telescope. THE PROCEDURE. Preliminary. Focus the cross- wires in the eye- piece. Set up the apparatus nearly in proper position as in Figure i, adjust- ing telescope, mirror and scale so that the scale is visible. The following description will assume the dis- position of the ap- paratus to be the most usual and sim- ple one, shown in Figure i : The scale A, B is horizontal with its middle point just below or above the optic axis of the telescope, which is perpendicular to the scale. The mirror (plane) faces the telescope and scale at a convenient dis- tance. The horizontal distance r between scale and mirror should then -4 P/ 3 Side. FIG. i. Silas W. Holman. 5 be roughly measured, say to I per cent., for use in the approximate computations in the several tests, etc. The selection of this distance is determined largely by convenience, and usually lies between one and three meters. It must be sufficient to insure to the smallest scale-readings to be observed, a sufficient number of divisions to be read with the requisite fractional precision. This means greater dis- tances and higher magnifying power for very small angles. Labor in computation may be saved by making this distance some simple round number, e.g., 1,000 or 2,000 scale divisions, but the work of this adjustment will usually outweigh that of computation. The distance must, however, be maintained accurately constant. The telescope may be at a distance different from that of the scale, if more convenient, but further removal reduces the magnification. As far as the principle of the method is concerned, the telescope may be at any position whatever relatively to the scale, consistent with having the line from the middle of the scale to the mirror nearly horizontal, as pointed out in test X. If there is annoyance from dupli- cate reflections from the glass front of the mirror house, the glass may be slightly tilted forward. The following tests and adjustments are then to be made in the order in which they are given, a repetition of any of them being sub- sequently made if disarrangement may have occurred : ADJUSTMENTS AND TESTS. A list is here given of the various adjustments and tests, together with a brief statement for convenient reference of the closeness neces- sary to correspond with an accuracy of one-tenth of one per cent, in the resulting value of tan cp, assuming a straight millimeter scale at a distance of I meter, with deflections not exceeding 500 mm. and not less than 100 mm., readings being taken to tenths of a millimeter by the eye. I. CROSS- WIRE Focus. By parallax test there must be no apparent motion of wires, or less than 2 L division. Telescope must also be perfectly focussed. Primary and secondary measurements the same. II. - OPTIC Axis OF TELESCOPE RADIAL. P. M. Cross-wires centered within i mm., or better, to 0.2 mm., if possible, when tested 6 The Telescope-Mirror-Scale Method. on scale at distance of i m. Optic axis must pass within i mm. of axis of suspension. III. SCALE GRADUATION. The average error in the distance of any ruling from the central ruling must be less than 0.03 mm. P. M. and S. M. the same. IV. COVER-GLASS THICKNESS. P. M. Glass negligible when less than i mm. in thickness. S. M. Glass always negligible. Glass must permit good definition. V. COVER-GLASS SURFACES PARALLEL. P. M. To be negli- gible, the maximum displacement must be less than 0.3 mm. on rotat- ing the glass 180 in its own plane. If more is found, the glass must be kept in the position of minimum displacement. S. M. The same. VI. COVER-GLASS CURVATURE. P. M. To be negligible, the focal length of the cover-glass must exceed 70 meters. This will be the case when an object at a distance of 10 meters from the telescope, and sharply in focus with the cover-glass interposed, does not require to be moved through more than about i meter towards or from the telescope to maintain the focus when the glass is withdrawn. S. M. The error is here always negligible. VII. MIRROR THICKNESS. P. M. Negligible thickness is o. 5 mm. S. M. Always negligible. VIII. MIRROR ECCENTRICITY. P. M. Negligible up to an eccen- tricity of 10 mm. S. M. Also negligible. IX. MIRROR CURVATURE. P. M. Correction depends on amount of eccentricity. It vanishes with no eccentricity. For specific case see later. S. M. Curvature has no sensible effect. X. MIRROR VERTICAL AND VERTICAL ANGLE TMO SMALL. P. M. Telescope and scale must not be separated vertically by more than 60 mm. S. M. Negligible, and distance need not be very small. XI. SCALE HORIZONTAL. P. M. Error insignificant when end of scale is not more than i or 2 mm. out of horizontal through middle. Negligible up to 12 mm. S. M. Negligible for several centimeters of tipping. XII. SCALE PERPENDICULAR TO OM AND PLANE. P. M. With- out reversals, the difference in distance of the ends of the scale (the 500 mm. marks) from the suspension fiber must be less than 0.6 mm. With reversals, 25 mm. is close enough. Scale must be plane within these limits. S. M. About the same as in primary. Hence advan- tage in reversals where practicable. Silas W. Holman. 7 XIII. NULL POINT. P. M. The scale once fixed in place can- not be disturbed without impairing XII. Hence accidental change of null reading must be remedied by turning the mirror, not by mov- ing the scale. With reversals, the null reading may be as great as 17 mm. without correction. Without reversals, the null reading must be less than 0.6 mm., even if allowed for. S. M. Same as in primary. XIV. DISTANCE OF SCALE FROM MIRROR. P. M. This must be measured and kept constant within 0.3 mm. This demands more than the usual attention, and renders some special device important. See XIV later. S. M. The same. XV. ESTIMATION OF TENTHS OF DIVISION. Nearest tenth is to be read in both P. M. and S. M. XVI. To COMPUTE THE DESIRED FUNCTION OF qp from the ob- served deflection d. Details are given under XVI later.- Five places of significant figures should be used in d, etc. 1 I. CROSS-WIRE Focus. The cross-wire intersection must be brought accurately into the focus of the eye-piece on each occasion of use of the apparatus, and by each observer for himself. Inattention to this point may easily give rise to an error as great as half a scale division. The focussing should be done by the parallax method, as follows : Focus the wires as sharply as possible by moving the eye-lens. Then focus the telescope very carefully on the scale or on some object showing some sharply marked point of reference. Move the head to and fro sidewise, so that the pupil of the eye shall travel from one side to the other of the aperture of the eye-piece. If the wires are not in focus they will appear to move over the scale. If so, refocus them, and then refocus the tele- scope on the scale. Continue until no apparent motion of the wires is perceptible. Good focussing is promoted by looking away from the telescope frequently ; also in some cases by fixing the attention of the other eye on a printed page held at the distance of most distinct vision beside the telescope. The accurate focussing of the telescope is not less important than that of the wires. The error from imperfect focus will be indeterminate in sign and magnitude. PRIMARY MEASUREMENTS. The negligible correction will then be 8d, whose magnitude must be such that 8d / d = 0.00030 for 1 Computation Rules and Logarithms. The Macmillan Co., New York. 8 The Telescope-Mirror-Scale Method. the smallest value of d to be used. Now deflections of less than 100 divisions are not employed in exact work for reasons shown in XV. Hence substituting d = 100 mm. we have d = 0.03 mm. The extreme motion of the eye in the parallax test will obviously produce a displacement of the maximum error, and more than double the aver- age error. Hence the focus will be good enough when the maximum displacement by the parallax test is less than about 0.06 divisions, or about JQ- f the smallest scale division. It is clear, then, that the utmost attention to this detail is necessary. SECONDARY MEASUREMENTS. This source of error affects pri- mary and secondary measurements equally. II. OPTIC Axis OF TELESCOPE RADIAL. This adjustment requires two operations : First, centering the cross-wire intersection, that is, bringing it into the optic axis ; second, directing the optic axis towards the axis of suspension, e.g., by focuss- ing the wire intersection upon the suspending fiber. The second operation is obvious enough, but must be repeated from time to time by way of precaution. It is not easily executed when the telescope draw-tube does not permit of focussing on objects as near as the mirror. The well-known method of centering the wires is briefly as follows : Lay the telescope in a pair of V grooves temporary wooden ones will suffice. Focus on the scale, or on a few rulings roughly equal to the scale divisions, placed at a distance about equal to TM. Rotate the telescope in the grooves. The wires are centered when the inter- section shows no motion on the scale when thus rotated. If adjust- ing screws are provided, the diaphragm carrying the wires should be centered until the motion is less than o. I or 0.2 mm., as demonstrated later. If, as is commonly the case, there is no means of adjustment of the wires, they must be remounted if the extreme apparent dis- placement on turning through 180 is more than about I mm., for work of the accuracy here assigned. DEMONSTRATION. We will take the simplest case, where the optic axis TF of the telescope is at right angles to the scale AB. If adjustments XII and XIII had been made the relative positions would be slightly different, but the result would be essentially the same as far as this source of error is concerned. Suppose the telescope so Silas W. Holman. 9 located that TF meets the mirror at c' instead of c, the latter being in the vertical axis of suspension. Let cc' be denoted by s. Then we seek an expression for the fractional correction Sd J d for the error which s will produce. If c were at c', as it should be, and were deflected through an angle y, the observed reading OA would be the true reading d desired. With c at c', the observed reading becomes OA', and A'A = Sd. As Tc' = r, we have 8d : c' c" = d : r. But c 1 c' 1 = z tan q> = 2 (d / 2r) approx. Hence d I d = 'zd I 2r* approx., for a deflection on the side consid- ered. For*-a reverse deflection the correction would obviously be of the same amount, but the opposite sign. Imperfect centering of the wires, besides misdirecting the optic FIG. 2. axis, introduces a slight error through the smaller magnification of the outer scale divisions, but this may be easily shown to be small com- pared with the foregoing. PRIMARY MEASUREMENTS. Without reversals. As the negligible correction above assigned is Sd /d = o.oo 030, the negligible value of s for the worst case, where r= 1,000 mm. and d= 500 mm., is to be found from z = (Sd / d) / (d / 2r*} = o.oo 030 X 2 X (i,ooo) 2 -r- 500 = 1.2 mm. Thus the optic axis must pass within about 1.2 mm. of the axis of suspension, which demands no inconsiderable care. As the adjustment requires two operations, both must be made closer than this limit, or one must be rendered imperceptible. The latter, by careful centering of the wires, is usually easier of accomplishment. IO The Telescope-Mirror-Scale Method. If this is done so that the displacement in the above test is less than ^ of , i.e., 0.4 mm. on a scale at the distance TM, or better to 0,1 or O.2 mm., this part of the error will be negligible. With reversal of deflections, this source of error vanishes when z does not exceed 2 or 3 mm. SECONDARY MEASUREMENTS. Substantially the same degree of care is needed as in primary work. III. SCALE GRADUATION. The graduation must be uniform to such an extent that the actual correction d to any observed deflection d shall be small enough to make &d / d less than the assigned limit. The only thorough way to test this is to measure out from the middle point of the scale to each successive smallest division by some comparator or dividing engine, thus determining every value of d / d, and then to apply the correo tions thus found unless they are negligible. If an error not exceeding J-0- mm. is admissible on a mm. scale, the test may be made by laying a standard (Brown and Sharpe) mm. scale against the unknown one, and inspecting the coincidences of the rulings. The first 50 or 100 divisions either way from the middle are not employed in work of even moderate accuracy, because the value of d is then so small that the unavoidable fractional error from the eye estimation and other sources is excessive. PRIMARY MEASUREMENTS. The requisite accuracy of graduation for the above assigned limit would be attained when for the smallest value likely to be used for d in such work, viz., r < The correction must therefore be applied when, as is almost always the case, the cover-glass is more than about i mm. in thickness. SECONDARY MEASUREMENTS. Since the fractional correction d I d is constant, the correction disappears in this work as in other similar cases. V. COVER-GLASS SURFACES PARALLEL. Defect with regard to this is common. Its effect depends so much upon the various possible angular positions of the surfaces relatively to O M that it is not readily reducible to a general expression, but an easy and sufficient test can be developed. Give the mirror a deflection to about the end of the scale. By clamping, or in some effectual way, hold the mirror so that this reading shall remain fixed ; focus sharply ; then using due care not to disturb the apparatus, place the cover-glass flat against the objective of the telescope, or better against some dia- phragm arranged to hold it not far from the objective ; read closely. Now rotate the cover-glass through 180 in its own plane, pressing it against the objective or diaphragm, and read again. A change of read- ing due to twice the refraction by the prismatic or wedge shape of the glass will be found. Turn the glass in its own plane into various (marked) angular positions and thus locate the diameter along which the displacement is greatest. This will locate the direction of the edge of the wedge, that is, the intersection of the two plane surfaces, since this must be vertical when that diameter is horizontal. Silas W. Holman. 13 PRIMARY MEASUREMENTS. It is more prudent not to use a glass which shows a maximum displacement 6" exceeding 0.3 mm. at a great- est deflection of d = 500 mm. ; for the negligible correction will be Be? i S = . o.oo 030 . . 5 = 0.3 mm. d 2 500 < If this limit is not easily reached, the wedge axis may be placed horizontal, which will sensibly eliminate the error. But this requires that during all use of the apparatus the cover-glass be continually inspected to see that it is in the proper position. In making this test the cover-glass may often, to good advantage, be rotated in position instead of being removed and placed in front of the objective. In that case special care must be taken not to disturb the mirror when the glass is being rotated through 180. The maxi- mum displacement observed in this method of test in either case is in excess of the worst error which the wedge would cause in the obser- vations, unless it was reversed in position from time to time which must be guarded against if the glass is poor. SECONDARY MEASUREMENTS. No relaxation from the foregoing requirement is admissible. VI. COVER-GLASS CURVATURE. Minor irregularities in the surface of the cover-glass of the mirror house produce merely a blurring of the image, such as is seen in look- ing through ordinary window glass with a telescope. The cover-glass must be sensibly free from this. If either surface of the cover-glass is systematically curved, the glass will act as a lens. The focus of points seen obliquely through the glass is then changed in both distance and direction. The change in distance is either unnoticed or is corrected by the focussing of the telescope. The change of direction causes an error in d. The glass may be equivalent to a spherical lens or to a cylindrical lens. The effect of the change in direction may be studied by the central ray of any beam ; we are concerned here with the curva- ture as revealed by a horizontal section only. For simplicity, suppose the glass to be equivalent to a spherical lens D G of focal lengthy with its axis coincident with O M. If the lens is convex, the true read- ing A- will be shifted to A' toward O ; if concave,, in the opposite direction. 14 The Telescope-Mirror-Scale Method. The correction required and the method of test may be developed as follows : The lettering of the diagram (plan) being as before, let O A = d = true deflection reading, O A' = d = observed deflec- tion reading, M G = e =p\ = distance of A' cover-glass G from mirror (roughly), J and M = conjugate foci of lens G when MG = /, the focal length of G, FIG _ 4- JG =p* AM,A'J = straight lines. As we have treated separately the thickness of the glass, A MO = a, A' JO = /3. tan a _ _ p<^ _ d pi -\- r e tan ft e r d^ d\ _ e . e d r /2 But by the law of lenses, inserting for convenience the negative sign because / x < f> ill ill = or = P\ Pi f * P* f Hence multiplying by e and transposing = i , d d The signs will obviously be the same for deflections on the opposite side of O. The correction is therefore d d d\ e But in all ordinary cases e < r, so that for the present purpose 10 is negligible compared with i, and . / Silas W. Holman. 15 This evidently applies to deflections in either direction, provided that the axis of the lens coincides with O M ; otherwise the correction would be unsymmetrical and would contain another term. The effect of this want of symmetry, however, would be detected under test V for wedge shape of cover-glass, and therefore need not be here discussed. If the cover-glass proves to be cylindrical in the test given later, the worst effect of the cylindrical surface will be produced when its axis is vertical. The correction is then the same as for the spherical surface. PRIMARY MEASUREMENTS. - - The negligible correction may be found from ~ 0.00030. d < f Assuming a value of e = 20 mm., not often exceeded, this yields f = 20/0.00 030 = 7. io 4 mm. 70 m. Hence the cover-glass, whether equivalent to a spherical or a cylindrical lens, must have a focal length exceeding 70 m. How the actual value of f is best determined will presently be shown. SECONDARY MEASUREMENTS. - - The fractional correction to the deflection is constant. Hence in secondary work the error introduced in the "constant" by neglect of the correction exactly offsets the errors entering into the subsequently observed deflections by the same cause, so that the systematic curvature has no effect. MEASUREMENT OF f. Direct the telescope (any good telescope other than that of the apparatus will answer if more convenient) upon any well defined object, such as a printed page, held at a distance of several meters (io or more if practicable). Interpose the cover-glass, placing it directly in front of the telescope. Focus sharply ; remove the cover-glass. If the glass is equivalent to a lens with spherical surfaces, the object will cease to be in focus. Without changing the focus of the telescope, bring the object towards it (concave) or move it away (convex) until the focus is again sharp. Let a denote the dis- tance from glass to object in the first position and b in the second. Then , as the two positions are conjugate foci. In oaf actual measurements it is, of course, better to measure (a b) directly with some care, and then a roughly (or b), thence computing b and f. To detect unequal curvature of different parts of the glass, it is well to measure f for each of its four quarters successively, covering its remaining surface. i6 TJie Telescope-Mirror-Scale Method. It may sometimes be better merely to test whether f is sufficiently great without actually measuring it. Thus if the limit is f > 70 m., then if the object be set up with a = 10 m., for example, i i , i 80 b 70 10 700 . b = 8.8 m. or a b = 1.2 m. If, therefore, b were distant from a by as much as i m., the glass would be good enough. Cover-glasses should never be found so poor as this on good instruments, but this cannot be relied upon, as makers frequently send out very unfit glasses. The glass should be tested by focussing through it separately on vertical, horizontal, and oblique lines. Difference of focus shows a cylindrical form which may or may not be superposed upon the spherical. The test should be applied to that value of f which makes a b the greatest. VII. MIRROR THICKNESS. The mirror usually consists of a thin plane-parallel piece of glass " silvered " on its rear surface. If in any case the mirror were very thick it would require special investigation as to corrections for want of parallelism or planeness of faces ; but with thin mirrors ordinarily used, and the quality which is insured x f by the requirement of good definition, x "" x d these points may be safely neglected. Inspection of the figure will show that a ray a b passing obliquely into the front surface of the mirror will pass out along c d, parallel to e f, the path it "^ a would have travelled had there been no glass in front of the reflecting surface. Also, the emergent ray c d is displaced from e ./by the same amount it would have been had it passed through a plane-parallel glass of twice the thickness of the mirror. Since by the process of adjustment the line O Mis sensibly normal to the mirror, the correction will be symmetrical with respect to O. If, then, /' rep- resents the mirror thickness, the correction to be applied to it will . . , , S d 2 / obviously be = . d i r FIG. 5. Silas W. Holman. 17 PRIMARY MEASUREMENTS. Requisite thinness of mirror to be negligible = 0.5 mm. SECONDARY MEASUREMENTS. - - The fractional correction being constant may be wholly neglected. VIII. MIRROR ECCENTRICITY. , If the vertical axis of suspension does not lie in the reflecting surface of the mirror, the eccentricity will cause an error in the scale reading. As the simplest case, let c M represent the reflecting ^A f surface (plane) in its normal -^ -- position, O M being perpendic- \ ular to it ; and let F be the A/% axis of rotation. When the mirror turns through an angle Y qp to the position M', the re- flected ray going to the tele- scope at O will be A 1 N instead of A N which it would have been had Y passed through M. The cor- rection to the observed reading d' = O A' is therefore A A' ; and for this an expression must be found. From the figure, '= M N tan 2qr, Ar FIG. 6. MN=eM tan qp = cM tan qc = JL YM tan 2 employed in de- I ducing it. It is, however, easy to modify the expression so that it may have any desired accuracy, by inserting in place of the approxi- mation the series expressing tan qp in terms of d/r given later. SECONDARY MEASUREMENTS. As the fractional correction to d has a constant value it vanishes for all values of d as in former cases. IX. MIRROR CURVATURE. This produces no error if the axis of suspension of the mirror is tangent to the spherical reflecting surface. If the curvature of the mirror is irregular, the definition will be impaired ; but if the telescope collects the light from the whole mirror, no sensible irregularity of reading will be caused. In the diagram, which is exaggerated for clearness, let Y be the vertical axis of suspension, MR the mirror undeflected (assumed con- cave), and O Y the line from F to the center of the scale, and to which the un- deflected mirror will be normal. Suppose the mirror to be turned through an angle cp about Y, then its new posi- tion will be M' R', FIG. 7. M having moved to M' through the angle M YM' = y. YM' C is now normal to the mirror, and let M' C be the radius p of curvature of the mirror. A ray along O Y would now meet the mirror at , whereas, if the Silas W. Holman. 19 mirror had been plane, the reflection would have been from F ; and if there had been no eccentricity, from M, since Y would then have passed through M. The error due to the forward motion from F to E would be capable of correction by the same expression as that for M F, but it may be shown to be so small as to be negligible except when p is very short, in which case, however, the mirror is not safe for use in accurate work even with the application of the correction. It is therefore at first necessary to deal only with the error which comes from the increased angle between the normal and O E. Owing to the curvature of the mirror, the normal to the mirror at E, instead of lying parallel to Ffand therefore making an angle q> with O E, makes with it a greater angle C E O = Now q>i =

for a convex mirror . ^PRIMARY MEASUREMENTS. The limiting value of p may be found from 8 ^i / i ^7 o.oo 030 =. X I p. 2O The Telescope-Mirror-Scale Method. Wjth X = zero, that is, no eccentricity, the correction vanishes. For X = i mm., p = 3300 mm. =3.3 m. For X = 10 mm., p = 33 m. and so on. An accidental curvature of less than 3 m. is not unlikely, and an eccentricity of I mm. is not uncommon and by no means always avoidable. It is therefore necessary to measure p roughly as shown below, but it is unlikely that the correction will be large when a presumably plane mirror is used, and a little consideration of the character of the error will show that a mirror requiring a large cor- rection cannot be employed. SECONDARY MEASUREMENTS. As the fractional correction is constant, it may be entirely omitted. To MEASURE p. Focus the telescope sharply on the reflection of the middle of the scale from M as in using the apparatus. Meas- ure MO within a few mm. Turn the telescope slightly so as to be able to look beyond the mirror, and place a printed page or other suit- able object in the line of sight at a distance about equal to MO. Without changing the former focus of the telescope, move the object towards and from the telescope until a point P is found at which the focus is again sharp. Measure M P within a few mm. Then the radius of curvature of the mirror is MO P ~ _MO MP in which the numerical values of both MO and M P are considered positive. If M P > M O, clearly p is positive and the mirror is con- cave. If MP < MO, p is negative and the mirror convex. If it is merely necessary to know whether p exceeds a specified limit, this expression may be transformed into MP = - 2MO P It is then merely necessary to calculate the value of M P and to see fhat the object is in focus at a distance less than this in the above test. X. MIRROR VERTICAL AND VERTICAL ANGLE TMO SMALL. Let Mff= horizontal line through M, M N= normal to mirror, T M O = line of sight when q = o. Silas W. Holman. 21 The reflecting surface of the mirror is here assumed to coincide with the axis of suspension, so that M H '= r. If the mirror is vertical and the telescope and scale are both at H, no correction will be required. The mirror may be made vertical or nearly so, but T must usually be either above or below the scale, hence the angle T M O cannot be zero. Figure 8 represents the case when neither M nor T M O are zero, and where N O < N T. For this case the fractional correction will be shown to be *d NH - NO d i* This obviously approaches zero as N If approaches zero, /. . Thus the normal M N will describe a conical surface with a vertical axis and having its vertex at M. This cone would intersect the vertical plane through the scale in an hyperbola with its vertex at M' . Let N' N" be a hori- zontal straight line ; then if N' N" be the horizontal projection on this plane of the part of this hyperbola described when M turns through qp, we have, tan y = N' N" / MH= N' N" / r. It is also true that as q> is small, the hyperbola will sensibly coin^ cide with N' N", but we need not make this assumption. Since the reflected ray will lie in a plane containing T and M N", the observed scale reading'when Mis deflected through (p will be at A', the intersec- tion of a straight line T N" (prolonged) with the scale. Since it is read by a horizontal scale having vertical rulings, the distance O' A f Silas W. Holman. 23 along the scale will be the horizontal projection upon the scale of the parabolic path of a ray from T reflected by M upon the plane of the scale. This sh'ort portion of the parabola of the ray will sensibly coincide with the horizontal straight line 5 5. Let A represent the point (unknown) of the scale at which the true reading corresponding to the observed reading A' would fall; then &/! = a A a A'. For values of qp so small that tan 29 = 2 tan q> nearly enough, &/! = 2 r tan g> O' A' = 2 N' N" O' A'. But also for larger values of qp, within the usual limits, although the value of tan 2

But, NO MO r cos (a -\- ft) cos a cos ft sin a sin ft NT M T r cos (a ft) cos a cos ft + sin a sin ft = i 2 sin a sin /8 approx. when a and /3 are small ; NH NO = i 2 . approx. 8 N T. XI. SCALE HORIZONTAL. To adjust the scale horizontal, focus T on the scale, with .#f swing- ing free. Note the height of the cross-hair intersection upon the divis- ions of the scale. Deflect M \.o the right and left. The intersection should remain at the same height. If it does not, raise one end of the scale. If the scale be tipped upward or downward from the horizontal through a small angle 7, remaining in the same vertical plane, the reading on either side will be shortened by the versed sine of the angle 7 ; that is, the fractional correction to d will be = versm 7=1 cos 7. d PRIMARY MEASUREMENTS. For the requisite closeness of adjust- ment, = i cos 7 ~~ o.oo 030, d < . , cos 7 = 0.99 970, and 7 = i.4. In making the test the observed change in the height Ji of the cross-hair intersection in passing from the middle to the end of the scale, would be &/i = 500 tan 7. Hence the requisite closeness will be 500 tan 7 = 500 tan i.4 = 12 mm. This adjustment can be made with perfect ease to i mm. or less, so that this error disappears. // SECONDARY MEASUREMENTS. As is constant so long as 7 is d constant, the amount of tipping of the scale within wide limits makes no difference so long as it remains always at the same angle. Silas W. Holman. 25 XII. SCALE PERPENDICULAR TO O M AND PLANE. To render M O A a. right angle, lay off or select two points A and B near the ends of the scale, exactly equidistant from O. Measure carefully the distances M A and M B from the suspension fiber of M. Adjust the scale until the two are equal. When this and the preced- ing adjustment have been completed, the scale should be rigidly and permanently fixed in place, and means provided to enable the fiber to be always brought back to its present position. A method of sup- port for the scale quite separate from the support of the telescope, as described later, contributes to stability and convenience. It may be noted that inequality of scale readings on reversal of. M is not a proof of imperfection in this adjustment, nor equality a sufficient proof of correct adjustment. The fractional correction will be shown to be = =F sin (3 tan 2 qp sin 2 /3 approx., d 2 where /3 is the small angle A O A' by which the scale is out of adjust- ment. The negative sign of the first term applies to deflections towards A, the positive to- wards B, used right-handedly. PRIMARY MEASURE- MENTS. If deflections are taken on one side only, i. e., without reversals of M, the requisite closeness in (3 will be attained when sin /3 tan 2 250 to 500 mm., a little less closeness will suffice, since the error at the deflection reading (whether with or without reversals) at which the calibration is made will enter into the value of the constant, and this will eliminate the error at the same point in subsequent readings. It will also reduce the average error about one-half if the calibration deflection is about 350 to 400 mm., but the difference in error of a small and of a large deflection will still remain the same. DEMONSTRATION. Let A and B be any points on the scale equi- distant from O and on opposite sides. Then O A and O B will be the true scale readings with the scale properly adjusted, and will be equal. O A 1 and OB' will be the observed readings. But Silas W. Holman. 27 The fractional correction 8 d\ / d\ on the side towards A 1 will be OA OA' _ OA OA' '' ~OA i O A sin (90 2 qp p) cos (2 y O A ! sin (90 -f- 2qp) cos 2 qp cos 2 qp cos /3 sin 2 or sin /3 = = /o <. = cos sin /3 tan 2 gc. cos 2 qp Now cos /3 = (i sin 2 = i sin 2 /3 approx., as '/3 is very small. 1 1-2/0 /O * L = - _ i = sm 2 /3 sin p tan 2 qp. ' OA' 2 Similarly on the side toward B, O R T = - _ -- i = sin 2 ft + sin ft tan 2 qp . C/ .Z5 2 Or, in general, = =p sin ft tan 2 qp sin 2 /3. d 2 XIII. NULL POINT. The "null point" or "zero reading," i.e., the reading when the mirror is undeflected, must be the middle point O of the scale, adjust- ment XII having been made ; and M must lie in a vertical plane perpendicular to the scale. In other words, the line of sight O M, when M is undeflected, must lie in a vertical plane which is perpen- dicular to the scale at its middle point O. This does not require either that any normal to M should lie in or parallel to that plane, or that the axis of the telescope should lie in that plane. Or, as less precisely expressed, the plane of the mirror need not be parallel to the scale, nor the axis of the telescope be exactly above or below O. The tele- scope may be at any position off at one side if more convenient, but ease and accuracy of adjustment, as well as other considerations, lead to> the customary location of the axis of the telescope more or less exactly above or below O. Provided that the axis of rotation (suspen- sion) and the reflecting surface of M are sensibly coincident, and that 28 The Telescope-Mirror-Scale MetJwd. X, XI, and XII have been completed, this adjustment may be accu- rately made for any position of the telescope thus : Focus the tele- scope sharply on the suspension fiber (axis of rotation) of M, with M swinging free. Turn the telescope, or shift it laterally without dis- turbance of the scale, until the cross-hair intersection falls upon the fiber. Still without disturbing the scale, tip the telescope about a hori- zontal axis parallel to the scale until the cross-hairs are approximately central on the mirror, and change the focus until the scale is sharply defined. The reading will in general not be exactly at the middle point, but the telescope and scale must now be clamped rigidly in posi- tion, and all subsequent adjustment of the null reading to zero must be made by the mirror, that is, by changing the direction of the suspended system until the null reading is exactly the middle point O from which A' and B' are measured off in making adjustment XII. This will be effected according to the nature of the apparatus; e.g., by changing the directive field, in a sensitive galvanometer ; by turning the torsion head or the whole instrument if it is an electrodynamo- meter or electrometer, or by twisting the mirror upon its suspension rod, and so on. If the axis of the suspension and the reflecting surface of M do not coincide, the above method of adjustment becomes inaccurate to an extent depending upon the eccentricity of the reflecting surface, and on the departure of the telescope from the vertical through O. If the eccentricity is but a few millimeters, its effect is wholly negli- gible if T is within a few millimeters of this vertical, as may be seen from adjustment VIII. In that case no attention to the adjustment of T and O beyond casual inspection is called for. If when this stage of the adjustment is reached it is found that the cross-hair intersection does not fall at the right height upon the scale for good reading, the telescope may be tipped slightly as a remedy. But if when this is done the illumination or extent of field is not all that the apertures of telescope and mirror should give, then the adjust- ments must all be repeated, beginning by raising and lowering the tele- scope and scale together, and tipping the former more or less, focusing centrally on the mirror and then on the scale alternately. During observations it is by no means always practicable to bring the null reading exactly to the middle point of the scale. This read- ing , approx., O" C as with ft very small, the factor cos ft is so nearly unity as not sensibly to affect the last term ; B d l OA O"C . . _* = sm 2 ft + sin ft tan 2 (f and in general = qp sin ft tan 2 qp sin 2 ft, d the -(- and signs in the first term applying respectively to right and left-hand deflections (jp. PRIMARY MEASUREMENTS. With Reversals of M. Disregarding the observational sign of d, the mean deflection used would be (d\ -f- 2 d%), and the correction would be = [sin ft tan 2 . o.oo 030 = sm 2 ft d < . . sin ft = 0.017 ; ft = i.o, or O" O = i ooo tan ft = 1 7 mm. Without Reversals, the correction would depend chiefly on

is proportional to that in r, but with the opposite sign. (See 32 The Telescope-Mirror-Scale Method. "Computation Rules," Proposition I, page xii.) The limiting value of Sr / r will therefore be the same, neglecting sign as 8d / d. Therefore S r / r = o.oo 030, Sr = o.3mm. Hence the requisite closeness in r is about 0.3 mm. ; that is, r must be measured and must be constant to about 0.3 mm. See remark in next paragraph. SECONDARY MEASUREMENTS. In secondary work r is not meas- ured, but must remain constant, and suitable means must be provided to see that it is so, within the same closeness of 0.3 mm. This requires much more attention than is usually given to this point, especially when the mirror hangs on a long suspension so that change of level of the instrument may easily change r by I or 2 mm. A special device is almost a necessity in careful work, to facilitate the test for constancy in the distance O M. In many instruments, especially those having a long suspending fiber, a slight difference of level alone produces considerable displacements of the mirror towards or from the scale or laterally, all equally objectionable. The direct application of XII and XIII for the elimination of these displacements on each day of use of the apparatus would be very laborious. This fact together with inadvertence frequently leads to the assumption that the distance, once adjusted, remains sufficiently constant. The foregoing figures as to the requisite closeness show the danger in such neglect. Devices will readily suggest themselves. The follow- ing will sometimes answer : Place the points of the leveling screws in positions from which they are not easily displaceable. Level the instrument until the suspension swings as it should. Then make upon the instrument, near the lower end of the suspending fiber, say on opposite sides of the suspension tube, reference marks so located that the fiber lies in the line of sight between them. Similarly locate another line of sight, nearly at right angles to the first and also pass- ing through the fiber. For convenience, one of these lines should be parallel to a pair of the leveling screws, and also to the scale. At each time of use, adjust the screws to bring the fiber into both of these lines of sight. Measure O M once for all, and at the same time meas- ure between two points chosen for convenience, one on the base of the instrument, the other on the scale. It is then necessary from day to day merely to bring the fiber into the reference lines by turning the Silas W. Holman. 33 leveling screws, and then to measure the distance between the chosen points, for which purpose a rod cut to the right length is convenient. XV. ESTIMATION OF TENTHS IN READING. Fractions of a division must be read, and this can be done only by estimation by the eye. With a little practice, so much facility in the estimation of tenths of a division is attainable that the error need never exceed one-twentieth of a division, with an average error of half this amount, or 0.025 division. PRIMARY MEASUREMENTS. The limit of attainable accuracy being fixed at d= 0.025 division, and the desired fractional accu- racy being & d / d = 0.00030, the minimum admissible value of d will be d= 0.025 / 0.00030 = 83, or in round numbers d= 100 divi- sions, whatever the size of the division. Smaller deflections than 100 must not be used in the most careful work, and preferably not less than, say, 200 divisions, whatever the scale distance. SECONDARY MEASUREMENTS. Same as in primary. XVI. To FIND qp, TAN qp, SIN qp, ETC., FROM d AND r. In the application of the telescope and scale method it is desired to find from the observed values of d and r the corresponding values of qp, tan qp, sin qp, etc., according to circumstances. For the most accurate work, especially in primary measurements, the best way is to compute tan 2 qp = (d / r) or log tan 2 qp, thence by tables of natural or log tangents to find 2 qp. This, of course, gives qp at once, and tan qp, sin qp, and other functions can then be found from tables. In some cases, notably in secondary work, it is, however, more convenient to have an expression for qp, tan qp, or other function, directly in terms of d and r. Such expressions generally take the form of a series. Several may be found in Czermak. The expression for tan qp is the one in frequent use, and therefore this alone will be given here. It is derived from a development of tan qp in series with ascending powers of tan 2 qp, viz. : tan qp = tan 2 qp | i (tan 2 qp) 2 -f- _ (tan 2 qp) 4 . . . . J. Substituting d / r for tan 2 qp gives 1 d r i , d \ 2 - i / d x 4 -| tan qp = - I ( ) 4- ( ) .... I. 2 r |_ 4 e \rJ % \ r ) 34 TJie Telescope-Mirror-Scale Method. This could be used directly, but may be modified into a more conven- ient form. Multiplying the terms of the parenthesis by d gives i r , i d* i d^ -| tan y = \d -\ ..... 2r L 4 r* r* J Then the quantity _L fll-L JL ^L ' ~4 r* ' ' T r* being very small may be conveniently treated as a correction to be applied to d. So we may write tan op = \d -r- 8] . 2 r The procedure would be to compute beforehand a table of values of this correction for the given value of r and for successive values of d (= 200, 250, 300, . . . 500) at sufficiently short intervals over the desired range. Then for a subsequent observed value of d', the cor- responding value of &' would be taken by interpolation in a table (or upon a plot), and applied to d'. This corrected value of d' would then be multiplied by i / 2 r to obtain tan (p. As the correction is small, only an approximate value of r is required in computing it, and slight changes in r during the progress of work would not vitiate the table, although they must be allowed for in the term i / 2 r. In secondary work any of the above methods of finding gr, tan |~/*f% i r\ 11 f\\l O ll 1 ** If wlU f\ LD 21-100? VClr U LU itUY Z me : , i UNIVERSITY OF CALIFORNIA LIBRARY