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CORNELL UNIVERSITY LIBRARY 
 
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 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
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 http://www.archive.org/details/cu31924074305891 
 
MISCELLANEOUS SCIENTIFIC PAPERS OF THE ALLEGHENY 
 OBSERVATORY — NEW SERIES. No. 10. 
 
 F. L. O. WADSWORTH, 
 
 DIRECTOR. 
 
 ON THE OPTICAL CONDITIONS REQUIRED TO 
 SECURE MAXIMUM ACCURACY OF MEAS- 
 UREMENT IN THE USE OF THE TELESCOPE 
 AND SPECTROSCOPE 
 
 F. L. O. WADSWORTH 
 
 PRINTED BY 
 
 THE UNIVERSITY OP CHICAGO PRESS, 
 
 CHICAGO. 
 
ON THE OPTICAL CONDITIONS REQUIRED TO 
 SECURE MAXIMUM ACCURACY OF MEASURE- 
 MENT IN THE USE OF THE TELESCOPE AND 
 SPECTROSCOPE. 
 
 By F. L. O. W A D S W O R T H. 
 
 Preliminary Note. — The main results of the present paper were 
 obtained in 1897 and were informally presented before the Astronomical and 
 Astrophysical Conference at the dedication of the Yerkes Observatory in 
 October of that year. 1 The general investigation on the "Conditions of 
 Maximum Efficiency in Astrophotographic Work," of which these formed a 
 part, 2 was interrupted by my departure from Yerkes Observatory in the win- 
 ter of 1897-8, and since then my attention has been so continuously devoted 
 to technical and engineering work that until very recently I have had no 
 opportunity to put any of the results obtained at that time in form for publi- 
 cation. This will explain, I trust, the delay in the resumption and continua- 
 tion of work begun more than five years ago. 
 
 In the use of any optical instrument there are three quanti- 
 ties which are more or less closely related and which together 
 determine what is ordinarily termed " optical definition." These 
 quantities are "resolution," "accuracy," and "contrast," and 
 the corresponding characteristics of the instrument with refer- 
 ence to them have been termed "resolving power," "metro- 
 logical power," and " discerning or delineating power." They 
 all depend directly on the form and distribution of intensity in 
 the physical image of the object under examination as formed at 
 "the focal plane of the instrument, and differ from each other 
 only in the way in which this form and distribution affect the 
 particular use to which the instrument is to be applied. Thus 
 in the discovery of double stars and general spectrum analysis, 
 resolution is of most importance ; in meridian circle and helio- 
 
 ' Yerkes Observatory Bulletin No. 2; Astrophysical Journal, 6, 150, 1897., 
 See also papers "General Theory of Telescopic Images," ibid., pp. 123, 127; and 
 " Effect of Atmospheric Aberration on the Intensity of Telescopic Images," ibid. 
 7» 7°- 
 
 "See Note on the Result Concerning Diffraction Phenomena, M. N., 58, 287 (b). 
 
 1 
 
2 F. L. O. WADSWORTH 
 
 metric work and in determinations of absolute and relative wave- 
 lengths, accuracy is the first consideration; while in the study of 
 planetary detail and what may be termed pictorial and chart 
 photography, contrast is the quality to be chiefly considered. 
 
 The question of the resolving power of instruments has been 
 considered for a number of general and special cases' by differ- 
 ent writers. The theory of metrological power and contrast 2 
 has received much less attention. In the present paper it is 
 proposed to investigate more fully the general conditions of 
 metrological power and accuracy in the optical measurement of 
 the relative or absolute position of points or lines. 3 
 
 Let us denote the resolution of an optical instrument by R 
 and its accuracy of measurement by A. These quantities are 
 the reciprocals respectively of a, the limiting resolving power, 
 i. e., the angular distance between two points or lines that can 
 just be resolved ; and of e, the limiting metrological power, or 
 the smallest angular distance that can be measured with cer- 
 tainty. These four quantities are connected by the general 
 relations 4 
 
 A = aR, 
 
 (0 
 
 ■;• \ 
 
 ■ Rayleigh, " Wave Theory," Enc. Brit., 24, §§ 11, 12, 13, and 14. " Investiga- 
 tions in optics with reference to the spectroscope," Phil, Mag., 8, g, 1879-80 ; with 
 reference to the microscope, ibid., 42, 167, 1896; Michelson, Phil. Mag., 31, 38s, 
 1891; 34, 280, 1892; Astrophysical Journal, 1, 1, 1895; Wadsworth, ibid., 52, 
 1895 ', 3» "7 and 3 21 J 4> 54, 1896 ; 6, 27, 1897 J Mem. Spet. Ital., 26, 2, 1897 ; Phil. 
 Mag., 43, 317, 1897; Astrophysical Journal, 16, 1, 1902. - 
 
 "The investigation of the general subject of delineating power and contrast has 
 been begun (Astrophysical Journal, 6, 119, 1897; 7, 70 and 77, 1898) and the 
 results applied in detail to some special cases. ( " Astronomical Photography," A. N. 
 144, 97; "Photography of Planetary Surfaces," Observatory, 20, 333, 365, 404; 
 " Visibility of Linear Markings on Planets," A. J., 18, 41.) The concluding part of 
 the general paper (Astrophysical Journal, 7, 70) and some other papers on addi- 
 tional special cases were in course of preparation when the work was interrupted as 
 explained in the preliminary note. They will be taken up again as soon as possible 
 
 3 The case of the measurement of position of sources of considerable angular 
 magnitude, such as the Sun, Moon, and major planets, involve different conditions of 
 measurement, and will for that reason be considered separately. [See M N c8 
 2SS (h).j ' " ' 
 
 * Michelson, " Measurement by Light Waves," Am. Jour., 39, 115. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 «ocsrSa»l 
 
 <%&v m -^SSSK^ 
 
 Fig. 
 
 where a is a factor whose value is not constant but varies 
 with the conditions of measurement. As has already been 
 pointed out, a fundamental condition for attaining a maximum 
 value of a is that the scale of the diffraction image of the 
 source whose position is to be measured shall be large com- 
 pared with the width of the 
 reference cross-wires or points 
 to which its position is referred. 1 
 When the diffraction image is 
 sufficiently broad with refer- 
 ence to the wire, x, as in Figf. 
 I, it is possible to locate the 
 position of the former with 
 reference to the latter with an" 
 error not exceeding one hun- 
 dredth of the total width 7nm= 
 w of the diffraction image. Since the resolving power a for fine 
 lines and points is (for rectangular aperture) equal to y 2 w, it 
 would follow that the maximum value of a might be as large as 
 50. Usually this degree of magnification of the image is not 
 attainable, and the corresponding value of a is reduced. Experi- 
 ence, however, shows that under best conditions a value of 
 from 10 to 15 can be attained. Thus, with a telescope having 
 an aperture of 4 cm, the limiting resolving power of which is 
 about 3", it is possible under favorable conditions to measure 
 angular differences of position as small as of20 either micromet- 
 rically or heliometrically. 2 About the same order of accuracy 
 is attained in setting on the images of spectral lines. Thus 
 Jewell finds that his probable error of a single setting on a line 
 in the solar spectrum obtained with Rowland's concave gratings 
 is about 0.00 1 tenth-meter. 3 The limit of resolving power of 
 the gratings is about 0.0 1 5 tenth-meters. * Similarly both Camp- 
 bell and Frost find that their average errors of setting on the 
 image of a line in the star spectra obtained with the Mills 
 
 1 Phil. Mag., 44, 83 ; Phys. Rev., 4, 96. 3 Rowland, A. and A., 12, 321, 1893. 
 -Jour. Franklin Inst., 138, I ; July 1894. - Phil. Mag., 43, 320. 
 
4 F. L. O. WADSWORTH 
 
 and Bruce spectrographs is about 0.0004 mm and 0.0003 mm 
 respectively. 1 With the cameras of 406 and 449 mm focal length 
 these linear errors correspond to an accuracy in angular measure- 
 ment of about o! 19 and of 14. The angular resolving power of 
 
 the two camera objectives (aper 
 tures 37.4 mm and 51 mm) are 
 only 2" 5 and 1 ' 8 respectively for 
 the photographic region of the 
 spectrum, X=4500. 
 
 In each of the above cases 
 the accuracy of setting (on the 
 image of the point or line) is 
 about fifteen times the resolution 
 
 obtainable with the instrument. 
 
 Fig. 2. 
 
 For this class of measurements 
 (i. e., telescopic) we may therefore assume 
 
 <*— ■ = X S (2) 
 
 c ^ 0.07a . 
 
 In order that these values of a and e may represent a real 
 and corresponding degree of accuracy in the determination of 
 the position of the source itself (as distinguished from the posi- 
 tion of the image) another very important condition is neces- 
 sary. This is that the distribution in intensity in the diffraction 
 pattern shall be symmetrical about the positio?i of the geometrical 
 image of the source. For if the distribution is unsymmetrical, as 
 in Fig. 2, the tendency will be to place the measuring wires to 
 one side of the position of the geometrical image, which is at 
 ao, by an amount 00' , depending on the excess of illumination 
 amm' on that side. The error thus introduced will correspond- 
 ingly reduce the accuracy of setting, so that in general we 
 shall have to write 
 
 " ' = &W. , 
 
 1 From unpublished observations communicated by these observers. In each case 
 the average error is that of the mean of four settings only. On the best solar plates 
 Campbell states that his average error of setting is less than one-third of this, i. c, 
 only about 0.00012 mm. 
 
OPTICAL CONDITIONS OF ACCURACY 5 
 
 where k is less than unity by an amount depending on the 
 asymmetry in the image. It becomes necessary, therefore, to 
 investigate the various causes of such asymmetry and the effect 
 produced by each. The general expression for the intensity, 
 / 2 , at any point,/, in the focal plane image is of the form 1 
 
 r =U$i™T {at - p)idxdy !- (3) 
 
 where p is the distance of the pointy from the element of the 
 wave-front, dx dy, whose amplitude of vibration is i. The inte- 
 gration is extended over the whole of the aperture through 
 which the wave-front passes. 
 
 The expressions for the distribution in intensity in the image 
 of a fine point which have been usually employed in discussing 
 questions of resolving power are 
 
 1 =W\}-* 1 A "~" cos V xdx \ = W h^i\ (4) 
 
 for a telescope with a circular aperture of radius R, and 
 
 p =WlJ * cos Tf xd *\ = w ~FW = ~F\ 
 
 w) w) 
 
 for a telescope of rectangular aperture of width d and length b. 
 These expressions represent respectively the distribution in 
 
 intensity at any distance a = -j from the center of the image 
 
 along a line parallel to the length b of the diffracting aperture. 
 
 In deriving the expressions (4) and (5) from (3) the following 
 assumptions are made : 
 
 1. That the amplitude of vibration i is uniform and constant 
 over the entire wave-front within the diffracting aperture, and 
 that the latter is symmetrical with respect to the line joining the 
 geometrical centers of the source and the image, i. e., with the 
 line of collimation of the telescope. 
 
 ■Rayleigh, "Wave Theory," Enc. Brit., 24, § II; Pop. Ast., 5, 534. 
 
 (s) 
 
O F. L. O. WADSWORTH 
 
 2. That the wave-front passing the diffracting aperture is 
 truly spherical and has its center at the center of the geometrical 
 image. This amounts to the condition that there is no aber- 
 ration. 
 
 3. That the wave-length X is constant within the limits of 
 integration, i. e., either the light is strictly monochromatic or 
 the telescope is strictly achromatic. 
 
 In practice not one of these assumptions is strictly correct 
 and the expressions (4) and (5) are not therefore strictly accu- 
 rate. In dealing with questions of resolving power the effect of 
 variations from these theoretical conditions has been investi- 
 gated in a number of cases and has generally been found to be 
 small. 1 We cannot, however, assume that the same conclusion 
 holds when we come to deal with questions involving the met- 
 rological power and accuracy, for we then have to consider 
 quantities of a. much smaller order of magnitude, and an amount 
 of disturbance or imperfection, particularly asymmetry, in the 
 image which is negligible in questions involving resolving power 
 (or in many cases of contrast) will introduce an error of meas- 
 urement considerably larger than the limit of accuracy e attain- 
 able under the best conditions. 
 
 The various causes which produce distortion and asymmetry 
 in the diffraction image may be divided into two general 
 classes : 
 
 (A) Those of a physical nature of a character already indi- 
 cated (1), (2), and (3) (pp. 5 and 6). 
 
 (B) Those of a more purely instrumental nature depending 
 on peculiarities of form or size or operation of the instrument 
 itself. 
 
 In considering both of these classes we shall in this general 
 paper assume the following conditions, which may nearly always 
 be fulfilled or at least closely approximated : 
 
 ■For special cases in which the effect is of considerable importance see "Theory 
 of the Objective Spectroscope," Astrophysical Journal, 4, 54; "Resolving 
 Power of Telescopes and Spectroscopes for Lines of Finite Width," Phil. Mag., 
 43,317; also paper, "The Effect of Absorption on the Resolving Power of Prism 
 Trains," to be published in the February number of the Phil. Mag. 
 
OPTICAL CONDITIONS OF ACCURACY 7 
 
 {a) That the aperture of the instrument is always rectangu- 
 lar and that one of the sides is parallel to the axis or line of 
 measurement, this axis being likewise the line along which the 
 distribution in intensity in the image has to be considered. The 
 assumption of rectangular aperture gives in general simpler ana- 
 lytical expressions, and the results obtained, in most cases, lead 
 to the same conclusions as would be reached if circular apertures 
 were assumed. 1 
 
 (b) That in addition to being parallel to one side of the 
 rectangular aperture the axis of measurement is so chosen that 
 the disturbing cause is symmetrical in respect to this axis. Spe- 
 cific cases in which this condition cannot be fulfilled will be 
 treated separately. 
 
 Case A (i) — Illumination over the incident wave-front 
 unsymmetrical. 
 
 The general expression (3) may be reduced to the form 
 
 p = w* [jt c ° s i ^ + v) * dx ay ] 
 
 +M//' in 0<*+^'***J- (6) 
 
 Under the conditions [a) and (b) assumed above we obtain 
 in this case for the intensity of the diffraction pattern at any 
 point on the axis £, (77 = 0), the expressions 
 
 ^^U^ cos W x ^l + U /{x)sin W xdx J \ (7) 
 
 = A (c* + s') , 
 
 where f{x) —i represents the amplitude of vibration at different 
 portions of the wave-front. 
 
 The only case of importance of this kind is that in which the 
 wave-front, before reaching the diffracting aperture, has traversed 
 an absorbing medium of varying density or varying thickness as 
 in the case of the prism spectroscope. This latter case has been 
 recently examined in connection with the question of resolving 
 
 ■The case of circular apertures is taken up in detail in some special cases where 
 the results differ appreciably from those obtained with rectangular apertures. 
 
° F. L. O. WADSWORTH 
 
 power of prism trains composed of very large or very dense 
 prisms. 1 
 
 The expression for I* which was found was 
 
 V 
 
 A 2 / 2 
 
 *+($' 
 
 (8) 
 
 where 5 is a constant depending on the coefficient of absorption 
 of the glass composing the prism train, and is determined from 
 the relation 
 
 ' = '>-*', (9) 
 
 Fig. 3. 
 
 i being the intensity of light transmitted through the axis of the 
 
 prism system. 
 
 The distribution in intensity represented by -(8) is shown in 
 
 1 . ■286 2 107 
 
 Fig. 3 for the particular values of B=— — and B= ' . For 
 
 these values of B the intensity of the transmitted beam falls off 
 75 per cent, and 89 per cent., respectively, from one edge of the 
 aperture to the other.. 
 
 The curve of intensities is symmetrical with respect to £=0, 
 the position of the geometrical image. The unsymmetrical 
 absorption therefore does not introduce any direct error of 
 
 ■"On the Effect of Absorption u.^ the Resolving Power of Prism Trains," see 
 footnote above. 
 
OPTICAL CONDITIONS OF ACCURACY 9 
 
 measurement at the focal plane. 1 In fact since the effect is in 
 general to increase the apparent width w of the line, the accu- 
 racy of setting will, if anything, be slightly increased. 
 
 It may be similarly proved that an asymmetry in the form 
 of the aperture itself will not affect the symmetry of the image 
 about the geometrical center. Indeed, we may choose particular 
 forms of aperture (other than rectangular) which will some- 
 what increase the resolving power and the accuracy of the 
 measurement. 2 
 
 CASE A (2). 
 
 Effect of asymmetry in the wave-front [aberration). — When the 
 wave-front which forms the image is truly spherical its equa- 
 tion is 
 
 *+f+*=f . (lo) 
 
 and the distance p from any point xy in this front from a point 
 £77 in the focal plane is 
 
 = ]?- 2x£ - 2y V + £* + rf J 
 
 (>■) 
 
 If the wave-front is not spherical we may express the coordi- 
 nates of the new surface with respect to the old as follows 
 
 x' = x + fix* + yx* + Sx* + ■ ■ ■ ) 
 
 y' =y + &f + y<f + W H [ (12) 
 
 3' =2 . ) 
 
 The expression for the intensity 7, 2 at any point £, r), in the 
 focal image will be found by substituting these new values for 
 x' , y' , 2' , in ( 1 1) and (6), and performing the indicated integra- 
 tions. But as before we considerably simplify the integrals with 
 which we have to deal by the assumption of conditions (a) 
 and (d), p. 6. Then all terms in odd powers of y higher 
 
 'The effect of asymmetrical illumination on the displacement of an image out of 
 the focal plane is considered under B. 
 
 = Discussion of this question will be taken up in a paper "On the Form of Tele- 
 scope Aperture Best Adapted to the Resolution and Measurement of Close Double 
 Stars," which is now in course of preparation. 
 
IO F. L. O. WADSWORTH 
 
 than the first disappear. Next, by changing/, equation (u), 
 to a new value /' (i. e., by readjusting the focus) we may 
 cause the terms in x* and y 1 to disappear. Finally, since any 
 disturbance which we need to consider is exceedingly small (as 
 will appear later) , terms in x 3 are relatively of much more impor- 
 tance than terms in x 4 , y*, or any higher powers, and all of the 
 latter may therefore be neglected. Besides this any aberra- 
 tional effect due to terms in x* will be symmetrical in its nature 
 and will not therefore affect the position of the center of area of the 
 diffraction image. The same would also be true of terms in x* 
 and y 2 whose effect, however, can, as already seen, be rendered 
 negligible by refocusing. 
 
 When we consider only terms in x 3 the expression for I* (6) 
 (since sin x 3 like sin x is an uneven function), becomes for rec- 
 tangular aperture 
 
 11 = A' j * cos 2 T, [x + yx 3 ] dx . (13) 
 
 z 
 
 This integral was first investigated by Airy ' and subsequently 
 more completely by Lord Rayleigh. 2 The latter has adapted 
 Airy's results to the consideration of special cases for which the 
 distortion of the wave-surface amounts to %, yi, and ^ of a 
 wave-length. 
 
 The values of I 2 for the value of yx 3 such that the extreme 
 displacement of the wave-surface at the edge of the aperture is 
 Y^ of a wave-length from the true spherical surface has been 
 calculated from Airy's and Rayleigh's results and is given in 
 column 2, Table I. The values of I 2 (from 5) are given in 
 column 3, for comparison. The abscissas .are expressed in 
 terms of a ol the resolving power of the telescope of aper- 
 ture b. 
 
 The form and relative position of the two curves I 2 and /,* 
 are shown in Fig. 4. The dotted curve represents the distribu- 
 tion in intensity in the focal plane image when there is no aber- 
 
 ■ Camb. Phil. Trans. 6, 402, 1838. 'Phil. Mag. (5), 8, 404, 1S79. 
 
OPTICAL CONDITIONS OF ACCURACY 
 TABLE I. 
 
 a 
 
 /,' 
 
 /* 
 
 "0 
 
 V 
 
 /* 
 
 — 2.0 
 
 0.0086 
 
 0.0000 
 
 O.u 
 
 0.7093 
 
 1 .0000 
 
 -1.8 
 
 .0061 
 
 .0108 
 
 -1-0.2 
 
 .9158 
 
 .8751 
 
 -1.6 
 
 .0012 
 
 .0358 
 
 +0.4 
 
 • 9137 
 
 ■ 5754 
 
 — 1.4 
 
 .0004 
 
 .0468 
 
 4-0-6 
 
 .6901 
 
 ■ 2545 
 
 — I .2 
 
 .0031 
 
 .0243 
 
 +0.8 
 
 .3629 
 
 .0547 
 
 — 1.0 
 
 .0018 
 
 .0000 
 
 + 1.0 
 
 .0999 
 
 .0000 
 
 -o.8 
 
 .0017 
 
 .0547 
 
 + 1.2 
 
 .0009 
 
 .0243 
 
 -0.6 
 
 •0399 
 
 ■ 2545 
 
 + 1.4 
 
 ■ 0395 
 
 .0468 
 
 -0.4 
 
 .1714 
 
 •5754 
 
 + 1.6 
 
 .1095 
 
 .0358 
 
 — 0.2 
 
 .4159 
 
 .8751 
 
 +1.8 
 
 . 1240 
 
 .0108 
 
 ±0.0 
 
 .709; 
 
 1 .0000 
 
 +2.0 
 
 .0762 
 
 .0000 
 
 ration (equation 5) ; the full curve, that when the aberration 
 amounts to % wave-length as assumed above. The effect of 
 the aberration is three- a. 
 
 fold : ( 1 ) it dimin- 
 ishes the intensity at 
 the center of the im- 
 age; (2) it renders the 
 image asymmetrical 
 with respect to its cen- 
 ter of area; and (3) it 
 displaces very sen- -, 
 sibly both the position 
 of maximum bright- 
 ness and of the center 
 of area. The effect of (1) is not serious, but the effect of (2) 
 and more particularly (3) on the accuracy of measurement is 
 most important. The displacement of the point 0' (Fig. 4) 
 from the geometrical center of the undistorted image is about 
 0.3a, while the limiting accuracy of measurement, as we have 
 already seen, is about 0.07a. The displacement 0-0' caused 
 by an aberration amounting to only % wave-length is there- 
 fore about four times the limit of accuracy e attainable under best 
 conditions, 
 
 An examination of the curve representing a double source, 
 Fig. 5, as viewed under the conditions assumed in Fig. 4, shows 
 
 W = not.,— 
 
 Fig. 4. 
 
F. L. O. WADSWORTH 
 
 that neither the resolving power, which depends primarily on 
 the form and "width" mm of the images of each component, 
 nor the contrasting power, which depends on the width mm and 
 the value o'a' of the maximum ordinate, are either of them 
 affected greatly by an aberration of the amount {y^ wave-length) 
 shown. Hence, it is generally assumed that the optical " defini- 
 tion" of an instru- 
 ment (which, as usu- 
 ally defined, depends 
 principally on resolu- 
 tion and contrast) is 
 not injuriously affec- 
 ted by aberrations 
 which introduce 
 differences of phase 
 
 ., in the wave-front of 
 
 Fig. 5. 
 
 period, and that therefore optical surfaces which reflect or 
 transmit wave-fronts at nearly perpendicular incidence need to 
 be accurate to yi and ]4 wave-lengths only. But as indicated 
 above, this degree of perfection is not sufficient in instruments 
 designed for accurate measurements. 
 
 The conditions of optical measurement of position are such 
 that we are concerned not so much with the absolute amount of 
 aberration as with the constancy of its effect. For in the great 
 majority of cases we are concerned with the "simultaneous or 
 successive determination of the separation or relative position 
 of two images in the same focal field. Under such conditions, 
 if the amount of aberration remained exactly the same for both 
 images, the distortion and displacement of both would also be 
 the same in amount, and in direction, and the measured distance 
 between their centers would be practically unaltered. To avoid 
 errors of measurement, therefore, we need only be certain that 
 the differential effect of the wave disturbance on the two images 
 is less in amount than the limit of accuracy e already estab- 
 lished. The degree of constancy required is readily determined 
 
OPTICAL CONDITIONS OF ACCURACY 1 3 
 
 from the results of the preceding investigation. The displace- 
 ment of the center of the image in Fig. 4 is, as already stated, 
 about 0.3a for a phase difference of J^ wave-length. This dis- 
 placement is very nearly proportional to the tilting of the actual 
 wave-front with reference to the true spherical wave-front, and 
 this tilting in turn is very nearly proportional to the phase 
 differences between center and edge. Hence, if the differential 
 displacement of the two images is not to exceed the limiting 
 value of e (0.070), the aberration must not vary by more than 
 
 1 °-°7 .x / \ 
 
 - • ; ^ 0.06X. (14) 
 
 4 °-3° 
 The amount of aberration (?'. e., the distortion of the wave- 
 front forming the image) will vary at any instant of time : 
 
 I, by reason of changes in position of an image in the focal field, 
 involving a change in the inclination of the axis of the wave- 
 front to the various optical surfaces and possibly a shift of the 
 wave-front as a whole with reference to those surfaces ; 
 
 II, because of changes in the optical constants of the media trav- 
 ersed by the wave before reaching the focal plane. We will now 
 examine the instrumental and operative conditions necessary to 
 keep the variations due to either cause within the limits given. 
 
 I. (1) If the two images are separated by a considerable 
 angular interval in the field of the observing telescope, the two 
 wave-fronts by which they are formed will meet the optical sur- 
 faces of the objective at different angles, and the resultant 
 distortion or curvature in each will vary by reason of this fact, 
 even though they were precisely similar before incidence. The 
 amount of aberration introduced in any given case is determined 
 by calculation of the relative retardations of the extreme, with 
 reference to the central rays, which meet at the focal plane. 
 
 Thus in Fig. 6 let abb' be the points at which the center and 
 extreme edges of the given wave-front x t x„ meet the front sur- 
 face of the optical system, and 0' , the best-defined center of 
 curvature of the wave-front x,x„ after its passage through the 
 optical system. Let hb be the distance from the point b to the 
 instantaneous wave-front incident at a, and ah' the distance from 
 
14 
 
 F. L. O. WADSWORTH 
 
 and 
 
 (*5) 
 
 Fig. 6. 
 
 the instantaneous wave-front incident at b' . Then for the pri- 
 mary plane 1 the retardation of the extreme with reference to the 
 central rays, i. c, the amount of aberration for the two halves 
 of the wave-front, will be represented by 
 ao' — {Jib +bo') | 
 ao' - (b'o' -li'a) \ 
 The determination of the value of these quantities in any 
 
 given case demands 
 a knowledge of the 
 indices of refraction, 
 radii of curvature, 
 and thickness of the 
 lenses forming the 
 optical system, and 
 is usually quite com- 
 plicated. We can, however, draw certain conclusions from the 
 conditions of the problem which are of more general interest 
 than the detailed consideration of many individual cases. 
 
 The displacement of the centers of two images o' o" will be 
 in opposite directions if they are on opposite sides of the prin- 
 cipal optical axis ao, and will be equal in amount if they are at 
 equal distances from this axis, and both the optical system and 
 the incident wave-fronts, x t x^ x[x'„ are symmetrical about their 
 respective axes. Under such conditions of symmetry the dis- 
 tortion of the field itself is symmetrical with respect to its center 
 o, and if these conditions could always be maintained, the proper 
 corrections could be determined once for all and then always 
 applied to the measurements. But the form of the incident wave- 
 fronts is different at different times, as shown at x" x" ' , x' v x°. 
 These changes may be due to influences which are outside of and 
 independent of the instrument itself; and the displacements 
 and the required corrections will therefore vary by an amount 
 which is indeterminate. Our only complete safeguard against 
 
 ' We confine our attention to the aberration at the primary focus because in the 
 case of lines perpendicular to the xy plane the best-defined image is formed at that 
 point. 
 
OPTICAL CONDITIONS OF ACCURACY 1 5 
 
 errors of this kind is to confine our measurements to a field 
 within which the aberrational distortion and displacement of the 
 lens system itself is negligible. We can form a general idea of 
 what this field is in different cases without investigating each 
 one in detail by the following considerations. 
 
 The requirements for obtaining the maximum practicable 
 accuracy demand, as has been shown above, that the effect of 
 aberration on the image shall not be more than one-fourth as 
 great as that which would begin to sensibly affect good " optical 
 definition ". If we assume that the aberration due to a good 
 lens system varies as the square of the distance from the center, 
 and if we call 8 the "field of good definition" (visual or pho- 
 tographic) of the given lens, then the "field of measurement," 
 on the above assumptions, should not exceed j4 8. 
 
 The value of 6 depends on the linear size of the objective, 
 
 b, on its semi-angular aperture, /? = — , and on the type of optical 
 
 2/ 
 
 construction. In general we may say that 6 varies approxi- 
 mately as —= and as -= for a given type of instrument. The 
 
 maximum values of 6 that have been so far attained with several 
 types of lenses of different apertures are given in Table II. 
 
 TABLE II. 
 
 Type of Instrument 
 
 Size of 
 
 Objective 
 
 b 
 
 
 4 in. 
 
 ': 
 
 4 
 
 I : 
 
 2 
 
 I : 
 
 3 
 
 I : 
 
 6 
 
 I : 
 
 15 
 
 I : 
 
 40 
 
 I : 
 
 Angular 
 Aperture 
 
 Field 
 
 Photographic * 
 
 1. Photographic doublet 
 (new form) 3 
 
 2. Photographic doublet 
 (symmetrical construction) 2 
 
 3. Photographic triplet (sym- 
 metrical construction) 
 
 4. Ordinary achromatic 
 (visual) 
 
 5. Ordinary achromatic 
 (visual) 
 
 6. Ordinary achromatic 
 (photographic) 
 
 7. Ordinary achromatic 
 (visual) 
 
 5 =0.20 
 
 5 =0.20 
 
 8 = 0.12+ 
 
 12 =o.oS3 
 
 18 =0.055 
 
 15 =0.007 
 1S.5 =0.054 
 
 about 2?5 
 about 2?o 
 about 2?o 
 
 about o°7 
 
 about 1 5 
 about 8?5 
 about 3° 
 
 about I?5 
 
 'The available good field of an objective is always slightly larger photographi- 
 cally than visually on account of the somewhat less rigorous standard of test that can 
 be applied. 
 
 2 See Annual Reports of the Allegheny Observatory, 1S9Q, p. 7 ; 1900, p. 19. 
 
i6 
 
 F. L. O. WADSWORTH 
 
 These values of 8 are based on experiments and tests made 
 by the writer and others on a large number of objectives of dif- 
 ferent sizes and types of the best modern construction. They 
 are considerably smaller in some cases than those claimed by the 
 makers of astronomical and photographic objectives, probably 
 because the standard of " good definition " is higher. Many 
 commercial photographic lenses, for example, " cover " fields of 
 40 or 50 with satisfactory results on an ordinary photo- 
 graphic standard. But 
 when the rigorous tests of 
 stellar photography are 
 applied, the definition of 
 these lenses breaks down 
 only a few degrees from 
 the axis ; in fact, I have 
 never found any of them 
 which give results at all 
 comparable with those de- 
 signed, by Hastings for 
 example, especially for astrophotographic work. A similar 
 statement applies to the longer focus type of objectives (4) and 
 (5). With a low-power eyepiece the visual definition is appa- 
 rently quite good over fields considerably larger than those 
 given, but with high powers and more careful tests the fields 
 are more restricted. 1 
 
 The available fields of measurement under the above stand- 
 ards vary therefore from a maximum of about 7 for (1) to 
 about ^° for (7). 
 
 1 The fields given above are -those obtainable in photographic use on flat plates 
 or in visual use without refocusing. These are the conditions that are generally 
 imposed in metrological work. If we use curved plates (or in visual work curved 
 micrometer slides) the field of good definition can be very greatly extended. Thus 
 with a symmetrical doublet of 4 '/ 2 in. aperture and 20 in. focal length (j3 = 4.4) it is 
 possible by the use of a curved plate to cover sharply a field of about 33 ° diameter, or 
 with a somewhat smaller angular aperture and a special construction of objective, a 
 field of over 40° diameter. See Annua/ Reports, Allegheny Observatory; also "Effi- 
 ciency of the Curved Plate for Charting Purposes," Observatory, 24, 274 and 419, 
 1901. 
 
 Fig. 7. 
 
OPTICAL CONDITIONS OF ACCURACY 1 7 
 
 There is one special case which has recently been examined 1 
 the results of which are of particular interest. This is the case 
 of the reflecting telescope and its associated spectroscopic 
 instrument, the concave grating. The character and amount of 
 aberration here depend on the form of the wave-front. We will 
 consider first the case in which this is spherical as in the ordi- 
 nary use of the concave grating. The general optical conditions 
 involved in this problem are shown in Fig. 7. 
 
 Let p be the position of the radiant point, 0' , that of its 
 image (either specular or spectral) ; and c, the center of curva- 
 ture of the spherical surface 2 dad' . Put pa = v ; o'a = u ; ac.= p; 
 pac = i; and cao' = 6. 
 
 Then from the general theory of optics we have always the 
 
 relation 
 
 cos 2 / cos 2 6 cos i -\- cos 6 
 
 = ■ (16) 
 
 V U p v 
 
 With Rowland's mounting this relation is satisfied by making 
 u = p cos 6 and v = p cos z , 
 
 and o' , c, p, all lie on the circumference of a circle whose diam- 
 eter is equal to p. The aberration at o' for this case has been 
 calculated by Rowland 3 and Glazebrook. 4 The latter finds for 
 the relative retardation for the central ray (u-\- v) relatively to 
 the edge ray pb + bo' , 
 
 E x ^pb -\- bo' — (u -f- v) = p sin /J (sin / — sin 6) 
 
 -\- — p sin* ft (sin 6 tan -f- s ' n z tan •* ) > ( 1 1) 
 
 8 
 
 where /3 is the semi-angular aperture of the surface bab' meas- 
 ured at the center of curvature C; therefore the angle bca. The 
 first term of this expression is that defining the relation between 
 the order of the spectrum m, the wave-length of the spectral 
 
 1 " Aberration of Mirrors and Concave Gratings at the Principal Focus," to be soon 
 published in Phil. Mag. 
 
 a The case of spherical surfaces only is considered here, because that is the form 
 of surface generally used for concave gratings. The case of parabolic and other 
 forms of surfaces is, however, considered at length elsewhere. See paper above 
 referred to. 
 
 3 Phil. Mag. (5), 16, 210. */6i,/., 377- 
 
1 8 F. L. O. WADSWORTH 
 
 line X, and the number of lines on the grating surface N, i. e., 
 
 p sin y3(sin i — sin 6) = - N?n\ . (18) 
 
 The second term is the aberration of the upper half of the 
 spherical surface ab. For the lower half of this surface ab' we 
 find similarly 
 
 Es = (pb' -\- b' o) — {it -f- v) = p sin /3 (sin i — sin 6) 
 
 + -p sin*/3 (sin 6 tan 6 -\- sin / tan i) = £ t . (19) 
 S 
 
 The aberration is therefore symmetrical, and its effect on the 
 image is simply to broaden it symmetrically without displacing 
 its center. 
 
 If the reflection is specular, 8 = i. Hence, if we make u = v 
 
 we always have 
 
 p ;= u cos 8 = v cos i . 
 
 The first term of (17) and (19) disappears, and the aberra- 
 tion (which is expressed by the second term) becomes 
 
 - sin* fi sin 6 tan 6 , (20) 
 
 4 
 
 which is likewise always symmetrical. 
 
 In case the images, either spectral or specular, are formed on 
 the circle passing through the center of curvature of the mirror 
 as above assumed, we can therefore use any extent of field 
 desired without introducing any error of measurement due to 
 the aberration of the spherical surface itself. The only limit to 
 the available field is that imposed by condition of good defini- 
 tion, i. e., that the aberration shall not exceed a quarter wave- 
 length. This condition applied to (17), (19), and (20) gives 
 
 sin 3 6 sin 2 / _. 2X 
 
 a H r = — ^-T~5 • ( 2I ) 
 
 cos cos 1 psin 4 /i x ' 
 
 In the case of Rowland's large gratings p= 650cm and 
 
 18=— =0.011. For the maximum value of z'=6o° we have 
 a 
 
 therefore 
 
 — _ - ^ 9 . s or = S 4 ° 15 • 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 I 'I 
 
 For gratings and mirrors of large angular aperture, such as 
 are sometimes used in stellar spectroscopic work, the maximum 
 value of both 9 and i are considerably reduced. Thus for a 
 grating of 150 cm radius, of curvature and linear aperture the 
 same as before, /3=o.050. We have then 
 sin 2 <9 
 
 + 
 
 0.106 
 
 Fig. 8. 
 
 cos 6 ' cos i 
 
 which shows at once that neither 6 nor i can in this case exceed 
 i8°20'. If we use the Rowland mounting for which 0=o for 
 the center of the field 
 and make e' = 15° we 
 have in the above case 
 Can 8 . 
 
 For the concave 
 grating the maximum 
 field is of course 
 obtained by making 
 2 = 0, and placing the 
 photographic plate or eyepiece directly above the slit. Under 
 such circumstances we can obtain good definition over a field 
 36 long (18 on each side of the center). 
 
 In the case of the reflecting mirror the maximum field is 
 determined by making & = i in (21). For a mirror of the dimen- 
 sions last considered we have therefore 
 
 or about % as large as obtainable with the concave grating. 
 
 Second case — incident wave-front plane. This case corre- 
 sponds to the ordinary use of the reflecting telescope, and to the 
 use of the grating as suggested by the writer in 1896.' 
 
 The optical conditions of this problem are shown in Fig. 8, 
 the notation being the same as before. 
 
 1 Astrophysical Journal, 3, 55-60. Since this article was written the concave 
 grating has been used in the manner suggested, by Poor and Mitchell (ibid., 7, 157 ; 
 10, 29), in stellar spectroscopic work, and by Jewell; Mohler & Daniel, (ibid., 12, 
 361); Frost (ibid., 12, 311), and by the writer (Report Allegheny Observatory, 1900, 
 p. 23), in solar-eclipse work. 
 
H — u cos 2 6 -\ cos 4 6 p cos 6 
 
 I 4 4 « 2^ j 
 
 20 F. L. O. WADSWORTH 
 
 The relative retardation of the central with reference to the 
 extreme rays is in this case 
 
 E T = u — (bo' -\- bh) for the upper half of the surface, , » 
 
 Ei = u — (bo' — ah') for the lower half of the surface. 
 
 The general expressions for E It E^ can be derived from the 
 
 geometry of the triangles bao' , ab' o' , bah, and a'b'h'. Using 
 
 the same notation as before we find as general expressions for 
 
 E z and E z 
 
 £ 1 = p sin 8 (sin 6 — sin i) 
 
 + - o sin' B • cos 6 + cos i — - cos" 0)1 — £ 
 
 2 I U ' \ 
 
 + - P - sin 3 8 -j sin 6 cos 6 (u - p cos 6)1 = C 
 
 ^p sin 2 6 cos 6- £ sin 2 6 cos 2 6 --s (^ 
 
 1 n 2 2 K 4 
 
 + -^-sin 4 /^ l = Z> 
 
 2 Z< 2 
 
 + -(> sin 4 /? (cos 6 -f- cos/) = F 
 
 8 
 
 and 
 
 ,£ 2 = psinj8(sin/-sin 6) + £ - C + &+F . (24) 
 
 In this case the aberration is unsymmetrical, the amount of 
 asymmetry being expressed by the term in sin 3 /3 ( C) . The value 
 of this term depends on the manner in which the grating is used. 
 Three cases may be distinguished, as has been already indicated: 
 
 (a) When the direction of reflection is parallel to the line 
 of incidence pa. 1 Then = — i, and from (16) we find for u, 
 {v— 00 ), 
 
 «„ = ^cos0 . (25) 
 
 In this case we have for the term C 
 
 p sin 3 8 sin 6 . (26) 
 
 (b) When we examine the spectra in the neighborhood of the 
 center of curvature. 6 = o for the center of the field, and we 
 have for u 2 
 
 «* = — ~ . . (27) 
 
 I + cos? V '/ 
 
 ' ASTROPHYSICAL JOURNAL, 3, 58, 59. 'Ibid., pp. 54-57. 
 
OPTICAL CONDITIONS OF ACCURACY 21 
 
 Then for (C) we find for small values of 6 
 
 p sin 3 /? sin #-cosz (i + cosz ) • ( 2 8) 
 
 (c) When we make the incidence on the grating normal. 
 Then z = o and 
 
 p cos 2 , . 
 
 "' = 7+^0' (29) 
 
 and for (C) 
 
 . , n ■ ,i/i + cos 8\ , . 
 
 pSin3/3sm ^(-^-) • <3o) 
 
 (d) Finally for regular specular reflection d =i, and 
 
 u = - cos 6 , (30) 
 
 just as in case («). 
 
 As the aberration is in all these cases unsymmetrical, the 
 available measurable field is found by equating (14) with (26), 
 (28), and (30) (neglecting terms in sin 4 B). 
 
 This gives us for cases [a) , (6) , and (</) 
 
 sintf.= \/ . (31) 
 
 i6p sin 3 /} 
 
 In the case of Rowland's gratings, already discussed, for 
 which p = 650 cm. and 8 =0.01 1, the value of 6 given by (31 ) is 
 
 =* 2 22 
 
 when the grating is used as in (a). When it is used as in (b), 
 and we make the maximum angle of incidence z — 6o°, we have 
 for 6. <^ 
 
 2 
 
 6 ™ x =e cos/(i + cos/)~°" 6 " 
 Finally, if we use the same grating as in (c), we find for 8 
 .2 cos" 0„ , , 
 
 I + COS "a 
 
 Another case of great importance and interest is the use of 
 mirrors as collimators or telescopes of spectroscopes. If the 
 mirrors are of long focus, very satisfactory results are attained 
 by their use. Thus in one of the instruments used by the writer 
 
2 2 F. L. O. WADSWORTH 
 
 the spherical mirror had an aperture of 6 cm, and a focal length 
 of about 175 cm. For such a mirror we have p = 350, /3 ^ — - . 
 
 With these values we get from (26) 
 
 sin i= 0.016 or / = 55' . 
 
 The diameter of undistorted field is therefore about i?8. 
 The diameter of the field of good definition (for qualitative 
 work only) is of course about four times this, or about 7? 5. 
 These results agree well with those actually found in practice, 1 
 
 For large angular apertures the results are much less satis- 
 factory. Thus for a surface such as was considered before for 
 which p = 1 50 cm and /3 = 0.05 we have 
 
 «, = «^«,a 4 o" • 
 By mounting the grating as in (d) and using a very large value 
 of i (z'= 75°) we may increase this field about six times, i. e., 
 
 0^2' . 
 We may likewise increase the field by diminishing the focal 
 
 length (keeping /3 the same) in the proportion - . For a grat- 
 
 r 
 
 ing of only 1 inch aperture and 5 inches focal length £=0.05 as 
 before and p= 12.5cm. The value of Q b for this case is 
 
 0^25' . 
 
 The field of good definition in all the above cases is of course 
 about four times as large as the field of negligible measurable 
 distortions. 
 
 The above results show clearly that the use of spherical grat- 
 ings or mirrors of very large angular aperture with parallel inci 
 dent light is impracticable for either quantitative or qualitative 
 work, except in the case of very small gratings used as in (<5). 
 If such mirrors or gratings are to be used of large size, it is abso 
 lutely necessary that their surfaces be specially figured for the 
 particular use desired. This question is considered at greater 
 length elsewhere. 
 
 I. (2) If the wave-fronts which form the two images are 
 
 "'An Improved Form of Littrow Spectroscope," Phil. Mag. (5), 38, 137-142, 
 1894. 
 
OPTICAL CONDITIONS OF ACCURACY 23 
 
 appreciably shifted in lateral position so as to fall upon different 
 portions of the optical train, it is at once evident from (14) that 
 the mechanical errors in the different surfaces involved must not 
 exceed the amount that would introduce a difference of phase 
 of 0.06X,. If t be the permissible mechanical error and i the 
 angle of incidence of the wave-front, we have at once 
 
 0.06X , 
 
 /= : for reflecting surfaces 
 
 2 cos 1 
 
 0.6X , 
 
 /= -, . for refracting surfaces 
 
 n cos / — cos 1 
 
 "r (32) 
 
 Two cases of importance that need to be considered in this 
 connection are those of the heliometer and the spectrograph. 
 In the first instrument the two images which are brought to 
 coincidence are formed by entirely different portions of the objec- 
 tive, and the optical surfaces of each portion must therefore be 
 correct to the limit indicated in (32). Since Zand i' are both 
 nearly 90 , the two surfaces of the flint lens must each be opti- 
 cally right to within about % a \ and the two surfaces of the 
 crown to within about y%\ each. The standard of workmanship 
 required in a heliometer objective is therefore very high, and 
 the difficulty of attaining the necessary degree of perfection is 
 rendered vastly more difficult by the division of the finished 
 object-glass into two parts. If there is the slightest strain or 
 difference in density in the material, the glass is likely to spring 
 slightly along the line of cutting, and this line is unfortunately 
 parallel to the axis of measurement. 2 
 
 In the case of the prism spectroscope and spectrograph the 
 wave-fronts which form the images of different spectral lines are 
 
 ' Rayleigh, "On the Accuracy Required in Optical Surfaces," Phil. Mag. (5), 8, 
 477- 
 
 2 The form of prism heliometer which has recently been devised and constructed 
 by the writer avoids this difficulty by rendering the division of the object-glass unnec- 
 essary. It can therefore be constructed of much larger size than has been possible 
 with the older form. See Allegheny Observatory Report for 1900, p. 23 ; also U. S. 
 Patents, Nos. 536493, 53&494, 536555- 
 
24 F. L. O. WADSWORTH 
 
 shifted laterally with reference to each other in passing through 
 the train by the amount 
 
 s 
 
 where / is the length of path traversed by the wave-front 
 between each successive refraction, beginning with the incident 
 side of the first prism and ending with the camera or view tele- 
 scope objective. For a prism train consisting of N similar 
 prisms with their corners in contact, we have for the shift from 
 the central ray which passes at minimum deviation 
 
 + 2tiB x + 4>tB,+ 6nB 3 + . . . . + 2Nn — 
 
 (34) 
 
 where B^B,, etc., are the lengths of the bases of the prisms. 
 For some of the larger instruments recently constructed the 
 value of 5 is quite large. For the Bruce spectrograph of the 
 Yerkes Observatory, for example, 
 
 N = 3 ; n ^ 1.67 ; B t = 12 cm ; B s — 13cm ; B 3 = 14cm . 
 The angular separation of the extreme images measured on the 
 
 plate is about 4 ; therefore -yr (one refraction) =^° = 0.0I2. 
 
 We obtain therefore for .S 
 
 S m!l% =s 2 cm . 
 
 The refracting angles of the prisms of this train are about 
 
 '63° 30', and n for the central ray is about 1.673. Hence i and 
 
 i' are about 6i° 45' and 31 45' respectively. For these values 
 
 we have for t, from (32) 
 
 0.06A. 
 
 'max = = 0.07X , 
 
 0.94 
 
 or the prism surfaces must be correct to within about l/ of a 
 wave over all those portions not common to the two beams, i, e., 
 
 2 cm -=- cos 6i° 45' ^ 4cm , 
 at each end of the prism faces. This would mean that the prism 
 faces should not depart from a perfect plane by more than 
 
OPTICAL CONDITIONS OF ACCURACY 25 
 
 0.000003 cm in a length of 6.5 cm, or that the radius of curvature 
 of the faces should be at least 7,000,000 cm. Such a degree of 
 accuracy is very difficult to attain in surfaces whose length 
 (13cm) is so much in excess of their width (5.1cm) as in this 
 case, but it is even more difficult to maintain when once 
 secured, if temperature conditions vary rapidly. 
 
 In case of the plane grating spectroscope the same portion of 
 the grating surface may always be used, but the angle of inci- 
 dence and reflection varies with the wave-length. As ordinarily 
 used with fixed collimator and observing telescope, the angles 6 
 and i are connected by the relation 
 
 i+$=y , 
 where 7 is the fixed angle between the axes of the two tele- 
 scopes, and the angles 6 and i are measured positively on 
 opposite sides of the normal as before. In this case any irregu- 
 larity of surface of height t introduces a retardation which is 
 
 evidently * 
 
 /(cos 2'+ cos 8) . (35) 
 
 For setting on the central slit image we must have cos i = cos 6 
 = cos 3^7, and the error of phase is 
 
 2/ cos-y . (36) 
 
 For setting on any spectral line of wave-length X we must 
 revolve the grating through an angle 8 such that 
 
 Nm\ 
 
 sin 8 = . - (37) 
 
 20 cos — 
 2 
 
 Also — - — 8 and /=8 + -, and (35) reduces at once to the 
 
 2 2 v ' 
 
 form 
 
 E = 2t cos 8 cos - . (38) 
 
 In order that the spectral image may not be displaced a 
 measurable amount from its true position with reference to the 
 central slit image, the condition is 
 
 2/ cos-(i — cos 8) s= 0.06 A (39) 
 
26 F. L. O. WADSWORTH 
 
 * y 
 
 If the angle between the two telescopes is 30 , cos- = 0.966, 
 and for the image of the D lines in the second order spectrum 
 of a grating with 5,000 lines to 1 cm, the value of B is 
 
 and for the fourth order spectrum 
 
 8^37°4o'. 
 
 And from (39) we obtain at once for / 
 ( 0.6 X for the first case, 
 ( 0.15X for the second case, 
 
 from which it appears that the irregularities in the surface of a 
 plane grating large enough to introduce a measurable displace- 
 ment of the image of a spectral line would probably manifest 
 themselves by a deterioration of the optical definition. 
 
 II. The effect of temperature variations in the media traversed 
 by the wave-fronts is more serious than those oi irregularities of 
 surface or varying obliquity which we have just considered. For 
 the latter would be constant in amount and direction, provided 
 the incident wave-fronts were constant in form, and under such 
 circumstances could be corrected for. But the effect of tem- 
 perature changes in the various optical parts of the instrument 
 is to change continually the form of the passing wave-front, and 
 thus not only render the corrections in question varying and 
 uncertain, as has already been stated, but also introduce errors 
 of displacement which vary with time alone. This latter class 
 of errors can be eliminated only in two ways : first, by making 
 the measurements on the two images absolutely simultaneous in 
 time ; or, second, by keeping the temperature variations within 
 the limits which will render the resulting displacements less than 
 the quantity e. The first condition can be satisfied in certain 
 cases by heliometric methods or by simultaneous photography 
 of the two images, the measurements being afterward made on 
 the photographic plate instead of directly in the focal plane. 
 The elimination of the effect of what may be termed time aber- 
 rational displacements in this way is one of the advantages of 
 
OPTICAL CONDITIONS OF ACCURACY 27 
 
 the heliometric and photographic methods of measurement that 
 does not seem to have been heretofore specifically recognized. 
 
 If the above time condition cannot be strictly fulfilled, then 
 the temperature variation must not exceed an amount that would 
 introduce an unsymmetrical aberration of more than 0.06X (see 
 (14)). We will next examine briefly what this condition of steadi- 
 ness implies. The change in the optical density (index of 
 refraction) of air and glass for 1° C. is 
 
 A«„ = — 1.1 X 10 " for air, 
 A// ir = -)- 4.0 X io -6 for flint glass 
 
 ." } (40) 
 
 AX 
 The total difference in phase — introduced in the wave- 
 
 front in traversing a thickness L raised to a difference of tem- 
 perature Zwill evidently be 
 
 AA 
 Under the condition that -r- shall not exceed 0.06 we have 
 
 A 
 
 at once 
 
 rm^- 0.06 A r . 1 
 
 LT^ , ^ 2.5 for air, 1 
 
 -1.1X10- 6 , 
 
 __ O.06A , , 
 
 S ; ^ 0.7 for flint glass. 
 
 — 4 X 10- 6 ' & 
 
 The focal length L a of the largest heliometers now in use is 
 about 250 cm, and the average thickness L g of their objectives 
 (aperture 16 cm) is about 1.5 cm. From (42) we obtain at 
 once 
 
 r,=£p? s c. f (43) 
 
 That is, in order to avoid an error of displacement greater 
 than the limit of accuracy 6 = 0.070 attainable with this instru- 
 ment, the temperature of the two halves of the object-glass must 
 
 ■J. O. Reed, "Einfluss der Temperatur auf die Brechung . . . einiger Glaser," 
 Inaugural Dissertation, Jena, 1897. Some varieties oE glass examined by Dr. Reed, 
 notably the barium silicate crowns and the light barium flints, have much smaller 
 temperature coefficients and would on that account be excellent for use in this con- 
 
28 F. L. O. WADSWORTH 
 
 remain constant to within y 2 ° C. and that of the air in the two 
 sides of the tube constant within o?Oi C, in the interval between 
 two successive sets of measurements. 
 
 In the case of the Bruce spectrograph considered above, L g 
 (for the prism train) is about 39 cm and the corresponding value 
 of L a (39 n) is about 65 cm. If the temperature of either the 
 air or glass changes alone, the limit of this temperature change 
 must be 
 
 r.So?o 3 8C. ) 
 
 7;^o?oi8C. \ K ' 
 
 If the two change together and by the same amount the limit 
 of the change must be (40) and (41) 
 
 ^g °'° 6 * =o?o I2 C. (45) 
 
 L s \n s — L a \>i a 
 
 For the camera objectives, focal lengths Z,j=45cm and 
 
 60 cm the limiting values of T are 
 
 T a :5o?o55C. and o?o4 C. (46) 
 
 The interval between measurements, which in this case is the 
 interval between the star exposures and the exposure for the 
 comparison spectrum, is so long and the temperature require- 
 ments so severe in the case of an instrument as large as the one 
 considered that the only way of avoiding error would seem to 
 lie in first keeping the temperature as constant as possible, per- 
 haps within o?i, and exposing continuously and simultaneously on 
 both the star and the comparison source. The method now general ly 
 used of making one exposure on the latter just before and 
 another just after the exposure on the star is not the equivalent 
 of a continuous exposure on both, because the temperature will 
 not change uniformly in the glass, even if it does in the air, and 
 the effect of an unsymmetrical or irregular change in the former 
 is, as shown above, about twice as great as in the latter. 
 
 Case A (3). Effect of variations in the value of X (non- 
 achromatism of image). 
 
 When the value of X in the general equation (3) varies, we 
 have to take into account not only the resulting change in the 
 
OPTICAL CONDITIONS OF ACCURACY 29 
 
 scale of the diffraction pattern for each wave-length, but also 
 several other effects as follows : 
 
 [a) The chromatic dispersion of the instrument and the rela- 
 tive inclination (if any) of the different incident wave-fronts 
 corresponding to different values of X. 
 
 (£) The spectral distribution in intensity in the different 
 wave-fronts. 
 
 (<r) The relative chromatic sensitiveness of the eye or photo- 
 graphic plate by means of which the position of the image is 
 determined. 
 
 (a) The effect of chromatic dispersion of the instrument is 
 to alter the inclination of the various wave-fronts passing through 
 it by an amount depending on the wave-length X and the optical 
 form and constants of the media traversed. 
 
 Let us denote this varying inclination of the different wave- 
 fronts by <f>. Then in general 
 
 4> = ^ (k-K) = -DS\ 1 
 
 d$=D'd\ 
 
 } (47> 
 
 I 
 
 where X denotes the wave-length of the wave-front which is 
 brought to focus on the axis of the geometrical image. 
 
 (6) The intensity of light in each wave-front will depend on 
 the spectral distribution in intensity in the source of radiation 
 itself, which we may denote by "^(X), and on the coefficient of 
 transmission for each wave-length, which we will denote by k K . 
 The relative intensities in the different wave-fronts will therefore 
 be proportional to 
 
 *a*(A) • (43)' 
 
 (c) Finally, the relative effect of the image formed by each 
 wave-front at the focal plane will depend on the "luminosity 
 curve," either visual or actinic, which we may denote by 
 Z(X). 
 
 Taking into account all of these factors we have for the dis- 
 tribution in intensity in the spectral diffraction pattern of a point 
 
3° F. L. O. WADSWORTH 
 
 or line at the focal plane of an achromatic telescope of rectan- 
 gular aperture the expression 
 
 H= P + °°/^(A)Z(a)(/=),A , (49) 
 
 «y _oo 
 
 where 
 
 sin 3 — (a — <j>) sin 2 — - (a — Z?AA) 
 
 ■ [^J = r "<-*"> ■ (SO> 
 
 from (5). 1 
 
 In the evaluating of the integral P s of (49) and (50) we have 
 to distinguish several cases : 
 
 1. When there is no chromatic dispersion, Z> = o, and if the 
 different wave-fronts all fall concentrically on the diffracting 
 aperture, the centers of the different spectral images will coin- 
 cide, or more strictly, will all fall on the optical axis passing 
 through the center of the geometrical image. Since each indi- 
 vidual image is a symmetrical function of X, the diffraction 
 pattern resulting from their centrical superposition will also be 
 symmetrical, and the result of the residual chromatic effects, (3) 
 and (c), as well as any outstanding longitudinal chromatic aber- 
 ration will be to change the form? but not the position of the 
 center of intensity of the physical image. This effect, like that 
 of A (1), will not therefore affect the accuracy of measurement 
 of positions. 
 
 2. When the chromatic dispersion is small the superposition 
 of the different spectral images will not be quite exact, but the 
 continuity of the image will be unbroken ; i. e., the chromatic 
 resolution will not be sufficient to enable us to isolate or deal 
 with any individual line or region of the spectrum. Under such 
 circumstances there is considerable uncertainty as to just what 
 part of the image corresponds in position to the geometrical 
 
 ■See "General Theory of Telescopic Images for different forms of Radiating 
 Sources," Astroi'UYSICAl Journal, 6, 124; § § A (l>) and A (<-). 
 
 3 As has already been indicated this change in form may fortunately be utilized 
 as the basis of a possible method of measuring stellar temperatures. Loc. cil., § A, 
 p. 125. 
 
OPTICAL CONDITIONS OF ACCURACY 31 
 
 center, and the error of measurement involved in the determina- 
 tion of the position of the source itself may be very appreciable. 
 Such a case as is here supposed presents itself in micrometric 
 and heliometric measurements of the relative positions of stars 
 and small planets at low altitudes. In every position except 
 exactly in the zenith (and even there under certain conditions of 
 barometric pressure) the Earth's atmosphere acts as a prism, 
 which not only results in the refraction, but also in the disper- 
 sion of the light coming from the star or planet. The necessity 
 of considering the dispersion effect was early pointed out by 
 Lee, 1 and was afterwards more fully discussed by Rambaut 2 and 
 Gill 3 , who do not, however, fully agree as to the magnitude of 
 the effect involved. Their disagreement can be explained in 
 part by their failure to consider fully the effect of the factors 
 k K , ^(X) and L{\) of (3) and (t) above, which are quite as 
 important in determining the apparent center of the image of the 
 star as is the term <j>=DA\ which represents the atmospheric 
 dispersion. Thus Rambaut shows that certain systematic differ- 
 ences in the measurement of the angular separation of the 
 components of /3 Cygni, made at different zenith distances, may 
 be explained if we take into account the factor Z>AX=Z>(X — X) 
 only, by supposing that the value of X for the two stars (or, as he 
 puts it, the mean refrangibility) differs by about 250 tenth-meters. 
 Differences of this amount are easily possible between stars of 
 such different types as the two components of the star in ques- 
 tion, but the change in apparent separation of their images due 
 to this difference in mean wave-length of maximum intensity is 
 considerably less than Dr. Rambaut assumes, owing to the influ- 
 ence of the omitted factors k K , ^(X), and L(X), in determining 
 the apparent center of the image when the latter is observed 
 either visually or photographically. This is very fortunate, 
 
 ■Phil. Trans., 1815. 
 
 ""Effect of Atmospheric Dispersion on the Position of a Star." M. A r ., 45, 
 123-125; ibid., 48, 256-280. 
 
 3" Effect of Chromatic Dispersion of the Atmosphere on Parallaxes." Ibid., 48, 
 53-76; ibid., 4I5-425- 
 
32 F. L. O. WADSWORTH 
 
 since otherwise, as Dr. Gill points out, observations of parallactic 
 displacement would be affected by errors or rather uncertain 
 corrections of so great a magnitude as seriously to reduce the 
 accuracy of such work. At the same time there remains a true 
 apparent displacement of the center of intensity of the image 
 which is, in many cases, greater than the limiting error of meas- 
 urement e. Whether this displacement of the center of intensity 
 affects heliometer measurements to the same extent that it affects 
 micrometer observations is, as Dr. Gill points out, a question that 
 can be settled only by individual experiment. 
 
 The whole question is of vital interest not only in connection 
 with parallax and double-star work, but also in connection with 
 meridian work and almucantur observations and (under certain 
 conditions as already indicated) zenith telescope measurements. 
 I have therefore evaluated the integral (49) for a number of 
 values of X and under a number of assumptions as to the form of 
 the functions k K , ^r (X), and L (X) for both visual and photo- 
 graphic telescopes. This work would occupy too great a portion 
 of the present paper and I will therefore present it in a separate 
 communication. 
 
 dQ 
 3. When — is large, as in the case of the spectroscope, we 
 
 no longer have to deal in general with the whole spectral image 
 of the source, but only with individual portions of it, i. e., with 
 individual spectral lines. In practically all cases the effective 
 "spectral width" AX of individual lines is so .narrow that over 
 
 this range the dispersion coefficient D=— may be regarded as 
 
 constant, and for the same reason the value of X itself may be 
 considered as constant with respect to the integration of (49) 
 in the functions k K , L (X) and in the denominator of (50). 
 Under such circumstances the expression for the distribution in 
 the intensity in the image of an individual spectral line becomes 
 
 sin 3 — (a — <f>) 
 
 J\ = A' J ^(^p^ —<i4> = A"^y,a.,a) , ( SI ) 
 
 !"^(a-*>]' 
 
OPTICAL CONDITIONS OF ACCURACY 33 
 
 or if the source (in this case the slit of the spectroscope) has a 
 finite width 1 <r , 
 
 (To 
 
 Il=A" f Vi*-*. «..*.! 
 
 M 
 
 In (51) and (52) -ty (<£)=t^ (X) represents the spectral law of 
 radiation not for the source as a whole, but for any individual 
 element. In a normal source whose mean wave-length is X the 
 distribution ^r (X) is generally assumed to be represented by the 
 
 law 
 
 ^(X) = #-*<*-*>■ , (S3) 
 
 which is symmetrical in form. For such sources /, is symmet- 
 rical and I 2 is therefore also symmetrical. This case has already 
 been considered. 2 
 
 When yjr (X) is unsymmetrical, as is generally the case when 
 the source of radiation is subjected to the effect of unusual 
 pressure or temperature conditions, or to abnormal magnetic or 
 electrical disturbances, the form of I\ and I\ likewise become 
 asymmetrical. Unfortunately our knowledge of the relations 
 between the physical conditions of molecular and atomic vibra- 
 tions and the resultant intensity of radiation are as yet too 
 meager and unsatisfactory to enable us to express ^r (X) in defi- 
 nite mathematical form in such cases. We can only consider 
 and endeavor to allow for the general effect produced, and this 
 will be to render the measurement of the mean wave-length, X ol 
 of a given line indeterminate, to a degree depending on physio- 
 logical and psychological causes rather than on physical ones. 
 That is, different observers will differ among themselves as to 
 the setting of a cross-wire on the mean center of intensity of 
 such an asymmetrically broadened image, but aside from this, 
 this kind of asymmetry cannot be said to be a real source of 
 error in the measurement of wave-lengths, since the latter can 
 only be considered in reference to the line itself, and not, at 
 
 'See Phil. M.ig., 43, 330, 33S; also Astrophysical Journal, 3, 336. 
 2 Loc. cit., p. 330 ff. 
 
34 F. L. O. WADSWORTH 
 
 least in the present state of our knowledge, in reference to the 
 free or natural period of the vibrations which have produced it 1 . 
 
 B. Causes of asymmetry in the image of Class B may be 
 subdivided as follows : 
 
 B (i). Lack of resolving power due to (a) small aperture, 
 (i>) imperfect definition, or (c) imperfect achromatism. 
 
 B (2). Errors of adjustment of focus. 
 
 B (3). Errors due to an asymmetrical arrangement or use of 
 the instrument itself. 
 
 B (4). Errors due to erroneous or imperfect design. 
 
 B (1). A frequent cause of error in the measurement of the 
 position of a point or line source is an unsymmetrical broadening 
 of the image by the superposition upon it of another image of a 
 second fainter source too close to the former to be clearly 
 "resolved." 
 
 If the two sources vibrate independently and have intensities 
 z,, i 2 , the distribution in intensity in the superposed images is 
 
 represented by 
 
 . sin s f . sin 2 (£ — k) 
 
 '■^-"'-^7-' (S4) 
 
 k being the angular interval between the centers of the two 
 sources, and £, the angular distance of any point in the image 
 from the center of the geometrical image of the principal source 
 S t . An example of the image of such a double source is shown 
 in Fig. 9. The dotted lines represent the separate diffraction 
 images of the two sources, the full line the resulting focal image 
 of the two together. 
 
 In measuring the position of such an image the reference 
 cross-wires are always set too far to the right in the figure, i. e., 
 toward the smaller component, by an amount depending on its 
 intensity in comparison with that of the principal source. Exam- 
 ples of errors of measurement of this kind are found in the 
 determinations of position, proper motion, and parallax of very 
 
 "In this connection see papers by Jaumann, "Zur Kenntniss des Ablaufes der 
 Lichtemission.'' IVied. Ann., 53, 832; 54, 178; and by Galitzin, "Zur Theorie 
 der Verbreiterung der Spectrallinien," ibid., 56, 78. Further developments of the 
 theory of radiation along the lines outlined in these papers seem probable. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 35 
 
 close double stars. 1 Similar errors in the determination of wave- 
 lengths of spectral lines have been frequently noted. 2 The only 
 way in which such errors can be detected and corrected is by the 
 use of higher resolving power, by which the two components 
 may be separated and measured separately. It appears probable 
 
 
 
 
 o 
 
 i 
 
 o' 
 
 1 
 
 T 
 
 - W 
 
 II 
 
 1 i 
 :0a s 
 
 
 
 
 
 !<— JP 
 
 
 
 ^lUk. 
 
 •-»! 
 
 
 
 
 ,n- /$mmy 
 
 iiiii 
 
 yJM 1 
 
 
 7U. 
 
 
 -»,o 
 
 
 - 1,0 
 
 a 
 
 1 
 
 i.O 
 
 
 s,o 
 
 Fig. 9. 
 from the work of Michelsons and the more recent work of 
 Perot and Fabry, 4 that by the use of-sufficietrtly- high resolving 
 
 1 Certain interesting jases of this kind which seem to have escaped attention will 
 be considered fully in ■* subsequent paper. (Note added January 1903.) One of 
 these cases was recently investigated by Comstock, whose paper, "The Motion of 85 
 Pegasi," was read on December 30, 1902, at the recent meeting of the Astronomical 
 and Astrophysical Sociecv in Washington. 
 
 "For example, see Rowland's "Table of Standard Wave-Lengths; " Kayser 
 and Kunge, " Ueber die Spectren der Elemente," Ab/i. d. K. Akad. d. W., Berlin, 
 1888-1892; Hasselberg, "Untersuchungen u. d. Spectra der Metalle," Kongl: 
 Svenska Vet. Akad. Handlingar, 1894-1898; Kayser, "Influence of Slit-Width on 
 Comet Spectra," A. N., 3217; A. and A.-P., 13, 367; Hartmann, "Wave-Length of 
 the Nebular Lines," Astrophysical Journal, 15, 29^; Wright, ibid., 16, 57, etc. 
 
 3" Application of Interference Methods to Spectroscopic Measurements," Phil. 
 Mag., (5) 34, 280. 
 
 *£ull. Astron., 16, 5; Astrophysical Journal, 9, 87; Jour. Phys., (3) 9, 369; 
 Astrophysical Journal, 15, 73, etc. 
 
36 F. L. O. WADSWORTH 
 
 powers a large number of the lines which are now regarded as 
 single will be resolved into two or more components, in many 
 cases of very unequal intensity. 
 
 In this case the error of setting the cross-wires is a real one, 
 for the reason that we desire to determine in our measurements 
 the position of one of the components of the double source 
 (generally the brighter one) and not the mean position of the 
 two. If we know the separation of the two sources and their 
 relative magnitude, we may indeed correct the results of the 
 measurements of the double source so as to give with a fair 
 degree of approximation the true position of either component. 
 In general the setting of the cross-wire on such an asymmetrical 
 image as is here being considered will be a compromise between 
 the tendency to set on the point of maximum intensity 0, Fig. 
 9, of the compound diffraction image and the point 0' , midway 
 between those points of equal intensity mm, which are just 
 perceptible to the eye of the observer. In the case represented 
 
 in Fig. 9, in which « = -a and /, = 24, the setting of the cross- 
 wire would be about r = <z -+- 0.4 k. 
 
 The corrected reading for 5 X is, therefore, 
 
 a = r — o.i,K , (55) 
 
 and for S^ similarly 
 
 a -\- k --=r -\- o.6k . (56) 
 
 B (2). Displacements due to imperfect adjustments for 
 focus. 
 
 In considering errors of measurement due to this cause alone 
 we shall at first assume optical conditions to be such that the 
 wave-front which forms the image is symmetrical and concentric 
 about the optical axis on which the image is formed ; i. e., that 
 there is no unsymmetrical aberration, either spherical or chro- 
 matic, and no eccentricity. 
 
 When these conditions are fulfilled a change of focus should 
 result simply in a symmetrical expansion of the diffraction disk 
 or band, which represents the physical image of a point or line. 1 
 
 "This is a common way of testing the excellence either of an objective or o£ 
 atmospheric conditions. 
 
OPTICAL CONDITIONS OF ACCURACY 37 
 
 The center of each image will remain fixed on the secondary 
 optical axis joining the center of the geometrical image with the 
 optical center of the lens system. Under such conditions the 
 separation of any two such images at the focal distance z from 
 the optical center of the objective is 
 
 £=2 2 sin- , (57) 
 
 where k, as before, is the angular separation of the two images. 
 Whence for any change in z 
 
 d£ = 2 sin — dz = ndz = dz- . (58) 
 
 This expression is entirely independent of f, the true focal 
 length of the instrument, and hence if z could always be main- 
 tained constant, dz and, therefore, <ff, would be zero, no matter 
 what the absolute value of z might be. This is only possible 
 under absolutely constant temperature conditions. When the 
 temperature changes, z will also change, first, on account of the 
 thermal expansion or contraction of the telescope tube, and, 
 second, on account of the mechanical refocusing which may be 
 necessary in order to compensate for the corresponding tempera- 
 ture change in/. We can best consider these two effects sepa- 
 rately. ( 
 
 1. The change in z, due to the thermal change in the length 
 of the telescope tube, for a differenced temperature A£-will be 
 
 8 
 
 — z = azt\t 
 dt 
 
 d£ 
 
 2 sin - 
 2 
 
 where a is the coefficient of expansion of the material of which 
 the telescope tube is made. But we have also a similar change 
 in the scale of the measuring or recording device at the focal 
 plane, which amounts to 
 
 — d£'—-a' (22 sin-) M . 
 
38 F. L. O. WADSWORTH 
 
 The apparent change in the separation of the two images as 
 indicated by the reading of the instrument is therefore 
 
 d$+(-d4') = 2Zs\n K -{a-a')\t = ±J . (59) 
 
 If the material of the micrometer screw or reference scale 
 is the same as that of which the telescope tube is constructed, 
 then a' = a and A,„lf = 0; i. e., in this case the measured separa- 
 tion of the images at the focal distance, z, remains the same at 
 all temperatures. 
 
 If the position of the images is first recorded photographi- 
 cally on a sensitive plate, and afterward measured by a com- 
 parator or similar device, we have to consider not only the 
 relative expansions of the glass plate and the telescope tube, but 
 also those of the plate and the scale or screw of the comparator. 
 In this case the total change in the measured distance as indi- 
 cated by the reading of the comparator at a temperature 
 2 r =4 + A7is 
 
 %d£ = 2Z%m-\(*-a.')\t+{a.'-a,")\'t\ . (60) 
 
 In order to make (60) vanish we must have 
 
 (a-a')A/ = (a"-a') A'/ . 
 
 This condition may be satisfied in one of three ways: (a) by 
 making a" = a and A.' t = At; i. e., by making the screw or scale 
 of the comparator of the same material as the telescope tube 
 and measuring the plate at the same temperature at which it was 
 photographed. This plan is generally neither convenient nor 
 practicable; {b) by controlling the temperature, T, of measure- 
 ment, so that for given values of a, a' and a" we always fulfill the 
 relation 
 
 a — a 
 
 This is also inconvenient and troublesome, although less so than 
 the preceding; (c) by making a = a' =a" ; i. e., by constructing 
 both the telescope tube and the comparator scale of glass or 
 platinum. This, while impracticable for large telescopes, is not 
 at all so for small ones, such as are used for spectroscopes. 
 
OPTICAL CONDITIONS OF ACCURACY 39 
 
 Instead of attempting to make the correction (60) vanish we 
 may simply determine its amount and apply it to the measure- 
 ments. In this case we need to determine the degree of accu- 
 racy with which the temperatures 4, t, and T must be known. 
 This is easily done by equating the differential of (60) with the 
 limiting value of e expressed in linear measure, i. e., 
 
 c = e/" = cz . 
 This gives at once from (60) and (14) 
 
 A}(a-a')Ar+(a'-a")A'/j,//=-J_. . (6l) 
 
 The values of a, a' , and a" for brass, glass, and steel (the 
 materials usually employed) are respectively 
 
 a ^0.000018 a'= 0.000008 a"=o.ooooio. 
 
 In case of the Mills spectrograph the aperture b is about 
 3.7cm. Hence for an angle k = i?5 = 0.027 * we have for (61 ), 
 
 dt = s 4 ° a ■ < 62 > 
 
 [0 Z, A '- 2 7, A7 
 
 and if we assume that it is equally easy to measure all three 
 temperatures, 4, /, and T, (62) indicates that the error (which 
 must be taken without regard to sign), in determining any one 
 of them must not exceed i?6. It is generally possible to be cer- 
 tain of all these temperatures to this or even a higher _degree_of 
 accuracy, and it is therefore usually better to observe them and 
 correct the observed readings of the comparator than it is to 
 attempt to eliminate the corrections by any of the methods indi- 
 cated in the preceding paragraph. 
 
 2. Effect of refocusing : In large telescopes the alteration of 
 the true focal length, /, for a given change in temperature is 
 so much greater than the corresponding thermal alteration in z 
 alone, that it is necessary to refocus the eyepiece or plate in 
 order to obtain good definition. Such refocusing is also essen- 
 tial, as will±>e shown later, even in small instruments, if there is 
 
 ' See Table II of this paper. 
 
40 F. L. 0. WADSWORTH 
 
 any unsymmetrical aberration or eccentricity in the incident 
 wave-front. In such cases we have 
 g 
 
 dz = — (z + dQ,) + dCl 
 
 = (z + SQ)aM-\-dil , , 
 
 d£ 
 
 2 sin - 
 2 
 
 where dQ represents the actual mechanical movement of the eye- 
 piece or plate in refocusing. The change in the separation of 
 the two images at the new focal plane, as indicated by the 
 micrometer screw or comparison scale, now is 
 
 A£=2(* + rfO)sin^j(«-a')A/+ r ^L i | , (64) 
 
 and in the case of comparator readings on photographs 
 
 2£=2(« + rffl)sin^j (a-a')A/+-J^_+(a'-a")A'/| . (65, 
 
 If we assume that the terms of (64) and (65) which represent 
 temperature corrections are computed as before and applied to 
 the measurements, we have left to consider only the effect of 
 the term 
 
 2dQ ■ sin- = KdQ, . 
 2 
 
 In order that this may be computed also with the same degree 
 of accuracy as already assumed we must have 
 
 d(dtl)^ = -±- , (66) 
 
 K 3O/BK V ' 
 
 /3 being the semi angular aperture. 
 
 The degree of accuracy which it is necessary to observe in 
 refocusing [i.e., in determining the amount of mechanical shift 
 of the eyepiece or plate with reference to the telescope tube) is 
 therefore independent of the size of the instrument and depends 
 only on its form [i.e., its angular aperture), and the angular 
 separation of the two objects measured. We will determine 
 what this is in a few cases of particular interest. 
 
OPTICAL CONDITIONS OF ACCURACY 4 1 
 
 For large visual telescopes and heliometers, /3 ^ 0.027. Hence 
 for k = o?35 = o.oo6i, 1 and \ = 0.000056 cm., 
 
 (/(i/fl)£o.oicms in. (67) 
 
 250 v " 
 
 For photographic telescopes and concave gratings of sufficiently 
 long focus to give full photographic resolution /3 ^ Vso = 0.0 13. 
 The field k is, however, correspondingly larger. For Rowland's 
 concave gratings, for example, k = 0.074 (more than 4 ? ). For 
 such cases therefore 
 
 d(dQ) :3 0.002 cm , (68) 
 
 or less than one one-thousandth of an inch. 
 
 From the above results, (66), (67), and (68), it would appear 
 necessary that the focusing scales of large telescopes designed 
 for micrometric or heliometric work should be provided with 
 verniers capable of reading to 0.1 mm, while those for the view 
 telescopes of large spectrographs and spectrometers should be 
 capable of reading to at least I /somm. 
 
 The accuracy of focusing demanded in all these cases is con- 
 siderably higher than is usually considered necessary, or is 
 attained. It is, of course, far in excess of either visual or photo- 
 graphic requirements on the standard of either resolution or 
 general optical definition. But as we have already stated, this 
 standard is an inadequate and unsafe one to apply as a test of 
 the accuracy of an optical instrument. 
 
 In the case of a telescope used ~as~a collimator the^accuracy- 
 of focusing necessary depends on the nature and method of use 
 of the instrument. 
 
 In general, any lack of exact collimation may be compen- 
 sated by altering the focus of the observing telescope, and if the 
 change in the latter is not objectionable and everything remains 
 symmetrical with respect to the axis of collimation, no error can 
 be thus introduced. In the case of prism spectroscopes, however, 
 the condition of symmetry with respect to the x axis cannot be 
 
 1 This is the limit imposed by considerations of aberrational distortion in the case 
 of the Lick and Yerkes telescopes (see Table II). This limit has, however, been 
 exceeded in a number of cases of individual measures with these instruments. 
 
42 F. L. .0. WADSWORTH 
 
 fulfilled, and any divergency or convergency in the beam of 
 light coming from the collimator objective will result in the 
 introduction of an unsymmetrical aberration in the wave-front 
 traversing the prism train, the amount of which will depend on 
 the curvature of the incident wave front, the optical perfection 
 of the prisms themselves, and the angles of incidence on the 
 successive faces of the prisms. 
 
 The simplest case to consider is that in which the axis of the 
 incident wave-front traverses the train at minimum deviation, the 
 faces of the prisms are optically plane, and the material of which 
 they are composed is of uniform optical density. The amount 
 of longitudinal aberration produced by the passage of a cone 
 of light of semi-angular aperture 6 through a single prism under 
 the conditions assumed above has been calculated by Rayleigh, 
 
 who finds ' 
 
 tan i («' — i) 
 
 hv = Viu — -j ■ - — - , (6o) 
 
 cos 2 / if v v/ 
 
 where i and i' are the angles of incidence on the first and 
 second faces of the prism and u is the radius of the spherical 
 wave-front. 
 
 We have also the relation 2 
 
 8 "=7^. (7°) 
 
 where /3is the semi-angular aperture of the cone of rays emerging 
 at the second face of the prisma — Hence-if we denote the~aper- 
 ture of the collimator by b and the length of path of the central 
 ray through the prism by L, we have 
 
 1 2z>~ 2(u+/,L)- \u-\-„L) ( - 11 ' 
 
 Also 
 
 1 [ I 
 
 f f+>if « 
 
 (72) 
 
 1 Rayleigh, Phil. Mag., (3) g, 40-49. See also "Theory of the Ocular Spectro- 
 scope," ASTROPHYS1CAL JOURNAL, 16, 2, July I902. 
 
 3 Rayleigh, Phil. Mag., (3) 8, 411. The result there given is for a symmetrical 
 aberration depending on x 4 . For an unsymmetrical aberration depending on x 3 the 
 numerical factor in the numerator is 3, not 4. 
 
OPTICAL CONDITIONS OF ACCURACY 43 
 
 whence 
 
 • ~-^Q. to) 
 
 and 
 
 „ bdf , 
 
 *'— 2{/« + (/-*Z)^j • (74) 
 
 In most cases the quantity (/— wi), which represents the dif- 
 ference between the principal focal length of the collimator and 
 the optical path of the central ray through the prism, is so small 
 that the second term of the denominator in (74) is vanishingly 
 small compared to/ 2 and may be neglected. 
 
 We then obtain at once from (69), (70), (71), and (74) 
 
 (^T tan , (,/■ - 1) 
 P f cos**' *■ ' (15) 
 
 where /3 as before denotes the semi-angular aperture of the col- 
 limator objective. 
 
 The limiting value of df will be found by equating (75) and 
 (14), which gives 
 
 , f _!_ » cos V \ /A / 6 x 
 
 It we neglect the successive changes of the second order in 
 
 the value of 6, which are indicated in (71), we have similarly for 
 
 N prisms 
 
 E N =NE , 
 
 • .df M =-M=. (77) 
 
 V N 
 
 In the case of a spectrometer which has a collimator of focal 
 
 length /= ioocmand aperture b= 5 cm, and a prism train consisting 
 
 of a single 6o° prism of light flint of index 1.6 for \ = 0.00005600, 
 
 we have /3= 0.025, 2 = 53 7' 48", and i' =30°. For this case 
 
 then 
 
 ^=±4-55cm , (78) 
 
 which shows that under the conditions assumed above a very 
 considerable range is permissible in focusing the collimators of 
 even very large spectrometers. 
 
 This conclusion is so at variance with the commonly accepted 
 
44 F. L. O. WADSWORTH 
 
 statement that an accurate focusing of the collimator is very- 
 important 1 that it is liable to be misinterpreted unless carefully 
 considered. It is only true when the minimum deviation condi- 
 tion is strictly fulfilled. It is necessary therefore to consider, 
 first, the degree of accuracy with which the various parts of the 
 spectrometer train may be adjusted and maintained in their 
 correct relative positions, and second, the amount by which a 
 given error in adjustment will affect the results given in (75) 
 and (76). 
 
 With any given instrument it is always possible by simple 
 hand adjustment alone to bring the prism to a position of mini- 
 mum deviation with an error not exceeding that which would 
 produce a displacement of the image equal to e, the metrological 
 power of the view telescope. The relation between the angular 
 displacement of the prism from the position of minimum devi- 
 ation and the resulting change in deviation of the central ray 
 has been determined by the writer, who finds for small values 
 of 0* 
 
 A=A o 4-sin- I 0*(i+-j 
 
 n cos 2 — A 1 — »' sin* — 
 2 \ 2 
 
 r sin/ O^m 
 
 l_cos 1 cos 1 >r J 
 
 -, .. •-.-. .a.;^'. I _-l - -. • { ^—^L\ - Cfr- 
 
 (79) 
 
 where A is the angle of minimum deviation of _ the ray, and A 
 the deviation when the prism has been displaced from the correct 
 
 position for \, by an angle 6. If, therefore, we put 8 = e ^ — - 
 we have for 6 
 
 = -ncost\\- ^—„ -, (80) 
 
 1 \^tini (+1- t \ ' 
 
 I 
 
 4 \ b tan i(tf — 1) 
 
 "See for example Schuster, Enc. Bril., article "Spectroscopy;" Hartmann, 
 ASTROPHYSICAL JOURNAL, 12, 36; FROST, ASTROPHYSICAL JOURNAL, 15, 15, etc. 
 
 'Astrophysical Journal, 1, 280. The expression there given is in error. The 
 
 / 
 
 term "\ I — « a sin a — should appear only in the denominator and not in the numerator. 
 
OPTICAL CONDITIONS OF ACCURACY 45 
 
 an expression very similar to (76). For the case of the spec- 
 trometer above considered 
 
 6 =0.0008 ^ 3' . 
 This is about as accurately as the prism can be maintained at 
 minimum deviation with the ordinary sliding link movement, 
 unless great care is taken in its construction. With such a 
 device therefore it' is useless to set the prism with any greater 
 exactness than is indicated above. 
 
 If, however, we use a pivoted link minimum deviation device 
 or, better still, a fixed arm prism train, 1 we can control the 
 movement of the prism much more accurately and can then use 
 a more accurate method of initial setting, for example a theodo- 
 lite, with advantage. We may thus with ease increase both the 
 initial and continuous accuracy of adjustment at least three times, 
 i. e., we can reduce 6 to not more than 1 '. 
 
 It is next necessary to determine the effect of a given value 
 of 9 on the aberration as given by (69) or (75). 
 
 The expression for the longitudinal aberration produced by 
 the passage of a cone of light through a prism placed in any 
 position with reference to the axis of the ray can be deduced 
 from the general fundamental equations given by Rayleigh in 
 the paper to which reference has already been made. The 
 expressions thus obtained are, however, complicated and cum- 
 bersome to reduce, and for small angles of inclination to the 
 position of minimum deviation the total aberration may be 
 obtained in a more simple way in the following manner : 
 
 Let caa , coo", Fig. 10, be the central and edge rays passing 
 through the prism. Owing to the difference in the angles of 
 incidence, the total deviation of the two rays in passing through 
 the prism will differ from each other by a small angle o' a' o" =/(#) 
 which we will call D. Since the wave-fronts are by suppo- 
 sition very nearly plane, the extreme edge aberration E between 
 the incident wave-fronts and the refracted wave front a' 0" will 
 
 be 
 
 £ = o'o" ^aoD = u6D = u6/{e) . (8i> 
 
 *Phil. Mag. (5), 38, 337, October 1894. 
 
4 6 
 
 F. L. O. WADSWORTH 
 
 When the central ray falls upon the first face of the angle of 
 incidence i for minimum deviation, we have from (79) 
 
 f(0) = S = C$- , 
 
 and 
 
 and from (70) 
 
 tan / (//* — 1) 
 
 Cn0 3 =6*it 
 
 tan/ (if— 1) 
 
 bv = v)u 7 ■ - 
 
 cos a 1 tr 
 
 (82) 
 
 (83) 
 
 Fig. 10. 
 
 the same expression as obtained by Lord Rayleigh. 1 
 
 The method above indicated c^f determining the aberration 
 of a prism applies not only to the position of -minimum devia- 
 tion but also to any other position within which equation (79) 
 holds. When the central ray is incident on the prism at an angle 
 other than that for minimum deviation, say at an angle i ± k 
 
 = A + CV . 
 
 ■It is perhaps worth while to note that the expression for the longitudinal aberra- 
 tion of a prism deduced by Abbot and Fowle, Amer. Jour. Sci., (4) 2, 253, and 
 Annals of the Smithsonian Astrophysical Observatory, i, 78-79, is based upon an 
 erroneous assumption. If equation (12) of their paper were correct there would be no 
 aberration, ;'. e., the angular deviation of the central and extreme rays would be the 
 same. 
 
OPTICAL CONDITIONS OF ACCURACY 47 
 
 And for the extreme ray incident at an angle ?± (k-\-6~) we 
 have similarly 
 
 Hence 
 
 Z> = C(2k6 + 6*) , (84) 
 
 and the wave-front aberration E is 
 
 E K =Cu»(i + *-£>! , (85) 
 
 As before we obtain the limiting values of df for a single prism 
 by equating (86) and (14). This gives us after reduction 
 
 ,,_ ^/__«cosz V i6k7 2 
 
 /r "7T + ~7/T \/Jtan/(/r- i) + « 2 cos 2 /' ' ( " 
 
 which for large values of k may be reduced to the form 
 
 7j- 1 X h" cos 2 /' . . 
 
 tif=^— ■ - ■ -&. r 7 — r . (88) 
 
 32 x fi- tan 1 (//- — 1) x ' 
 
 When the prism train contains .A^ prisms, each of which is out of 
 the position of minimum deviation by the same amount k, the 
 total aberration E N (to the same degree of approximation as 
 before) is N times that for a single prism. If we call the first 
 term under the radical in (87) A and the second term B, we 
 have for N prisms 
 
 For k = o we have as before 
 
 #v=i ■ (9°) 
 
 I N 
 
 and for large values of k 
 
 <#v = f. (91) 
 
 i. e., the effect of adding to the number of prisms in the train is 
 greater as the departure from the position of minimum deviation 
 increases. 
 
48 
 
 F. L. O. WADSWORTH 
 
 The permissible values of ^"both for a single prism and for a 
 train of three prisms have been computed for the spectrometer 
 already considered (in which _/"= ioo cm, /3 = 0.025 and /x = 1.6 
 for X= 5600 tenth-meters) for a series of successively increas- 
 ing values of k from tc = 1' to k = io°. They are given in 
 columns 2 and 3 of Table III. 
 
 TABLE III. 
 
 K 
 
 <//«. - v =' 
 
 <//«. -V=3 
 
 K 
 
 df K ,N=i 
 
 df K , N= 3 
 
 
 cm 
 
 -4.55 
 
 -3-54 
 
 — 2.25 
 -0.57 
 —0.26 
 -0.15 
 
 — 0. 10 
 —0.07 
 —0.066 
 
 — 0.049 
 
 —0.029 
 —0.016 
 
 cm 
 2.62 
 I. 71 
 0.90 
 U.2I 
 0. 12 
 0.06 
 0.035 
 0.023 
 0.022 
 0.016 
 
 0.010 
 u.005 
 
 
 cm 
 
 cm 
 
 
 
 
 
 
 
 
 
 
 
 
 
 57 
 26 
 
 IS 
 
 10 
 
 07 
 03 
 
 
 "° 3' n " 
 o° 15' 0" 
 o° 30' 0" 
 1° 0' 0" 
 i° 30' 0" 
 ?° ' 0" 
 1° 15' 0" 
 3° 0' 0" 
 
 
 
 
 
 0° 30' 0" 
 
 I" 0' 0" 
 
 1° 30' 0" 
 
 2° 0' 0" 
 
 0.21 
 0.12 
 
 0.06 
 
 3° 0' 0" 
 4° 0' 0" 
 
 0.035 
 
 5° 0' 0° 
 
 10° n' n" 
 
 
 10° 0' 0" 
 
 0.010 
 
 As k increases, the permissible values of df decrease, at first 
 very rapidly and then much more slowly. For large values of k 
 df must be very small ; 1. e., the focusing must be very exact, 
 especially with several prisms. This is the reason why Schus- 
 ter's method of focusing (in which the prism is turned far out 
 of the position of minimum deviation, first on one side and then 
 on the other) gives such accurate results. Thus Table III 
 shows that when the displacement k is 10°, errors of focusing of 
 only 0.016 cm, or of 1 part in 6,000, become manifest with a 
 single prism. It is probably for this reason also that the state- 
 ment already alluded to on p. 44 has been so often made. 
 
 Further consideration of the values of df for such values of k 
 as may be likely to occur because of errors of adjustment, shows, 
 however, that so far as this cause is concerned we still have a 
 very considerable latitude in the adjustment of the collimator. 
 Thus if the error of adjustment k does not exceed 3', as assumed 
 on p. 45, the focusing of the 100 cm collimator need not be 
 
OPTICAL CONDITIONS OF ACCURACY 49 
 
 closer than 2 cm for one prism or 0.9 cm for three prisms. For 
 k = i', errors of focusing as large as 3.5 cm and 1.75 cm for 
 one and three prisms respectively may be permitted. 
 
 In the case of the spectrograph, in which a considerable 
 angular field is under simultaneous observation, it is of course 
 impossible to adjust the prisms to minimum deviation for all 
 parts of the field at once. If we suppose that the prisms are 
 accurately adjusted to minimum deviation for the central image 
 in the field (wave-length X ) and that the maximum semi- 
 angular field, corresponding to spectral images of wave-lengths 
 A. and X.', is K, the amount k by which a single prism is out of 
 minimum deviation for these wave-lengths is 
 
 \{K±h )=\\K±c{*)\ 
 
 and «~ l -K. &>*> 
 
 2 
 
 The required accuracy of focusing is obtained as before from 
 (87). Values of df for ^=30', i°, 2°, 3 , 4 andio" (« = I5' 
 to 5 ) are given in Table III, column 5. 
 
 When the prism train consists of N prisms and the maximum 
 semi-angular field is K N as before, the amount by which each 
 prism of the train is out of minimum deviation is : 
 
 1 
 
 for the first prism, k, ^ — - X 2 
 
 2N 
 
 •■A - I 
 
 for the second prism, «„ = — - K- N ; - 
 
 for the third prism, k, =— - K N ; 
 
 2N- 1 „ 
 and for the iVth prism, « v = -^ A jV . 
 
 2JV 
 
 The total aberration, E N , in the train is, under such circum- 
 stances [see (85) ], 
 
 '(' + ¥)■ 
 
 (93) 
 
 E N = Cufr 1 
 
 f { bdf 2 \ 
 
SO F. L. O. WADSWORTH 
 
 and the permissible errors df for the train are therefore the 
 same as for a train of N prisms each of which has an error of 
 adjustment k equal to 
 
 2 
 
 The value of df for this case for three prisms and for the same 
 semi-angular fields as before are given in the last column of 
 Table III. 
 
 It will be seen that even for semi-angular fields as large as 
 2° (total 4 ) the permissible range in focus of a i meter col- 
 limator is nearly */£ mm, a comparatively large quantity com- 
 pared to that which expresses the permissible range in focus of 
 the view telescope. The statement already made that the 
 accuracy required in focusing the collimator is much less than 
 that required in focusing the view telescope is therefore gener- 
 ally true, not only for single prism spectrometers, but for large 
 multiple prism spectrographs with fields of 3° or 4°. 
 
 In this latter case, however, the range of focus is not suffi- 
 ciently great to permit us to collimate over more than a limited 
 range of wave-lengths or to disregard altogether changes in 
 temperature in the apparatus. In a well-designed objective of 
 the usual two-lens type, of the size and angular aperture 
 assumed above, the change in focus, df K , is between 0.0004/ and 
 0.0005/for a change of wave-length of 600 tenth-meters on each 
 side of the minimum focus. For the three-prism spectrograph 
 having a field of 4° (total) the permissible range of focus, as 
 stated above, is about y± mm or 0.00023/ on each side of the 
 point of exact collimation. By setting the slit at a point about 
 this distance outside the minimum focus we can therefore col- 
 limate the rays with the required degree of accuracy over a 
 maximum total range of nearly 1,200 tenth-meters. 1 In this 
 case, in order to maintain the accuracy of collimation over the 
 entire range we must refocus the collimator for every chancre in 
 temperature. On the other hand, if we set the slit at the exact 
 
 1 This is a little less than the results claimed by Hartmann with the Potsdam 
 three-prism spectrograph. Astrophysical Journal, 12, 39. 
 
OPTICAL CONDITIONS OF ACCURACY 5 1 
 
 focus of the collimator for any given wave-length and given 
 temperature, t, the required accuracy of collimation will be 
 maintained for that particular spectral region over a range of 
 temperature A^ such that 
 
 or, since —(z) = az and similarly — (f) =a"'f , 
 
 0. 00023 
 
 ■V^- -r4 . 94 
 
 a — a 
 
 For objectives of the usual crown and flint glass the second 
 term of the denominator a'" is about 0.000025 for 1" C. For 
 brass and steel the values of a are about 0.000018 and O.OOOOIO 
 as already given. Hence we have for A/ max : 
 
 A/ ma:[ ^ 15" C. for steel a tube 
 i/„„ ^ 30° C. for a brass tube. 
 
 In general the most satisfactory procedure is to limit the 
 range of wave-lengths for which we attempt to secure simulta- 
 neous collimation so as to leave a certain range of focus for 
 changes in temperature. If we content ourselves with collima- 
 tion over a range of 600 tenth-meters (which in the case of most 
 three-prism spectroscopes represents an angular field of about 
 four degrees) we have left to take care of temperature changes 
 a range of focus of about 0.00012/, which permits a maximum 
 temperature variation 
 
 A/ max = T-5° for a steel tube 
 
 A7„, = 15° for a brass tube 
 
 without refocusing. 
 
 It is very evident from (94) as well as from our previous 
 considerations with reference to the focusing of the view tele- 
 scope that so far as avoiding the effect of temperature changes 
 is concerned, brass is a much more satisfactory material than 
 steel to use for telescope tubes. Indeed it is possible by using 
 this metal in conjunction with suitable varieties of glass to obtain 
 
F. L. O. WADSWORTH 
 
 almost perfect equality between the temperature coefficient of 
 expansion a' and the corresponding temperature coefficient of 
 focal length a'" so that the range At of (94) may be increased 
 and the quantity dVL of (63) reduced to zero. Two troublesome 
 adjustments and errors are thus avoided. The more detailed 
 results of this investigation will be given in a subsequent paper. 
 
 The effect of curvature of the prism faces or of varying opti- 
 cal density in the material of which they are made has alreadv 
 been investigated in part for plane wave-fronts (pp. 26-29). So 
 far as this effect is individually considered it will be practi- 
 cally the same with spherical wave-fronts of small curvature 
 as with plane wave-fronts, but it may happen that in the case 
 of spherical wave-fronts the unsymmetrical aberration due to 
 either or both of these causes may be in the same direction as 
 that due to the prism train itself, or it may be in the opposite 
 direction. In the first case the two effects are additive and the 
 amount of permissible aberration due to each is correspondingly 
 reduced, i. e., for a given error of setting of the slit the amount 
 of permissible temperature variation in the prism train is less 
 than that indicated in (42), (44) and (45) by an amount depend- 
 ing on the sphericity of the incident wave-front. In the latter 
 case the effects are to a certain degree compensatory, but, on 
 account of the different form of distortion given to the wave- 
 front by the two causes, the compensation can be exact onlv 
 under certain conditions. Thus in the case of a single prism, 
 which has also been investigated by Lord Rayleigh, 1 it is found 
 that the aberration due to sphericity of the incident wave-front 
 and the aberration due to curvature of the prism faces can 
 destroy each other only when the curvature of the two faces is in 
 opposite directions {i. e., one face convex and the other face 
 concave) and very nearly equal. In the case of varying optical 
 density it is necessary to secure compensation to have at least 
 two prisms in the train whose variations are in opposite directions. 
 
 From these general considerations it is evident without further 
 investigation that, so far as the limiting aberration in the beam 
 
 1 Phil. Mag., g, 46-48. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 53 
 
 emerging from the spectroscope train is concerned, we cannot 
 rely upon compensatory effects in enabling us to reduce the 
 rigor of the conditions already imposed in (42) and (66), since 
 any one or all of the effects involved are liable to change in 
 amount, and even in sign with changes of temperature. The 
 best that we can do in any case is to so adjust the focus of the 
 collimator and the order of succession in the prism train, that 
 under what might be termed normal conditions of temperature 
 and adjustment the wave-front issuing from the final prism face 
 
 Fig. 11. 
 
 at minimum deviation is most nearly plane; or, in other words, to 
 determine by one of the methods which will be mentioned later 
 the accurate focal length of the collimator and prism train com- 
 bined, not of the collimator alone as usually done, and then to set 
 the slit as indicated on pp. 50 and 51. 
 
 When the incident wave-front is not symmetrical with refer- 
 ence to the axis of the instrument, the displacement of the image 
 of a point or line due to a change in focus will depend on the 
 distribution in intensity in the diffraction image in planes out- 
 side the principal focal plane. The detailed investigation of this 
 problem in the case of a beam affected by unsymmetrical aberra- 
 tion or eccentrical incidence is a matter of considerable com- 
 plexity. In the present case we may assume that for small dis- 
 tances df from the true focal plane the center of intensity of the 
 diffraction pattern will lie on a line which passes through the 
 central point, o, of the image at the principal focus, and the 
 point 0, which marks what may be termed the mean center of 
 
54 F. L. O. WADSWORTH 
 
 illumination of the incident wave-front. Let the angle between 
 this line oO, and the secondary optical axis, oa, i. e., the angle 
 aoO, be t. Then if the ano-le oac be k as before, it is evident 
 that the distance £, of an image from the central point c of the 
 plane in which it lies, will be 
 
 £=/sin K,-f(z — /)sin(/c, — t/) (95) 
 
 The separation of two images which lie at equal distances on 
 opposite sides of the optical axis and having a total angular 
 separation k will be 
 
 £=2£,=/sin k + (z—f) \ sinf^ — r\ + sin (^ + r,J t , 
 
 or since k and t x , t 2 are all very small angles 
 
 « = s{«-(r 1 -Oj+/(T I -r = ) (96) 
 
 and 
 
 where S is, as before, the total linear separation of the mean 
 centers of illumination of the incident wave-fronts. 
 
 Comparing (97) with (58) we see that in this latter case the 
 change in the separation of the two images involves the con- 
 sideration of the change in both z and/! It is necessary there- 
 fore to know the law of variation of /with t to the same degree 
 of exactness as that with which we know the coefficients of 
 linear expansion a, a' , and a" involved in a change in 2. 
 
 Expressing the values of ds and df in the same form as before 
 [(63) and (94)], and taking into account the expansions of the 
 micrometer screw and photographic plate, etc., as in (59) and 
 (60), we finally obtain in this case: 
 
 First, for direct micrometric or heliometric measurements 
 at the focal plane/= z -\- dfl, 
 
 A£ = k (z + Jty (a - a') A/ + ,/fi [k — (r, - r : )] 
 
 - (t, - t 3 ) (; + tin) (a - a'") A/ , (98) 
 
OPTICAL CONDITIONS OF ACCURACY 55 
 
 and second, for comparator measurements on photographic plates 
 
 2£ = k (z + dQ) [(a - a') A/+ (a' - a") A '/] 
 
 + dQ, [k - (r, - t,)] - (r, - r 2 ) (z + d&) (a - a" ') A/ . (99) 
 
 When t 1 —t 2 , i.e., when the mean centers of illumination S t and 
 S 2 are incident at the same point on the objective, the measured 
 separation between any two images is exactly the same in the 
 case of eccentrical incidence as in the case of centrical incidence 
 (64) and (65), previously considered. From this it follows that any 
 unsymmetrical diaphragming or unsymmetrical absorption (the 
 effect of which is the same) is without effect on A£, provided 
 only that the wave-fronts in each case fill the whole aperture. The 
 position of each individual image, however, will be shifted by an 
 amount indicated by differentiating (95), and the application of 
 this principle forms the basis of the ingenious methods that have 
 been described by Cornu, 1 Newall, 3 Hartmann, 3 and others for 
 determining the exact focal length of a telescope. 
 If the focusing were always exact we would have 
 
 da, = (a-a'")fM r N 
 
 (100) 
 =* (a-a'")(z + dQ)\t , V ' 
 
 and under such circumstances (98) and (99) would again reduce 
 to forms identical with (64) and (65). Exact focusing, in the 
 sense in which the term is used in metrological work, is, how- 
 ever, never possible by the ordinary standards of definition; 
 hence it is better to consider these terms separately. 
 
 The quantities a, a' , a" , and a" ' , which appear in the first 
 and last terms of (98) and (99) are all small and may be accu- 
 rately determined once for all. Hence since d£l is always small 
 compared to c, and k is known from the measurements them- 
 selves, we can, as we have already shown, determine the value 
 of the first term of these corrections with all requisite accuracy 
 if we know the value of At within 3° (± i?6). In order to do 
 the same for the other two terms we must first of all determine 
 the values of the angles t, and t,. 
 
 ■ Ann. de V ' Ecole Normale, (2) g, 2 1. 2 M. N., 57, 572, 1897. 
 
 3ASTROPHYSICAL JOURNAL, 12, 37, 1900. 
 
56 F. L. O. WADSWORTH 
 
 The first and most interesting case to be considered is that of 
 the prism spectrograph. Here the separation of the axial pen- 
 cils of the two extreme wave-fronts is produced bv the dispersion 
 of the spectroscope train. The amount of this separation has 
 already been calculated [see equation (34)]. 
 
 If there were no absorption in the prism train, and the prisms 
 and view telescope had apertures sufficiently large to trans- 
 mit the entire beam of the two extreme lateral pencils, the 
 separation 5 given bv (34) would be that required in (97). 
 Owing to the unsymmetrical absorption, however, the mean cen- 
 ter of illumination will not be on the axis of the wave-front, 
 but will be displaced toward the refracting edge of the prism by 
 an amount proportional to the coefficient of absorption. With 
 the same notation as that already employed in (8) and (9) we 
 have for the total quantity of light transmitted 
 
 ^ =J7''- '-**'*•■ 
 
 (101 
 
 Hence the mean center of illumination will lie at a point S a 
 such that 
 
 f 
 
 2 
 
 (102) 
 
 which by comparison with ( 1 1 ) gives at once 
 
 Bi 
 
 ■So = -^ nap log- j .. (103) 
 
 „ 1.3S6 . 
 
 Jt , or.o= — - — . as previously assumed, 
 
 and therefore 
 
 S "-~- b b ' ( io 4) 
 
 / 3 
 
 i. e., the center of illumination is displaced one-sixth the entire 
 diameter of the wave-front toward the refracting edcre of the 
 
OPTICAL CONDITIONS OF ACCURACY 57 
 
 prism train, and the angle t, for the wave-front passing at mini- 
 mum deviation is one-third the semi-angular aperture of the view 
 telescope. 
 
 In order to avoid displacement of the central image (for 
 which k = 0) with change of focus, the center of the objective 
 must coincide with the mean center of illumination 0; i. e., the 
 principal axis of the view telescope should not lie on the axial 
 line of the ray transmitted at minimum deviation but should be 
 shifted toward the refracting edge of the prism train by the 
 amount indicated in (103). This is a point generally overlooked 
 in the design of spectrographs. 
 
 In the case of lateral pencils the value of the middle term 
 representing the effect of the mechanical change in focus will be 
 less or greater according as the signs of t, and t 2 are the same 
 or opposite, and according as the sign of their differences is the 
 same or opposite to the sign of k. If t, has the same sign as k 
 we may make the term k — (t,— t 2 ) zero by properly controlling 
 .S, and 6" 2 , the points of incidence of the centers of illumination 
 of the lateral wave-fronts. In the case of the spectrograph this 
 can be accomplished by so proportioning /, the focal length of 
 the view telescope, and L, the total length of path through the 
 prism train, that the value of 5 in (97) and (34) is identical with 
 the value of f, the separation of the two images at the focal 
 plane. This gives us, from (34) and (35), 
 
 /(r, - r s ) =/k = ^ Q j A + 34 + 5 Lf+ ■ ■ -(2JV -"i) L w I 
 
 >-\ Z I + 2Z 5 + 3 Z 3 -t- ■ •• ■-<V , 7 V |] - 
 
 S d°5) 
 
 I 
 
 or since the total angular displacement k is 2.V times the dis- 
 placement -jr for one refraction, we have for three prisms 
 
 For the Bruce spectrograph already considered 
 
 f= 10.08 -f- 16.42 = 26.5 cm , 
 instead of 44.9 cm and 60.7 cm, as adopted for camera A and 
 camera B. 
 
5S F. L. O. WADSWORTH 
 
 Since (34) is deduced on the assumption that the prisms and 
 objective of the view telescope are placed as close together as 
 possible, (105) and (106) express the minimum value of /"that 
 will satisfy the above condition for the elimination of the term 
 k — (t,— t 2 ). We can, however, satisfy this condition for any 
 larger value of f by separating the prisms or by withdrawing the 
 camera objective so that the total path L through the spectro- 
 scope train is correspondingly increased. It is so easy to do this 
 in the original design of the instrument, and the advantage result- 
 ing from it in the entire elimination of the effect of errors in the 
 scale reading D, is so great, that it is singular that this particular 
 point of design seems to have been previously overlooked. 
 
 We may accomplish the same result in the case of the plane- 
 grating spectrometer by making the distance from the grating 
 to the objective of the view telescope equal to the focal length 
 of the latter. In this case as before the objective of the view tele- 
 scope must be increased in size sufficiently to receive the entire 
 cross-section of the lateral beams. In this latter case the use 
 for which the instrument is intended, as well as the form of view 
 telescope adopted, must be considered before we can determine 
 in any case whether it is worth while to introduce this modifica- 
 tion of the usual design. With reflecting view telescopes the 
 condition is very easily and almost necessarily satisfied, and this 
 is another advantage of this form of grating spectroscope. 1 
 
 In the case of the heliometer we may satisfy the same condi- 
 tion of zero displacement with change in the position of the eye- 
 piece by proper diaphragming. With a full semi-circular aperture 
 the total quantity of light in the transmitted wave-front is pro- 
 portional to -*■>-=. If we diaphragm one edge of the aperture 
 by an amount mR the amount of light transmitted will be 
 
 ! R* - .v= dx 
 
 - -R 
 
 R ,r- i _. r ■ ( io 7> 
 
 = — (1 — ")) i 2111 — m' + sin -1 (1 — m) -\- - 77 
 
 1 Phil. Mag., 38, 137; Astrophysical Journal, i, 232, etc. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 59 
 
 The mean center of illumination of the diaphragmed aperture 
 lies at a point 5 such that S e =R(l — n) and 
 
 /-.ff(i-«'i 
 _, 
 and therefore 
 
 -R 
 
 ■ x- dx — — A 
 
 2 
 
 [(;/ — i) y 2ii — if -f- sin - ' (« — i)] -f- - 
 
 : (i — Hi) ■) 2vi — m" -\- sin ' (i — ;//) = — A- 
 
 ~=A m . (108) 
 
 ■w 
 
 Fig. 12. 
 
 Let A =-irR* be the amount of light transmitted when there 
 is no diaphragm. Then we have from (107) and (108) 
 
 A A,„ £ 
 
 A„ TZ 2 
 
 (109) 
 
 Equation (108) serves to determine m, the amount of dia- 
 phragming required for any given value of n, and (109) gives 
 the ratio between the illumination secured with this diaphragm 
 and the illumination with full (half) aperture. To determine ?i 
 we have also in this case 
 
 S i =/T l =f- = R(i-u) 
 
 whence 
 
 2/3 
 
 (no) 
 
<5o 
 
 F. L. O. WADSWORTH 
 
 Assume as before that for a 16 cm heliometer /3 3* .03. Then for 
 
 two objects separated by an angular interval of o?5, - = 0.00436 
 
 and « 3^ 0.855, *, e., the center of illumination of the incident 
 
 wave-front should fall upon each half of the divided lens 0.145 
 
 of the radius from the center. This is accomplished by giving 
 
 to m a value obtained by solving (108) for «=o.855. We thus 
 
 obtain 
 
 ;« = o.47 5 , 
 
 i. c, a little less than one-fourth of each half of the objective 
 should be covered, as in Fig. 12, the screens W W being placed 
 so as to shut out in each case the light from the edge which is 
 on the opposite side of the optical axis from the image formed 
 by that half. 
 
 By expressing k as a fractional part of /3 we can determine 
 the general relation between k and m for any heliometer. This 
 
 has been done for values of „ varying by tenths from o to 1 
 
 si 
 and by 0.2 from 1 to 2. The results for A,„ , — , and m are given 
 
 in Table IV and plotted in Fig. 13. 
 
 With this relation it is easy to devise a mechanical arrange- 
 
 TABLE IV. 
 
 K 
 
 
 
 A 
 
 
 J 
 
 « 
 
 A m 
 
 ~Ta 
 
 in 
 
 
 
 1 .00 
 
 1.5708 
 
 I .00 
 
 
 
 0. 1 
 
 0.95 
 
 I.3706 
 
 0.9363 
 
 O.23O 
 
 0.2 
 
 .90 
 
 I . 1726 
 
 ■8732 
 
 • 369 
 
 0-3 
 
 .85 
 
 0.9738 
 
 .S099 
 
 .490 
 
 0.4 
 
 .80 
 
 •7774 
 
 •7473 
 
 .600 
 
 0.5 
 
 •75 
 
 .5808 
 
 • 6S49 
 
 •705 
 
 0.6 
 
 .70 
 
 .3876 
 
 •6233 
 
 .805 
 
 0.7 
 
 .65 
 
 .1994 
 
 •5634 
 
 .900 
 
 0.8 
 
 .60 
 
 .0144 
 
 •5046 
 
 .9S8 
 
 0.808 
 
 .596 
 
 .0000 
 
 . 5000 
 
 I .0O0 
 
 0.9 
 
 • 55 
 
 — .1664 
 
 • 4471 
 
 I.085 
 
 1 .0 
 
 .50 
 
 - -3422 
 
 ■ 3911 
 
 I. 172 
 
 1 .2 
 
 • 40 
 
 - -6770 
 
 .2845 
 
 1-345 
 
 1.4 
 
 ■30 
 
 - -9784 
 
 . 1SS6 
 
 1.512 
 
 1.6 
 
 .20 
 
 -1.2424 
 
 ■1047 
 
 1.678 
 
 1. 8 
 
 . 10 
 
 -'•4542 
 
 •0372 
 
 1 .S40 
 
 2.0 
 
 .00 
 
 -1.570S 
 
 .0000 
 
 2.000 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 6r 
 
 merit which will automatically move the screens W W of Fig. I 2 by 
 the required amount when the two halves of the object-glass are 
 adjusted to bring two images separated by the angular interval k 
 into coincidence. The great advantage of such an arrangement 
 in the case of the heliometer 
 
 2,0 
 
 is that when it is adopted 
 the eyepiece may be moved 
 in or out at will to suit the 
 convenience and personality 
 of the observer without 
 thereby introducing any dif- 
 ferences or errors in the set- ' 
 tings of the instrument itself. 
 The only drawback is that 
 the light is considerably re- 
 duced for objects having 
 considerable separation. 
 
 In this case, as in the 
 preceding, the values of the 
 angles t z and t 2 , which ap- 
 pear in the third terms of (98) and (99), are determined by 
 the relations 
 
 (t.-t,) /=/(«) , 
 
 and these terms reduce to the form 
 
 K/(a — a"') A/ , (in) 
 
 in which all the quantities are known or may be measured, and 
 which, like the first terms of these equations, may be computed 
 with all necessary accuracy when we know At for ordinary object- 
 glasses to within about 3 . Temperatures of the object-glass 
 and telescope-tube may be easily observed with this degree of 
 accuracy, but we may by the special optical construction already 
 mentioned on page 52 eliminate this term entirely by making 
 a — a =0. 
 
 B (3). It would be impossible and, in fact, quite unnecessary 
 to enter here upon a general discussion of the methods of optical 
 measurement best suited to individual cases. We shall endeavor 
 
62 F. L. O. WADSWORTH 
 
 to point out only some general principles of working which the 
 previous investigations have shown that it is necessary to regard 
 if we aim at the highest attainable degree of accuracy in our 
 work, and to investigate in this connection one or two individual 
 sources of error in the use of particular instruments which seem 
 either to have escaped previous attention, or at least to have 
 been considered less carefully than their importance demands. 
 
 In general, the causes will produce the maximum effects in 
 the optical distortion and displacement of images, and the ones 
 therefore which we must most carefully guard against are those 
 dealt with in sections A (2), A (3), B (1), and B (2). 
 
 In order to eliminate, or minimize as far as possible, the 
 effects of asymmetrical, spherical, or chromatic aberration, A (2) 
 and A (3), we must employ instruments and methods of work 
 planned with special reference to avoiding the following sources 
 of error: (1) the use of too large fields or of too great angles 
 of incidence, particularly when using short focus mirrors or 
 gratings at the principal focus ; (2) the use of methods of com- 
 parison or of forms of instruments and of optical construction 
 which necessitates that the different images whose relative 
 positions are to be measured are formed by different portions of 
 the surfaces of the optical train ; (3) variations in temperature 
 conditions during the interval of measurement or of record ; 
 (4) the effect of small chromatic dispersions, particularly when 
 the spectral curve of radiation is different for the different 
 sources; (5) the effect of varying chromatic absorption and 
 unsymmetrical broadening of the spectral lines when the chro- 
 matic dispersion is large. 
 
 In the case of errors of Class B we must especially guard 
 against (6) the use of too small resolving powers, and (7) the 
 continued effect of temperature variations and asymmetrical 
 illuminations. 
 
 Of these various effects those which have heretofore received 
 most attention are (3) and (7). Temperature variations are. 
 indeed, our greatest source of trouble and error in the oreat 
 majority of physical measurements of all kinds. It appears, 
 
OPTICAL CONDITIONS OF ACCURACY 63 
 
 however, from the results of the investigations on this point 
 (pp. 26-28), that these effects are of even larger order of magni- 
 tude than is ordinarily assumed, and that in spectroscopic work 
 particularly we must guard against them with the greatest care. 
 As was pointed out on pp. 12, 13 and 26-28, the best method 
 of eliminating residual effects, after reducing the temperature 
 variations to the smallest possible range, is to measure the sepa- 
 ration of the images and determine the so-called constants of 
 the instrument, i. e., focal length, pitch of measuring screw or 
 scale, etc., simultaneously with each observation, instead of 
 relying on values determined once for all at considerable inter- 
 vals, as is usually done. 
 
 This method of simultaneous record is carried out most com- 
 pletely at present in determining star places by the chart photo- 
 graphs of the Astrophotographic Survey, in which the constants 
 of reduction for each plate are determined from the photo- 
 graphic record on the plate itself. The next best example is 
 that of the spectrographic method of determining absolute wave- 
 lengths and motions in the line of sight by the use of compari- 
 son spectra. The only criticism to the present practice of the 
 method in the latter case is that the record of the comparison 
 spectrum and of the spectrum to be measured is not simul- 
 taneous, and in the case of star spectra, in which an interval of 
 sometimes two hours intervenes between the records, the resid- 
 ual errors of intermediate temperature_ variations, smalL though 
 they may be, are in certain cases quite sensible. -It would be 
 much better, as already pointed out, if the exposure on the 
 comparison spectra were made continuously instead of inter- 
 mittently at the beginning and end of the exposure, and this 
 could be arranged without any great difficulty. 
 
 It must be farther noted, however, that this method of simul- 
 taneous record and individual reduction of each set of measures, 
 although it may eliminate almost completely the effect of small 
 temperature variations, will not eliminate the effects of (1), of 
 (2), of (4), of (5), nor of the latter part of (7), unless, indeed, 
 the conditions of observation are precisely the same in all of 
 
6_| F. L. O. WADSWORTH 
 
 these respects for both the objects upon which the direct measure- 
 ments are made and the objects upon which the comparison meas- 
 urements for the "constants " of reduction are executed. It is 
 physically impossible always to fulfill this requirement in the case 
 of (4) and (5), and frequently very difficult in the case of (1 ), (2), 
 and (6) because of the different parts of the field in which the two 
 sets of images (direct and comparison) are situated, or because 
 of necessary limitations in the size and power of the observing 
 instrument. 
 
 For these reasons it is unsafe, if we aim at the highest 
 degree of accuracy in optical measurements, to neglect the 
 causes of optical displacement considered above, or to assume 
 beforehand that their effects are vanishingly small. In this 
 connection it is necessary to insist again on the point alreadv 
 made, that the ordinary tests and standards of optical definition 
 are not sufficiently rigorous to serve as criteria in determining 
 questions relating to the limiting accuracy of measurement. 
 Nor are we safe in considering our results free from constant 
 errors which may arise from instrumental causes, simplv because 
 they agree consistently among themselves, or are checked bv 
 occasional measurements of known quantities. Our onlv safe 
 way of procedure would seem to be to investigate in each indi- 
 vidual case, as we have already done in the general one, the 
 maximum individual effects produced by possible disturbances of 
 theoretical conditions, and either to reduce each of these maxi- 
 mum individual effects (unless there may be two or more which 
 are always ?iecessarily compensatory) to a quantity less than e, 
 the limiting metrological power; or to determine the magnitude 
 of each effect to the same degree of exactness, in order that it 
 may be taken care of as a known correction. In such individual 
 cases there may exist possible causes of disturbance of a special 
 character not included in the general classes of errors alreadv 
 discussed. As stated at the beginning of this section it is 
 unnecessary and undesirable to take up in detail a complete 
 discussion of any number of such cases. 
 
 There is, however, one individual case in which an error of 
 
OPTICAL CONDITIONS OF ACCURACY 65 
 
 measurement due to a cause not yet considered is very likely to 
 be introduced, and which on account of its importance in one of 
 the leading fields of astrophysical research we will consider at 
 length. This is the case of the slit spectroscope or spectrograph 
 as used for determinations of velocity of motions in the line of 
 sight, or more generally of absolute wave-lengths. Here, in 
 order either to secure greater intensity of spectrum, or to 
 localize determinations with reference to different parts of the 
 sources under examination, images of both the latter and of the 
 comparison sources are first formed on the slit of the instru- 
 ment, and we measure the relative positions, not of the spectral 
 images of the sources themselves, as they would be formed by 
 an objective prism or grating, but the relative positions of the 
 spectral images of the slit as illuminated by the light concen- 
 trated upon it by the image-forming objective. The distinction 
 is an important one, for, as we have seen in the case of A (3), 
 any asymmetrical distribution in intensity in the source of radia- 
 tion will result in an asymmetrical distribution in intensity in 
 the image, and a consequent displacement of its center of 
 maximum brightness. Hence if the slit is not uniformly illumi- 
 nated across its entire width, its spectral image corresponding 
 to any particular wave-length in the light falling upon it will be 
 displaced from the position it would occupy under conditions of 
 uniform illumination, no matter what the conditions in the source 
 itself may be. The asymmetrical illumination of the slit may 
 arise from (1) asymmetrical distribution of intensity in the 
 image of the source itself on the slit, due either to a real asym- 
 metry of radiation or to asymmetrical distortion or aberration 
 of the image-forming objective ; (2) displacements of the center 
 of symmetry of the image with reference to the center of the 
 slit. As the effect on the measurements will be the same in 
 both cases we may consider them jointly. 1 
 
 'This effect is entirely different in nature from that produced by lack of uniform 
 illumination of the collimator objective, which has already been considered in B (2) 
 So far as I have been able to ascertain, the effect of asymmetrical illumination of the 
 slit itself it had escaped attention previous to the time when the preliminary results of 
 this paper were informally announced at the Astrophysical Conference at the Verkes 
 Observatory in 1897. It was then expected that these results would be at once inves- 
 
66 F. L. O. WADSWORTH 
 
 The general expression for the distribution in intensity in the 
 spectral image of a slit uniformly illuminated across its entire 
 width by radiation from any individual spectral line has already 
 been given [equation (52)], and this expression has been evalu- 
 ated for a number of special cases. 1 If the illumination across 
 the width o- is not uniform, we must add another factor, /(£), to 
 express this variation in intensity, which gives us in (52) 
 
 tr 
 
 r l , l =A"jl*f{£)t 1 \£-<j„o. a}dt . (112) 
 
 — 2 
 
 As in the consideration of a uniformly illuminated slit, we 
 will first examine the case in which the light on the slit is abso- 
 lutely monochromatic and the spectroscope itself is free from 
 aberration. Then at the focal plane of the instrument the func- 
 tion t/t x becomes simply 
 
 7ra 
 
 sin 2 — 
 
 (?)" 
 
 and we have for I 2 ,, 
 
 sin' - (a — f ) 
 
 h„=a" / /(£) — ^ dt , (ii 3 ) 
 
 which is the same form as that expressing the distribution in 
 intensity in the image of an individual spectral line [equation 
 (51)], for an infinitely narrow slit. 
 
 It is evident that (113) like (51) will be symmetrical or 
 asymmetrical according to the form of the function /(£). We 
 will examine a few cases of particular interest in detail. 
 
 First, suppose 
 
 f(t) = A + ££. („ 4 ) 
 
 This represents a uniformly progressive increase in the intensity 
 of illumination across the entire width of the slit, such as is 
 
 tigated in detail by others more directly interested than the writer in line of sight 
 work, but their bearing on this line of research seems to have been overlooked or not 
 fully realized by those present at that time. 
 
 1 Loc. cil. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 6 7 
 
 graphically illustrated in Fig. 14. Taking the origin of co-ordi- 
 nates at the point corresponding to the geometrical image of 
 one edge of the slit (instead of the center as before), the integral 
 (1 13) becomes 
 
 r 
 
 {A+Bi) 
 
 sur — ■ a 
 
 <*0 / 
 
 "(« + j) 
 
 Ei-(«+on 
 
 ■A- ("s) 
 
 Fig. 14. Fig. 15. 
 
 I have evaluated this integral by mechanical quadrature for 
 the special case A = -, B = — and o- = a [i. e.. the case in 
 
 r 2 2a " v 
 
 which the angular width of the slit is equal to the resolving 
 power of the view telescope), and the intensity of illumination 
 at one edge, £ = 0, is one-half that at the other, £ =a = cr . 
 The results to three places of decimals are tabulated in terms 
 
 of fractional parts of — in Table V and plotted in Fig. 15. In 
 
 both the table and the figure the origin of co-ordinates has been 
 shifted back to the center of the geometrical image. 
 
 An examination of these results shows that the position of 
 maximum brightness in the spectral image of the slit is displaced 
 in this case about 0.08 a from the center of the geometrical 
 image. This is just a little more than the limiting accuracy e of 
 the instrument, i. e,,a. uniform variation of 50 per cent, in inten- 
 sity of illumination from edge to edge of the slit will displace 
 
68 
 
 F. L. O. WADSWORTH 
 
 the center of the physical spectral image by an amount some- 
 what exceeding the limiting accuracv of measurement. 
 
 TABLE V. 
 
 a 
 
 /...=/(£) 
 
 — /.. 
 
 =/«) 
 
 ! » 
 
 /■••=/(« 
 
 "o 
 
 
 «o 
 
 
 a 
 
 
 -2.45 
 
 O.OII 
 
 -0.75 
 
 235 
 
 +0.95 
 
 0.14S 
 
 -2.35 
 
 .011 
 
 — 
 
 65 
 
 3'7 
 
 +1.05 
 
 .090 
 
 — 2.25 
 
 .012 
 
 — 
 
 55 
 
 429 
 
 +1.15 
 
 .054 
 
 -2.15 
 
 .014 
 
 — 
 
 45 
 
 552 
 
 +1.25 
 
 .036 
 
 — 2.05 
 
 .015 
 
 — 
 
 35 
 
 072 
 
 + L35 
 
 .030 
 
 -1.95 
 
 .018 
 
 — 
 
 25 
 
 79S 
 
 +1-45 
 
 .030 
 
 -1.85 
 
 .020 
 
 — 
 
 15 ! 
 
 S9S 
 
 + L55 
 
 ■ 032 
 
 -i-75 
 
 .023 
 
 — 
 
 05 
 
 q6S 
 
 +1.65 
 
 ■ 033 
 
 -1.65 
 
 .026 
 
 - 
 
 - 
 
 °5 ! 1 
 
 000 
 
 +L75 
 
 .031 
 
 -i-55 
 
 .028 
 
 - 
 
 - 
 
 15 j 
 
 9S7 
 
 + 1.85 
 
 .027 
 
 -1-45 
 
 .030 
 
 - 
 
 - 
 
 25 
 
 033 
 
 + 1 95 
 
 .022 
 
 -i-35 
 
 .032 
 
 - 
 
 - 
 
 35 
 
 844 
 
 +2.05 
 
 .017 
 
 -1.25 
 
 .038 
 
 
 
 45 
 
 727 
 
 +2.15 
 
 .013 
 
 -LIS 
 
 .048 
 
 " 
 
 " 
 
 55 
 
 595 
 
 +2.25 
 
 .011 
 
 -I.05 
 
 .068 
 
 - 
 
 _ 
 
 65 
 
 462 
 
 +2.35 
 
 .010 
 
 -O.95 
 
 . 102 
 
 
 
 75 
 
 337 
 
 +2-45 
 
 .010 
 
 — O.85 
 
 •153 
 
 
 
 85 
 
 231 
 
 
 .... 
 
 Second, suppose 
 
 f{i)=C^ $ . (116) 
 
 This represents a progressively increasing" illumination from 
 edge to edge as graphically illustrated in Fig. 16. 
 
 With the origin of co-ordinates at one edge of the slit as 
 before, the integral (113) reduces to the form 
 
 ' ,in- If. -(* + *Y| 
 
 H, = A 
 
 GM+i)}J 
 
 ("7) 
 
 This integral has been evaluated for a slit width a = a as before 
 for the special case C=l . The results are tabulated and 
 
 plotted in terms of - as in the preceding case in Table VI 
 and Fig. 17. 
 
 The position of maximum intensity, o' is, in this case, dis- 
 placed about 0.13 a Q from the center of the geometrical image, 
 i. e., about twice the limiting error of measurement. 
 
 Cases in which the illumination of the slit is of the nature of 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 69 
 
 that just investigated are frequently met with in spectrometric 
 work on the Sun and planets when the image of these bodies is 
 placed eccentrically on the slit or when particular portions of 
 the surface, such as the Sun-spots or solar prominences, are 
 under examination. In the latter case the illumination is likely 
 to be of the nature of that assumed in (114) and (115), and the 
 effect of the variation in intensity of the image on the measured 
 
 + S,0 
 
 Fig. 16. 
 
 Fig. 17. 
 
 wave-lengths of the spectral lines can be disregarded, as we have 
 seen, only when said variation is less than 50 per cent. 
 
 In the case of measurements of the velocity of rotation the 
 
 TABLE VI. 
 
 
 
 
 a 
 
 
 
 — 
 
 /,,. 
 
 /■=/(?>' 
 
 — 
 
 /.., 
 
 /■=/(*)> 
 
 ■0 
 
 
 
 a 
 
 
 
 — I .0 
 
 0.062 
 
 0. 1 1 
 
 
 -0.00 
 
 0.969 • 
 
 1. 00 
 
 -0.9 
 
 .090 
 
 
 
 -0. 1 
 
 1 .000 
 
 
 -0.8 
 
 .136 
 
 0.24 
 
 - 
 
 -0.2 
 
 0.9S5 
 
 0.92 
 
 -0.7 
 
 .204 
 
 
 - 
 
 -0.3 
 
 .925 
 
 
 -0.6 
 
 .295 
 
 0.45 
 
 
 -0.4 
 
 .827 
 
 0.71 
 
 -0.5 
 
 .406 
 
 
 
 -0.5 
 
 .702 
 
 
 —0.4 
 
 •533 
 
 0.71 
 
 
 -0.6 
 
 .564 
 
 0.45 
 
 -0.3 
 
 .665 
 
 
 
 -0.7 
 
 .426 
 
 
 — 0.2 
 
 .784 
 
 0.92 
 
 - 
 
 -0.8 
 
 .301 
 
 0.24 
 
 —0. 1 
 
 .896 
 
 
 - 
 
 -0.9 
 
 .198 
 
 
 — 0.0 
 
 .969 
 
 1 .00 
 
 H 
 
 -1 .0 
 
 . 120 
 
 u. II 
 
 'For the purpose of comparison the values of I', which expresses the distribution 
 in intensity in the image of a slit of the same width uniformly illuminated, are given 
 in columns 3 and 6. See Phil. Mag., 43, 323, Table I ; AstroI'Hysical Journal, 
 3.333 'tscq. Thecurvel' —f(y)\& plotted in dotted lines in both Fig. 15 and Fig. 17. 
 
F. L. O. WADSWORTH 
 
 slit is generally placed tangentially to the limb of the Sun or 
 planet, and the conditions of illumination are then more gener- 
 ally such as are represented by the distribution assumed in ( 1 1 6 ) 
 and (117). For in the case of the Sun the variation in the 
 intensity of radiation from the center to the edge is of the nature 
 shown by the curves of Fig. 1 of the preceding paper of this 
 series, 1 and may be represented by the general expression 
 
 The effective intensity /(f) at any point on the axis 7/=o will be 
 found for this case by integrating the expression 
 
 P 1 ^- sin' *(«-„) 
 f(i)=/^{^ + yf),—^-. -<h, . (II 9 ) 
 
 At the edge of the Sun the value of rj increases very rapidly 
 with respect to f, and for large values of r/ the integral 
 
 -1 
 sin 2 .r 
 
 ■tx 
 ^0 x' 
 
 rapidly approaches a constant value. Hence for the purposes of 
 the present general investigation in which we are concerned 
 more with the form of /(f) than with its absolute value, we may 
 neglect the second factor of the term under the integral sign 
 of (119) and simply write 
 
 /tf)S J<t>(£* + f)d n . (120) 
 
 This integral was evaluated by the writer several years ago (in 
 connection with another research by Professor Michelson), for 
 the particular value of <f> (f 2 — ?? 2 ) corresponding to wave-length 
 \=5790 tenth-meters, as given in Vogel's tables. The results 
 were published in the form of a curve (Fig. 5) in Michelson's 
 paper in the Philosophical Magazine? An examination of this 
 
 'F, \V. Very, "The Absorptive Power of the Solar Atmosphere." Misc. Sci. 
 Papers, Allegheny Observatory, No. g; Astroi'Hysical Journal, i6, 73. See also 
 VOGEL, " Spectralphotometrische Untersuchungen," Monal. d. K. Akad. d. Hiss. 
 Berlin, March 1S77 ; pp. 23, 24, Plate II. 
 
 ""Application of Interference Methods to Astronomical Measurements," 30, I. 
 
OPTICAL CONDITIONS OF ACCURACY 7 I 
 
 curve will show that for small values of £ the function /(£) can 
 be very closely represented by the parabola 
 
 which is the form of function used in (116) and (117) already 
 examined. 
 
 The effect of this inequality of illumination is, as we have 
 already seen, to displace the spectral image of any line by an 
 amount considerably in excess of the limiting accuracy of 
 measurement. The exact amount of the displacement will 
 depend (a) on the width of the slit in comparison with the 
 diameter of the solar image, (b) on the position in which the 
 latter is placed with reference to the slit opening. In the case 
 already investigated in (117) we have assumed that the photo- 
 spheric edge of the solar image is placed tangent to the outer 
 edge of the slit so that when £ = 0, /"(?) is also zero. 1 If the 
 image overlaps the slit, the conditions approximate more nearly 
 those assumed in (114) and (115) and the displacement is less; 
 if the edge of the image is within the slit, the displacement is 
 even greater. Thus when the edge is coincident with the center 
 of the slit, the position in which it is generally placed for veloc- 
 ity of rotation measurements, the displacement of the central 
 point of intensity from the geometrical center will be nearly 0.6 
 of the half width of the slit, i. e., about 0.3 a, or four times the 
 limiting accuracy e. 
 
 The direction of the displacement in the spectrum will depend 
 on the position of the solar image with reference to the direction 
 of deviation of light by the spectroscope train. If the edge 
 of the image is on that side of the slit corresponding to the 
 red end of the spectrum, the displacement will be toward the 
 violet, and the wave-length derived from the measurement will 
 be too small; if turned in the opposite direction the wave-length 
 will be correspondingly too large. The sign of the displacement 
 will, therefore, change not only as we shift the solar image so 
 
 'This of course is only an approximation to the actual conditions at the edge of 
 the Sun. There is, however, such a sharp demarcation in intensity of total radiation 
 between the photosphere and the chromosphere and outlying corona that the approxi- 
 mation is in most cases not far from the truth. 
 
72 F. L. O. WADSWORTH 
 
 that first one edge and then the other falls on the slit; but also as we 
 reverse the direction of deviation of the spectroscope train, "right" 
 or "left," leaving the position of the solar image unchanged. If 
 we therefore attempt to determine the velocity of rotation of a 
 body by the direct measurement of the separation of the spectra 
 from the opposite edges of the slit image, the total error in any 
 one position of the spectroscope will be twice the error of dis- 
 placement noted above. If we then take a corresponding series 
 of measurements with the direction of dispersion reversed, the 
 errors of displacement will be of the same magnitude as before, 
 but reversed in sign; hence the total difference in measurement 
 will be four times the individual displacement in any one -posi- 
 tion of image and spectroscope. 
 
 Errors of such magnitude as these cannot, of course, be 
 disregarded, even though they may be eliminated by the exact 
 reversal of the relative positions of image and spectroscope. I 
 have, however, failed to find any reference to this particular 
 source of error in the papers of the investigators who have given 
 us our best spectroscopic determinations of velocity of solar 
 rotation. Crew, indeed, in his second investigation 1 notes that 
 the results of his measurements differ with grating "right" and 
 grating "left," but he ascribes this difference to an entirely 
 different cause, i. e., the local heating of the slit jaws. This 
 will undoubtedly produce an effect unless the greater part of the 
 solar image is cut off by a suitable screen interposed in front of 
 the slit, but even when this is not done the total displacement of 
 the slit jaws due to this cause can hardly, in the opinion of the 
 writer, be as great as it is necessary to assume to explain the 
 observed differences of measurement. To do this it is neces- 
 sary, as Crew has shown, to assume a change of temperature of 
 about 15" C. 2 of each jaw between each measurement, in an 
 interval of only about one minute, and to assume, further, that the 
 expansion due to this change in temperature is all toward the 
 
 1 " On the Period of Rotation of the Sun," Am. Jour. Set., 38, 204. 
 
 2 The assumption of a difference of 10° C, made in Crew's paper, p. 210, 
 explains only about two-thirds of the observed differences of equatorial velocities 
 with grating " right " and grating " left." 
 
OPTICAL CONDITIONS OF ACCURACY 73 
 
 slit opening. Crew observed actual differences of temperature 
 of 10° between two thermometers placed on the two sides, but 
 the change in the slit jaws themselves would be probably much 
 less on account of their own mass and the masses of surround- 
 ing and attached metal. Likewise, the jaws when heated, would, 
 if of the usual double-motion construction, expand in both 
 directions. Lastly, if the change of temperature explanation 
 were the correct one, there would be a continual and progressive 
 shift of the spectral image during the entire interval of meas- 
 urement, and this effect would not have been likely to escape 
 the attention of so careful and accurate an experimentalist as 
 Professor Crew. 
 
 On the other hand, if the width of the slit and the setting of 
 the edge of the solar image corresponded to the conditions 
 assumed in the previous integration of (117) the resulting 
 displacement of the spectral images "right" and "left" would 
 fully explain and account for the observed differences of meas- 
 urement. For, as we have shown, the total difference of meas- 
 urement to be expected is four times the individual displacement 
 of any one image, and this latter is 0.13a,, for the conditions 
 assumed. In this instrument the aperture and the focal length 
 of the view telescope are 4 inches and 94 inches respectively. 
 Hence for the D lines the linear value of a is 
 
 0.0005896 
 
 a„/= :!— 1_ x 4 = 0.014 mm . 
 
 94 
 
 Hence the total difference = 4X0. 13^=0.5 2a is in linear 
 
 measure 
 
 0.014 mm X 0.52 ■- 0.0073 mrn • 
 
 The total observed difference in measurement amounted to about 
 
 0.015 revolutions of the micrometer screw, or, since the latter is 
 
 of 0.5 mm pitch, to 
 
 0.0075 mm , 
 
 or almost exactly the quantity required by theory. When, 
 therefore, the cause of displacement investigated above is taken 
 into account in Crew's measurements, there is no discrepancy 
 remaining to be explained. 
 
74 F. L. O. WADSWORTH 
 
 The results already obtained indicate the care which it is 
 necessary to take in determining the exact distribution of light 
 in the slit image when the latter is necessarily variable, as in the 
 preceding case. If this cannot be determined, the only way to 
 avoid sensible errors in the wave-length measurements is to 
 secure and maintain an exact centering of the image on the slit, 
 so that the distribution of light over the slit opening is symmet- 
 rical about a center line. A case of great importance in this 
 connection is that of determinations of wave-length in stellar 
 spectra for the purpose of measuring motions in the line of 
 sight, etc., with the compound slit spectroscope. The image of 
 the star, as formed on the slit by the main telescope objective, 
 is, if the latter is free from unsymmetrical aberration, a symmet- 
 rical diffraction pattern. If the slit is placed in the focal plane 
 of the main objective, the theoretical distribution of light in 
 this image is, for any given wave-length, represented bv the 
 usual expression (4), i. c, 
 
 1\=A 
 
 (?)" 
 
 In long photographic exposures, the actual effective distribu- 
 tion of light in the slit image is somewhat different from this, 
 owing to atmospheric and instrumental disturbances which 
 broaden the image into what Newall has very aptly termed a 
 "tremor disk." The determination of the exact distribution of 
 light in such a disk is a matter of considerable uncertainty, since 
 the broadening results not only from vibrations of the image, 
 but also from temporary changes of focus and of chromatic 
 dispersion and aberration due to passing air-waves of variable 
 intensity. Under good conditions of "seeing," however, the 
 principal cause of the broadening may be regarded as due to 
 vibrations, and under such conditions we may derive an expres- 
 sion which will represent at least a closely approximate distribu- 
 tion of light in the tremor disk, from the law of probability. 
 
 The general expression which represents the probable law of 
 
OPTICAL CONDITIONS OF ACCURACY 75 
 
 distribution of errors, or, in this case, of displacements from a 
 central position, is 
 
 y — e -A*(> ( I2I ) 
 
 and the most probable error (7. e., displacement) is that for 
 which _y = 0.5 and is given by the relation 
 
 l^P=—j— (l22) 
 
 In the present case it seems a fair assumption to consider 
 that the most probable displacement of the image will be about 
 y 2 a o< i- e., half the resolving power of the main telescope objec- 
 tive. This assumption gives us for h 
 
 ,. o-9S38 
 
 ( I2 3) 
 
 which substituted in (r2i) gives us ior y 
 
 °.9°<)7- = ( I2/ 
 
 y = e 
 The resulting distribution in intensity in the tremor disk will 
 
 then be represented by the integral 
 
 or if we assume a rectangular aperture as before, by 
 
 <#=/(*) . («S) 
 
 *=sin 2 -£ 
 
 —0.9097-3 
 
 I; / ' a> TTy' /i =At) > (126) 
 
 and the resulting distribution in the spectral image of the slit of 
 width o- will be 
 
 /^ + "' sin°-(a — i) 
 
 In the integration of (126) and (127) it is convenient to 
 express a a in terms of <r , the width of the spectroscope slit. 
 
7 6 
 
 F. L. O. WADSWORTH 
 
 The usual practice is to make this width equal to 2 a a ' so as to 
 just include the central diffraction disk of the undisturbed star 
 image. Substituting this value of <r in the above equations we 
 obtain 
 
 -3.64- 
 
 m 
 
 d£ . 
 
 (128) 
 
 This has been integrated by mechanical quadrature as before. 
 The results are tabulated in Table VIII and plotted in Fig. 18. 
 
 TABLE VIII. 
 
 a 
 
 '/ 
 
 »-*■«' 
 
 /,■-.-<*■*■ 
 
 ±0.00 
 
 I .000 
 
 1 .000 
 
 U.000 
 
 ±0.10 
 
 0.972 
 
 0.973 
 
 — .001 
 
 ±0.20 
 
 .888 
 
 .892 
 
 — .004 
 
 ±0.30 
 
 • 77i 
 
 ■773 
 
 — .002 
 
 ±0.40 
 
 .632 
 
 ■633 
 
 — .001 
 
 ±0.50 
 
 .490 
 
 .490 
 
 ± .000 
 
 ±0.60 
 
 .361 
 
 ■35S 
 
 +.003 
 
 ±0.70 
 
 .254 
 
 .247 
 
 +.007 
 
 ±0.80 
 
 .171 
 
 .161 
 
 + .010 
 
 ±0.90 
 
 .ni 
 
 .099 
 
 + .012 
 
 ± I .00 
 
 .069 
 
 • 057 
 
 + .012 
 
 The full curve, //, closely resembles the exponential curve 
 e~ A £*. We have therefore substituted for (128) a second 
 empirical curve of this form having such a value of A' that the 
 two curves coincide at the point x = a, 
 for A' 
 
 ^' = 2.854 , 
 and for/'(£) 
 
 y 2 a . This gives us 
 
 -=.8S4^ 
 
 e < =* /; 
 
 (129) 
 
 The values of /' (£) from (129) are tabulated in the third 
 column of Table VIII and plotted as the dotted curve in Fio- 
 18. Over all that part of the tremor disk which is likely to fall 
 within the slit opening, i. e., from f = to f = ± 2 a , the coin- 
 cidence between the two curves, // and /'(£), as given bv 
 
 ■This is twice as large as the value ci a assumed in the previous cases. Com- 
 pare Figs. 14, 16, and IS. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 77 
 
 (129), is closer than the uncertainty as to the exact form of 
 /; itself. 
 
 Assuming that the effective size of the tremor disk is denned 
 by the limits mm such that P m — 0.04/0% we have under the con- 
 ditions assumed above 
 
 m fit = W^ 4.5a = 2.25/«„ m ; l^ ) 
 
 Fig. 18. 
 
 i. e., the tremor disk is about two and one-quarter times the size 
 of the true central diffraction disk of the star. When the tremor 
 disk is centered on the slit opening, the percentage of the total 
 quantity of light in the image which enters the slit is of course 
 given by the expression 
 
 
 '°et$^o.j6 ; 
 
 (1 3 
 
 i. e., the time of exposure is increased about one-third by the 
 effect of atmospheric and instrumental vibrations of the previ- 
 
 ously assumed probable magnitude of -o„ . 
 
 This result agrees fairly well with the actual exposure times 
 which Campbell finds necessary in obtaining well defined spectra 
 with the Mills slit spectroscope as compared with those required 
 
7S 
 
 F. L. O. WADSWORTH 
 
 on the same stars with the Harvard objective prism spectrograph, 
 when we allow for the difference in the resolving power and 
 light efficiency of the two instruments. It is lower than the 
 estimate given by Frost in his description of the Bruce spectro- 
 graph, but for some reason the latter is somewhat less efficient 
 in regard to exposure times than the Mills instrument. 1 
 
 Substituting the value of /'(£) given by (129) in (127), and 
 expressing a in terms of <x as before, we obtain 
 
 2r , 
 
 /;„ 
 
 t= sin 
 
 ■ («-*) 
 
 ["(-*>] 
 
 -ft 
 
 ( : 32) 
 
 If, as before, we take the origin of co-ordinates at the point 
 corresponding to the geometrical image of one edge of the slit 
 and consider the most general case in which the center of the 
 tremor disk may be located anywhere within the slit opening, 
 we have as before for £ 
 
 ■Compare result on p. 20 of Frost's paper, Astrophysical Journal, 15, with 
 those given by Campbell, Astrophysical Journal, 8, 141. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 79 
 
 where Act is the amount by which the center of the tremor disk 
 is displaced from the center of the slit. Substituting, this gives 
 
 n„ = 
 
 + Ao-) 2 
 
 -2.854- 
 
 „,=[._(« + : + *,)] 
 
 <ie ■ ('33) 
 
 The only case for which /;„, (133). is symmetrical is that 
 for which A<r = o. The center of intensity is then at the point 
 
 £ =-o- ol and there is no error of displacement. 
 
 In order to find the relation between the error of centering, 
 Ao-, and the resulting displacement Af of the spectral image, I 
 have integrated (133) for different values of Ao- from Ao-= 0.i<r o 
 to A<r = o.5o- . The resulting values for If,, are given in Table IX 
 
 TABLE IX. 
 
 
 
 /, ,. for Different Values of — 
 
 
 a 
 
 
 '// 
 
 O" 
 
 
 a 
 
 
 
 
 
 
 
 Ao = o.io 
 
 Ao = 0.2 o Ao 
 
 = 0.30- 
 
 A7 " D 4(T Aff 
 
 = 0.5(r 
 
 — I. 00 
 
 
 
 
 018 
 
 
 
 — O.9O 
 
 
 
 021 
 
 
 
 - .80 
 
 
 
 034 
 
 
 
 - -70 
 
 
 
 058 
 
 
 
 - .60 
 
 
 
 107 
 
 
 
 - -50 
 
 
 
 191 
 
 
 
 - .40 
 
 
 
 296 
 
 
 
 - -30 
 
 
 
 422 
 
 
 
 — .20 
 
 
 
 550 
 
 
 
 — .10 
 
 
 
 678 
 
 
 
 - -05 
 
 O.948 
 
 
 742 
 
 
 
 ± .00 
 
 ■977 
 
 O.909 
 
 803 
 
 u.677 
 
 545 
 
 + .05 
 
 ■995 
 
 .94S 
 
 85S 
 
 
 74' 
 
 611 
 
 + .10 
 
 1. 000 
 
 • 975 
 
 903 
 
 
 799 
 
 672 
 
 + -15 
 
 0.988 
 
 .986 
 
 935 
 
 
 S44 
 
 727 
 
 + .20 
 
 •959 
 
 •978 [ 
 
 946] 
 
 
 872 
 
 765 
 
 + -25 
 
 .912 
 
 .948 
 
 934 
 
 
 877 
 
 783 
 
 + -30 
 
 
 • 893 
 
 896 
 
 
 855 
 
 776 
 
 + -35 
 
 
 .S17 
 
 832 
 
 
 So 7 
 
 742 
 
 + -40 
 
 
 .724 
 
 746 
 
 
 732 
 
 6S2 
 
 + -50 
 
 
 
 532 
 
 
 534 
 
 519 
 
 -j- .60 
 
 
 
 315 
 
 
 
 + -70 
 
 
 
 150 
 
 
 
 + -So 
 
 
 
 061 
 
 
 
 + -90 
 
 
 
 024 
 
 
 
 + 1 .00 
 
 
 
 016 
 
 
 
80 F. L. O. WADSWORTH 
 
 and the curves for Acr = o.3 <r (full) and Aer = o.l cr and 
 0.5 <j (dotted) are plotted in Fig. 19. 
 
 The positions £ of the points of maximum intensity in the 
 curves I~ n , are found to be as follows : * 
 
 For Ao = o.io OJ £,= 0.59500 Af =■ 0.0950-,, — °- r 9 a o 
 
 Ao = o.2o o; £,=-0.6550-,, A^ ^ 0.1550-0 = 0.3100 
 
 Ao = o.3o , £,= 0.7000-,, ^ = °-2o°«f« = o.4oa 
 
 Ao- = o.4o , £, = 0.7350-,, A| =• 0.23500 = o.47a D 
 
 Ao = o.5o , £=-0.76500 A^^ 0.26500 = 0.53^ . 
 
 The relation between Acr and A£ is plotted in Fig. 20 (full 
 curve). As will be seen by the dotted curve it can be repre- 
 sented very closely by the empirical equation 
 
 In ineasuring the apparent position of the displaced diffrac- 
 tion image the tendency will be, as before, to compromise 
 between the point of maximum intensity, o' , and the point mid- 
 way between the apparent edges, m. m (defined as before) , of 
 the line. The resultant setting, r, for the cross-wire will be at a 
 
 2 
 point not far from - Af . We may therefore write 
 
 2 
 A„,f = measured displacement =■ - A£ 
 
 3 
 
 --»-4(t)' 
 
 (■35) 
 
 In order that the displacement shall not be great enough to 
 affect the accuracy of measurement we must have as before 
 
 or in this case from (14) and (135) 
 
 o.o?ao. 5 sJ^j , (136) 
 
 or Ao 
 
 — = -°4 • 
 
 i. e., the star image must be centered on the slit with an error 
 not exceeding 4 per cent, of the slit width. 
 
 'The points of maximum lie between the numbers in heavy type in the table. 
 
OPTICAL CONDITIONS OF ACCURACY 
 
 Si 
 
 This is a condition which has apparently never before been 
 recognized as necessary to accuracy in the use of the slit spec- 
 troscope. Care has of course always been taken to keep the 
 star image as nearly central as possible in the slit opening, but 
 the object aimed at has been simply to utilize the greatest amount 
 of light and shorten the time of exposure as much as possible. 
 But the above investigation 
 shows that the requirements 
 of accuracy and the avoid- 
 ance of constant errors re- 
 quire a far greater degree 
 of care in centering and 
 following than is usually 
 deemed necessary, greater 
 in fact than many of the 
 devices now used for follow- 
 ing make possible. 
 
 This question is one of such great importance in view of the 
 minuteness of the displacements upon which determinations of 
 motions in the line of sight depend, that it may be interesting to 
 calculate the error in kilometers per second produced by an 
 
 error of say 10 per cent, in the centering of the star image on 
 
 Ao- 
 the slit. From the previous results we find, for — =o. 10 , 
 
 A ra f=°-i3< 1 o • (137) 
 
 Since the angular resolving power a a is the same for both 
 
 collimator and view telescope, we can readily establish a rela- 
 
 dd 
 tion between a ol — = the dispersion of the spectroscope train, 
 
 and v s the motion in the line of sight of the body under exami 
 nation. From the well-known relations 
 
 f k f- X 
 fa < = b f —p 
 
 and 
 
 Aa = 
 
 ,/e 
 Ik 
 
S2 
 
 F. L. O. WADSWORTH 
 
 we get at once 
 
 _A| V_ 
 v '—d9'T 
 
 dk 
 
 and for the value of A,„£ assumed in (137) 
 
 0.13F o. 1 3 V 
 
 bV s : 
 
 ,d0 
 
 b Tk 
 
 3S0OO 
 
 R 
 
 R 
 
 km per sec. , 
 
 (138) 
 
 where R is the spectroscopic resolving power of the prism train. 
 The following table contains the approximate resolving 
 powers of the more important instruments that have been used 
 in line of sight work and the corresponding error 8v s produced 
 by a 10 per cent, error of centering of the star disk from (138): 
 
 TABLE X. 
 
 Name of Instrument 
 
 c ... . Approximate R 
 Spectroscope 1 rain lor Hy 
 
 Kilometers 
 per Second 
 
 Potsdam 1 1 (Vogel) 
 
 j about 
 f- 2 Compound Prisms , 40,000 
 
 about 
 
 Pulkowa ( Belopolsky) 
 
 0.9 
 
 Bruce 1 (Newall)' 
 
 
 Mills (Campbell) 2 
 
 3 " Prisms 74.000 
 
 2 " " J2.000 
 
 0.5 
 
 McMillin (Lord)3 
 
 0.9 
 
 Potsdam III (Vogel)* 
 
 Bruce 1 1 ( Frost )S 
 
 3 " 
 
 3 " 
 
 63,000 
 1 10,000 
 
 0.6 
 0.2 
 
 
 
 An examination of this table shows that even for this small 
 error in centering the errors in the resulting; determinations of 
 v s are sufficiently large to account for a large part of the differ- 
 ences of measured velocities between some of the different 
 observers. When we remember the difficulty of keeping a faint 
 star accurately centered on the slit, and remember, also, that any 
 constant asymmetry of the tremor disk itself due to peculiar 
 instrumental conditions will notably increase the effect of a given 
 displacement from the center, wc can readily see that we have 
 here what may be a constant source of error of the most serious 
 kind, and in some cases of even greater magnitude than indicated 
 in the above table. If, for example, the error of centering were 
 
 1 AsTRoriiYsiCAL Journal, 3, 266. 3 /<W., 4, 50. 
 
 *Ibid., 8, 123. *lbid., n, 393. s //,/,/., I5> 1. 
 
OPTICAL CONDITIONS OF ACCURACY 83 
 
 for any reason constant in direction with reference to the large 
 telescope, any reversal of the spectroscope with reference to the 
 latter would introduce differences in observed positions of the 
 lines, and hence of v s of twice the amount indicated in Table X. 
 As this error is purely one of manipulation, we cannot in 
 general make any correction or allowance for it after the photo- 
 graphic record has been taken. The utmost care, therefore, 
 must be exercised in order to attain the accuracy of centering 
 and following requisite to eliminate it completely, as indicated 
 in (136). To do this we should, as far as possible, fulfill the 
 following general conditions : 
 
 1. Have a sharp, well-defined, and symmetrical image of the 
 star formed on the slit of the spectrograph. If the main image- 
 forming objective is a visual refractor, this will necessitate in 
 general the use of a correcting lens, and this should be so 
 designed as to be free from spherical and chromatic aberration 
 for the region of the spectrum under examination, and so 
 mounted that its principal optical axis coincides with that of the 
 main objective and the axis of collimation of the spectroscope 
 itself. The light from the other portions of the spectrum should 
 be cut out from the following eyepiece by use of suitable screens 
 or screening devices * placed between the slit and eyepiece. On 
 account of the perfect achromatism of the reflecting telescope 
 this form of objective has great advantages over the refractor in 
 this connection; another reason why such instruments should be 
 given the preference for spectroscopic work. 2 
 
 2. The guiding device should use as a reference mark for 
 
 'Such, for example, as the optical color screen described by the writer in this 
 Journal, 3, 169. 
 
 2 The importance and value of the reflecting mirror as compared with the refrac- 
 tor in spectroscopic work have been urged upon the attention of astrophysicists by 
 the writer for many years. See Phil. Mag., July and October, 1894 ; A. and A., Decem- 
 ber, 1894 ; ASTROI'HYSICAL JOURNAL, January, 1S95 ; ibid., March, 1895 ; ibid., March, 
 1896; ibid., May, 1896; ibid., October, 1896; ibid., February, 1897; ibid., February, 
 1898; Pop.Astron., February, 1898, etc. The advantages of the reflector in this line 
 of work are now being more generally recognized, and a number of large instruments 
 of this type are planned or in actual course of construction by our large astrophysical 
 observatories. 
 
84 F. L. O. WADSWORTH 
 
 centering the star image, some point on the slit itself, in order 
 to avoid any errors of parallax or relative displacement of the 
 slit and guiding cross-wires. For this reason I believe a modifi- 
 cation of the Huggins reflecting slit and following device is 
 better adapted to this purpose than any other form yet invented. 
 The use of an auxiliary independent guiding telescope such as 
 is proposed with the new Potsdam instrument would seem to be 
 especially dangerous. 
 
 3. If the required accuracy of following cannot be attained 
 with the usual slit width ff,,=2« ol then the width should be 
 reduced and the time of exposure correspondingly increased, or 
 else greater spectroscopic resolving power should be used. The 
 effect of decreased slit width will be to reduce the limits of 
 integration in (127) and (133), and correspondingly reduce the 
 resultant asymmetry and displacement of the center of intensity 
 of the spectral image. The result of increasing R will be to 
 decrease the value of Sv s in (138) for a given value of Act. 
 
 In this connection it may also be noted that " bad seeing," 
 which results in an increase in size of the tremor disk, will reduce 
 
 Act 
 
 the effect of a given percentage error of centering — by flat- 
 tening the curve // of Fig. 18, and thus reducing the difference 
 between the intensity of illumination at different parts of the slit 
 image. When the "seeing" is "bad," therefore, the slit may be 
 opened wider without increasing the effect of errors of following. 
 B (4). It was the intention of the writer to take up also, in 
 connection with the present paper, the discussion of effects of 
 errors of mechanical design and construction, with reference 
 particularly to the avoidance of flexure and strain resulting from 
 changes in position or changes of temperature of the instrument 
 during use. The present discussion has, however, so far exceeded 
 the limits originally set that it seems better to reserve this part 
 of the subject for a future paper in which the description of the 
 spectrographs and some other astrophysical instruments of the 
 new Allegheny Observatory will also be taken up. 
 
 Allegheny Observatory, 1901-1902. 
 
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