-jp d = JR cos q = 2.99215 -r^- tan z cos q L J 4oo -{- r (I) By applying the fundamental formulae of spherical trig- onometry to the spherical triangle, Pole Zenith Sun, and differentiating the equations, we find : dA = cos o cos q cosec z (2) Eliminating cos q between these equations, we obtain d = [2.992151 ,.f F sec d sin z tan z ^ (3) L J 4oo -\- r at where z, A, s and t represent respectively the zenith dis- tance, azimuth, decimation, and hour angle of the sun. dA The numerical value of ~^- varies with the position of the sun in the heavens, but may be readily determined at any time as follows : Let the horizontal circle of the solar compass or transit be set to read some integral 10' and the telescope be than pointed upon the sun by rotating the in- strument about the lower motion. The sun having been brought into the field of view, the earth's diurnal motion See Johnson's Theory and Practice of Surveying, pp. 47, 48. COMSTOCK STUDIES IN ASTRONOMY 65 will carry the sun across the vertical thread of the instru- ment, and the time at which one edge of the sun is just' tangent to the thread should be noted to the nearest sec- ond upon a watch. Let the instrument be now turned upon the upper motion, keeping the lower motion clamped, in the direction of the sun's movement, and the vernier set at the next integral 10'. The time at which the sun's edge again becomes tangent to the vertical thread should be noted as before. If we represent by n the interval, in seconds, between the two observed times, we shall have : dA _40 dt ~'~ n If desired, the transit may be set so that the second vernier reading is 20', 30', etc. , greater than the first read- ing, and we shall then have: dA 80 120 jj = = etc. and dt n z n s n = \ n z = n 8 . . . . etc. This value of the differential coefficient enables us to express equation (3) in a form adapted to field use, but since for this purpose an error of even several seconds in the value of d is of small consequence, we shall introduce some modifications in the formula which will render it more convenient without seriously impairing its accuracy. o The declination of the sun can never exceed 23.5, and we therefore write in the place of sec d its mean value, 1.051. We also put in place of the temperature r a mean value, 50 F., and assume for the barometric pressure 30 inches of mercury. With these modifications equation (3) be- comes : , _ [3.3854] Fain z tan z n We may put the numerator of this fraction equal to 100 N and tabulate the values of N with the argument the sun's altitude, h = 90 z, as follows: 66 BULLETIN OF THE UNIVERSITY OF WISCONSIN h N h N 10 131* 30 36' 45 14 15 86 40 22 24 9 20 62 50 13 15 6 25 47 60 7 11 4 30 36 70 3 We now helve for the refraction in declination : d = 100 n The altitude of the sun, 7i, should be noted on the verti- cal circle of the instrument to the nearest half degree at the time of determining n. The tabulated values of N correspond to a temperature of 50 F. and a barometric pressure of 30 inches. They may be adapted to any other temperature by diminishing d by one per cent for each 5 by which the temperature exceeds 50, or by increasing one per cent for each 5 be- low 50, but this correction and the correction for varia- tion of the barometer can usually be neglected. At great elevations the barometric pressure becomes so much re- duced that its variation must be taken account of, and this may be done by diminishing d by one per cent for each 300 feet of elevation above the sea. The following examples will serve to illustrate the ap- plication of the formulae above developed. On the after- noon of May 12, 1894, at a place in latitude 43 5' N., lon- gitude approximately 90 west of Greenwich, I took the following observations with an engineer's transit: COMSTOCK STUDIES IN ASTRONOMY 67 Vernier. 170 170 10 Watch. h. m. s. 4 5 30 4 6 27 Vertical Circle = 32 8' d = 58" Vernier. 170 10 170 20 Watch. h. m. s. 4 23 18 4 24 17 Vertical Circle = 28 46' ^=39 By a direct computation from the formula 1 d = 57" cot (S + N) where N denotes the Bessel auxiliary, I find for the refrac- tion in decimation at the time of these observations 59" and 67" respectively, thus showing an agreement far within the limits of error permissible in surveying practice. If, as is often the case, an accuracy of 20" is sufficient, and the altitude of the sun is not less than 10, we may dispense with the tabular values of N and write d = 2000 -r- hn where h is the altitude in degrees and the value of d is given in minutes of arc. The error of this formula in the preceding cases is 7" and 4", respectively. i Chauvenet, Spherical and Practical Astronomy, Vol. I, p. 171. 68 BULLETIN OF THE UNIVERSITY OF WISCONSIN III. DETERMINATION OF THE ANGULAR EQUIVALENT OF ONE DIVISION OF A SPIRIT LEVEL. The methods most in use in this country for the deter- mination of the value of one division of a level require that the level should be attached either to a level-trier or to a telescope provided with a good micrometer. In field astronomy it frequently happens that neither of these aux- iliaries is available and the following method, which in respect of precision is not inferior to either of the others, may be employed with advantage since it requires no aux- iliary apparatus other than a theodolite or engineer's tran- sit. The original suggestion of this method is supposed to be due to Braun. 1 Let the spirit level be firmly attached to a theodolite which is thrown out of a level so that its vertical axis makes an angle of from 1 to 3 with the true vertical. It is prac- tically convenient to so attach the level that the radius of curvature drawn through the middle point of its scale shall be approximately parallel to the vertical axis of the theod- olite, i. e. the level shall be in adjustment. As the theod- olite is turned about its vertical axis the level bubble will run from one end of its tube to the other and back again during a complete revolution of the instrument, and two positions, two readings of the azimuth circle, may be found in which the bubble will stand near the middle of its scale. A small turning of the instrument either way from one of these positions will produce a corresponding small motion of the bubble in its tube, and this turning of the theodolite and resulting motion of the bubble may be made to furnish not only the value of a division of the level, but also a test of the uniformity of its curvature. To determine the relation between the readings of the l Astronomische Nachrichten, No. 2490. COMSTOCK STUDIES IN ASTRONOMY 69 azimuth circle of the theodolite and the readings of the level bubble upon its scale, let the accompanying figure represent a portion of the celestial sphere adjacent to the zenith, Z, and let V and S be the points in which the axis of the theodolite, and the line drawn from the center of curvature of the level tube through the middle of the bubble, respectively, intersect the sphere. The arc SV is the in- tersection with the celestial sphere of a plane passing through S, V, and the center of curvature of the level tube, and if the adjustment of the level above referred to is approximately made, VS may be considered as the inter- section with the sphere of the plane in which the curva- ture of the level tube lies, so that as the bubble moves in its tube its successive positions, when projected upon the sphere will lie along VS, and any position may be identi- fied by its distance from V, represented in the figure by p. Since the bubble always stands at the highest part of the tube, its position, S, and the corresponding value of p are found by letting fall a perpendicular from the zenith upon tne arc VS, and in the right-angled spherical triangle thus formed we have the relation, tan p = tan y cos t where r, as it appears from the figure, is the angle by 70 BULLETIN OF THE UNIVERSITY OP WISCONSIN which the axis of the theodolite is deflected from the true vertical. Since the level tube turns with the theodolite when the latter is revolved in azimuth, while the positions of the points V and Z remain unchanged, it appears that the angle t must vary directly with the readings of the azimuth circle, since it measures the inclination of the plane of the level tube to a fixed plane passing through the vertical axis of the instrument. If we represent by AO the reading of the circle when the arc VS is made to coincide with VZ, we shall have corresponding to any other reading A' : tan p = tan y cos (A A') (1) The value of A in any given case may be determined by finding two positions of the instrument, circle readings Ai and A 2 , in which the bubble stands at the same part of the tube. Since the values of p corresponding to these two readings are equal, we must have: A -A 1 =A S -A and A = ^(A 1 +A t ) If A' and A" denote slightly different readings of the azimuth circle, and &' and b ff the corresponding readings of the middle of the bubble on the level scale, we may write two equations similar to equation (1), and taking their difference obtain: sin (p' p") A' A" . / . A'4-A"\. ,~. , - 7 = 2 sin - jr - sin I A a -- ^ - I tan y (2) cos p cos p 2 \ 2 / Since p' p" is the distance moved over by the bubble, we may write p' p"=(b' &")$, where d is the value of a division of the level, and transform (2) into 2 tany cos*p sin 4 (A' - A") sin L^ ~ * (A ' + sin I- b' - b" In this equation cosPp may usually be put equal to 1, or its actual value may be found from the average value of p given by equation (1). Every other factor in the second member of this equation is known with exception of tan y, and the determination of y will determine d. COMSTOCK STUDIES IN ASTRONOMY 71 For this purpose the instrument should be carefully lev- elled at the beginning of the work and the telescope di- rected at some object, approximately at right angles to the line joining two of the leveling screws of the instrument. Let the zenith distance, ', of this object be determined from readings of the vertical circle taken Circle Right and Circle Left. The vertical axis is now to be deflected toward the object by turning the leveling screws, and the zenith distance of the object, reckoned from the vertical axis of the instrument, z", is to be determined from circle readings in the same manner as z'. We then have, ob- viously, y = z' - z" To make sure that the deflection of the axis lies in the plane passing through the object sighted upon, it is well to note the position of the bubble of that level of the in- strument which is at right angles to the telescope tube. The leveling screws must be so turned that the reading of the bubble of this level on its scale is approximately the same after deflection as before. By comparison with micrometric apparatus, this deter- mination of y and the resulting value of d may seem crude, but with a vertical circle reading to minutes only, the values of z' and z" can be determined within 30", and if y be made 3, d will be determined with a probable error of one part in four hundred, an accuracy quite sufficient for even the most delicate level. The value of y should be between 1 and 3, a coarse vertical circle and fine hori- zontal circle corresponding to the larger limit, and the reverse conditions to the smaller one. To illustrate the method, I select the following partial investigation of the microscope level of a small universal instrument, Bamberg No. 2598. The level was investigated by means of the circles of the instrument to which it was attached, without removing or in any way disturbing it: 72 BULLETIN OF THE UNIVERSITY OF WISCONSIN DETERMINATION OF Y. Instrument. Circle R. Circle L. Levelled ........ 180 26' 49" 358 4' 13* Deflected ....... 179 27 3 359 3 48 = 59' 40.5* z. 91 11' 18" 90 11 37.5 After the level readings which follow were completed, these circle readings were repeated with the instrument deflected and subsequently leveled, giving a second deter- mination of y = 59' 42". I adopt : y = 59' 41' The following are the bubble observations in the de- flected position of the instrument: BUBBLE. CIRCLE. BUBBLE. BUBBLE. CIRCLE. BUBBLE. 25.3-0.8 111 34' - 0.7 25.6 26.3 0.2 291 6' 0.4 26.0 27.7 1.6 24 1.8 28.1 28.1 2.3 16 1.9 27.7 30.2 4.1 14 3.9 30.1 30.2 4.2 26 3.9 29.7 32.6 6.5 111 4 6.4 32.6 32.3 6.4 36 6.2 32.1 34.8 8.6 110 54 8.7 35.0 34.0 8.1 46 8.5 34.2 37.0 10.9 44 10.8 37.0 36.2 10.2 56 10.6 36.5 The observations began with the level bubble at one end of its scale, circle reading 111 34', and the instrument was turned through successive intervals of 10' until the bubble reached the opposite end, when the settings were repeated in the inverse order to eliminate the effect of any slight change in the instrument or level. The instrument was then turned into the position corresponding to the second set of circle readings which were taken with the bubble running from one end of the tube to the other, in both directions. The mean of the four readings of the ends of the bubble corresponding to any circle reading may be adopted as the COMSTOCK STUDIES IN ASTRONOMY 73 corresponding reading of the middle of the bubble, and these mean readings are given in the following table: CIRCLE. BUBBLE. CIRCLE. BUBBLE. T 26 DlFF. 11134' 12.35 291 6' 13.22 8946' 25.57 4.23 24 14.80 16 15.00 89 56 29.80 4.28 14 17.08 26 17.00 90 6 34.08 4.72 111 4 19.55 36 19.25 90 16 38.80 4.18 110 54 21.78 46 21.20 90 26 42.98 4.32 44 23.92 56 23.38 90 36 47.30 Since the bubble readings which stand on the same line in the second and fourth columns of the table are approxi- mately equal, it is apparent that the corresponding circle readings lie on opposite sides of A and equally distant from it. A may, therefore, be determined by taking the mean of any pair of circle readings which stand in the same line, and the angles A A', A" A , which we shall designate by T, may be found by taking half the difference of corresponding circle readings. Values of T are given in the fifth column of the table. The quantities 2b are the sums of the numbers in the second and fourth columns, and their differences given in the last column show that any irregularities which may exist in the curvature of the level tube are very small, and we may determine a mean value of d to be used over the whole extent of the level tube. Since the values of r dif- fer so little from 90, we may assume in equation (3) and taking the differences between the first and fourth, sec- ond and fifth, third and sixth lines of the table, we shall have A' A" constantly equal to 30', and equation (3) be- comes 74 BULLETIN OP THE UNIVERSITY OF WISCONSIN 4 tan y sin 15' [1.7959] ~ 2 (6' - 6") sin 1* " 26' - 26 from which we obtain the following three values d = 4. 72 = 4. 74 = 4. 73 the mean of which may be adopted. COMSTOCK STUDIES IN ASTRONOMY 75 IV. THE SIMULTANEOUS DETERMINATION OF FLEXURE, INEQUALITY OF PIVOTS, AND VALUE OF A LEVEL DIVISION FOR A "BROKEN" TRANSIT. In a "broken" transit, i. e. one in which the rays of light are bent at right angles by a reflecting prism placed in the axis, it is well known that the bending of the axis under the weight which it has to carry produces an effect upon the observed times of transit of a star, which may be represented by the expression f.cos z sec S, where / is a con- stant peculiar to each instrument, and z and 8 denote the zenith distance and declination of the star. Since this ex- pression has the same algebraic form as the corrections for inclination of the axis, and for inequality of pivots, they may all be united into a single term: (V 4- *+/) cos z sec d where (i+f) is a constant correction which must be ap- plied to the value of &' directly determined with the spirit level. If i +/ is positive for Ocular West it will be nega- tive for Ocular East, and the sign is, therefore, prefixed to it. Since it is not necessary in the use of a broken transit to separate the constant correction i+f into its constituent parts, it will for the present be treated as a single unknown quantity whose value /? is to be determined in connection with T, the angular value of a half division of the level used for measuring &. In a straight transit / is zero, but i has usually an appreciable value and the cor- rection ft must, therefore, be determined, and may be con- veniently determined by the method here developed for a broken transit. If from the general equation of the transit instrument 1 sin c 4- sin 8 sin n cos d cos n sin (r m] = (1) i Chauvenet, Spherical and Practical Astronomy, Vol. II, 123. 76 BULLETIN OF THE UNIVERSITY OF WISCONSIN the quantities m and n be eliminated by means of the rela- tions (78), l we have the following: sin c + cos z sin b sin z cos b sin (a + A) = (2) where 90 a and b represent the azimuth and altitude of the point in which the rotation axis of the instrument, produced toward the west, intersects the celestial sphere. A and z are the azimuth (reckoned from the north toward east) and zenith distance of a star at the instant of its transit over a thread whose collimation is c, i. e. the point 90" a, b is the pole of the small circle traced upon the celestial sphere by the thread in question when the instru- ment is rotated about its axis, and the distance of this circle from its pole equals 9(P + c. Since in practice b and c are never so great as 10', equa- tion (2) may be written without sensible loss of accuracy : c + cos z . 6 = (a -J- A) sin z (3) Substituting in this equation for b its value as given by the spirit level, and writing a similar equation for the case in which the object observed is not the star, but its image reflected from mercury or some other level surface, we have: Dir. c' + cos z' (n' T -f- /3) = (a + A') sin z' (4) Eef. c" cos z" in" T + /3) = (a +A") sin z' where n' and n" are the measured inclinations of the axis expressed in half divisions of the level scale. We now put z' = z + x z" = z x and introducing these values into(4) find by substraction : c' c" + (n 1 + n') cos x cos Z.T -f- 2 cos x cos z fi = (A' A"} cos x sin z + (2a + A' -f- A") sin x cos z (5) In practice the object observed will usually be a circum- polar star, and owing to its slow motion the quantity x = J (z' z") will be so small that we may assume cos x = 1 sin x = cos S sin t sin ^ ( T T'} where T' and T" are the observed times and t is the hour angle of the star at the instant (T' + T"). * Loc. cit. COMSTOCK STUDIES IN ASTRONOMY 77 For the coefficient of the last term in equation (5) we ob- tain from (it) with sufficient precision 2a -\- A' -\- A* = (c + c") cosec z and introducing these values into (5) we have (ri + w") r -f 2/? = (X A") tan z (c' c") sec z -\- (c' + c") cos S sin t cosec z sin % (T T") (6) If the star is near the meridian or is observed near the collimation axis of the instrument, the last term in this ex- pression will be very small and may frequently be neg- lected. Putting P = (A' A") tan z Q = (c -f c") cos d sin t cosec z sin (T T) we obtain from the equations sin z sin A = cos sin t sin z cos A =. cos q> sin S sin q> cos $ cos t (7) reduced by means of the relations furnished by the as- tronomical triangle, the equation P = cos d cos q sec z . 2 sin | (T - T") 206265 where q is the parallactic angle of the star. Introducing Bessel's auxiliary JVinto this equation, substituting in the last term of (6) in place of cos 3 sin t cosec z its equivalent, sin A, and collecting in a form convenient for computation the equations necessary for the reduction of a series of ob- servations, we have the following: tan N cot (p cos t P = [-5.615161 cosS Sin \ (T - T "^ (8) L J sin z tan (N -{- d) Q = (c' + c") sin A .sin $ (T - T") ( n ' 4. n ") T -f 2/5 = P + Q (c' c") sec z The zenith distance and azimuth of the star, z and A of the formulae, may either be derived from the instrument at the time of observation, or may be computed from the latitude and the co-ordinates of the star, = 43 4' 38* d == 88 44' 53*. 5 Chronometer A T == + 3.9s. Jog [5.61546] cos 6 = 3.95484 OCULAR. WEST. WEST. EAST. EAST. h. m. s h. m. s. h. m. s. h. m. s. T 20 46 20.7 21 5 52.0 21 32 8.2 22 3 54.0 R' 11.172 8.090 15.274 15.720 n' n" + 39.4 + 38.3 - 42.9 - 42.7 - 48.9 - 49.4 + 40.7 + 40.4 T" 20 35 57.4 21 14 16.7 21 40 39.3 21 53 50.7 R" 12.800 6.289 17.431 13.060 t 19 20 17.8 19 49 13.2 20 15 32.6 20 38 1.1 z 46 30 15 46 21 35 46 14 46 8 5 A 1 37 1 32 1 26 1 20 log cos t 9.53558 9.66155 9.74628 9.80355 N 20 9 25 26 7 56 30 48 25 34 13 43 log cosec z 0.13941 0.14045 0.14136 0.14208 logcot(N+d) 9.53463 n 9.66630n 9.75362 n 9.81213 n log sin | (T T"} 8.35530 8.26364 n 8.26911 n 8.34113 log sin A 8.452 8.429 8.400 8.369 log (c' + c") 2.541n 2.955 n 2.193n 1.848 logP 1.98618 n 2.02523 2.11893 2.25018 n logQ 9.348 n 9.648 8.862 8.558 log (c 1 c") 1.97185n 2.01571 2.09405 2.18508 n log sec z 0.16222 0.16107 0.16007 0.15929 (c' c") sec z - 136.17 + 150.24 + 179.52 - 220.99 P - 96.87 + 105.98 + 131.50 - 177.90 Q - 0.22 + 0.44 4- 0.07 + 0.04 80 BULLETIN OF THE UNIVERSITY OP WISCONSIN The preceding computation furnishes the absolute terms of the following equations: + 77.7r + 2^ = +39.08 - 85.6 r + 2/5 = -43.82 - 98.3 r 2/3 = - 47.95 + 81.1 r - 2fi = + 43.13 A least square solution of these equations furnishes the Talues : if jr r = + 0.506 ft = - 0.600 Prom numerous determinations with the spirit level, the inequality of the pivots is known to be i 0".64, which, combined with the value of A gives for the flexure the Talue/= + (T.04. COMSTOCK STUDIES IN ASTRONOMY 81 V. DETERMINATION OF TIME AND AZIMUTH FROM TRAN- SITS OVER THE VERTICAL, OF THE POLE STAR. In a development of the formulas for determining the time from transits over the vertical of a circum-polar star, published in 1828, Bessel says by way of introduction: " That this may not appear futile I remark, what Hansteen and Schumacher have properly noted, that the most ap- propriate use of a portable transit instrument for a time determination consists in mounting it, not in the meridian, but in an azimuth which admits of an observation of one of the polar stars, wherever this may be with respect to the meridian, closely followed or preceded by a transit of a fundamental star." The obvious advantage which this mode of observing possesses lies in the shorter period of time during which the observer depends upon the stability of his instru- mental constants. For meridian observations this period is rarely much less than half an hour, while by the method suggested it need never exceed five minutes. Nevertheless, the general opinion of two generations of field astronomers seems fairly represented by the words of Chauvenet, who, after devoting a score of pages to a discussion of the method, remarks in closing: "The methods which have here been given * * * are intended for the use of ob- servers in the field who have but little time to adjust their instruments and wish to collect all the data possible, re- serving their reduction for a future time. The greater labor of these reductions, compared with those of meridian observations, is often more than compensated by the saving of time in the field. " This greater labor of reduction is now obviated through the simplifications introduced into the method by the Russian astronomer, Dollen, who main- tain s with equal zeal and cogency the greater precision and 82 BULLETIN OF THE UNIVERSITY OF WISCONSIN at least equal convenience of his method for all purposes of field astronomy. Under Dollen's influence the method has, within the last quarter century, come into consider- able use in eastern and central Europe, and from an ex- tended practical application of it the writer of these pages is satisfied of the justice of the claims made in its behalf. This section of the present paper is an attempt to bring to the attention of American teachers of practical astronomy, in substance, the theory of Dollen's method, but it cannot be considered a substitute for the precepts and discussion contained in the elaborate introduction to the Stern Ephem- eriden zur Bestimmung von Zeit und Azimut, published annu- ally by Dollen since 1886. As indicated by the above title, the observations for time are equally available for a determination of azimuth, and reduced to their simplest terms these observations are as follows: Let the transit (universal instrument, or the- odolite, in case a determination of azimuth is also desired) be pointed at Polaris, and the chronometer time, S', at which the star appears bisected by the middle vertical thread, noted. Then revolve the telescope about the hori- zontal axis without disturbing the azimuth of the instru- ment and observe the time of transit, S, of a clock star over all of the threads, and measure the inclination of the axis, &, with a spirit level, if possible both before the observa- tion of Polaris and after that of the southern star. Reverse the instrument, point again upon Polaris, and observe it and a clock star, as before. If the instrument possess a graduated horizontal circle, which is read in connection with the observations of the stars, these data will deter- mine the zero point of the circle, i. e. its reading when the telescope points north, and the azimuth of any terrestrial point toward which the telescope may be directed. We proceed to consider the theory of the method and adopt as a basis for the investigation the fundamental equation of the transit instrument, 1 i Chauvenet, Vol. II, Eq. (79). COMSTOCK STUDIES IN ASTRONOMY 83 sin \in (r m) = tan n tan d -J- sin c sec n sec 8 (1) together with the equations tan n = sin b sec n cosec q> sin m cot
-f- sin
PA = 90 - n ZA = 90 - b P == 90 - m Z = 90 + a The symbol r represents the east hour angle of the star at the instant of transit over the middle thread, and we have obviously the relation r=a-S-JT (4) Since each star observed furnishes an equation of the types (1) and (4\ it appears that if the instrumental con- stants 1} and c are known an observation of the transits of a circum-polar star and a southern star suffice for the de- termination of the unknown quantities AT, m, n, a, and our problem consists solely in so transforming the preceding equations as to facilitate the determination of AT and a. Denoting by the subscripts 1 and 2, respectively, quanti- ties pertaining to the polar and the southern star, we write equation (1) for each of these stars as follows: sin (z-j m 3) = tan d l tan n \ 1 -f cosec S : cosec n sin (c -\- x^) I (5) c i sin (r 2 m 5) = tan S 2 tan n ] 1 -f- cosec d 2 cosec n sin (c -f- #3 where 3, x lt and x 2 are small arbitrary quantities subject only to the condition that they must be so determined as to satisfy the equations. Since this is equivalent to only two relations among the three quantities we are at liberty to impose a third relation, for which we choose sin (c -f- #1) sin S 2 = sin (c + # 3 ) sin t 84 BULLETIN OF THE UNIVERSITY OF WISCONSIN which makes the bracketed factors in the two equations equal. Presupposing that $, x ly and x 2 are small quantities we differentiate equations (5), and eliminating x lt and x 2 find, when quantities of the order en 2 are neglected, ^ = __ (1 - sin d 2 ) c _ cosd 2 sin 6 2 cotS 1 cos (r l m) If for $ 2 we substitute the polar distance, P 3 = 90-S 2 , this equation becomes, very approximately, 3- = c . tan \p z \ I + cot d 1 tan S 2 cos (T m) I (6) Dividing the first of equations (o) by the second, we obtain : ' tan * (,, + r.) --- + * i (r, - r.) (7) We now assume the auxiliary quantities, 2r = (a - S') - (a a - S) U ' = a 2 - S - AT - m - 5 (8) and introducing them into (7) find , , ,.,, sin ($.. + 2 ) . ton(r+ t 7)=- i||(a ' i + a ->tor whose solution is tan U = cot 6 ^ tanS, tin 2r 1 cot d l tan 6 2 cos 2r In equations () dT-\- m is now the only unknown quantity, and to determine m we apply (1) to the polar star and sub- stitute in it the value of tann given by (2) and the value of T^ m given by (4) and (8), and find sin m = cot di tan g> sin (2r -f- U + $) 4~ sin b sec (p + sin c tan
) cot \ 1 cot d^ tan % p 2 cos (2r + C7) j- (14)
If K and M denote respectively the reading of the azi-
muth circle corresponding to the star observations, and to
that position of the instrument in which the rotation axis
lies in the plane of the prime vertical (collimation axis in
the meridian), we have, obviously,
M = K + a' + b tan cp + C'c (15)
where C' is an abbreviation for the coefficient of c given in
the preceding equation.
Since the collimation constant, c, changes sign when the
instrument is reversed, an observation of Polaris and a
southern star in each position of the instrument, W. and
E., will suffice for the determination of 4 T and c from the
observed times of transit, and also, if the instrument is
provided with an azimuth circle, for the determination of
M and c, from the circle readings. The agreement between
the two values of c thus determined furnishes a valuable
control upon the accuracy of the observations and their
reduction.
In the preceding investigation the effect of flexure, ine-
86 BULLETIN OF THE UNIVERSITY OF WISCONSIN
quality of pivots and diurnal aberration has been neg-
lected. These quantities may, however, be taken into ac-
count, as in the case of meridian observations, by applying
to the observed level constant, &, a correction, p, for
the first two sources of error, and by applying to S a cor-
rection,
s.
- 0.021 cos (p. C
for the aberration.
The formulae requisite for the reduction of observations
in the vertical of the pole star may now be collected,
slightly simplified and arranged as follows :
Data known independently of the observations:
8.
cot 8 t sin (t -f U}
tan a' = tan m' cosec (p
A T + Cc = sin (t -f- U")
log a' == log (15 cosec q> ) + log m' -\- 2 6 (m') 26 (a')
In equations (16) the quantities h, I, C, C' are analogous
to the transit factors A, B, C used for the reduction of me-
ridian observations, and (7,0" may be tabulated for a given
latitude and assumed constant for a period of several years.
The quantities U and m' must be computed anew for each
observation, and a' must also be computed in case the azi-
muth is required. To diminish the labor of this computa-
tion Dollen tabulates for a selected list of 180 stars certain
General Constants, through which these computations are
considerably shortened.
With assumed values of the coordinates of the stars and
an assumed interval S 8' 4 m put
TT i 206*265 __
U = x -\ cot d sin (t-}- U) = N Q
10
We shall then have
- (U+ m'} = x + p N = t a' = p' N
where p and p' are functions of the latitude which differ
from tan + 6 ( N tan
The values of 0,r *-,/* andD involve only the coordinates
of the stars and are given among the general constants for
each star of Dollen's list.
The values of fc, G and Ad are as follows:
A a = a 2 - (or 2 ) g = - a^ - (ajo -
G -
Ad = S 2 - (5,) fc = - S -
where the subscript denotes the tabular values of the co-
ordinates corresponding to x , N^. These assumed values
are given as a part of the table of constants for each star,
and an ephemeris of g and log k precedes the table of con-
stants.
The actual reduction of a set of observations by means of
these general constants will not often be made, but recourse
will be had to the General Ephemerides constructed from
them for 93 of the 180 stars. These ephemerides give at
intervals of ten days throughout the year the instantaneous
values of ^V and T, T = a% -f- x, and from them the observer
should construct a local epherneris of the values of and
a' for a few of the tabular dates near the epoch of his ob-
servations, using the relations
9 = T + pN a' = p'N
Values of and a' interpolated from the ]ocal ephemeris
will be immediately available for the reduction of observa-
tions in which the observed interval S S' equals the
90 BULLETIN OP THE UNIVERSITY OF WISCONSIN
value 4 m assumed in the computation of X Q and N Q The ob-
servations should be so arranged as to secure at least a
rough approximation to this interval between the observa-
tion of Polaris and the clock star, but a deviation of even
f
several minutes from the prescribed amount may be very
simply corrected.
Since the interval S S' affects U, m' and a' precisely as
does ^ a 2 whose effect is represented in the term EG,
we apply to S and K the corrections
R J S - (S' + 4) ^' j S - (S' -f 4m)
and the reduction of the observations takes the very sim-
ple form :
r
8 = S + R r + 6 - CH AT Cc = 6 - S
K = K -f- R' Q r + B'b - C'x M T C'c = # - a'
The level corrections Bb, B'b are most conveniently taken
from a table of multiples of
- sec 9.97082 log cosec in m'
8.29440
8.27372
AT Co
- 14.08
- 14.28
6
3
3
M T C'c
332 59 16.4
152 59 16.9
tan a'
8.46008
8.43939
s. "
AT + 1.47 c = - 14 08 M - 20 7 c = 16.4
AT - 1.51 c = - 14.28 If + 20.7 c = 169
s.
AT = - 14.18 J/ = 16.6
c= + 0.07
c = 0,01
COMSTOCK STUDIES IN ASTRONOMY
93
1891, SEPTEMBER 4.
logp = 9.97085 log p' = 1.31247
Star Oc.
s Cygni W.
C Cygni E.
Equations:
R R
-0.196 -14.85
-0.350 -19.85
s.
C C'
1.471 20.46
1.513 20.42
JT+1.47c = - 14.08
B
20 43 8.79
21 10 0.06
T- 1.51 c = - 14.29
b
- 7.2
+ 5.1
AT = - 14.18
S' + 4
20 42 56
21 9 39
c = + 0.07
8
20 43 23.59
21 10 14.03
M" - 20. 5 c = 16.3
CK. R Q r
- 0.02 - 0.05
- 0.02 - 0.12
Jlf" + 20.4c = 16.7
b sec cos it -|- cos g> sin it cos (T + r)
(1)
= sin (p cos p + cos == cot $ (p -{- K) cos T cos r cot (p TT) sin T sin r (2)
We introduce into this equation the auxiliaries
I cos A = cot ^ (p -}- 7t) cos r I sin A = cot ^ (p ft) sin r (3)
and obtain
lcos(T- A) = tan = y. z/ (p = - 2 cosec 2 ?> cof N. A }1 *.*i O
placed by an expression which is most conveniently treated
in connection with the thread intervals.
98 BULLETIN OF THE UNIVERSITY OF WISCONSIN
The southern star, and occasionally the polar star, will
be observed on several threads, and from the several ob-
served times the time of transit over the middle thread
may be found by Bessel's method, 1 or as follows: The re-
duction of any thread to the middle thread is given by the
equation
'
m . , tf sec & cosec A z cot A 2
= T + i sec g> cosec A 2 + - -- : - -? - 2 (15)
2 sin z tan q
where q is the parallactic angle of the star when on the
middle thread.
When the star is observed at its transit over the almu-
cantar passing through the pole, we have rigorously
q = t z = 90 q>
and since the last term of (15) is very small we may in
most cases substitute these approximate values in it.
From the observations on the first and last threads we ob-
tain, approximately,
/ = sec q> cosec A% = (T" T) -t- (i" i') (16)
Applying (15) to each observed thread and taking the
mean of the resulting equations, we obtain
The last term rarely amounts to more than a few hun-
dredths of a second, and if the star observed is near the
prime vertical, or near elongation, it may be neglected. It
should be noted that owing to the factor cosec A 2 , f is posi-
tive for stars west of the meridian and negative for stars
east of the meridian.
Effect of Diurnal Aberration. The effect of the diurnal ab-
erration is to displace every star toward the east point of
the horizon by the amount
s.
D = 0.021 cos = 0.3020
Hourly rate of chronometer = 0.04
COMSTOCK STUDIES IN ASTRONOMY
101
Star.
C Leo.
49 Here.
ju Here.
S- Leo.
Retfn to Middle Thread
s. s.
+0.25 +0.00
s. s.
-0.27 -0.02
s. s.
+0.25 +0.00
s. s.
-0.27 +0.02
Chron. Rate. Level
4-0.01 +0.54
+0.01 +0.40
0.00 -0.07
-0.01 +0.07
T*
13 40 44.04
13 50 4.68
14 1.65
14 8 19.53
a z
10 10 49.05
16 47 17.55
17 42 20.71
11 8 42.21
* - T &
-3 29 54.99
+2 57 12.87
+3 42 19.06
-2 59 37.32
<*i ~ TI
+5 58 20.6
+5 44 7.6
+5 33 21 6
+5 25 3.6
t
217 56 6
318 16 20
332 14 22
233 49 47
P
66 3 13.9
74 51 7.6
62 13 18.7
73 59 31.1
log sinp
9.96091
9.98464
9.94682
9.98283
log cos t
9. 89692 n
9.87292
9.94689
9. 77099 n
log a
8.29400
8.27027
8.30809
8.27208
log sin t
9. 78871 n
9. 82321 n
9. 66818 n
9.90702 n
co log (1 a cost)
9.9933111
0.0060815
0.0078831
9.9952305
log cos M
9.9999692
9.9999657
9.9999798
9.9999516
H
-0 40 57.1
-0 43 13.9
-0 33 8.7
-0 51 21.2
X
51.3
43.6
55.3
44.3
log tan (p x)
9.8128460
9.8837244
9.7805448
9.8769541
\ cos M -T- (1 a cos t) I
9.9932803
0.0060472
0.0078629
9.9951821
log cos N
9.7770484
9.8606937
9.7593298
9.8430583
N
+53 14 20.4
-43 28 54.2
-54 55 54.1
+45 50 8.0
M+N
+ 3 30 13.55
- 2 56 48.54
- 3 41 56.19
+ 2 59 55.12
log cot N
9.8733
0.0230 n
9.8463 n
9.9873
C
+1.497
-2.113
-1.407
+1.946
s.
s.
s.
s.
AT+C A cp
+18.56
+24.33
+22.87
+17.80
CAq>
- 2.32
+ 3.27
+ 2.18
- 3.02
AT
+20.88
+21.06
+20.69
+20.82
102 BULLETIN OF THE UNIVERSITY OF WISCONSIN
From the equations
AT+ 1M Aq> = 18.56 v = +0.02
AT-Z.\\A(f> = 24.33 + .20
AT- 1.41 Aq> = 22.87 - .17
jr+1. 95 ^ = 43 4 36.7
From Comparisons = +20.80 = 43 4 36.5
This excellent agreement is due, at least in part, to the
reversal of the instrument, one-half of the observations
having been made Circle Right and one-half Circle Left,
thus eliminating the effect of error in the assumed thread
intervals.
In order to secure the convenient observation of stars it
will be advantageous to prepare in advance an observing
programme showing the time at which the several clock
stars cross the almucantar of the polar star, and their cor-
responding azimuths. If only a few stars are to be in-
cluded in the programme this can be most conveniently
done by putting T 2 = T^ in equations (13) and solving with
COMSTOCK STUDIES IN ASTRONOMY 103
four place logarithms the following approximate equiva-
lents of those equations :
a sin t
t = or 2 a tan M =
a = sin TT cosec j) cos N =
1 a cos
tan CD tan 4
\-acost
T! = T 2 == a 2 - AT + M + N
When the sidereal times ^ and T 2 are known the zenith
distances and azimuths of the stars may be directly com-
puted from the fundamental formula for the transforma-
tion of coordinates, but the following method will usually
be found more convenient:
In the spherical triang]e formed by the polar star, the
zenith and the pole, we represent the east hour angle of the
star by r and find
cos z = sin cp sin d^ -f- cos (p cos d^ cos T
= cos (d 1