-m- UNIVERSITY .OF CALIFORNIA. Receded ^fytu _', i Accessions No. ^^>^^/ ' Shelf No. AZIMUTH. A TREATISE ON THIS SUBJECT, WITH A STUDY OF THE ASTRONOMICAL TRIANGLE, AND OF THE EFFECT OF ERRORS IN THE DATA. BY JOSEPH EDGAR CRAIG, LIEUTENANT-COMMANDER, U. S. NAVY. NEW YORK: JOHN WILEY & SONS, 15 ASTOR PLACE. 1887. COPYRIGHT, 1887, BY JOSEPH EDGAR CRAIG. CONTENTS. PART I. FAGBS INTRODUCTION ------- . - - - ._.- i_6 PART II. DEDUCTION OF FORMULAS. FOR THE DETERMINATION OF AZIMUTH. Altitude-azimuth 7-8 Time-azimuth ---__-____---- -_. 8-9 Time-altitude-azimuth ---____-_.____- 9-10 Horizon-azimuth ----------------- 10 Time-altitude-latitude-azimuth ------------- 10 PART III. DIFFERENTIAL VARIATIONS IN THE ASTRONOMICAL TRIANGLE WITH REFERENCE TO AZIMUTH. General remarks upon errors and changes - _-.-__-.-- 11-12 Deduction of differential equations in the problems of Altitude-azimuth ---------------- 12-13 Time-azimuth --..--_-___--_-_ 14-15 Time-altitude-azimuth - - - - - - - - - - - - - - -15-16 Horizon-azimuth ---------------- 16-18 PART IV. CONSIDERATIONS AFFECTING THE EQUATIONS OF MAXIMUM AND MINIMUM ERRORS, AND RESPECTING THE CURVES OF THESE ERRORS. Method of reckoning the angles /, Z, and q __.___. 19-20 Remarks on the terms of the equations, co-ordinates, and diagrams ------- 20-21 Explanation of the terms maximum and minimum as used in this treatise 21 dZ dZ dTj Tabular statement of equivalent expressions for the value of -JT, r , -rr, etc., in the several azi- muth problems ---------------- 22 Other useful equations --.-___-__----- 23 PART V. DETERMINATION, BY INSPECTION, OF THE MOST FAVORABLE AND THE LEAST FAVORABLE POSITIONS OF A GIVEN BODY FOR OBSERVATION IN A GIVEN LATITUDE. Altitude-azimuth - __.__-___-- 25-26 Time-azimuth ____.__--____---- 26-27 Time altitude-azimuth - - _____-____-_-- 27-28 Remarks on the algebraical determination of the expressions of maximum and minimum errors in the computed azimuth -__-___-- _____ 28-29 Remarks on tracing the curves and a summary of forms of equation to the loci 30 iv CONTENTS. PART VI. EQUATIONS TO THE LOCI OF MAXIMUM AND MINIMUM ERRORS IN THE COMPUTED AZIMUTH DUE TO ERRORS IN THE DATA. PAGES Locus No. i. Altitude-azimuth error in h -'.-'.*,_* - 31-34 Formulas of transformation to obtain the equations of the projected curves - - - - - 34-36 Locus No. 2. Altitude-azimuth error in/,------------ 36-37 Locus No. 3. Altitude-azimuth error in co - i |B/a Z ^ c * From the fundamental formula of trigonometry, FIG. i. FIG. 2. cos b = cos a cos c -f- sin a sin c cos B, we have ) sin d = sin h sin L -f- cos h cos L cos Z f sin d sin h sin L cos Z = -. = ; (2) cos h cos L ^ ' 8 AZIMUTH. from which are derived cos s cos (s p) 1fr f \ when * = i(L + *+t): (3) /cos s sin (s d ) = A /- -v when j = #co-L + h-\- d) ; ... (4) y sin co-L cos A; /sin (s L) sin (s K) taniZ = \/~ rr^ when s = $(L -4- h -\- p). ... (5) y cos s cos (sp) Small errors in the data will have the same effect on the computed azimuth, whichever formula is used. So far as inexactness in the logarithmic tables is concerned, formula (5) will give the result nearest to precision, since the tangent of an angle varies more rapidly than either the sine or the cosine. \iZ~> 90, (3) will be better than (4); if Z < 90, (4) is preferable to (3); since the cosine varies more rapidly than the sine for an angle greater than 45, and less rapidly for an angle less than 45. 19. TIME-AZIMUTH. Given /, L, and d, to find Z. From the fundamental formula, we have sin A cot B = sin c cot b cos c cos A sin t cot Z = cos L tan d sin L cos t cos L tan d sin L cos t cot Z = : ; (7) cm / v/ sn from which are derived tan = cot d cos / ; cotz= cos(0+)c gi-'- sin *: i, - sin(0 + L) sin d For the solution of an astronomical bearing the altitude of the heavenly body not be- ing observed formula (10) will be needed for finding the true altitude, from which the required apparent altitude will be determined. It will be convenient to accept the value of less than 90, taking care to give it the proper sign. There will be no ambiguity in the value of Z found from its cotangent. If the latter is positive, the azimuth is less than 90 ; if negative, greater than 90. AZIMUTH. 9 The following formulas, derived from (7), are given in some text-books to the exclusion of the preceding: Case /.When 5 < 90 S = l>(p + co-L) .......... , . (11) D = \{p ~ co-L), the greater minus the less;. . . (12) tan X =. sin D cosec 5 cot \t; ..... .... (13) tan Y = cos D sec 5 cot \t ; .......... (14) Z = X Y. ........... . . . (15) llp>coL,X+Y=Z ........... (16) lip 90 : (.r- F) = Z. ........... (18) These formulas are convenient when a series of observations is to be taken, as in the case of serial time-azimuths, therefore convenient for preparing a table of azimuths ; because sin D, cos D, cosec S, and sec S have constant logarithms (in the case of the sun regarding the declination at the mean of the times of observation as constant) ; but for a single observation they give more labor than (8) and (9) exact, and some trouble in freeing the result from ambiguity, whereas (9) is perfectly clear. If, however, besides Z the angle q were required to be computed, formulas (11) to (14) would be the most convenient. If / > coL, for (12), write D = % (p coL) ; for tan X, write tan % (Z q) } whence sum = Z; for tan F, write tan \(Z-\- q) f whence difference = q. lip < coL, for D, write (coL /) ; for tan X, write tan (q Z}, ) whence sum gives q ; for tan F, write tan %(q-\- Z}, } and difference gives Z. 20. TIME-ALTITUDE-AZIMUTH. Given /, h, and d, to find Z\ sin a sin B = sin b sin A | cos h sin Z = cos d sin / ) sin t cos d . , sin Z = -- : .............. (20) cos li ^ ' io AZIMUTH. Of the two values of Z given by (20), supplements of each other, the proper one may be known from the conditions of the case. If Z is very near 90, and it is doubtful on which side of the prime vertical the body lies, the ambiguity may not be removed ; but the method is inadmissible then, since a small error in any given part may make Z impossible, sin Z being greater than unity; and in any event such error will produce in Z a. very large error (see art. 14). 21. HORIZON-AZIMUTH. Given h = o, d, and L, to find Z: sin d From (2), cosZ:= ^o7Z Attending to the sign of d, no ambiguity can arise. 22. A fifth method, which the writer has never seen alluded to, may be employed. It may be termed TiME-ALTITUDE-LATITUDE-AZIMUTH, to distinguish it from that of art. 20, which, to be precise, should be called Time-altitude-declination-azimuth. Given t, h, and L, to find Z. This method may be resorted to in the remote contingency of declination not known, as, for instance, the mutilation of the ephemeris so far as concerns the declination only. From trig., sin B cot A = sin c cot a cos c cos B\ ) , . sin Z cot / = cos L tan h sin L cos Z; ) whence cot $ = tan A cos c ; \ cos $' = cos $ tan c cot a ; V ..... . ... (23) B = & & ; ) and cos Z = Attending to the signs, there will still be two values found for Z. But, sine* negative values of Z and values greater than 180 are excluded (art. \7\ the proper value will be known. When d is given it will be more trustworthy than is usually the Z., hence this method is not considered farther, though having a place in the problem of azimuths. cot L tan h ; ......... (3, PART III. DIFFERENTIAL VARIATIONS IN THE ASTRONOMICAL TRI- ANGLE WITH REFERENCE TO AZIMUTH. 23. The error in the computed azimuth arising from small errors in the data may be trans- lated into the change in the value of the azimuth corresponding to small changes in the data. Since with three parts given in the astronomical triangle and not less than three parts the remaining parts can be found, we have in the problem of azimuths four parts in each of the methods proposed. In order that the astronomical triangle shall change its aspect, not more than two parts can remain constant all the others must vary.* In the fundamental formulas employed, differentiating for any two parts of the triangle as variable, the other two being constant, the change in one variable may be found in some terms of the other variable and the two constants, and in a simpler form, sometimes, by employing parts of the triangle that do not enter the problem. For the investigation of maximum and minimum effects of errors, these equivalent expressions may be used with advantage. 24. In this problem, since Z is regarded always as one of the variables, we shall find the expression for the approximate error in Z corresponding to a small error in each of the given parts, taken separately, the other two parts being regarded as constant for that occasion. The expression for the total error in Z will be, for practical purposes, the algebraic sum of the three errors thus found ; that is, the total differential equals the sum of the partial derivatives. 25. In each case the expression for error depends on the particular parts of the triangle used, both the variables and the two constants ; the remaining two parts, though they do not enter the formula, vary with the variables employed. For illustration, the error in the computed azimuth owing to a small error in latitude will not have the same value when the method of time-azimuth is employed that it will have in the altitude-azimuth computation excepting as stated in art. 13, for one point of obser- vation of the celestial body ; namely, when either q or Z equals 90, whichever may be possible ; at e or / the numerical values of the resulting errors will be the same by both methods, but the signs will be contrary. Therefore, in practice, having determined the azimuth, and afterwards finding that the latitude used is slightly erroneous, care must be taken in correcting the result that dZ is not computed from the expression derived from the formula of the method not used to compute the azimuth. The foregoing applies as well to the case of error in declination ; but in practice the occasions will be few when the declination can be regarded as having any appreciable error. For discussion, however, a sensible error in declination affords interesting study, and it will not be omitted in showing the relations of the errors in the data. Though the expressions for the errors are not strictly true excepting when the incre- * Except in the one peculiar case of two sides and the angles opposite them remaining constantly equal, each to 90; while the remaining side and its opposite angle are the only variables, varying by the same amount, being always equal each to the other, in angular measure. 12 AZIMUTH. ments of change are infinitely small, yet they are nearly enough true provided the change in circular measure is sufficiently small to be regarded equal to its sine ; therefore, since an angle that does not exceed one degree may be regarded equal to its sine to insure accuracy to the nearest second in the result, the small errors in the data usually met in practice may be dealt with. dZ may, "however, fall beyond the limits allowable as a finite difference for numerical use as a correction. 26. The increment of change is really length in arc. If given in seconds or minutes it may readily be reduced to circular measure, the radius being unity ; but, in practice, this need not be done, for, the error sought being expressed in the same unit as the given error (in seconds or minutes), the factor for reduction common to both will divide out. 27. The expressions are often useful to find the actual value of the error in a result aris- ing from the employment of erroneous data ; or, speculatively, to determine the restrictions to be imposed on probable errors in the data, in order to keep within the limits of error allow- able in the result. The expressions derived are for errors; hence the sign must be changed to make them corrections to be applied to the computed azimuth. The most important use, however, for these expressions is their application to the in- vestigation of the best and the worst conditions for observation of the celestial body. 28. For convenience, before deducing any equation of maximum and minimum errors in the computed azimuth, we shall derive all the expressions for the resulting error arising from errors in the data taking each datum separately as variable, the other two given parts as constant in the several methods of ascertaining the azimuth. It will be convenient to adhere to the familiar common notation of spherical trigonometry to facilitate the changing of expressions into others equivalent until the result for each error in the data is reached. Therefore we shall have with the astronomical triangle a companion triangle, employing the latter to obtain the partial derivatives (see Figs. I and 2, art. 18). In the final ex- pressions it is deemed unnecessary to replace dZ, dh, etc., by AZ, Ah, etc., to represent finite differences. 29. I. ALTITUDE-AZIMUTH. Given h, L, and d, to find Z. 1st Case. Error in Z owing to error in h : Z and h (B and d) variable ; L and d (c and b) constant. cos b = cos c cos a -|- sin c sin a cos B (25) o = cos c sin ada-\- sin c cos a cos B da sin c sin a sin B dB ; sin c sin a sin B dB = (sin a cos c sin c cos a cos B] da ; = sin b cos C da ; , D sin b cos C art = ; : : = da ; sin c sin a sin B , D cot C sin b sin c <*& = aa, since -: ~ = >=.; sin a sin B sin C AZIMUTH. 13 - *) ; . ' ' I. 2d Case. Error in Z owing to error in L : Z and L (B and c) variable ; h and d (a and b} constant. By interchanging c and a in the preceding case we derive, in the same way as in the ist case, dL ....... (27) ^ tan t cos L I. ^d Case. Error in Z owing to error in d\ Z and d (B and b) variable ; h and L (a and c] constant. From (25), sin b db = sin c sin a sin B dB ......... (28) JD sin b J, dB = -. ---- : -- : 5 do ; sin c sin a sin B sin b sin a .'. since -. ~ -*, sin B sin A . . sin c sin A -- j. - cos L sin t A ^ cos Z, sm t Total error, dZ = dZ K -f- dZ L -\- dZ & ; ^ : dd ............ (.29) dh + - T 1 - - dL - -^-- l - dd ..... (30) - - -- T - T - - T - -^-- - tan q cos h tan t cos L sm / cos L 14 AZIMUTH. 30. II. TIME-AZIMUTH. Given t, d, and L, to find Z. 1st Case. Error in Z owing to error in t: Z and t (B and A) variable ; L and d (c and b) constant. sin A cot B sin c cot b cos ccos A ......... (31) cos A cot B dA sin A cosec" B dB = cos c sin ^4 a^4 ; multiplying by sin B, cos A cos ^ ^4 sin A cosec .5 dB = cos sin A sin .# L4 ; cos A cos ^ cos c sin ^4 sin B , dB - - : -f ^ -- dA ; sin A cosec B cos C dB -- : sm A cosec B cos C sin .5 . cos C sin , dA ; . - sin A sin cos ^ cos d dZ t = . dt ............... (32) cos h II. 2d Case. Error in Z owing to error in L: Z and L (B and c] variable ; / and d (A and b) constant. From (31), - sin A cosec 2 BdB = cos c cot b dc + sin c cos A dc ; A i DJD cos c cos b -4- sin < sin c cos y2 , sin A cosec Z&/Z? = -- ! -- - -- dc ' sm <7 sin A cos # ~slr?"^^ : = lirr^^ ; .......... (33) sin* sin* sin^ sin* trig., . ' ........... (34) AZIMUTH. 15 . . multiplying the first and second members of (33) by the first and second members of (34), respectively, dB = cot a dc ; sin B dB = cot a sin B dc ; dZ = tan // sin Zd (go L) ; dZ L = tan h sin ZdL .............. (35) II. $d Case. Error in Z owing to error in d\ Z and d (B and b) variable ; L and / (c and A) constant. From (31), sin A cosec 2 B dB = sin c cosec 2 b db\ multiplying by sin B sin b, sin A sin b sin c sin B - . 5 dB = - : 7 do ; sin B sin b sin A sin a sin c sin C by trig., - = = ~^ /< and -: -, = -. D ; sin ^ sin sin b sin /? ' /. sin 0d!Z? sin C db ; cos ^ dZ = sin ^ ; cot d . ' . from (49), dZ d cot d cot Z dd. ............. (50) Total error dZ = tan L cot Z dL cot L, at the upper transit on the meridian Z passes through o, from Z < 360 to Z > o ; increases to a maximum value < 90, when q = 90, then decreases to \ f- > at lower transit ; ( 3 ) from 360, decreases to a minimum value > 270, when q = 270, then increases to j ^ [at upper transit. If d < L, Z at upper transit passes through 180, from^> 180 to Z <, 180; de- creases through 90 to \ ^ o f at lower transit ; from 360 decreases through 270 to 180 ( 3 ) at upper transit. If d^> L, Z at upper transit passes through 180, from ^> 180 to 2T< 180; de creases to a minimum > 90, when q = 90, then increases to 180 at lower transit ; increases to a maximum < 270, when q = 270, then decreases to 180 at upper transit. q will be reckoned to a possible 360 as follows : If -f- af> L, q = 180 at upper transit ; decreases through 90 to < ,- [ at lower transit ; from 360 decreases through 270 to ( 3 ) 1 80 at upper transit. If d < L, q o at upper transit ; attains a maximum value < 90, when Z 90 then decreases to \ ,- [at lower transit; from 360 decreases to a minimum > 270, when ; ( 3 > Z = 270 then increases to j 3 a t upper transit. If d> L, q o at upper transit, increases, passing through 90, to 180 at lower ( 360 ) transit ; continues to increase through 270 to j * [ at upper transit. Briefly, whenever the given star crosses the meridian of a given place, every one of the 20 AZIMUTH. angles /, Z, and q passes through either o or 180; rot necessarily the same value for all at the same time. But each angle is < 180 or > 180 according as the other angles are, each, < 180 or > 180; the sides remaining as noted in article 16. 35. For use in a given observation the higher equations giving points of maximum and minimum errors will be made to consist of terms in some trigonometric functions of L and d and some one function of / (or of h}. Substituting the values of L and d that belong to the particular occasion for finding the azimuth, / (or h} can be determined, to know when to observe the body so that the resulting error shall be a minimum ; or, to know what point of observation and its neighborhood to avoid as giving a maximum error. For convenience in tracing the locus of the equation, and for the best exhibition of it, L will be regarded as fixed, with d and / (or h) variable : hence, giving different values to d, those of t (or h}, corresponding, will be found. 36. In the equation to the locus, with given values of L and d, and calling, for instance, sin h = x, the roots found from the numerical equation embracing both real and imaginary ones, even though every real root correspond to some point of a plane curve regarded as purely algebraic, in which x may have any value, not being restricted to the value of sin h lying between -f- I and I, yet the locus on the sphere may not contain every point repre- sented by the real roots. For if one of these gives sin h > I, it will present an impossible case. Again, the root may have a value not impossible for the trigonometric function, yet one that sin h never attains. For illustration, when d= o the greatest value that h can have is (90 Z), when sin h = cos Z; hence, in this case should the root have a numerical value greater than cos L it will be inadmissible. Therefore, among several real roots there may be but one value that is admissible. The problem may also be materially simplified on account of positions of the body that are impracticable for observation ; for example, the diurnal circle may cross a curve of minimum errors below the horizon ; then (theoretically) the best practicable position of the body is in the horizon. 37. While equations in the terms mentioned must be resorted to for finding the time or the altitude at which most favorably to observe a given body in the observer's latitude, and though incidentally useful in constructing the curves of maximum and minimum errors, yet, for the latter purpose, equations in terms of Z, L, and h are far simpler.* By them, the point tracing the locus is virtually referred to polar co-ordinates instead of the more compli- cated bi-polar co-ordinates employed with the equations in terms of t, Z, d, and of //, Z, d. But, more than this, the system of polar spherical co-ordinates may readily be trans- formed into a system of plane rectangular co-ordinates for the projection of the curve. By employing the equations to the latter, the difficulties inherent in all the other equations for determining all the properties of the curve are removed, and analysis may be made. With- out a knowledge of the properties of the curve of projection, a true conception of the curve on the surface of the sphere is unattainable. There remain two other forms of equations that are interesting and incidentally useful ; namely, (i) the equation referring the point to the system of rectangular spherical co-ordi- nates, the origin being the zenith, and the spherical axes the meridian and prime vertical ; (2) the polar equation to the plane curve of the projection in which 6 is the complement of the azimuth and r the linear distance corresponding to the projection of the zenith distance. 38. The diagrams are stereographic projections on the primitive plane of the horizon, * Insisting on the use of the horizon as the primitive plane, for the best view of the curve (article 38). AZIMUTH. 21 the point of sight being at the nadir. The parts lying above the horizon are drawn in full lines ; those below the horizon, in broken lines. The distortion in the projection of the lower hemisphere gives an appearance of asymptotic properties to branches of the curves passing through the nadir, and of lack of symmetry in the branches of the loci above and below the horizon. But symmetry exists, and may be more easily perceived if the lower hemisphere is revolved 180 about a tangent at the east point, the point of sight being moved to the new pole of the primitive circle. This can be easily sketched without attempt at great accuracy ; but the stereographic projection is considered preferable for the cases presented. In the projection itself the asymptotic properties do exist. No attempt is made to construct diagrams on a sufficient scale, or with the precision necessary, for use in graphically finding the point to accept as the best, or to avoid as the worst, in observing the body. 39. Tables should be prepared giving the most favorable time for observation, when error in h in the altitude-azimuth and error in / in the time-azimuth are considered, in the case of bodies that cross the prime-vertical in their diurnal course ; and tables to give the dividing line between altitude-azimuth and time-azimuth being the better method when error in latitude alone is concerned. Tables applicable to other cases would be useful, but they are not so much needed as are those specified. 40. Since it is purposed to determine, from investigation of the expression for error, the best and the worst conditions for observation numerical maxima and numerical minima, ir- respective of sign, will often be called max. and min. ; notwithstanding that when found alge- braically the numerical max. may be an algebraic min., and the numerical min. an algebraic max., according to the sign. In general, in this treatise, if the terms maximum error and minimum error occur unqualified they will mean numerically greatest and least errors, re- spectively. Algebraic max. and min. corresponding to true max. and min. where true is the term used by mathematicians respecting max. and min. found by giving to the first differential co- efficient the value zero. Inspection of the given expression will usually show whether it is susceptible of algebraic max. and min. If these exist it will sometimes further be seen at exactly what position of the heavenly body they occur. If not obvious, recourse must be had to the analytical method of finding the precise conditions giving them. But inspection may show that, though no true max. or min. exists, yet values of o and oo do occur, and, for our purpose, these are the truest maxima and minima, sacrificing English to emphasis. To distinguish these from true (algebraic) max. and min., and from finite nu- merical max.. and min., the term absolute will be used in this treatise ; dZ = o, an absolute min.; d Z '= oo , an absolute max. 41. To facilitate comparisons to be made hereafter, it will be convenient to collect the several expressions for errors and to give to each one some equivalent forms from trigonom- etry not given in what precedes. 22 AZIMUTH. I. ALTITUDE-AZIMUTH. dZ I cos q cos ^ No. I. -77- = - -- 7 = -- 7 = . f ; L and ....... (67) cos ^ = cos a cos b -\- sin a sin b cos C. > Differentiating the first of (67), all parts being variable, sin a da = sin b cos db cos b sin cdc -f- cos b sin c cos A db -f-sin b cose cos A dc sin sin csin A dA. . (68) { (sin b cos t cos b sin ccosyi) d# \ / sinacosCdb \ (sin ^cos # cos sin dcosA)dc > = < sin # cos ,# afc V. . (69) sin b sin cs'm A dA ' sin a sin b sin Dividing out sin a, da = cos C db cos B dc sin b sin " L at its elongation ; but, for these bodies, no algebraic max. or min. For all bodies whose d < Z, algebraic max. and min. occur, both being numerical min. Where the min. occurs is not obvious, though text-writers have given overwhelming preponderance to tan q, treating cos h as insignificant, and thus have erroneously assigned the min. error to the position on the prime-vertical. (See articles 5 to 9.) In the case of d > Z, when Z '= 90 is never attained, incidentally the min. occurs when the body is nearest in azimuth to the prime-vertical. But for d < L recourse must be had to solving d\- r] = o to obtain the conditions giving algebraic max. and min., \tan q cos h] which are numerical min. Though the exact point desired (on either side of the meridian) is not obvious, yet it is seen to lie on that side of the prime-vertical towards the nearer pole, since for equal values of q on each side of the prime-vertical h will there be less and cos h greater. This branch of the curve of min. errors from zenith to nadir through the east and west points together with the branch derived from q = 90, giving the entire curve of min. errors, is shown in locus No. I, the meridian being the locus of max. errors. It will readily be seen that the text-writers would have erred to a less degree had they assigned the most favorable position to the six-hour circle instead of to the prime-vertical. For, if d < L, 2 6 AZIMUTH. when Z = go c tan q = cot L cos h' tan q = - , cos d and (55) becomes dZ TT = tan L. . . . (tf) dh When / = 90 cot L tan Z cos = tan L sin Z. . . . (b) (b) < (#) excepting for d = o, when / = 90 and Z = 90 occur at the same time. (See arts. 5 and 15.) 44. The following brief memoranda result from an inspection of the several coefficients of error, L and d being now constant, even though the coefficient is that for error, or change, in one of them ; hence that one a variable in getting the expression for error. No. 2. From ($6), -,-=r = = , for error in L, absolute max. when t = o or 180 ; 3 ' dL tan t cos L absolute min. when t = 90 or 270. No algebraic max. or min. The locus of max. errors, the meridian ; that of min. errors, the six-hour circle. No. T,. From (^7), ,= : f, for error in d, absolute max. when t = o or dd sin t cos L 1 80; numerical min. when / 90 or 270. Loci: the meridian for max.; the six-hour circle for min. The numerical min. is obvious, but may be verified by finding the algebraic max. and min. The loci being the same as for error in L (No. 2), but the min. error > o. 45. II. TIME-AZIMUTH. dZ cos q cos d No. 4. From (58), -,- = ~ , for error in t. dt cos h For d > L, absolute min. when q = 90, 270. For d < L, numerical min. erroneously assigned to the prime-vertical by some text- writers. For all bodies algebraic max. and min. generally giving numerical max. when q = o, 180. Solving d f- . j = o, algebraic max. and min. will be found, giving numerical max. generally when q = o, 180 ; and for d < L numerical min., which, though the point is not obvious, proves to lie on that side of the prime-vertical towards the nearer pole. In very high latitudes, for some declinations, this point is very near the meridian and very far from the prime-vertical, in bearing; hence the importance to arctic explorers to know the truth, not to be misled by false teaching. As in No. i (see last paragraph of art. 43) the prime-vertical gives a less favorable posi- tion of the body than that on the six-hour circle. For if d < Z, AZIMUTH. when Z 90, When t = 90, dZ cos, d ' dZ cos = COS Q r 77= COS 7, ~~ V*WO I/ 7, \^V/^ */ 7 at cos # at * cos # tan af cos ^ cot Z, it is obvious that there is one max. on each side of the meridian, occurring between the times of the upper culmina- tion of the body and its elongation ; since for equal values of sin q on each side of unity (q = 90), cos h will be less for the greater altitude. Loci : Absolute min., the meridian ; numerical max. and min. given by curve No. 6. III. TIME-ALTITUDE-AZIMUTH. 7^ 46. No. 7. From (61), TV = tan Z tan h, for error in h. Absolute max. when^= 90, 270. Absolute min. when Z = o, 180 also when k = o. Numerical max. from algebraic max. 2 8 AZIMUTH. and min. found by d(tan Z tan h) = o. There will he a max., then, for any star that does not cross the prime-vertical ( d> Z,), on each side of the meridian above and also below the horizon, if a rising-and-setting body. 'If the body is always visible, a max. above the horizon on each side of the meridian ; if always hidden, a max. on each side of the merid. below the horizon. For bodies that cross the prime-vertical above the horizon (-f- d < L), a max. be- low the horizon on each side of the meridian ; for those that cross below ( d < L), a max. above, on each side. Loci : Absolute max., the prime-vertical ; absolute min., the horizon and the meridian ; numerical max. by curve No. 7. sf "7 4- *7 No. 8. From (62), - r = - , for error in /. Absolute max., Z = 00 (when / is at tan r not 90) ; absolute min., t = 90 (when Z is not 90) ; numerical max. or min. when Z = /cos d cos /\ o, 180, and t = o, 180, from algebraic max. and min. found by d\ -- ~ -- 7) or -- ~ -- 7 vcos cos til tan T)=* '(7 \ tan Loci: Absolute max., the prime-vertical; absolute min., the six-hour circle; numerical max. and min., the meridian. N dZ sin / sin d No. Q. From (63), -TT = tanZtantf = ~ 7, for error in #. Absolute mm.. ' dd cos Zcos h when Z = o, 180 ; absolute max., when Z = 90, 270; numerical max., when q = 90 (limited to bodies having */> Z), obvious, but may be derived by algebraic max. and min. jf sin * \ from d \ ~ 7) = o. \cosZcos /z/ Loci : Absolute min., the meridian ; absolute max., the prime-vertical ; numerical max., the curve of q = go . [NOTE. In the Horizon-azimuth, inspection of (48) and (50) shows that, at a given place, d = o is the best con- dition for either error in d or error in L ; and that for a given d, the lower the latitude the better.] 47. The determination of the expressions of maxiimim and minimum errors in the computed azimuth, due to small errors in the given parts of the astronomical triangle. In No. I, error in 7z, putting d \ T) = o, it would appear that the second differ- & \tan<7cos^/ ential is usurping the functions of the first differential coefficient ; but it must not be so re- garded for this problem of the variation of the triangle. The equation that the first differ- ential is derived from is the general equation giving the relations existing at all times among the parts of the triangle represented, the constants as well as the variables, and it is not an equation conditioned on algebraic max. and min. existing. The first differential coefficient gives the ratio of change of bearing to change of altitude for a given body in a given latitude. But, this ratio having different values for different positions of the body in its diurnal course, the maximum and minimum values are sought, L and d remaining constant. Hence, regarding the first differential as an isolated factor, legiti- mately derived, which multiplies the error in altitude to produce the error in the computed azimuth, we dispossess it of its character as a differential, regard it as a quantity susceptible AZIMUTH. 29 of true max. and min. values, and seek these by putting its own first differential equal to zero. The new variable q thus enters, but its differential is eliminated, knowing the relation ex- isting between it and that of the variable h, while L and d are constants ; leaving dh a factor in every term of the derived equation, thus dividing out. The equation with any assumed latitude is that of a curve, on the surface of the sphere, which is intersected by parallels of declinations within certain limits. The intersections show where the bodies having the declinations corresponding to the parallels must be observed to give algebraic max. or min. errors in the azimuth. To obtain this equation, in whatsoever terms of the astronomical triangle it may be expressed,/- and d are assumed constant not only while deriving the partial differential, but while finding its true max. and min. values ; hence these, in this case, are found by making the second differential coefficient in the orignal problem of finding simply the general expression for error in altitude equal to zero. Not so, however, in the cases of error in L and in d. That we are dealing with variations of the astronomical triangle is forcibly shown in these cases. Taking (59) as an instance, dZ -Tf = tan h sin Z, the expression for error in L is the partial differential regarding t and d constants ; that is, no error in the declination or in the recorded time of observation. But, having obtained this first differential coefficient, it is not the second differential that is put equal to zero to obtain algebraic max. and min. errors in the azimuth. We do, indeed, put d (tan h sin Z} = o ; but this must be solved regarding L, now, as constant as well as d. For the error in L will have its maximum effect if the star is observed when tan h sin zT attains its maximum value, as the star moves with its unchanged declination viewed by the observer in his fixed latitude even though his assumed latitude is slightly in error. Similar considera- tions obtain when error in d is concerned. It is imperative, then, that in putting the differential of \hzfactor of the error equal to zero, as, for example, (59) d (tan h sin Z} = o or (60) d ( ~-\ = o, L and d shall be regarded C.OS / j constant in the solution, notwithstanding the factor multiplying the error is derived from a variable L or d ; for the change in the aspect of the astronomical triangle is caused by all the parts, other than L and d, varying. 48. The preceding conditions being fulfilled, to determine the relations existing among the several parts of the astronomical triangle to give a max. or min. effect to the coefficient of the error in any given part in the problem, when any star is observed at any place, the latitude and declination both remaining fixed, while the star travels along its path, it is evi- dent that the resulting equation expressed in terms of L, d, and any other one part, will give the value of this part required to fulfil the condition of most favorable or least favorable ob- servation. ' ' ] * Hence the most convenient parts to form the equation are L, d, and /, for determining the time to observe. L and dfbeing known, t can be found from the properly selected root of the equation, taking care that t is represented by the same trigonometric function throughout. The conditions, good for the particular star, hold good for any star whatsoever ; hence, giving successive values to d, while L remains constant, the roots of the equation give the hour- angles (/) corresponding ; and the curve, on the surface of the sphere, of max. and min. values 30 AZLM-UTH. of the error in the computed azimuth is the locus of the intersections of the parallels of declination with the hour-circles given by the imposed conditions. Hence, in this sense, we may say that / and d vary together without denying the unvarying character of d so far as concerns the establishment of the relations causing the coefficient of error to be a maximum or a minimum. Giving different values to the arbitrary constant L, we shall have on the sur- face of the sphere a system of curves. 49. For tracing the curve, in any particular latitude, the equation to the curve may be turned into terms of Z-, fixed, and any other two parts varying together. Thus the terms may be in trigonometric functions of Z, d, and h, one function adhered to for^, the next best con- dition to L, d, and / terms for determining at what point to observe in any given case. But, for curve-tracing, the most useful originally derived equation is that consisting of terms in Z, h, and Z the function, for Z being the cosine for from this equation we can de- rive the equation to the projection of the curve referred to plane rectangular co-ordinates. The foregoing remarks apply to all the cases, in the several methods of finding the azi- muth, remaining unmemioned. Briefly, the coefficient of error once legitimately derived, its max. or min. effect must be sought by considering that the latitude and declination do not vary while the aspect of the astronomical triangle changes with the progress of the star along its path the diurnal circle = parallel of declination. 50. Forms of equations to the loci. Summary. (l) In terms of Z, d, t; ] , . (( For use in finding position to observe in any given case. \2 ) JL^, ^Z, rt, ] (3) " " " L,/i,Z; referred to spherical polar co-ordinates. (4) derived from (3); referred to spherical rectangular co-ordinates. (5) derived from (3); referred to plane polar co-ordinates. (6) " (3) or (5); referred to plane rectangular co-ordinates. (l), (2), (3), and (4) all refer to the curve on the surface of the sphere ; (5) and (6) to the stereographic projection of the curve. (4) and (5), though interesting, are of little use in this treatise. The next step will be to deduce the equations of algebraic max. and min. in the several cases, before discussing the curves they represent for numerical max. or min. The equation to equal numerical effect of error in L, by both time-azimuth and altitude-azimuth, will also be deduced. PART VI. EQUATIONS TO THE LOCI OF MAXIMUM AND MINIMUM ERRORS IN THE COMPUTED AZIMUTH DUE TO ERRORS IN THE DATA. 51. Locus No. i. Altitude-azimuth, error in h (art. 43). The equation to that part of the locus of numerical mininum error, not obvious, being the locus of algebraic max. and min., when d < L. 1st Equation in terms of Z,, d, and /. Of the several equivalent forms of (55) any one may be differentiated to give the equation, but care must be taken that a factor essential to give some part of the entire curve shall not be divided out during the operations of simplifying ; otherwise, a true branch may be missing in the result : so, also, in the remain- ing cases (56), (57), etc. The labor of reduction varies with the different forms used. In these pages, the expression giving the least labor, in each case, is retained, after experimenting with the several forms to ascertain which is preferable, as well as to check .the result. (cos o\ tl = sin LI sin / sin q dq cos q cos / dt = o (77) cos Z sin q Substituting from (65), dq = : - at; cos Z sin 8 q cos q cos t = o; (78) .*. cosZ(l cos 8 q) cos q cos t = o, turn into /, L, d; cos (i cos 8 C) cos CcosA = o, turn into A, c, < sin c cos b cos c sin b cos A By trig., cos B - . ; (80) sin a sin b cos c cos b sin c cos A. COS C = : ' (ol) sin a cos a = cos b cos c -J- sin b sin c cos A; (82) sin 8 a i cos 8 a (83) 32 AZIMUTH. Substituting (80) to (83) in (79) and reducing, cos 4 A -j- tan b cot c (cot 2 b 2} cos 3 A 3 cot 2 cos" ^ -|- tan b cot c (2 cosec 2 b -\- cot 2 c) cos A cosec 2 b = o ; cos 4 * -I j (tan 2 d 2} cos 3 * 3 tan 2 L cos 2 / 1 tan d v + >(2 sec 2 */-|- tan 2 Z) cos t sec 2 d = o. 1 tan */ v ' (84) 52. 2d Equation in terms of L, d, and /z, may be derived from the foregoing, but more conveniently from tan q - -- 7 ] = cos h) (85) /. tan q sin h dh cos h sec 2 q dq = o. (86) From (64), tan Z cos (87) Substituting and dividing out dk, tan q sin ^ 2 tan Z cos ^ = o. (88) /. sin q cos q tan Z sin ^ I = o. (89) Substituting sin q = sin Z cos L cos d we have cos q sin 2 Z cos L sin cos ^ cos Z I = o. (90) cos ^ (i cos 2 Z) cos Z sin // cos Z cos d = o, turn into //, Z, cos C (i cos 2 .Z?) sin cos # cos B sin & = o, turn into rt, , & " } (91) By trig., cos / = cos b cos c cos a : -- . - . sin a sin ^ (92) cos 6 = cos c cos a cos sin sin b (93) AZIMUTH. 33 Substituting (92), (93), and (83) in (91), and reducing, sin 4 h : 7 (sin 2 d-\- 2) + 3 sin 2 L sin 2 h 4- ' -7 (2 cos 2 d sin 9 Z-) sin h cos 1 d = o. (94) sintfP sm*/ v Note the difference between (84) and (94) in the functions of L and d. 53. 3d Equation in terms of h, L, and Z. From (79), cos B (i cos 2 C) cos C* cos A = o, turn into /?, , r. Multiply by sin 2 b. sin 2 cos B sin 2 ^ cos B cos 2 C sin b cos 7 . sin b cos ^4. ... (95) From trig., sin b cos C = sin cos t: cos # sin c cos .# (96) sin b cos yi = sin c cos tf cos c sin # cos B (97) sin 2 b i cos 2 (98) cos # = cos c cos # -f- sin sin cos /? (99) Substituting (96) to (99), as required, in (95) and reducing, cos 3 ^-f- cos a sin a cot c cos 2 ^ (i -j- sin 2 cot 2 c -f- cos 2 a) cos .Z? -|- sin a cos a cot c = o. (100) . . letting A represent tan Z., cos 3 ^4" A sin h cos ^ cos 2 Z (i -\- A 2 cos 2 ^-f- sin 2 ^) cos ^"-j- A cos h sin /t = o. . (101) Or, putting zenith-distance in place of the complement of the altitude, cos 3 Z -\- A cos z sin # cos 2 Z (i -|- A 2 sin 2 z + cos 2 z) cos Z -\- A sin ^ cos^r = o. . (102) 54. From (102) may readily be derived the equation to the stereographic projection of the curve, referred to plane rectangular co-ordinates. The primitive plane being the horizon, the rectangular axes X and Y are, respectively, the projections of the prime-vertical and meridian ; the origin being the centre of the primi- tive circle, representing the zenith ; the point of sight situated at the nadir. To agree with the accepted reckoning of / and Z to the westward, x should be reckoned positive to the west- ward and negative to the eastward. Hence, in north latitude x would be positive to the left, negative to the right hand, and the reverse in south latitude ; the latter conforming to the conventional reckoning for plane rectangular co-ordinates. 34 AZIMUTH. So far as tracing the curve is concerned, since there are always branches symmetrically situated on each side of Y. the meridian, it does not matter which direction for x is arbitrarily chosen as positive ; therefore, for convenience, x will be taken positive to the right hand, negative to the left, adhering to the conventional reckoning, y is reckoned positive towards the elevated pole, negative towards the depressed pole. 55. To obtain the formulas of transformation of equation (102) to the equation of the pro- jected curve. The radius of the primitive circle being unity, z the zenith distance of any point of the celestial sphere, and r the linear distance, on the projection, of this projected point from the centre of the primitive circle, we have, from the principles of stereographic projection, r = tan%z; (103) 2 tan ^z and from plane trig., sin z = - ~- (104) i + tan \z I tan 2 4ar coz * = r+Twn^ - (105) 2 tan \s tan z = - rr~ ............ (106) i tan z Since the angle Z on the sphere and in the projection has the same value, y cos Z = sin co-Z - ; and, since r 2 = x* -\-y\ ............ (108) we have for direct substitution in (101) or (102), to obtain the equation to the projected curve, 2r ~ / IOQ \ x* y , . (no) (in) i r ' i .*' y cot z = tan h = - = - ; --- ^_ . . . (\ 12) * 2r = ' ' (U3) AZIMUTH. Let A represent tan L ; B cos A whence |/7~^B 2 = sin Z ; C " sin A " 4/r^C 2 cos L. (114) [NOTE. A, alone, is required in the cases discussed in this treatise ; but based on the principles contained herein, a large number of curves interesting to mathematicians may be found, when considering all the variations of the astronomical triangle, even though the problems lack utility in astronomy. The writer has found, in the case of Time-altitude-latitude azimuth (art. 22), the necessity of employing either B or C, in order to introduce but one arbitrary constant (see Appendix).] It will be found convenient to retain r until its elimination is effected as far as possible before substituting for r\ r\ etc., x* + y, x* + 2.x*f + y, etc. 56. To obtain the equation to No. i, for the projected curve, we have from (101) or (102), substituting, as required, the equations of art. 55, _ _ r 3 "* r\i + V) a ~\ l h -f _ 2Ar 4 (i - r 2 ) = o. (116) Writing out r" x 3 = o. (117) (118) Substituting, as required, (118) in (117); arranging in order of powers of y, and changing signs throughout, to give positive sign to the first term, we have : -f 4A/ -f 4A/ = O. ... (i 19) Rearranging in order of highest-degree terms, + 4 Ay - 2y + 8Avy - 4Aj 4 - 6A;tr 2 y - +y (120) This is the equation to the locus of algebraic max. and min., giving numerical min., and applies only when d<^L. 57. For this case, the equation referred to spherical rectangular co-ordinates is of too high a degree to be useful ; but in some of the subsequent cases that form will be of interest 36 AZIMUTH. if not adding anything to aid in the analysis of the curve. The equation in this form will be introduced in Locus No. 4. The equation referred to plane-polar co-ordinates may be easily obtained by omitting to y substitute for cos Z, in the foregoing work, =. But little additional interest will be yy -{- x* lent in any case, and for this locus the equation is too complex to be used advantageously. 58. Although not a locus of algebraic max. and min., the curve q = 90 is a branch of the locus of numerical min., giving absolute min. = o, applicable only when d > L. This curve, for our purposes, may be called the curve of elongations, while known to mathema- ticians as the spherical ellipse.* The equations are as follows : tan L sin L tan L cos / = - - j ; sin h = . -,\ cos Z = - - 7- ...... (121) tan a sin a tan h Substituting, from art. 55, in the last of (121), we have for the equation to the projection : / + 2A/ .? + .*> + 2A** = o; ........ (122) or, y -\-sfy-\- 2Ay -\-2.Kx* y o ......... (123) 59. The remaining branch, giving absolute max., the meridian, is clearly defined as the axis Y. No equation is needed ; but we have, as a matter of interest, cos" t I = o, cos t = i ; i {cos (L d, or cos (L -f- d; cos* Z I = o, cos Z = i- (124) y ' * = o ..... ... (125) For analysis of (120) and (123) see arts. 92, 93. 60. Locus No. 2. Alt.-az., error in L (art. 44). There is no branch of algebraic max. and min. For absolute max., the meridian, as in No. I (art. 59). For absolute min., the six-hour circle, which is clearly defined by simple construction according to the principles of stereographic projection ; yet, as a matter of in- terest, we have the equations * For the knowledge th,at the curve of q = 90 is the spherical ellipse, the writer is indebted to Prof. W. W. Hendrickson, U.S.N. AZIMUTH. 37 cos t = o ; sin h = sin d sin L ; cos ^ = ^ . (126) tanZ. y -j- 2A.y -f- x 1 i = o, or, y -1- x* 4- 2Ai/ i = o. Whence, y A ^A a + i x> (128) (129) For analysis see art 94. 6l. Z-^J .Afa. 3. Alt.-az., error in d (art. 44). The meridian is a branch, giving absolute max. (see art. 59). The six-hour circle is a branch giving algebraic max. and min., always numerical min. This is obvious from (57), dZ i / i \ -jj = -- : - -- r ! but, to derive the equation, we have d\- -- .- = o : dd sin t cos L \sm q cos h' .'. sin ^ sin ^^ cos & cos g dg = o ......... (130) By (64), dq = -. . ^ sin Z cos cos q cos ^ sin 0- sin h -- * ~ = o sin Z sin ^ sin Z" sin h cos ....... (132) sm C sin B cos # cos C cos .5 = o. ) By trig., the first member of (132) = cos A-, . * . cos A = o ; cos t o, t = 90 or 270, whence (126) to (129) recur (art. 60). 62. Locus No. 4. Time-azimuth^ error in t (art. 45). Curve of q = 90 is a branch of absolute min (see art. 58). Curve of algebraic max. and min. giving generally numerical max., the meridian ; numerical min. on (?) to be found. The equation in terms of t, L, and d. , /cos q sin Z\ From (58), d\- -) =o. \ sin t I sin t sin g sin Z dg sin t cos $ cos Z dZ -{- cos 4 sin Z cos t dt = o. . . . (133) 3 8 AZIMUTH. cos Z sin q ,_ cos q sin Z From (65) and (58), dq = ^- f dt ; dZ ^-^ (W ; . . _ cosZsin 2 ^sin Z -\- cos 2 ^ sin ZcosZ-f- cos q sin Z cos / = o. . . (134) Divide out sin Z and substitute i cos 2 q for sin 2 q, ( J 35) and we have - cos Z(i 2 cos 2 q) -J- cos q cos t = o ; cos .5(1 2 cos 2 7) -j- cos (7 cos A = o. Now turn (136) into /, Z, d; A, c, b. [NOTE. Though the factor sin Z, which is divided out, gives the meridian when Z = o, yet enough remains to give it still. But, employing d 1 -^ I = o, (cos d being constant), sin Z also divides out and leaves nothing to represent the meridian (see art. 51).] In (136), substituting (80), (81), (82), and (83), and reducing, 3 2 7. cos 4 A 2 tan b cot c cos 3 A -4- . , , r-^ cos 2 A -J- 2 tan cot ^ cos ^ sin # sin c sin 2 sin 2 b cos 3 sn cos 2 L cos 2 d cos 4 / 2 cot d tan cos s *-| 5-7 r-r- cos 2 / + 2 cot i, the positive sign of the radical is inadmissible. This branch is that of numerical minimum. 64. The equation in terms of h, L, and d. AZIMUTH. 39 sin h sin L sin d In (139), substitute cos/= j , (142) v JW cos L cos d The first factor becomes (sin h sin L sin d}* cos" L cos" d= o; (H3) . . sin h sin Lsind = cosL cosd; ( 1 44) ( sin Z-sin d-\- cos L cos Z gives an imaginary result ; therefore this branch exists only for I, the positive sign of the radical is inadmissible. This branch Sill & gives numerical min. The whole curve is given by [(sin h sin Z sin d}* cos 2 L cos 2 d~\ [sin 2 h sin d 2 sin Z sin h -f- sin d~\ ; (148) or sin* h ^-^ . \, S1 " ^ sin 3 h 4- (5 sin 2 Z + sin 2 //) sin 2 sin -f x* - x* = o ; ....... (194) or, / *' + 2A/ + 2A^r> / + x* = o ...... . (1940) For analysis, see art. 98. 76. Locus No. 7. Time-alt. -azimuth, error in h (art. 46). The meridian and the horizon are branches giving absolute min. ; the prime-vertical giv- ing absolute max. The branch of numerical max. given by algebraic max. and min. as follows : From (61), cos L sin 2 ? cos dcosZ-\- sin A cos g cos L = o ( ! 97) Since AZIMUTH. sin* q =. i cos*^, 45 cos */cos Z cos d cos Z cos* q 4- sin h cos ^ cos Z = o ; turn into *, Z, L, and it is obviously AZIMUTH. 49 dZ the curve q = 90, as shown in (63), -77 tan Ztan d. Though obvious, we may find this branch analytically, as one of algebraic max. and min., as follows : choosing from (63), we have . , sin q d\ %,) = o; cos cos Z cos q dq -f- sin q sin Z dZ = o. (228) By (66), dZ = cos q cos d -- = -- =*& ; cos Z cos Z, z , sin q sin .Z cos q cos ^ . . cos Z cos # -j ^ -f = o. cos Zcos L (229) Substitute _ sin Z cos L sin o : cos d and clear of fractions, when we have cos" Z cos q -(- sin" Z cos q o ; /cos* Z + sin 4 Z\ ( _ j J cos ? = o. (230) cos ^ = o, tan cos / - sin / = tan d' sin L sin d ' tan L tan h' . J90 C (230 For the equation to the projection of q = 90, see art. 58, Locus No. I, where this curve ex- ists for an absolue min. Eq. (123), y* -j- x*y -f- 2 Ay -f- 2A.T 2 y = o (see art. 101). 85. Locus No. 10. Time-azimuth and altitude- azimuth giving equal numerical values to the error in the computed azimuth, arising from a small error in the latitude ; whence the limits within which the one method is more favorable than the other, so far as the error in lati- tude is concerned. By inspection of (56) and (59), taking for investigation the single case of a rising-and- setting body having -\- d> L, we see that in the method of altitude-azimuth the error when the body is in the horizon at rising, t lying between 180 and 270, has a positive finite value, decreasing to zero when t becomes equal to 90 as the star travels in its diurnal path ; by the So AZIMUTH. time-azimuth, when the star is in the horizon the value of the error is zero, and then it in- creases negatively to a negative finite value when t = 90 ; Z throughout this time remaining between 360 and 270. .Therefore, (l) for some point of observation (e') in the star's path, between the horizon and the six-hour circle, the values of the error in the azimuths, computed with both methods separately, will be numerically equal, with contrary signs ; (2) it is obvious also that before the star reaches the meridian, after passing (*'), some point of observation (/"') will give identical values to the errors, that is, numerically equal, having the same sign. The time-azimuth then will be preferable from the time of rising of the star until e is reached ; thenceforward, passing through t = 90 and up to (/'), the altitude-azimuth the better method; thereafter to the meridian we should employ the time-azimuth. We may, in a similar way, follow the star west of the meridian. Our object, now, will be to find the equations to the curves of equal errors with opposite signs, and with like signs ; whence, constructing the curves, the limits within which either method is preferable will be graphically given. 86 (a). First. Identical errors signs alike. Putting (56) and (59) equal to each other, f = tan h sin Z\ tan / cos L cot / tan h sin Z cos L = o ; ) I cot A cot a sin B sin c o. ) Multiplying by sin B, sin B cot A cot a sin 2 B sin c = o ........ (234) By trig., sin B cot A = sin c cot a cos c cos B ; } sin 2 B = i - cos 1 B. \ ...... Substituting (235) in (234) and reducing, cos" B cot a sin c cos B cos c = o ; ) cos 2 Z i an h cos L cos Z sin L = o; > ....... (236) cos Z(cos Z tan h cos L sin L = o. ) The first factor of (236) gives cos Z = o ; ............ (237) .'. Z=go, 270, and the prime- vertical is a branch of the locus. Putting the second factor equal to zero, we have tan L AZIMUTH. 51 which value can exist only when q = 90, 270 ; hence the curve of elongations, q = 90, is a branch of the locus. Hence the prime vertical is a branch for all bodies having d < L, and we have for con- ditions, tan d sin d cos t = - f, sin h = j ; ......... (239) tan L sin L and the curves of elongations are branches for d > , giving tan L sin L cos t = - - ->, sin It -- -g .......... (240) tan a sin d From (238) we have, as in (123), the equation to the projection of the curve of elongations, / + *?y + 2 A/ -f 2A* 8 y o ; and from (237) for the prime-vertical y= o equation to the projection. 86 (&). The branches found in art. 86 (a) may be determined with less labor by selecting from (56) and (59) as follows : cos t sin h sin Z sin q - r sin t cos L sin t cos L but both methods are retained as checks upon each other. From (232)^ we have cos t sin h sin Z sin q = o ; cos A cos a sin B sin C = o. By trig., - cos B cos C (233)^ ; . . cos B cos C = o ; ) _ ..... ..... (236)* cos Z cos q o. ) . * . cos Z = o (gives the prime-vertical, and) cos q = o (gives the curve of elongations.) 87. Second. Equal errors, numerical, opposite signs. Changing the sign of one member in (232)^, and cancelling sin t cos Z, we have cos / == sin h sin Z sin q ; j cos A -f- cos a sin B sin C = O. ) 5 2 AZIMUTH. For the equation in terms of t, L, d\ A, c, b. sin A sin b sin A sin c By trig.. sin B = - : -- and sin C = - : - : sin a sin a cos a sin b sin c sin* A substituting in (241), cos A -) --- -- = o. ol 1 1 c Substituting sin* a I cos" a, and sin 8 ^4 = I cos* A, and clearing, we have cos A cos" a cos /2 -f- sin sin t cos # sin sin c cos # cos a A = o. . . (242) Since cos # = cos b cos r -j- sin b sin cos A, we have cos ^4 cos 8 b cos 8 cos A 2 cos # cos *: sin sin c cos 8 y2 } sin 8 b sin 8 cos 3 A -f- sin sin ^ cos cos -j- sin 8 sin 8 c cos ,/2 > (243) sin b sin cos b cos cos 8 A sin 8 sin 8 c cos 3 ^4 = o. ) Arranging in order of powers of cos A, collecting terms, 2 sin 8 b sin 8 c cos 3 A 3 sin b sin c cos # cos c cos 8 y4 ) -f- (i cos 2 b cos 8 c -j- sin 8 # sin 8 ^) cos ^4 . -j- sin b sin c cos b cos = o. f The coefficient of cos ^ reduces to sin' b -f- sin 8 r, and dividing through by 2 sin 8 b sin 8 c we have cos 3 A-\-% cot # cot *: cos 8 A % (cosec a c -\- cosec 8 b) cos A % cot b cot c = o; l cos 3 * -f f tan d tan Z cos 8 / - \ (sec 8 Z + sec 8 d) cos / - tan ^ tan Z = O. j 88. The equation in terms of h, L, d\ a, b, c. T /^..W U *.-.. i.- /i COS COS ^ COS In (242), substituting cos A = sin b sin c performing the operations, collecting terms, and simplifying, we have s , sin 8 c -f- sin 8 b \ cos a f cos b cos c cos 8 a -- -J -- cos + cos cos <: o ; | V (246) sin 3 h | sin d sin Z sin 8 h \ (cos 8 Z + cos 8 d} sin ^ + $ sin L sm d o. ) AZIMUTH. 53 89. The equation in terms of Z, L, h ; B, c, a. Changing the sign of the second term of (233), cot t -f- tan h sin Z cos L = o ; ) cot A -j- cot a sin B sin c = o. J Multiplying by sin B, sin B cot A -j- cot sin 8 .Z? sin = o. ........ (248) Substituting (235) in (248), and reducing, cos a B -f- cot c tan # cos B 2 = o ; ........ (249) cos 9 Z-{-tan jr cot h cos Z 2 = o ......... ( 2 5) 90. The equation to the projection of the curve of equal numerical errors, contrary signs. Substituting in (250) from art. 55, as required, and reducing, we have -2 = ........... (250 2x* + 2x" = o ; ...... (252) or, rearranging, y -f 3^y + 2;r 4 + 2Ay + 2AA-jj/ y 2^r 2 = o ...... (253) For analysis see art. IO2. PART VII. ANALYSIS OF THE EQUATIONS TO THE LOCI, AND THE TRACING OF THE CURVES. 91. To Lieutenant H. O. Rittenhouse, U.S.N., the writer is greatly indebted for assistance, both mathematical and constructive, in analyzing the equations and tracing the loci. In the progress of delineating the latter, Lieutenant Rittenhouse gave much aid in removing diffi- culties encountered. The drawings for the accompanying plates were made by him, and they represent the curves faithfully for the particular latitudes given, to serve as illustrations. For the determination of the most favorable and the least favorable conditions for obser- vation of the star, by finding the algebraic maximum and minimum effects of the errors in the data, the writer claims originality. To the fact that text-writers have been content to ascer- tain (?) by inspection of the first differential of the particular equation employed in the prob- lems for time, latitude, and azimuth what conditions are the best, is due the fallacy of ac- crediting the prime-vertical the place of best observation, when error in h and error in t are concerned, in the altitude-azimuth and the time-azimuth. Having undertaken this work, the problem of finding, in any given case, the best position for observation demanded the equation in /, L, d, or, as an alternative, in h, L, d, for L and d must be known and used. By means of these two equations considerable progress was made in describing the curves : where one equation failed to clear some doubtful point, the other would sometimes come to the rescue. But, still, there always remained some points in ob- scurity; inciting guesses where reason failed: some guesses being very good, as ultimately proved ; others as bad as could be. All the curves had been defined (?) by these equations, and the writer was continuing a great amount of " dead work" among them, in the hope that some one curve, at least, simpler than its fellows, would prove accommodating when it oc- curred to him that the tracing of the curve need have no dependence on the problem of find- ing, in a given case, the time to observe at, or the altitude to observe ; but that, in a given latitude, by varying the altitudes for points of the curve, the corresponding azimuths might be found and therefore the curve could be traced. Consequently, the equations were de- rived in terms of Z, L, and h ; and it was then discovered that these equations could be easily transformed into the equations in terms of x, y, and one arbitrary constant, tan Z, for refer- ring the projection of the curves to plane rectangular co-ordinates.* If this were not accom- plished, still the equations in Z, L, h, were much more useful than the others, for they re- * See foot-note to art. 37. AZIMUTH. 55 ferred to the more simple system of spherical polar co-ordinates ; and, at the same time, in general, they dropped one degree lower than those in t, L, d and h, L, d. The equations to the projection, once obtained, the tracing of the curves became com- paratively easy, notwithstanding some of the equations are of high degree and involve some nice points. It is not believed that all these points have been discovered, and the writer will be gratified to see newly discovered truths respecting the curves presented. Considered as plane curves, simply, losing sight of the problem from which they are evolved, they possess very interesting features. Based on the same principles giving these curves, a great number of others may be discovered by considering all manner of variations in the astronomical tri- angle outside of the utility problems. (For this subject, see appendix.) Each curve is given for latitudes 30, 45, 60, thus fairly showing the alterations of form between these stages as the curve gradually changes between the limiting latitudes of o and 90. In the growth of the curves few unlooked-for changes of form occur. In No. I and No. 4 some surprises are met with. In No. 4, apart from analysis, the change in form between 30 and 60 of latitude graphically hints at a great change in higher latitudes. 92. Locus No. i. Alt.-azimuth, error in h. Articles 43 and 51 to 56. Branches I. Algebraic max. and min., giving numerical min. 2. Absolute min., curve of elongations. 3. Absolute max., the meridian. 1st Branch. (d] By (120), / + 4*y + s*y + -f 4A/ + ioA*y + 8A*y + 2 A* 8 + 4A 2 / - 2/ -f 4 A/ 6A*y +/ + 2x*y = o ................. (254) / From terms of highest degree, y (j? + *")" (y -|- 2x*) = o .......... (, 2 55) Hence, there exists an infinite branch having an asymptote parallel to X\ and no other real infinite branch. Coefficient of highest power of x\s2y-\- 2A ; . ' . y = A for the asymptote, (256) which is the projection of the parallel of declination, d = L ( 2 57) 56 AZIMUTH. For y= tan L, and disregarding sign of y, let ? = arc from the origin to the point a where the asymptote cuts the meridian, Y ; then tan \z y tan L, .'. z 2L .......... (258) . * . z = L -\- L. On Y the arc from the origin to the equator = L ; from equator to the point a, the arc = dec. of point a = the remaining L, . * . resuming sign, d = L. (b] For points of locus on Y, put x = o, . . o; ...... (259) = o .......... (260) First factor gives y* = o, the origin. Second factor gives y = A 4/A 2 -f- i (261) N. and S. poles, P and P'. (Imaginary, see (/)). (c) For points on X, put y = o. = o; . . x* = o, the origin ; x* = i, E. and W. points (262) (d) For form at origin, from terms of lowest degree : y -f- ^y = o, or y (y -f- 2x*} = o, (imaginary) (263) But from 2yx* 2Ax* = o (264) y = tv? (265) V' And the curve is of the form 'g- 4). (e) Form of curve at E. point. Moving origin to (i, o) the coefficient of x is found to be 4A; the coefficient of y, 4 A' . . y = -- T- a x is the tangent at E ........ (266) I -f- A _A Similarly, y = . /rs is the tangent at W .......... (267) AZIMUTH. 57 (/') To ascertain the character of the locus at P and P' ; moving origin to P, putting for y,y A -(- 4/A 3 -f- I, the coefficient of y vanishes. We have for determining the sign of the coefficient ofy, |2A 4 + 3A 1 -f i - 2A 3 |/A 2 + i - 2A 4/A a + i ; } similarly for x\ |2A 4 + 3 A 2 + i - 2A 3 l/A^+1 - 2A The signs of these two terms, therefore, will be always the same, and the form at P is imag- inary. P is therefore a peculiar point, and, from the symmetry of the locus, P' must also be a peculiar point. () Though no real branch of this curve passes through P or P', curves to follow do pass through these points; therefore, to prove that the N. and S. poles -are given by y A 4/A 2 -\- i, we have for P, the N. pole, co-L = arc from the origin (zenith) to the pole. By stereographic projection, if r is the linear distance from origin to P, then r = tan \ co.-L ........... (268) i cos co-Z, i sin L , ., N By trig., tantco-= = -- ........... (269) sin L /sin 2 L -f- cos* L . . 1 ' 5-r- ( 2 7) - Y . cos L Similarly for P', I sin L . . y ........... (271) _ i _|_ sin L or -j = A + fA 2 -f i= ~; - - ( 2 7 2 ) r' = distance from origin to P' = tan (90 + L). i cos (go -r L i + sin L By trig, tanl(90 + )= "-' ..... (273) r'=-j ............. (274) (^) The curve does not cross its asymptote, for, combining (256) with (254), we have (A' + 2AX + 2A(A'+i)y-fA 3 (A' + i)':=Q, ..... (275) which gives imaginary roots only. 58 AZIMUTH. (/) For limiting form of curve, L = o, put A = o in (254), y = o divides out ; ........... .-. axis of X, the prime-vertical, is a part of the locus. There remains y + 4*y + 5 *y + 2* 6 2/ 2^y + y + 2** = o ..... (277) Infinite branches are imaginary and form at origin imaginary. If y = 0, S = O, ................ (278) and x = V i, imaginary. ........... (279) If x = o, y = o, ................ (280) and (y i) a = O, # = I, double point at N. and S. . . (281) Moving origin to N., putting y = y -f- r > the f rm * s given by *>+y = o; ............ (282) .'. imaginary branch at N. point, the same at S. point ; N. and S. are isolated points. Arranging (277) as a cubic in x*, we have + ( 4 y - 2 y + 2)^+y(y-2/+i) = o ..... (283) These coefficients are all positive, therefore x* can have no positive root, and therefore x can have no real root. The locus is therefore imaginary, excepting the line y o (that is X, the prime-vertical which is the equator), and the peculiar points at zenith and N. and S. (k) For limiting form L = 90. Put A = oo in (254) and we have = o; (284) /. y = o, giving X, the prime-vertical ; (285) (x* '-}- y) s = o, giving origin, zenith, a double conjugate point. . . . (286) (/) Not considering detached points, it is seen from (/) and (K) that starting with Lat. = o, the axis of X is the locus ; and ending with Lat. = 90, again the axis of X is the locus; while between these limiting latitudes there is always one branch passing through the origin AZIMUTH. 59 and E. and W., and always having an asymptote/ = A. On the sphere it is a continuous branch through zenith, E., nadir, W., to zenith. The question arises : How does the curve change its shape with the change of latitude, so that it shall return to its original form? [NOTE. For the following elucidation and the determination of the envelope, the writer is much indebted to Lieutenant Rittenhouse and Professor Hendrickson.] The equation (254) to the curve contains A and A 2 . Arranging it as a quadratic in A, we have = o. (287) Now, for any values of x and y, two values of A can be found to satisfy the equation. That is, suppose the curve drawn for a given latitude, take some point on it, and this reason- ing declares that for some other latitude, also, the curve will pass through the chosen point. The fact that the curve returns to its initial shape while the latitude increases to 90 is prim a- facie evidence that the curve in some way reaches a limit and then returns. This taken in connection with the quadratic character of A points to the envelope of the curve as a means of obtaining the limit sought. The envelope is obtained from the condition fi* = ^AC (in the general quadratic equation) ; that is, it is the condition that will make A have equal roots in the locus (287). Squaring the coefficient of A and putting the result equal to four times the coefficient of A 3 into the absolute term and simplifying, we obtain 4/V -f is/* 4 -f- Factoring, x" -f 8j>V + 26/V + -f ) - x? = o. i + ^]=0.. (289) (ni) The equations given by the several factors of (289) make up the envelope. The first factor, x*, gives the line Y ; the second factor, (x* +y) 2 , gives the point (o, o), the origin : neither is of use to us. But the remaining factors give a locus shown in Fig. 5. [NOTE. The part for the 3d factor alone will be discussed.] From which we see that the line ZME may form a boundary within which the curve No. I FIG. 5. may pass through its phases. All facts ascertained respecting the curve develop nothing to conflict with such a conclusion, and all facts are satisfied by such conclusion. 60 AZIMUTH, (ri) Taking the 3d factor of (289), (*>+/) (27 + *) + 27 -* = a . . . . . * i . (290) The only infinite branch is given by 2^-j~ ;c ( 2 9 ! ) x and its asymptote is jp = , ( 2 9 2 ) 2 since there are no terms of the degree next to the highest. For points on X and Y, Put y o, x 3 x = o, x (X* i) = o, .# = o, origin ; \ x- i, E. and W. j ' (293) Put x = o, 2y* 4- 2y = o, r ( V s 4- i) = o, y = o, origin ; > (294) ^ = l/_ i, imaginary. For tangent at origin, 27 JT = o, y ~ ........ ( 2 95) 2 For tangent at E. point, put x-\- I for ;r. x Lowest-degree terms are 4/-f~ 2;tr I 'J /= -- ....... ( 2 9^) 2 The curve crosses its asymptote only at the origin. A (o) By (266) the tangent to the curve No. I at E is found to bej = -;*'. This I -f-A expression has a limiting value i when A = i, L = 45. Now this limit of the inclination of the curve at E is exactly equal to the inclination of the envelope at E by (296). (/) To find the maximum ordinate of the envelope (for the point M). The equation to the envelope (290) expanded is -f */ -f 2;r> + * 8 -f 2j/ * = ......... (297) dx 6/ + zxy -\- For maximum ordinate we have -~ = o. Putting the numerator = o, we have f + W + 3** - i = o ........... (298) AZIMUTH. 61 Solving (297) and (298) for x and y, we obtain from (298) y= 2x Vi -\-x* (299) Substituting (299) in (297), we have, after squaring and simplifying, the cubic 2$x e -\- \6x* + I2x* 4 = 0; (300) From which an approximate root is found ^ = .238126 + ; .-.* = 0.483 5422+ (301) Substituting (301) in (299) we find y = 0.1468784 + for max. ordinate (302) (q) To ascertain the character of the curvature of No. I at E. In (254) move the origin to (i, o). Ax The terms of first degree give only the line y = : v-j, (266). Taking the terms of 1 H- A second degree in connection with those of the first degree, we have the conic whose curva- ture near the origin is similar to that of the curve. The equation of the conic is A/ -f 8 (i + A a ) xy + 9 A* 2 + 2(1+ A> -f 2 A* = o. (hyperbola). . . (303) If y = o, we have gAx* -f- 2 Kz = o ; whence for all values of A, x = o, x = f (34) If x = o, we have Ay* -f- 2 (i + A") y = o; 2 (i + A 9 ) y = 0, y= J i (305) The second value of y always negative for positive values of A and never less numerically than 4. For asymptotes, {- 4 (A'+ i) - l^i6A 4 + 2 3 A'+ i6| x - ( 4 A* + 8A a + 5) y = ~K~ |- 4(A a + i) + VI6A* + 2 3 A'+ i6| x f (4A* + 6A a + 3) . n . y = ^ . . . . (307) 62 AZIMUTH. (306) and (307) are the asymptotes of the form c mc m m and These asymptotes always intersect in the 2d angle ; and since the curve always goes through the origin and has the two negative intercepts, the curvature at E is always of the form shown in Fig. 6. ^^ (r) The conclusions regarding the envelope are supported by accurately - constructed curves for lat. 30, 45, and 60, and also for a high latitude, A = 4', lat. about 75. For L < 45 the curve is not tangent to the envelope ; L = 45 it is tangent at E. point ; L > 45 it is tangent at some point S. of the envelope, but the curvature diminishes at E. As L increases, the point of tangency S. moves towards Z, and the curve flattens more and more rapidly until it coincides with the prime-vertical. From a mathematical point of view the latitude to which most sig- nificance should be attached would appear to be that at which the curve ceases to swell at E. and begins to swell towards the origin, that is, lat. 45. But, for our purposes, the lati- tude that gives the broadest swell to the curve is significant, for the curve will then have the widest departure from the prime-vertical. The curve constructed with any latitude will have its own greatest ordinate. In the system of curves the maximum of these greatest ordinates must be the maximum ordinate of the envelope ; and the curve will be tangent to the envelope at the extremity of this ordinate. The writer has not yet been able to find, analytically, the exact latitude which will give this maximum ordinate. The approximate numerical value of the latter, and that of the corresponding abscissa (found in (/)), when substituted in the equation of the curve expressed as a quadratic in A, (287), will give an approximate value of A, whence the desired latitude. Assuming the roots equal (the numerical solution verifies the equal roots), we have A __ V -^ Substituting the values of x and y, the computation gives A = 1.40242 = (tan L = 54 30' 33-" ' 93. Locus No. i. 2d Branch. (a) By (123), / + *> + 2A/-f2A**-.y = o (310) By terms of highest degree y(y*-\-x*} = one infinite branch whose asymptote is parallel to X. AZIMUTH By highest power of x, y -J- 2A = o ; /. y 2A . . gives asymptote. The curve does not cross its asymptote. This asymptote cuts the meridian, Y, at twice the distance from the origin that the asymptote to the 1st branch of this locus cuts it, for by (256) y = A. (b) To find where the curve cuts Y, put x = o ; /. y -f- 2A/ y = o, or y (y 9 -|- 2 Ay i) = o ; y = o origin, and y = A 4/F+~A a ; P and P' (313) To find points on X, put y = o; .-.x y = o (314) Tangent at origin, y = o, axis of X. > Tangents at P and P', parallel to X. )' (3* 5) (c) Limiting form for L = o, A = o. - y = o, or y (y* -J- & * J ) = '> y = o, axis of X, prime-vertical ; xs o , prme-vertca ; ) r ('* i ( = i, primitive circle, horizon. ) ' y -f- X* (d?) Limiting form for L = go, A = oo . 2Ay + 2 A^ a = o, y -f- #* = <> origin, zenith ...... (317) The 3d branch the meridian, axis of Y. 94. Locus No. 2. Alt. -azimuth, error in L. Articles 44 and 60. Branches I. Absolute min., the six-hour circle. 2. Absolute max., the meridian. 1st Branch. Though a great circle, easily constructed, yet, as interesting, a brief analysis is given. By(l26), / + 2Aj-f^ l = o ........... (318) There is no infinite branch. Points on X, x= I, E. and W. ; . ........... (319) Points on Y, y = A V\ -+- A 2 , P and P' ......... (320) 64 AZIMUTH. Tangents at E. and W., j = -r-, y = -^ . . . . (321) Tangents at P and P' parallel to X. 94a. Locus No. 3. Altitude-azimuth, error in d. Articles 44 and 61. Branches the same as in No. 2, but the six-hour circle a curve of true max. and min., giving numerical min. 95. Locus No. 4. Time-azimuth, error in t. Arts. 45 and 62 to 67. Branches i. Alge- braic max. and min., giving numerical min. 2. Absolute min., curve of q = 90. 3. Curve of algebraic max. and min., giving generally numerical max., the meridian. (a) By (155), / + 2*y-f*> + A/ + 2A*'/ + A* 4 - A/- Ax*+y = o. . (322) Highest-degree terms, y (y -}- ^r 2 ) 2 = o, ........... (3 2 3) gives one infinite branch only, that having an asymptote parallel to X. The coefficient of highest power of x, A -\-y = o ; /. y = A, asymptote, ........ (324) the projection of the parallel of declination, d = L (see art. 92 (#)). (b) Points on X. If y = o, x* (l * a ) = 0, x* o, origin ; x = i, E. and W Points on Y. \ . \ + A/ = o, origin ; and y -f- Ay Ay -j- i = o, _?.} ' (326) r , (See (^ to (0-) (<:) Tangent at origin, axis of X. Form of curve at origin, y = A*", same as in No. i (see art. 92 (d}} ...... (3 2 7) (d) For tangent at E. , g , /. for tangent at E., y = A* ; } similarly, tangent at W., y = Kx. \ AZIMUTH. 65 (e) For points on Y other than the origin. Returning to (326), we have y + Ay - A>/ + 1 = o ........... (330) Not having been able directly to factor (330), recourse is had to the equation to the curve on A sin h cos h the surface of the sphere, (153), cos Z = - . a , . The condition required to give , i. ~] sin /? points, other than the origin, on Y is Z o or 180 ; /. cos Z i, and we have (i -j-sin 1 ^) = A sin h cos h .......... (33 1 ) (/") From (330) we see that, with L = o, y is imaginary, and no point on Y ; and with L = go , when A = oo, we have_^ (y i) = o; y = o and y = I ; .'. the two points on the primitive at its intersections with Y ; that is, on the sphere, at the intersections of the meridian with the horizon, giving h = o for each point. Now, by (331), for any permissible values of L < 90 to give points on the meridian, sin h, hence h also, must be positive for cos Z = -f- i ; and negative for cos Z = -- i ; that is, -\- y cannot exceed unity, and y cannot be less numerically than unity. (<") Squaring both members of (331), we have I + 2 sin 2 h -f sin 4 h = A 2 sin 2 h cos 2 h = A 2 sin 2 h (i sin 2 /i) = A 8 sin 8 h A 2 sin 4 h, (332) 2 A 2 i whence sin 4 h -f -pqp^i sin 8 h = - t , A ........ (333) Solving (333) as a quadratic, (A 8 - 2) A - (334) . , A 8 - 2 I/A 8 - 8 , . sm h -- - - ....... (335) By trig., cos # = -j - a~r", which by the principles of stereographic projection gives I r" tclll 0*3* i y coss =-- i; ........... (336) whence, taking the positive sign of the radical in (335), A 2 - 2 -f A ^A 8 -8 i - / j r+? = 66 AZIMUTH. squaring both members of (337), clearing, and dividing by the coefficient of y\ 6A a 2A 4/A 3 - 4 + A 3 - I. (338) Solving (338) as a quadratic, _ 3A 8 + A I/A' - 8 |2 (A 4 + A 3 VA a - 8 - A a + A i/ATTg - 2)|* 4 + A' - A |/A^8 5 ' (339 ' whence the two values of j, rejecting the inadmissible signs, to conform to the conditions in (/), are y = 3A 2 + A i/A 3 - 8 - 2[2 (A 4 -f A 3 V A a - 8 - A 3 -f A V A 3 - 8 - 2)]* ) * A 2 -A4/A 3 -8 }*(340) 3 A a + A VA* - 8 -f 2 [2 (A 4 + A* i/A 3 "^ - A 3 + A l/A 3 "^ - 2^1* - ~ (/) There remain two other values of y found by taking the negative sign of the radical in (335)> whence TT?^ ( A 3 - 2 - A V A 3 - 8 2(1+ A 3 ) (342) Solving, as in (K), rejecting inadmissible signs, we have 3A' - A I/ A 3 - 8 - 2 [2(A 4 - A 3 VA* - 8 - A'- A 8 - 2)] * * ( 4 _j_ A 3 + A V A a - ( 3A a - A t/A' 8 + 2 [2 (A 4 - A 3 (343) 8 - A* - A 4/A 3 - 8 - 2)] * ) * - [ (344) (^) It is obvious that these values of y are imaginary for any value of A* < 8 or A < 2 t/2 [also seen from (335)]. Hence, with the lower latitudes, the curve does not touch the me- ridian at any point other than the origin. For the latitude whose tan is 2 4/2~ we have two pairs of equal roots, for then (340) and (343) become identical, and (341) and (344) identical. What have been in the lower latitudes continuous branches of the curve on each side of the meridian, passing through the zenith and E. and W. points (Fig. 6), at the instant A = 2 4/2, cross on the meridian. (Fig. 7, for one pair of equal roots above the horizon.) AZIMUTH. 67 FIG. 6. FIG. 7. FIG. 8. Similarly below the horizon for the other pair of equal roots. But when A becomes greater than 2 4/2 by the smallest increment, the equal roots separate, and we have two different roots, and the branches of the curve separate as in Fig. 8. (/) To investigate for the points on the meridian, Y, where the equal roots occur: I st. The altitude of the point. In (334) substitute A = 2 4/2, A 2 = 8; whence sin' h = sin h = V$ = =. (345) Hence two branches of the curve cross each other, above the horizon, on the meridian, at an altitude whose sine equals -f- =; the zenith-distance of this point having the direction v$ towards the elevated pole. And the lower branches cross on the meridian at a negative alti- tude numerically equal to the positive, reckoned towards the depressed pole. (;) 2d. The declination of the point. To find the parallels of declination passing through these points. Taking the equation in terms of /, L, d (139), the second factor gives 2A cos d i cos t : - cos t -\- _ ^ j A 2 = o ; sin d cos 2 d (346) cos / = i, whence i 2A cos d i sin d cos'' d -A 2 =o;. . . . (347) 2A cos d i T~ A I -\- - , .TV - sin d cos 2 d' (348) Substituting A = 21/2, 4 4/2 cos d cos 8 d' (349) 68 AZIMUTH. 32 cos* d 14 i Squaring both members, -^^r- =49-^7^ H-^r clearing of fractions, 32 cos 6 */= 49 sin 2 ^ cos 4 d 14 sin" ). (o) As a matter of interest the approximate values of the latitude, etc., for the occurrence of equal roots may be found. Lat. 70 32', co-L = 19 28', A -p = 35 16, z = d= 54 44', - d' = 54 44', *' = (180 - d'} = 125 16', - h' = (180 -/) = 35 16'. Conclusions, looking at the change in the character of the curve in high latitudes. In Lat. 70 32', the most favorable position of the star whose dec. is 54 44' is on the meridian at lower culmination, as far as possible from the prime-vertical in bearing. Where this star's parallel of declination crosses the meridian is the point of intersection of two curves of algebraic max. and min. ; namely, the curve under present discussion and the meridian. It has been asserted that the meridian is generally the locus of numerical max. ; though the numerical maxima would not be equal numerically for the two culminations of the body, yet at both culminations we shall have maxima as compared to the numerical minima on the first branch of curve No. 4, at the points of the star's crossing it, for stars whose d < L. 7 AZIMUTH. This holds good for all such stars with latitudes less than 70 32' (approx.). With higher latitudes there may be a belt (a belt above the horizon, also one below) crossing the meridian, within which stars having certain declinations between the limiting declinations of the points at which the two branches of the curve intersect the meridian, will not touch the curve of numerical min. For such stars there will be a numerical max. and a numerical min. on the meridian, corresponding to algebraic max. and min.; for there exists no other minimum with which to compare. From (58) we see that the numerical min. will occur at that culmination of the star that has the less altitude, whether positive or negative. But the sign of (58) being negative, the numerical min. will be the algebraic max., regarding the error in t as positive. We therefore see how erroneous may be the statements italicized in arts 5 and 9, when ap- plied to observations in very high latitudes ; for, with declinations permitting the star to cross the first branch of the curve, the best position may be very near the meridian in azimuth. (/) For limiting forms of the curve, L 90, A = oo. From (322) we have (x* + y) (i x* y 1 } = o; ......... (360) X* -j~y = O, the zenith, a point ; := I, the horizon. L = o, A = o, ^r> + 2^y+/ = o; j(*'+y) = o; . . . (362) y = o, the prime-vertical ; ) x* + y = o, the zenith, a point, f ' 96. Locus No. 4. Second branch, q =. 90, for absolute min. The same as in Locus No. I, art. 93. <)6a. Third branch. The meridian for algebraic max. and min. ; numerical max., for all latitudes below that for which tan L = 2 4/2, at both culminations of the star ; the max. at the culmination having the less altitude, whether positive or negative, being the less. For higher latitudes than tan L = 2 4/2, stars within certain declinations will have one numerical min. and one numerical max. on the meridian, and will not touch the first branch (see art. 95 (0)). 97. Locus No. 5. Time-asimuth, error in L. Arts. 45, 68 to 71. Branches I. The horizon for absolute min. 2. The meridian for absolute min. 3. The branch given by (179), for algebraic max. and min., giving numerical max. (a) y + *y - *y - * + 2 A/ + 4 A*y + 2 A*> - ** = o. Highest-degree terms give (y x) (y + x)(f + #*)' = O .......... (365) Hence infinite branches having asymptotes. AZIMUTH. 71 To find asymptote parallel to (y x} = o we have from (364), retaining only highest- degree terms and those of the next degree, (366) Also, .*. y x = A and y -f- x = A for asymptotes. . . . (368) There are no parabolic infinite branches. (b) Points on X. Put y = o. -x'-2x<-x* = o; .-.*(*'+ ,)' = o; (369) whence x 3 o, origin, zenith, and (V -f i) a = o, imaginary (370 (c) Points on Y. Put x = o. y 4- 2Ay 2y 2Ay -f y = o ; -i) = o. > (37I) y = o, origin ; y i, N. and S. points. \ (372) y A Vi 4- A 2 , P and P 7 , poles. (d) For tangents at origin, y -*" 18 = o, jj/ = ^r and ^/ = x tangents (373) ( 2, when A < 2. Hence when A > 2 the form is shown in Fig. 9; and when A < 2, as seen in Fig. 10. FIG. 9. FIG. 10. For latitudes 30, 45, and 60 the form as in Fig. 10. For S. point put y = y I. The coefficient of y is 4A; always . The coefficient of x* is 2A 4; always . /. the form is always as in Fig. 10. (K) For form at P and P'. Moving origin to P and P', y = y -f- b, in which b is A -f- I/A* -f- I, for P, and - A I/ A 2 i for P', we have, Coefficient of y is \6P + ioA 4 - 8 3 - 6A' + 2b\. Coefficient of x* is \V -f 4A 3 4^ 2A i }. For latitudes 30, 45, and 60, substituting the values A = l/f, i, and VJ, and the corresponding values of b, we find form at P and at P 7 for all these lati- tudes, as shown in Fig. n. FIG. n. (/) To determine where the curve cuts the asymptotes. AZIMUTH. In (364) let y = x A, and we obtain (4A a + 6)* 4 - (8A 3 + I2AX+ (8A 4 -f 8A'+ 2)x*- 73 2A a - 2A)* - (A 1 - A 8 ) = o. (378) When A = V\, this equation has one positive and one negative root, both less numerically than unity, and has no other real roots ; when A = i, one positive root less than unity, one root equal to zero, and no other real roots ; when A = 1/J, there are no real roots. Similarly for the other asymptote, substitute y = x A. (K) For limiting forms of curve. If and by (364), L = 90, A = oo, (379) x 1 -|- y = o, conjugate point at origin, zenith; x* -\- y = i, horizon. If L = o, A = o. - 2 -*' = 0. . . ( 3 80) The curve is symmetrical with regard to both X and Y ; tangents at origin, y = x ; asymptotes, y = x. The curve crosses asymptotes at origin only, has inflexion at origin and is shown in the accompanying diagram. 98. Locus No. 6. Time-azimuth, error in d. Arts. 45 and 72 to 75. Branches i. The meridian for absolute min. 2. The curve of algebraic max. and min., giving numerical max. and min.; given by equation (194). (a) Highest-degree terms give 2.d Branch. 2 A / + x 1 - o. (381) (382) whence two asymptotes parallel to y - x, and there are no parabolic infinite branches. 74 AZIMUTH. For asymptote parallel to y = x, =-A (383) -, = * For asymptote parallel to y = x, y = A i/A a + i, poles (c) Tangents at origin, y = x. For tangents at E. and W., differentiating (381), x whence y = x A, and y = x A are asymptotes. . . . (385) (b) Points of curve on X, put^ = o. x* x* = o ; /. x* = o, origin, zenith ; (386) x = i,E. and W. points (387) Points of curve on Y. Put x = o ; f + 2 Ay y = o; -'-f = o, zenith ; , /J4I A ** ___ ^ * A A *VM/ T uy f\x AX A.t\xy " : _|_ faRni dx 4.y 2y -\- 2Kx* -i- f^A^ \*= i ~ - A ' WSV ' y = -T- are tangents at E. and W, (39) (d) To ascertain the form of the curve at Pand f, put y -\- b for_y and move origin to (o, b\ The coefficient of y becomes 4^' + 6A a 2b (391) The coefficient of x* becomes 2 Kb -J- I . (392) For point P, b = 4/A' + i A, and the AZIMUTH. coefficient of y is always -J-, coefficient of x* is always -j- ; .-. at .Pthe form is always as in Fig. n, art. 97 (h). 75 For point P', The coefficient of y is always . The coefficient of x* is -(- when A < |/T giving form The coefficient of x* is - when A > ^ g j v i ng f orm same as at P. 0) For curve crossing its asymptote. Combine first of (385) and (381). Eliminating/, we have x = t u- u for which I + A' |/i -A 4 2A (393) (/) Therefore, when A < I, asymptote crosses curve in two real points ; when A = i, asymptote is tangent to curve (and coincides with the tan- gent to the curve at E. and W. points) ; when A > i, asymptote does not cross the curve. (g) Combining second of (385) and (381), we find A 2 ) VT^"A 4 2A _ i - A a Vi - A 4 . , (394) and the same condition as by (/). (h) For limiting forms of curve. If L = o, A = o, equation (381) becomes y" x" f-\-x*=o', (395) (396) 76 AZIMUTH. Two line^ = x,(Z = 45 and 135), and primitive circle, horizon. (i) If L = 90, A = oo, (381) becomes 2Ay -(- 2A;r 2 j o ; j(y + *') = o; (398) whence axis of x and conjugate point (zenith). 99. Locus No. 7. Time-altitude-azimuth, error in h. Arts. 46 and 76 to 79. Branches I. The prime-vertical for absolute max. 2. The meridian for absolute min. 3. The hori- zon for absolute min. 4. For algebraic max. and min., giving a numerical max. by equation (207). The first three branches are defined. Branch. By equation (207), *y + 2 Ay - 4A*y - 2AS +/ = o. (399) (a) Highest-degree terms, y* (y* -f- 2^y -f-^" 4 ) ; .......... (400) gives no real infinite branches. The terms 2Ajr" -j-^V 4 , put equal to zero, indicate a parabolic branch, (40i) (b] To determine contact of curve with axes. Points on X. Put y = o ; 2Ay 2A^r 4 = o; .. x" (x* i) = o ; (402) Whence x" = o, ze nith ; (403) x i, E. and W (404) Points on Y. Put x = o ; y + 2 Ay 2y 2 Ay -i-y = o; AZIMUTH. 77 = o, zenith ; \ y = i, N. and S. ; >- (406) y = - A Vi+A\ P and P'. ) /_.... (407) For form at origin y* = 2Ajr 4 . . . . (c) For tangents at E. and W. dy-\ - -- = - ---- = A; ........ (408) j/ = A^r, tangents at E. and W ......... (49) (d~) To determine form of curve at N. and S. Put y y -\- i, and move origin to N. Coefficient of y is 4A, always -}-. Coefficient of x? is 2A 4 ; if A < 2 ; -f if A > 2. Hence, when A < 2 When A > 2 From next higher terms when A = 2, same as A > 2. Put y i for jj/, and move origin to S. Coefficient of y is 4A. Coefficient of x* is 2A + 45 ' at S. always as A < 2 for N. (e) To determine the form of curve at P and, P'. Put y -(- b for y, and move origin to (o.#). The sign of the coefficient of y is found then to depend on 3|, (410) 7 8 AZIMUTH. and the sign of the coefficient of X* to depend on (411) Substituting the values of A and of b for the cases of the curves drawn, lats. 30, 45, 60, we find from the signs of (410) and (411) the form of the curve at P and at P' to be the same as at N. for A > 2. The values of A and b being as follows : For P, since b = A + 4/A a + i, Lat. 30; A = l/, b = Vl>\ Lat. 45; A= i, b = V~2 I ; Lat. 60 ; A = 4/J, b =2 V$. For P', since b = A ^A' -f i, Lat. 30 ; A = VI b = - V$- Lat. 45 ; A = I, b = \/2 i ; Lat. 60 ; A = i/J, b = - i/J - 2. (/) To find circles of altitude tangent to locus. Let *'+y = c' ............. (412) Substitute in (399) X* = c* y* to eliminate x, and we have a cubic in y : y (i + <7-4X- 2^(1-0 = 0; ....... (413) ~ - / x Or > /- ( Tqr?7^- - a - 2x" = o ; . . . . (419) i) = o; ......... (420) whence peculiar point (/ + * 8 ) = o, zenith ; .......... (421) x 3 +y = I, circle, horizon .......... (422) (K) L = o, A = o, in (399) / + 2^y + ^y-2y-6^y-44r>+/ = o y + 2^> 4 + jry - 2j/ 4 - 6^y - 4** +/ = a Highest-degree terms y 1 (y + ^ a ) s = o, gives infinite branches whose asymptotes, parallel to X, by coefficient of highest power of x are y-4 = o, y = 2 ........... (424) For form at origin y 1 = 4%* ; i y = *> - ~V^- ...... (425) (*) For contact with X and Y. 4 = o, at origin only, zenith ......... (426) y_ 2 y + y =0; (427) 8o AZIMUTH. y = = o, origin i, double 1 o . . . (428) points. ) (/) For form at (o, i) move origin, and lowest-degree terms give tangents, x* = o ; ) = x. \ (429) (/) Combining (424) with (423), we find that the curve does not cross its asymptotes. Form when A = o. 100. Locus No. 8. Time-altitude- azimuth, error in t. Arts. 46 and 80 to 83. Branches I. Absolute max., the prime-vertical. 2. Absolute min., the six-hour circle. 3. Algebraic max. and min., giving numerical max. and min., the meridian. We have, also, the equator for a branch giving a constant error. Though all branches are great circles, and therefore directly defined in the projection, yet it may be of interest to analyze the equation to the equator ; that of the six-hour circle having been treated in art. 94. The equation to the equator, by (223), is (430) From y -f- x*, there is no infinite branch. Contact with X, put Contact with Y, put = i, E. and W. (431) 'y = A (432) AZIMUTH. 8 1 For tangents at E. and W., move origin to ( i, o), y = ^f hx, tangents at E. and W. . . ...... (433) Limiting forms, L = o, A = o, y = o, axis of X, the prime-vertical ; L = 90, A = oo, y* -\- x* = i, primitive circle, horizon. / 101. Locus No. 9. Time-altitude-azimuth, error in d, Arts. 46 and 84. Branches i. The meridian for absolute min. 2. The prime-vertical for absolute max. 3. The curve of elonga- tions, q = 90, for algebraic max. and min. giving numerical min. The third branch has been treated in art. 58. It is noteworthy that this curve, q = 90, though it has occurred frequently in the loci preceding No. 9, yet in no case heretofore giveh has it been the locus of algebraic max. and min. 102. Locus No, 10. Time-azimuth and altitude-azimuth, for error in L, giving the same numerical error in the computed azimuth. Arts. 85 to 90. Branches i. Errors equal, hav- ing signs alike, the prime- vertical and the curve of q = 90. 2. Errors equal numerically with contrary signs. id Branch. By equation (253), / + 3*y + 2* 4 + 2 A/ -f 2hx*y - / - 2*' = o ...... (434) (a) Highest-degree terms have no real factors; therefore, there is no infinite branch. No real factors in terms of lowest degree ; hence, no branches through the origin. (b) For contact with X and Y. Put j = o, 2* 4 - 2*' = o, **(** i) = o; ...... (435) x* = o, conjugate point at origin, zenith; ) *=i,E.andW. f ...... (436) Put x = o, y + 2A/-/ = o, or ?(f + 2hy- i) = o; . . . (437) y = O, conjugate point at origin ; y = A Vi -\- A", P and P 7 . . . (438) (c) To find form of curve at E. and W. Moving origin to (i, o) in (434). Terms of lowest degree, 82 AZIMUTH. 2.X , _ give y = x for tangent at E. ; 2.X similarly y = -r- for tangent at W. y Numerical values of for different latitudes: x (439) 3.47 in lat. 30, 2 in lat. 45, 1.15 in lat. 60 (440) ( \ V2. For the curves drawn for lat. 30, 45, and 60, the smallest value of A is V~$ (lat. 30), which is greater than J ^2 ; hence, for the latitudes used, the coefficient of 3? is in all cases minus and the curve at P is of the form For P', substitute y A VA* + I for y, and we find the coefficient of y to be always , and the coefficient of x* to be always -f- ; hence, form at P' always as shown for P above. (e) For limiting forms of curve. If L = o, A = o, equation (434) becomes f -f $x*y* -f 2;tr 4 - / 2x* = o, = o, ) = o. i + ^ = i, .primitive circle, horizon; | -[- 2.T 1 ) o, conjugate point at origin, zenith, f ' If L = 90, A = oo, AZIMUTH. 83 the equation becomes 2y 3 -}- 2x*y = o, or y (}? -J- x*} = o ; . = . . . . . (443) y = o, giving X ; jj/ a -f- x* = o, conjugate point at origin. 103. Though this treatise is called Azimuth, and though an attempt has been made to present the subject in a new light on some points, no attempt has been made to put into the hands of the worker in the field the practical methods of observation for the elimination of errors. The Coast Survey office, in its appendices* to the reports, leaves nothing to be de- sired on the score of precision in work with refined instruments. For the navigator, provided with the sextant, there is, it is hoped, something gained by the investigations pursued, appli- cable as well to more refined work. Even with the sextant, when the observation on shore for determining the direction of the meridian cannot be taken at the most favorable instant, errors in the data may sometimes be eliminated by observing at the same relative points both east and west of the meridian. Even though the error -by the observation on one side of the meridian may be large, the mean of the results on both sides may leave no resulting error in the computed azimuth. But the azimuth problem has also been particularly useful in studying the variations of the astronomical triangle. For this, the problems on time and on latitude do not bring out so many truths. Inspection of the terms in which the errors are expressed usually suffices in the two problems mentioned. Not so, often, in the azimuth problem, though much will be obvious. .,, 104. A single instance will suffice. Taking the case dZ - ' -j dt (58) for error in Z, due to a positive error in the hour angle, which, by art. 34, employing the general astro- nomical triangle, is error in time. Having regard to the signs of trigonometric functions and to the resulting sign before the whole expression necessitates a knowledge of the approxi- mate values of the particular parts of the triangle found in the expression within the limits of quadrants. We look to the resulting sign of dZ to determine whether it is an algebraic max. or min. at particular points. If -j- and dZ changes from an increasing function to a decreas- ing function, we shall have an algebraic max. If -|-, and the change is from a decreasing function to an increasing function, we shall have an algebraic min. If and dZ changes from a numerically increasing function to a decreasing function, we shall have an algebraic min., though a numerical maximum negative. If and change is from a numerically de- creasing function to an increasing function, we shall have an algebraic max., but a numerical min. From (58), by inspection alone, we cannot tell whether there is any point, other than on the meridian, where the function changes from increasing to decreasing or the contrary ; but if there be such a point for -|- d < L, for instance, we know from the change that occurs on the meridian at upper culmination that here will be an algebraic min. (numerical max. neg.) ; and at the point of change before lower culmination we shall have an algebraic max. (nu- merical min. negative) ; at lower culmination an algebraic min. (numerical max. negative), not having, however, the same numerical value as at upper transit, the latter being greater; at the next point east of the meridian an algebraic max. (numerical min. negative). The curve * By Charles A. Schott. 84 AZIMUTH. passing through W, Z, E, as found in No. 4, determines for a given declination whether the star reaches any point on it ; then we can discriminate between the algebraic max. and min. If there is a point, we have already seen how to discriminate. If there is no point on the curve, as for certain declinations, when the observer is in very high latitudes, then we have the meridian alone to consider ; d being less than Z, at upper culmination we shall have an algebraic min. (numerical max. negative), and at lower culmination an algebraic max. (numeri- cal min. negative), for cos h will be greater for lower transit ; hence dZ numerically less but negative. 105. In respect to the elimination of errors by observing at symmetrically situated points on both sides of the meridian, notwithstanding a poor point of observation for one side alone: In No. I, No. 2, and No. 3, if the body is observed at the same altitude on both sides of the meridian, the error due to error in h, that due to error in Z, and that due to error in a? will be eliminated. Therefore, since in No. I, even though the error in h should not be eliminated, yet if the star is observed at q = 90 the error will reduce to zero, we should endeavor to observe at that point on both sides of the meridian ; therefore select a close circumpolar star. But if the star crosses the prime-vertical we should observe at its crossing the curve of min. errors for error in h on both sides of the meridian. In No. 4 the error in t cannot be eliminated by the mean of the results, but if q = 90 for d > L, it will be zero, and if d < L there is a point of min. error ; this point, then, or q = 90, should be selected on both sides of the meridian, for then, by No. 5 and No. 6, the errors in L and d will be eliminated. Therefore the value of a close circumpolar star to apply these conditions. APPENDIX. A SCHEME FOR DERIVING CURVES FROM THE VARIATIONS OF THE SPHERICAL TRIANGLE. I. The curves already obtained from the problems on azimuth suggest the possibility of obtaining a great variety of loci derived from the variations of the spherical triangle ; and, from the astronomical triangle, by regarding always L and d constant in finding the maximum and minimum effects of variation in any part, due to the variation in any other part, we may find many interesting curves, though the number will be less than if not restricted to L and d constant. 2. The ultimate equation in x and/ may be disengaged or not, at one's pleasure, from any idea of the sphere, since A (or B, or C) will be a single arbitrary constant (see art. 55) ; but the sphere furnishes the astronomical triangle as the means of obtaining the equations. Therefore there may be curves of interest to the mathematician, though having no utility. 3. To adopt a system for obtaining the equations to the loci, we have the following : 1st. Any four parts of the triangle involved, and any three of these given, the remaining part may be found. 2d. Form all possible groups of four parts. 3d. In each group, each part may be taken in turn as the one to find. 4th. In every case, the partial differential coefficient, for each given part in error and the part to be found, may be taken. 5th. Each partial differential coefficient* will have some kind of max. and min., as defined for the purposes of this treatise ; assuming always, in determining these max. and min., that L and d are constant, in order to conform to the nature of the case in the problems used in practice, even though L or d, or both, may have been variable in the problem giving the partial differential. 6th. If algebraic max. and min. occur, the differential coefficient's own first differential put equal to zero will show their existence from the equations derived, all parts except L and d varying. 7th. If there were an instrument to measure q accurately, and if our instrument for measuring 2T gives precision, and if, in nature, there were problems demanding Z and q as given parts, then all possible problems based on the astronomical triangle might be of utility. 8th. Lacking utility, all possible cases will still give examples in curve-tracing. 9th. Great circles of the sphere will probably recur many times in the loci of the preced- ing article, as they have done in the curves Nos. I to 10 defined in this treatise ; and possibly the same novel loci that those curves give, or that may be found in the future, will also recur. loth. Making the combinations, we have 90 individual cases; each one having a recipro- cal, which need not be considered, thus making 180. * Excepting when it consists of functions of the constanis. only. 86 APPENDIX. nth. In these ninety cases are included the equations to curves found in this treatise, and equations for additional useful problems (time ; latitude ; computation of the altitude in the problem of astronomical bearing; and the time-altitude-latitude-azimuth, in distinction from time-altitude-declination-azimuth, called in this work time-altitude-azimuth). I2th. In making the cases for partial differentials, the two constants will, of course, be given parts in the original problem, and either one of the variables will be another given part ; the remaining variable, a part to be found. 1 3th. For illustration : 1st A, b, and c, being given, to find B. A and b constant, c and B variable, we shall indi- dB^ dB_ dc ' dc. cate by writing A, b, -j\ 7- being the coefficient of error in B due to a small error in c. For max. and min. effects, we have d \~~r~] = O regarding b and c (corresponding to co-Z- and co- \d * din the companion triangle), always as constant in this part of the work. 2d. For the reciprocal : dc Given A, b, and B, to find c. A and b constant, B and c variable, A, b, -T~, error in c, due to error in B. The reciprocal will, in all cases, be omitted as giving the same locus as the first form, but having max. and min. interchanged ; and to simplify the indication we shall omit the symbol . B a, writing only A, 0, . C I4th. For combinations of four of the six parts A, B, C, a, b, c of the triangle we have 1. A,,C,a (6) 2. A, B, C, b (7) 3 A,B, C, c (8) i 4. A, , a, &. (3) 5- A,B,a,c (4) 6. A,B,b,c (2) 7- A, C,a,& (9) 8. A, C, a, c (10) 9. A, C, &,c (11) 10. A, a, b, c (5) 11. ,C,a,6....- (12) 12. B, C, a,c (13) 13- B,C,b,c (14) 14. B, a, b, c (i) 15. C, a, b, c. (15) APPENDIX. 87 The numbers (i), (2), and (3) show the parts taken for the curves described in this treat- ise. (4), the parts for the time-altitude-latitude-azimutlt ; and (5) those for the time-sight, the problem of latitude at any time, and the computation of h for use in the astronomical bearing. 1 5th. From the astronomical triangle, A = t, B Z, C=q, a = co-h, b = co-d, c = co-L ; we derive the following cases for loci : (1) Parts involved, , a, b, c; Z, h, d, L. D ^ a, b, ; h, d, -j \ alt.-az., error in L (a) C J_^ B Z a, c, T ', h, L, -j : alt.-az., error in d. (b) B Z b, c, ', h, L,-r : alt.-az., error in h . . . (c) Ct rL h // B,a,-^;Z,h,- (d) B,b,-;Z,d y j^ (e) a h B, c, -T', Z, L,, (/ ) (2) Parts involved, A, B, b, c\ t, Z, d, L. B Z A, b, ; t, d,j: time-az., error in Z- (g) C JLs X" B Z A, c,-,; t L, -j : time az., error in d. (h) b a B Z b, c, -.;d,L,-\ time-az., error in / (i) A t B,b,-; Z,d,j. (*) c L, 88 APPENDIX. (3) Parts involved A, B, a, b ; /, Z, h, d. B Z A, a, -T ; t, h, , : time-alt.-dec.-az., error in d. () B Z A, b, ; t,d,j-: time-alt.-dec.-az., error in h (o) , B Z a, b, -;, A, a,-: time-alt.-dec.-az., error in t (fi) *i T A,B, a r , t ,Z h j fr) A t_ B,b, -; Z,d,- H ( S ) (4) Parts involved, A, J3, a, c\ t y Z, h, L. B Z A, a, ; t, h, j : time-alt.-lat.-az., error in L. T> rp A, c, ; t, L, j : time-alt.-lat.-az., error in A. T> ^ a, c, -7; h, L, : time-alt.-lat.-az., error in t. jfi t r> rr i ( ,\ B,a,-; Z, h, - L B, f ,^;Z,L,^.. a h (5) Parts involved, A, a, b, c; /, h, d, L. A t ( time-sight, error in L. j c ' L ( reciprocal, for lat. problem, error in t. r - (A) A . t a, c, ; A, Z,, : time-sight, error in d. n$\ , A t ( time-sight, error in h. b, c, a, L, -7 : { a n ( recip., alt. computed, error in t. c L A, a, -r ; /, h, -, : lat. problem, error in d. (D) . , c L ( lat. problem, error in Ji. ) , ' a ' ' h ' \ recip., alt. computed, error in L, } a r h A, c, T ; t, L, = : alt. computed, error in d. APPENDIX. (6) Parts involved, A,, C, a; t, Z, q, h. *9 a n A C B , Z - A, (-, - , t, g, fi. B , Z A >V / M A, a, c , *>**- B C^- Z a*- B > 6 ' a ' * 9 ' h' A , t ,,, B, a, -~; Z, k,-. (^) s* A r * (M\ I SI /7 h ....... \LVJ. ) Ixj t*, TJ > ff) ft-t *y / -D ^ (7) Parts involved, A, , C, &; /, Z, ^, - ^ 6 'f ; '*! w 5 Z A,b,-\t,d,- (P) B,C,j\Z,i 9 j (Q) B,b,^',Z,q,- (R) A t C, b, -^ ; q, d, ^ A? ^ (8) Parts involved, A,B,C,c\ t, Z, ?, Z. ^^ ?. /z ?: -,^, -; r,^, r A,C,-', t,g,j. (^ c A A c - t L Z - -(V) **} c t S" i /7 " B C *' Z a- ' ' '( W ) Js, \s i i "> Y< 7* ^r -t> ^4 t B,c, --, Z,L,-. A t C,c, B \ q, L, g. 90 APPENDIX. (9) Parts involved, A, C, a, b\ /, q, h, d. a h a A (10) Parts involved, A, C, a, c, ; t, q, h, L. (n) Parts involved, A, C, b, c\ t, q, d, L. . b d C (a) C q % A ' "' ~b ' tj k ' d ( b ) (e) (0 a h A ' C ' c ; *' ?> L C Q A, a, - f, h, \ (h) /-* -; AA| . (o ^; g.hSj. (k) - L* m a ' 9 ' L ' h' A , t (m) (n) A L C J 9 A, b, - ; /, d, ^ (o) C q A,c, j-; t,L,j ( p ) A t I*, 0, = ; q, d, -j (q) (s) APPENDIX. 91 (12) Parts involved, B, C, a, b; Z, q, h, d. a B,C, V Z,q,' d . . . ..,. -,_. . . . . . . . (t) B, a, c 7 , q . . . (u) c q B,b, z ' d >'h- * (v) C,a, B b' Z . . . (w) B z C,b, a ' . . . (x) a, d, B Z . . (y) (13) Parts involved, B, C, a, c; Z, q, h, L. - ^. B,C, - ; **i- . ... (A) B,a, c ' Z ' h ' L' ' ' ' ' . . . (B) ' ' . . B,c, a '' Z L q ^^ h' . . . (C) C,a, B 7 ; q ' h 'L . . . (D) C, c, B Z . . . (E) a, c, C' Z ' V . . . (F) (14) Parts involved, B, C, b, c ; Z, q, d, L. B C b d . . . (G) ' c ', y, B,b, C , q . . . (H) 6 ** B c c Z L q ... (I) b' ' ' d' C,b, B qd Z - . . . (K) t. ** C, c, B b'' Z d . . . (L) b, c, ?. d, L, . . . (M) -^ , t*, J-r, 6 q 92 APPENDIX. (15) Parts involved, C, a, b, c \ q, h, d, L. b d C,a,-;q,k, 7 ............. (N) C J-* C,b t l; q.dfj. ............ (O) a r h C, c, - ; g, L, j ............. (P) >-^ a,b, -; A, 4 | ............. (Q) x- a, c, -; h, L, ............. (R) (S) 4. Fundamental equations from trigonometry for the solution of the various problems of finding any part of the triangle, three parts being given ; needed for transforming, in find- ing equations to the curve of max. and min. variations, in any three terms ; and used, if de- sired, for differentiating directly for the general expression of error in one part due to an error in some other part. For convenience, these equations are written here, and the parts involved written at the left hand ; (i), (2), etc., conforming to the groups given in the I4th and 1 5th sections of art. 3. B, a, b, c. cos b = cos c cos a -f- sin c sin a cos B ........ (i) A, , b, c. sin A cot B = sin c cot b cos c cos A ........ (2) A,B,a,b. sin a sin = sin b sin A ............. (3) A, B, a, c. sin B cot A = sin c cot a cos c cos B ........ (4) A, a, b, c. cos a = cos c cos b-\- sin c sin b cos A ........ (5) A, B, C, a. cos A = cos B cos C -\- sin B sin C cos a ...... (6) A, B, C, b. cos B = cos C cos A -j- sin C sin A cos b ...... (7) A, B, C, c. cos C = cos A cos B -f- sin A sin ^ cos ...... (8) A, C, a, b. sin C cot ^ = sin b cot # cos b cos 7. ....... (9) A, C, a, c. sin sin A = sin # sin C. ...... ...... (10) A, C, b, c. sin A cot C= sin ^ cot c cos cos A ........ (i i) B, (7, a, b. sin f cot B sin cot C = sin cot c cos # cos B ........ (13) B, C, b, c. sin b sin C = sin c sin B ............. (14) C, a, b, c. cos c = cos a cos -f- sin a sin ^ cos C. ....... (15) APPENDIX. 93 5. Additional formulas involving five parts, useful in transforming. A, B, a, b, c. sin a cos B sin c cos b cos c sin b cos A (16) B, C, a, b, c. sin b cos C = sin a cos c cos a sin c cos Z? (17) ^[, 7, #, b, c. sin c cos A = sin # cos a cos sin a cos ..... (i 8) A,C,a,b,c. sin # cos C = sin b cos cos b sin cos y2 (19) A, B, a, &, c. sin # cos yi = sin c cos # cos c sin # cos B (20) .#, C a, b, c. sin ^ cos B = sin # cos b cos sin b cos ....-. (21) A, B, C, a, b. sin A cos # = sin C cos Z? -j- cos C sin Z? cos (22) ^4, Z?, (7, b, c. sin Z? cos c = sin ^ cos C ' -\- cos y2 sin C cos (23) A, B, C, a, c. sin C cos # = sin B cos y -j- cos B sin y4 cos c. ... (24) y4, /?, C, a, c. sin ^4 cos c = sin Z? cos 7+ cos Z? sin C cos . ... (25) A, B, C, a, b. sin B cos # = sin C cos ^4 -f- cos C sin ^4 cos b (26) ^4, ^, ", b, c. sin C cos b = sin ^4 cos B -(- cos ^4 sin B cos c (27) 6. The following differential equations will furnish all the partial differential coefficients required, without recourse to differentiating the equation of the particular problem considered : A, a, b, c. da = cos C db -\- cos B dc -j- sin b sin C dA (28) B, a, b, c. db = cos A dc -j- cos C da -(- sin c sin A dB (29) f, 0, b, c. dc = cos B da + cos ^4 db -\- sin sin B dC. (30) Or, corresponding, t, k, d, L. dh = cos q dd -\- cos ZdL cos d sin q dt (31) Z, h, d, L. dd = cos t dL -f- cos q dh cos L sin / dZ. (32) q, k, d, L. dL = cos Z dh -f- cos t dd cos /z sin 2Td^ (33) 1st. Error in /. , , -. LOCI IN TiME-ALTITUDE-LATITUDE-AZIMUTH. t, h, L, to find Z ; A, a, c, to find B. Group (4) B sin B cot ^4 = sin c cot cos c cos .Z?. Eq. (4) sin Z cot ^ = cos L tan ^ sin L cos Zl 94 APPENDIX. From (28), o = cos C db -\- sin b sin CdA. ... ... (34) From (29), db = sin c sin A dB. ... . . . . . . , . (35) dB sin b sin C sin Eliminating 0, -r^ = ~ = ^ 7*= - 7^ (3 6 ) dA. sin c sin yi cos o sin # cos C ^? = cos ^ ( . 7) dfr COS // COS <^* S**' It is obvious that the curve of elongations is that of absolute max. Putting d (2d member) = o, eliminating dh and dq, and simplifying, we obtain sin Z (cos d cos q sin h -j- cos L cos 2T) = o (38) sin Z = o, the meridian one branch. The other branch by sin b cos C cos # -j- sin c cos B = o (39) To turn into B, a, c, substitute from (17) sin b cos C, sin a cos cos c cos 2 # sin c cos .# -j- sin c cos ^ = o (40) Substitute cos 2 # = I sin 8 a. sin c sin 2 # cos B = sin # cos # cos c (41) cos = -r- cot <2 cot c (42) cos Z = tan A tan L. Equator (43) No novel curve, and (43) proves to be a curve of constant error; for, when d = o, by substi- . . dZ i tuting (19) in (37) we have -r- = 7-. dt sin L Meridian for algebraic max. and min. gives numerical min. at that culmination of the star having the less altitude, and numerical max. for greater altitude. But both these are numeri- cal min. for stars whose d > Z, as compared with absolute max. for q 90, 270. The equator divides the meridian into branches of algebraic max. and min. 7 D 2d. Error in L. t, /i, j ; A, a, ^. By (28) and (29), db = cos A dc + sin c sin A dB; (44) o = cos C db -\- cos B dc . (45) Eliminating db, cos C cos A dc -f- cos C sin c sin A dB -(- cos B dc = o ; . . . . (46) dB _ cos C cos A -\- cos B dc cos C sin c sin A sin C sin A cos b , . and by (7), = - ^-r : z , . . . ........ (47) cos C sm c sin ^ ^2T tan q sin B l/i - B 2 (i - r 2 ) 2r 4 = o. (65) By - 6Bvy + Bvy - 6B 4/1 - BV/ + 6B 4/i - B 1 X r* 2B4/I -B f xr 4 2B i/ 1 - B 2 X r 8 = o. APPENDIX. Substituting and and we obtain, -f / - 6B'>V - 6B 2 / + B>V + 2BV/ + B 2 / - 6B i/i - - 6B Vr 6B I/ 1 - B>V + I2B 1/i^ + 6B 1/i - B 2 / + 4*> -f 8#y + 4/ + 2B Vi - BV + 46 l/i - BV/ + 2B Vi - By - 2B 4/i - BV-6B i/i -BV/ - 6B i/r^^BV/ - 2B i/7^=~By = o Collecting coefficients of /, /, etc., 97 (65) + 6B 4/1 - B 3 - 6B 3 - 6B 4/i - B 3 + "D* J3 - 6B 4/1 - BV + 4*' - 2B 4/i - B a -f 2BV 4- I2B 4/i -- BV - 6BV 4- 6B 4/i BV (66) + 4 + 2B 4/1 -- B 3 4- BV 4- 46 4/1 BV 6B 4/i - BV 4- 8*' - 6B 4/1 - BV 2B ^i - BV, 26 V i -BV, absolute terms, reducing, B 1 / 4- 48 4/i - B 3 / - 6B 3 / 4- 2BV/ 4- 4/ - 46 4/1 - B'/ 4- 6B 4/i - BV/ 4- B 3 / - 6B 3 * 3 / 4- BV/ (67) 2B 4/i BV/ 4- 4*> 4- 2B 4/i BV 2B 4/i BV = o. Dividing by the coefficient of/, and arranging in order of terms of highest degree, 2 . 5 , 4 3 ,4 < /i B a e 6 4/i B 2 , 24/1 --B 3 6 *> + */-* B >* + ~*y /rirs v/ t~ g x y 4---S B 4 i -t/ 3 o 44/1 -- B 3 4 24 2 4/1 - B 3 B B B i ^ u > . . (68) which is the equation to the stereographic projection of the locus of algebraic max. and min. for error in h in time-alt.-lat.-azimuth. 8. TIME-SIGHT. Given h, L, */ d, ^ find t. cos # = cos b cos 4" s ' n ^ sm ^ cos ^ sin ^t = sin d sin Z. 4~ cos d cos Z cos /; . . (69) 9 8 APPENDIX. from which are derived the well-known formulas, /cos s sin (s ft) . , , . T .. /' = y C os Z sin/ ' m which * ==l(-f / -f *)? (?o) /sin j sin fa z) = V sinco-Xsin/ m wh ' ch -* /.ji****.^*.** i * *' / 1*1 1 / 7" I *. l 7 \ /\ cos ^ = A / - -, . /, m which s = i (co-Z +/ + A); . . (71) cos ^ sin * i / r i / i ru^T)75rTr-^)' * = *(+*+/). . (72) Small errors in the data will have the same effect on the hour-angle computed by each of these formulas. So far as inexactness in the logarithmic tables is concerned, (72) will give the result nearest to precision. If t > 90, (71) will be preferable to (70). If t < 90, (70) will be preferable to (71). 9. ist Case. Error in t owing to error in h. t A Group (5), d, L, ^ ; b, c, - . Differentiating (69), sin a da = sin b sin c sin A dA ; . . . . . . . . (73) dA sin a sin b i da ' sin A sin b sin c sin B sin b sin c sin B sin dt i i cos .; (74) rr- (75) d% sin ^ cos Z sin q cos d? sin t cos sin t sin h dh -)- cos h cos t dt = o ; (77) dt sin / sin h cos h " dh ~ cos t cos h ~~ sin / cos d cos Z ^ ^ ' ' ^ ' /. sin 8 t sin ^ cos Z cos d cos cos 8 h = o (79) From trig, substitute, i sin 2 h for cos 2 k, and sin Z sin d-\- cos Z cos ^ cos t for sin & Performing the operations and collecting terms, sin Z sin d cos Z cos L ........... (85) Lclll w^ Though in the problems of azimuth both of these have been found to be branches of numerical min. at the same time, yet both have never been derived algebraically from the given expression for error. II. 2d Case. Error in t owing to error in L. , , t A n, a, -j^', a, b, - . From (69) we obtain o = cos b sin c dc -J- sin b. cos c cos A dc sin b sin c sin A dA. . . . (86) dA cot b sin c cos c cos A Dividing by sin b --- ~ ~' ........ sin A cot B cot B And from (2), = -- : -- = ^- = -- : -- ; ......... (88) sin c sin A sin c dt i dL .'. -77 = r - ^ -- T dt^ = 7 - ^ -- T ....... (oQ) dL tan Z cos L tan Z cos L Inspection shows that the prime vertical and the curve of elongations form the locus of minimum errors in the computed time ; the former for absolute min. when d < L, and the latter for algebraic max. and min., giving numerical min., when d > L. It will not be necessary to derive the equation to the latter. The lower the latitude the better. 12. ^d Case. Error in / owing to error in d. t A ' ' ~d ' a ' c ' ~b ' Interchanging b and c, d and L in the preceding, we derive j* dt d = - - ............ (90) tan q cos d APPENDIX. 1 01 Inspection shows that the locus of numerical min. consists of the same branches as for error in L ; but here the curve of elongation gives absolute min. when d > L ; and the prime- vertical gives numerical min. from algebraic max. and min., f or d < Z,. Total error, dt = : ~ 7 dh 4- - - = ~dL -f- - ; L (which will not come on the prime vertical) and observe at greatest elongation, q = 90, when dt d = o. If observed at this instant, it matters not what the declination may be, between the limits d = 90 and d > L ; but, for failure to seize the star exactly at q = 90, the less the declination the better. (#) Supposing the star given us does cross the prime vertical, then [i] is true but [2] is false. (c] Since errors in d are less likely to occur than errors in L and h, and since these two errors will give the least dt when Z= 90, we would better select a star that crosses the prime vertical, when dth = cosec Z sec L dh = sec L dh, and dt L = sec L cot Z dL = o. (d} Therefore, for error in h and error in L it matters not what the declination is, pro- vided the star is observed on the prime vertical. (e) But for error in d, the most eligible star of those that cross the prime vertical is that having greatest declination, not as asserted in [2] preceding. (/) Therefore, for errors in all the data, theoretically, the most eligible star is the one crossing the prime vertical, that has the greatest declination ; that is, d = L, making Z = 90 and q = 90 when the star is on the meridian. Practically we should avoid stars on the me- ridian and take one whose d Z., provided it is observed at q = 90, then dt d = o. This excludes stars that cross the prime vertical. 2d. But Chauvenet makes his given star come on the prime vertical. Therefore d < L, 102 APPENDIX. and q cannot = 90; the error dt d cannot reduce to o, but it will be least for q nearest to 90, which will occur when Z = 90 ; that is, when the star is on the prime vertical. 3d. Of several given stars observed on the prime vertical the most favorable one will be that whose cos d tan q, when on the prime vertical, is greatest. And this will be for d = L. For notwithstanding the greater d is, the less the cos d, yet the greater the d the nearer will q be to 90, and tan q will be the greater; and tan q will preponderate, making cos d'by its side insignificant. Singularly enough, in (92) Chauvenet gives cos d overwhelming preponderance over tan q, provided the star is on the prime vertical ; whereas in dZ = 7 dh, of like form, he tan q cos h permits cos h to have no influence, and assigns the star to the prime vertical on the strength of tan q alone. Proof of Fallacy. Changing the form of Chauvenet's expression into one equivalent, we have, since sin Z cos L = sin q cos d, dt = . ^ Y dd. . . . (93) sm Zcos L Therefore, for any given star the nearer q and ^both are to 90 the more favorable; and as one nears 90 so does the other. For any one of several given stars restricted to d < L and to observations on the prime vertical, (Z = 90), cos L L being fixed, to select the best one of these stars we have to consider q alone, and take that one giving q nearest to 90 which will make cos q the least. sin Z cos L Now, since sin q = - j , * cos d ("*("\C j when Z 90, sin q -> (95) cos d Cos L being constant, q will be nearest to 90 when the declination is greatest ; or directly from (95) q = 90 when d '= L, and the star that is farthest from the equator, instead of nearest (as asserted by Chauvenet), is the most eligible. APPENDIX. 103 15- LATITUDE PROBLEM. Latitude by an altitude observed at any time. Given t, h, and d, to find L. Given A, a, and b, to find c. cos a = cos b cos c -f- sin b sm c cos A ; sin h = sin ^/ sin Z. -|-cos ^ cos L cos / ; (96) from which are derived the well-known formulas, tan =. tan d sec sin sin cos == sm # L = T 0'. (97) Attending to the algebraic signs of and 0', there will still be two values of L. The proper value will be determined by the known approximate value. 16. ist Case. Error in L owing to error in h. From (96), A and b being constant, sin a da = cos b sin cdc-\- sin b cos c cos A dc ; (98) da cos b sin c sin.# cos c cos A dc sin a (99) and from (16), sin a cos .Z? sin = cos (100) ~j~ da cos dh cos ^' By inspection, the most favorable position of the body is on the meridian, and the least favorable on the prime vertical, giving absolute max. If the star does not come on the prime vertical, the least favorable position will be on the curve of elongations, q = 90, 270. The latter condition is obvious, but it may be found algebraically, there being algebraic max. and in in. 104 APPENDIX. 17. 2d Case. Error in L owing to error in /. This is the reciprocal of (89), /. = tan Z cos L, ........... ( IO2 ) and the max. and min. are interchanged, as compared with art. n. 18. 3^ Case. Error in L owing to error in d. From (96), o cos c sin b db cos b sin c dc -j- cos b sin c cos A db -f- sin b cos c cos A dc ; (103) dc sin b cos c cos b sin c cos y4 sin c cos cos c sin cos (104) and by (16) and (19), sin a cos {7 cos C (105) sin a cos B cos B cos (106) The meridian is found to be the locus of algebraic max. and min., and this gives numeri- cal max. for d > L and numerical min. for _ d < L, since for q = 90 there is absolute min. and for Z = 90 an absolute max. In the curves discussed in this treatise heretofore, q = 90 and Z= 90 have been com- panions very often in giving, alike, numerical max. or min. But here they part company, one giving absolute min., the other giving absolute max. 19. THE COMPUTATION OF THE ALTITUDE. Given t, L, d, to find h. (a) For error in h owing to error in t we have the reciprocal to (75) and an interchange of max. and min. (b) For error in h owing to error in L we have the reciprocal to (101) and max. and min. interchanged. (c) For error in h owing to error in d, - a t, L,, A, C, Y. a n APPENDIX. I05 Differentiating (69), \ t sin a da = cos c sin b db -}- sin cos A cos # d# ; ( IO 7) / dfo -j- cosV sin b sin c cos # cos A db -\- sin # (108) , . / sin # cos C and by (19), = -5- = cos C. (109) Therefore absolute min. when q = j 9 [ and numerical max, when q = | 8 I from al- gebraic max. and min., the maximum being equal to unity. Numerical min. > o and < i for Z = 90 from algebraic max. and min. Loci. The curve of elongations and the prime vertical for min. The meridian for max. EXPLANATION OF THE PLATES. The numbers shown on the plates correspond to the loci numbered in the text. Each locus is shown separately for the latitudes of 30, 45, and 60, marked respectively A, B, and C. Each plate shows the stereographic projection of the sphere on the plane of the horizon ; the prime-vertical being the axis of x, and the meridian that of y. On each plate are drawn diurnal circles, i.e., parallels of declination, marked at their inter- sections with the meridian, the axis of y, as follows : d' for a star having -f- d > L ; d" for a star having -f- d < L ; d'" for a star having d < L d lv for a star having d > L. Diurnal circles are shown also for d = o, the equator ; and for the parallel d = L, as a right line passing through the nadir at infinity, and parallel to the prime-vertical, the axis of x. The diurnal circle -f- d = L is not drawn, but it may readily be imagined as tangent to X at the zenith, Z. The astronomical triangles are not constructed, since their lines would encumber and ob- scure the plates for their more important use, that of showing the curves of max. and min. errors. The triangles may easily be completed, mentally, by imagining the vertical circles, through Z, and the hour-circles, through P, as intersecting on the parallels of declination at the points discussed as giving max. and min. errors thus with co-Z,, already drawn, forming the triangle. Lines above the horizon are drawn full ; those below, dotted. The loci are discussed with regard only to the theoretically most favorable and least favor- able positions of the body (see Introduction, art. 3) ; and equal respect will be paid to points below the horizon as is paid to points that would be visible to the observer on the earth. Again, indetermination may occur at the point that otherwise would be theoretically the best, and the close vicinity to this point will be a favorable region. Therefore the point that theory declares to be the most favorably situated may be re- ferred to one that \* practicably the best from the theoretical point of view. The terms prac- ticably and practically should not be confounded, as the best practical conditions may be left for the observer to select, after presenting those that in theory are the best. Since true maxima and minima are such with respect to the values of the function that lie immediately on each side of them, neither the several maxima being equal to one another, nor the several minima equal to one another ; and since, also, of alternate true maxima and EXPLANATION OF THE PLATES. 107 minima, a particular maximum need not be so great as a minimum that is not alternately near; and since, moreover, disregarding signs, out of several numerical maxima of the func- tion, one may be the greatest, and one the least, in numerical value (and the same respecting numerical minima), therefore we shall avoid, in defining the points in the first instance, un- less there can be no mistaking them, the use of the terms most favorable and least favorable ; and shall say favorable and unfavorable, these terms corresponding respectively to numerical min. and numerical max., whichever algebraic sign the function may have. The terms most and least favorable may afterwards be used with discrimination. In no case of these loci has it been found necessary to have recourse to the second differ- ential of the expression for the error, in order to discriminate the maxima and minima. In No. I the method of discriminating is given, for illustration ; but it is deemed unnecessary to repeat this proceeding in detail in all cases, since what is obvious to the writer will be obvious to the reader. A positive error in the datum is assumed for the discussion of maxima and minima values in the error of the computed azimuth. Evidently, to make the expressions serve in practice as corrections, the proper sign of the error found to have existed in a given part must be used ; and, also, the sign of the expression for the error must, itself, be changed. It will be necessary to keep in mind the method used in this treatise in reckoning the angles /, Z, and q, and the explanation of the terms maximum and minimum, as here used. The reckoning of Z differs from that of astronomers, generally, who reckon from the south point of the horizon. But the writer considers the method used in this treatise the rational one for the purpose in view ; and there seemed no way of escape from making terms with the maxima and minima. (See pp. 19-21.) Though P in the diagrams may be either the north or south pole, whichever is the elevated pole, so long as d of the same name as the latitude is regarded as positive, and if of the contrary name as negative, yet, for convenience and uniformity, the plates are for north latitude. Therefore with respect to the plates we refer to points north and south, as well as east and west. In referring to circumpolar stars we may have those with d < L, possible only when L > 45; one with d = X, = 45 as the single case ; those whose d > L, which may be called close circumpolar stars, for they will possess ihe general characteristics of stars very near P, / being small, and they cannot have a d < 45. There will be stars to consider whose d > Z, yet not circumpolar stars, possible only when L < 45; and the stars whose d < L cannot be circumpolar ones when L < 45. In referring to the plates, only a brief of the matter from the text will be given, and abbreviations will be largely used. Locus No. i. ALTITUDE-AZIMUTH ERROR IN h. [Summary of pp. 25,26,31-36,55-63-] = ^ ^ (55) Branches: I. Absolute max., the merid. NFS. 2. Absolute min., curve of elongations Pa' Za" and e'P'e". 3. Num. min., from alg. max. and min., curve c'Eb'Zb" We". It is obvious that the merid. and q = 90 curves give no alg. max. or min.; for the sign of (55) depends on tan q, since cos h is always positive. (55) therefore changes sign when q passes through the values o, 90, 180, 270. At b", W, and c", q lies between o and 90; therefore tan q is (-}-), and alg. max. occur with respect to alg. min. at b ', , and c 1 ', where tan q is ( ), since q > 270 and < 360. But on a given par. of dec. these alg. max. and min. are equal numerically, and both are num. min. as compared with abs. max. on merid. Following the star in its diurnal course, beginning at lower cul., we find : For -j- d > L, at d\ abs. max. oo ; a ', abs. min. o favorable ; d', abs. max. oo ; a", abs. min. Q favorable. For -\- d == L, at low. cul. abs. max.; at Z indeterminate, but akin to a min. The slight- est departure from the zenith gives a determinate case, and a very small error, which increases continuously during the progress of the star towards low. cul., where oo occurs. For -\- d < Z., at low. cul. d", abs. max. oo ; b', alg. min., num. min. ( ), favorable; d", abs. max. oo ; b", alg. max., num. min. (-[-). favorable. For d= o, on merid. abs. max. oo ; at E and W, num. min., ( ) and (-J-) respectively, favorable. For d <. L, num. min. at c' ( ) and c" (-J-), favorable, but practicably in the horizon. For d= L. In the projection, this is the asymptote to the curve of alg. max. and min., meeting it as a tangent at infinity, the nadir. Practicably, the horizon gives its favor- able points. For L, abs. min. o at e' and e" , favorable ; practicably in horizon, if star rises. The unlettered line parallel to d = L is the asymptote to the lower curve of elonga- tions. In the projection it crosses the merid. at the distance 2 tan L from the origin ; and, from the origin, d = L lies at the distance tan L. The lower asymptote is, on the sphere, a small circle inclined to the equator, and is tangent at the nadir to the p. v. and to the curve of elong. For all bodies whose d> L, and d= o, the most favorable practicable positions are in the horizon. The constant (instrumental and personal) error in h will be eliminated in the mean of the results of observation of a star east and west of the meridian, if taken on the same parallel of altitude. The algebraic max. and min. curve, coinciding with the p. v. in L = o, swells, with in- creasing latitudes, until the max. ordinate occurs (or widest departure from the p. v.), in about latitude 54 31'; it then returns to its first form by the time that latitude 90 is attained. (Compare with No. 4.) Altitude -Azimuth, error ink Altitude -Azimuth, error in L Altitude Azimuth, error in // s If... Locus No. 2. ALTITUDE-AZIMUTH ERROR IN L, [Summary of pp. 26,36,37,63,64.] d = J. . . . (56) Branches: i. Absolute max., the merid. NPS. 2. Absolute min., the six-hour circle Pa'b'E . . . . P' . . . . W .... P. Neither branch gives algebraic max. and min., because tan t changes sign when the given body transits the six-hour circle and the meridian. The least favorable positions for any star are those on the meridian, the error giving . The most favorable positions are on the six-hour circle, giving o for error, as follows : + d> L L o L L L at at at at at at a' and V and E and c' and cutting e' and But for all rising and setting bodies having negative declinations the best practicable points are in the horizon. In determining the direction of the meridian on shore, the error in latitude will be elimi- nated in the mean of the results of observation of the given body east and west of the meridian on the same parallel of altitude. Alcilude -Azimuth, error in L Altitude -Azimuth, error in L. M2 \liiludi' -Azimuth, error in L,. -7 1 ' 1 ' 1 ' 1 f-d = LJ { \ \ \ i i i i .- i Locus No. 3. ALTITUDE-AZIMUTH ERROR IN d. [Summary of pp. 26, 37, 64.] = : (57) 1 dd sin / cos L Branches: I. Absolute max., the meridian NFS. 2. Num. min., from alg. max. and min., the six-hour circle Pa'b'E . . . P', etc. These branches, with their- kinds of max. and min., are obvious ; for sin t changes sign on passing through o and 180; but does not change sign when passing through 90 and 270. From the sign of (57) we see that the num. minima, having the same numerical value east and west of the meridian, are alg. min. on the west side and alg. max. when east. The least favorable position for any given body is when on the merid., error = oo, The most favorable, the error having a finite numerical value = sec Z,, is on the six-hour circle, as follows : -f- d > L, at a' or a" ; -f- d = Z, at points on the 6 h circle ; + d < L, at V or b" ; d = o, at E or W; d < L, at c' or c" ; d = Z, at points on the 6 h circle ; d > Z, at e' or e". The points for max. and min. are the same as in No. 2, but the num. min. in No. 2 is absolute, o, for all bodies ; while in No. 3 it is equal to sec L for each of all bodies. For all rising-and-setting bodies having negative declinations, the best practicable points are in the horizon. If the given body is observed east and west of the merid. on the same par. of altitude, the error will be eliminated in the mean of the results of computing the azimuth ; and since this condition is the same regarding error in h and error in L, we should observe as follows : If the error in L is likely to have a more pernicious effect than the error in h, owing to uncertainty of the geographical position while our instrument and the conditions for observa- tion of the altitude are good, we should observe the star east and west, on the six-hour circle : then, failing to seize the observation exactly at the points aimed for, the failure to eliminate the error whoMy wiirhot have much effect. But if the conditions are reversed, and error in h is more to be apprehended, try to seize the body east and west when on the curve of min. error in No. I. In moderately high latitudes a close circumpolar star will give q 90 and / 90 very near each other, so that whichever is chosen it will be favorable for eliminating the errors in both L and h, and also d. Besides, the>akrtude~ will be favorable for the practical problem aside from the theoretical. Altitude -A.zttnu.ih. error in A Al/i/in/f A'/J//iu//i, ('//<;/ i// d M.3 B 'Altitude -Azimuth, error in. rf Locus No. 4. TIME-AZIMUTH ERROR IN /. , dZ cos a cos d [Summary of pp. 26, 27, 37-41, 64-70.] = - (58) Branches: i. Absolute min., the curves of elongation Pa' Za" and e'P'e" . 2. Alg. max. and min., giving num. max. and min., the merid. NFS and the curve c'Eb'Zb" We"- The latter gives always numerical min., and pertaining, as it does, to d < L, all those stars will cross it unless the latitude is very high. So long as all stars having d < L cross this curve c'E .... c", the merid. will give num. max. at each culmin. for all stars. Because, for d> L the values on the merid. will be numerically greater than the valge o on the curves of elong. ; and for d < L they will be greater on the meridian than on the other curve. But in latitudes > 70 32' (about) some d's L, alg. max. and min. both occur on the meridian. At upper transit, q 180, cos^ does not change sign when passing through this value, but is ( ) on each side ; therefore the sign of ~ is (-f ) Ctrl and we have an alg. max. At lower transit q, having passed through the value 90 (causing cos q to change sign, and so giving no alg. max. or min., though the value o is found) is equal to o c ; and in transit cos q 7 r? does not change sign, but is (-f) on each side, making the sign of 7- ( ), whence an alg. min. To attain upper culmin. again, q passes through the value 270, cos^ changes sign, and, though we get o for the error, no alg. max. or min. occurs until the max on the merid. recurs. Therefore for J s-d> L, the num. max. and min., if governed solely by algebraic max. and min., would correspond to the latter, not only numerically but in the algebraic signs ; for the (+) error is the greater numerically since cos h at the upper transit is less than at the lower. But at a' and a" absolute min., o, intervene and the alg. max. and min. on the merid. both become num. max. For d > L similar conditions prevail, but not exactly like conditions, for the alg. max. and min. are reversed, both in numerical value and sign ; for cos h is then the greater at upper transit, and q there passes through o, and at lower transit through 180. Taking d < L, q on one side of the meridian remains always between o and 90. and on the other be- J "7 tween 360 and 270. Therefore the sign of cosy is always (-}-) and the sign of y always ( ). Therefore alg. max. and min. occur alternately ; the minima on the meridian (unequal numerically) and the maxima on the other curve (equal numerically). But our numerical maxima and minima (unfavorable and favorable points^ interchange with the algebraic minima and maxima. To follow the star in its diurnal course, beginning at the lower culmination : For -j- d > L. At d', alg. min., num. max. ( ), unfavorable ; at a', abs. min. = o, favorable; d ', alg. max., num. max. (+), itnfavorable ; a' 1 , abs. min. = o, favorable. For -\- d < L. At d' 1 , alg. min., num. max. ( ), unfavorable; b' , alg. max., num. min. ( }, favorable; d" , alg. min., num. max. ( ), unfavorable ; b" , alg. max., num. min. ( ), favorable. For -\-d-L. Practicably, favorable very near Z, the zenith, and unfavorable at low. culm. At Z the value is indeterminate. For d = o. At E and W favorable, on the merid. unfavor- able. For d < L. At d'" , low. culm., alg. min., num. max. ( ), unfavorable; at c', alg. max., num. min. ( ), favorable ; at d"' , alg. min., num. max. ( ), unfavorable; at c", alg. max., num. min. ( ), favorable. For d > L. At low. culm., alg. max., num. max. (-}-), unfavorable; at e', abs. min. = o, favorable ; at d IT , above horizon, alg. min., num. max., unfavorable ; at e" , abs. min. =o, favorable. For d= o and for negative d's the horizon furnishes the best practicable points if the body rises and sets. The error in Z due to error in t will not be eliminated by observing the star east and west, symmetrically situated with respect to the meridian. The asymptotes are the same as those in No. I ; one at the distance from Z equal to tan Z, and the other at the distance 2tan L. Both No. i and No.-2 have the spherical ellipse for giving points of absolute min.; and the other branches look alike in low latitudes ; but, both starting with the p. v. in latitude o, No. i swells out until reaching its limit of growth, when it returns to coincide with the prime-vertical in latitude 90. No. 2, on the other hand, continues its growth until the branches meet on the merid., then separate, and maintain separation in very high latitudes, untiljnjat^ 90 one branch becomes the horizon (equator), and the other a point at the zenith (pole). Time -Altitude -Azi//t nth, erro/' /// , / 7'i/nr A'/J//tul/t, , m error in t Tim.e-Aliitude- Azimuth, error in t C - -I.-- Locus No. 5. TIME-AZIMUTH ERROR IN L. /? 7 [Summary of pp. 27, 41, 42, 70-73.] - = tan h sin Z = etc (59) di-f Branches: I. Absolute min., the horizon NESW. 2. Absolute min., the meridian NFS, 3. Algebraic max. and min., giving numerical max., the curve Pa'"Z^c"^Se"'c"'b'"ZaP, together with the infinite branches b'a Na^b^ and e'P'e" 1 . Governed by the signs of sin 2Tand tan h, the alg. max. and min. are alternates. Leaving out of consideration the abs. min. on the horizon and on the meridian, we find for -f- d > > when not a circumpolar star (Plate A) at a', alg. max. ; at a'", alg. min. ; at a iv , alg. max. ; at Z., already used. But taking in the absolute min. to complete the discrimination in max. and min. of the various kinds, we have : At d' (lower transit), abs. min. = o ; at a', aig. max., num. max. (-(-) ; at a", abs. min. = O; at a'", alg. min., num. max. ( ) ; at d', abs. min. = o ; at a [v , alg. max., num. max. (-f-) ; at a v . abs. min. = o; at a vi , alg. min., num. max. ( ). Taking -(- d > L when the star is a circumpolar one (Plates B and C), we see that it is most favorably situated at lower culmination ; since for a given error in azimuth in the seizure of the star on the meridian, tan h sin Z will be less than its value close to the upper culmi- nation. Comparing No. 4 with No. 5, it will be seen that the most favorable position considering error in t is near the most unfavorable with respect to error in Z, in the case of stars whose L, even if not circumpolar; if a circumpolar star, these antagonistic points approach closer; and if a close circumpolar star, they are very near together. But the error in latitude will be eliminated if the star is observed both east and west at the same relative points to the meridian. Therefore select q = 90. and 270 to reduce the error in t to o if the star is seized exactly at those points, and nearly zero if near those points. Time -Axi/niitli, error in L. A Time Azimuth, error in L B ime -Azimuth, vrrvr in L. Locus No. 6. TIME-AZIMUTH ERROR IN d. dZ sin q [Summary of pp. 27, 42-44, 73-76.] = - J. . ...... (60) [NOTE. The investigation of Locus No. 6 (time-azimuth error in d) by its equations (186) and (187), or inspection of the corresponding diagrams, indicates that for some latitudes and declinations there maybe three points on either side of the meridian where the diurnal path of a body may cross the locus. These three points are determined by the three roots of the cubic equations referred to above. By reference to the diagram for Locus No. 6 (A), it is readily seen that for latitude 30 some bodies cross the locus three times on each side of the meridian, while others cross it but once, and it is clear that for some particular declination the diurnal circle will be tangent to the locus. The relation between a given latitude, L, and the declination, d, causing the diurnal circle to be tangent to the locus, is found by determining the condition that will give equal roots to cos / and sin h, in their re- spective equations. The equation (186) is the more convenient for this purpose. Writing A for tan L, and B for tan d, the equation becomes cos 3 /+ (A* + B* - i) cos t- 7.AB = o ............ (a) Should either A or B be greater than unity, the coefficient of cos / will be positive ; and by the theory of equations two roots will be imaginary. Thus at the outset we are limited to the consideration of latitudes and declinations not exceeding 45. The condition for equal roots in (a) is (A* + *- I) 8 + zjA'JS* = o ; ) -, or A 1 + B* - i= - 3 V^ 3 ^'. f ' Expanding (b) and arranging, we have (A* - i) 3 - o. This is a cubic equation in B*; and if A (tan L) is known, we can solve for IP by Horner's method of approximation, and thence deduce B, The two values of B thus found, equal numerically, with opposite signs, determine the two declinations (North and South), giving tangency to the locus. Having found B, we may easily deduce the value of cos t corresponding to the point of tangency; for, substituting from (b) 3V^ 2 2? a for A 9 + B* i in (a), we have cos 3 1 - 3V^g 2 cos / - 2AB = o ; the roots of which are always *^AB, -*\/AB, and 2*\/AB ............. (c) Therefore cos t = \/AB gives the value of t corresponding to the point of tangency ; and from (c) we may observe that, for the point of cutting the locus, cos / is opposite in sign and numerically twice as great as for the point of tangency. A particular case of interest is that in which the declination causing tangency is equal to the latitude. Put B = A in (b). We have SA' + 15^* + 6A* - i = o; ............. (d) whence (8A* - i) (A 9 + i) (A 9 + i) = o. The equation in A* has one positive root, A 3 = , and two negative roots, A* = i. The two last give imaginary values for A. From the first we obtain A = J V^z. Log A = 9.5484550. Whence L = d = 19 28' i6".39, giving tangency to the locus.] Branches: i. Abs. min., the merid. NFS. 2. Alg. max. and min., giving num. max. and min., the curve c'Eb'Za"Pa'Zb"Wc", and the infinite branch e'P'e" . In lats. 45 to 90 every par. of dec. will cut the curve of alg. max. and min. (on either side of the merid.) once, and only once. In lats. < 45 each par. of dec. cuts at least once ; some one north par. in the given lat. intersects once and at another point is tangent ; so also some one south par. of dec.: some parallels cut the curve three times. Therefore Plate A presents more features than do B and C. Looking at one side of the merid. only, if L > 45, the curve meets its asymptote at infinity only, the nadir; if L = 45, the asymptote is tangent to the curve at ^and then meets it again only at infinity; if L < 45, the asymptote crosses the curve once above the horizon, once below, and then meets it at infinity. Alg. max. and min. points for parallels of dec. that interse'ct the curve once on each side of the merid. may easily be discrimi- nated, since they depend on the sign of sin q alone. In case the parallel cuts thrice, the num. value of COS /f must be considered in connection with its sign. But, since the points are of alternate max. and min., no difficulty will be experienced in distinguishing the max. from the min., though from inspection of (60) the relative numerical values are not obvious. Recollecting that sin q is (+) west of the merid. and (r-) east, let us follow the star characterized by m', m", etc.: at low. cul., abs. min. o ; at m', alg. max., num. max. ( + ) ; at m", alg. min., num. min. (+) ; at m'", alg. max., num. max. (+) ; at upper cul., abs. min. o; at w lT , alg. min., num. max. ( ) ; at m", alg. max., num. min. ( ) ; at m vl , alg. min., num. max. ( ). The tangent par- allels of dec. are not drawn. At their points of cutting the curve will occur num. max., and at tangency no max. or min., but a decreasing or increasing function as may be, algebraically. For + d> L, max. error at a' and a": if the star is not circumpol., practicably the most favorable position is exactly on merid. at upper cul., but in horizon is good ; if a circumpol. star, then at low. cul. For + d = L the choice is lower cul.; at Z the error is indeterminate, but close to Z very large. For + d < Z., if a circumpol. star, at low. cul. best ; but if not circumpolar, at upper cul., though the horizon may be good ; the max. lying at b' and b". For all stars with (-) dec. observe on merid. Comparing No. 4 with No. 6, the remarks on Nt>. ^ app^^iere, by substituting for error in L the words error in d; and considering all the errors in ;he data, the conclusions are the same as given in No. 5. JVo. 6 a -Azimuth, erro/- in-d A T/mc : Az/,mut/-t, e/ror ui d ,M.6 B Time -Azimuth, error in d C LOCUS No. 7. TlME-ALTITUDE-AziMUTH ERROR IN k. J*T [Summary of pp. 27, 28, 44-46, 76-80.] = tan Z tan A (61) an Branches: i. Absolute max., the prime- vertical WZE. 2. Absolute min., the merid. NFS. 3. Absolute min., the horizon NESW. 4. Alg. max. and min., giving numerical max., Pa'"Zd lv and Na'b'Ec'"e"'S W JrttfaiuU'/V 1 . Considering only the curve of alg. max. and min., it is evident that on either side of the merid.: If -\-d> L and circumpolar, the star will cross the curve once, above the horizon only; if not circumpolar, once below and once above the horizon, \i-\- d = L > 45, one contact only, in the zenith ; if -f- d = L = 45, a point of contact in Z and one in N\ if -\-d = L < 45, contact at Z and a cut below horizon. If -f- d < Z., and circumpolar star, no points of con- tact ; if not circumpolar, one cut below horizon. If d = o, the equator at cuts the curve. If d < Z-, one point of intersection, above horizon. If d> L, once above and once be- low the horizon, if a rising and setting body ; if not, there is one point, below only. Now, including the branches of absolute max. and min., and attending to the signs of tan Z and tan h, we have : For -|~ d > L (not circumpolar, Plate A), at d', low. cul., abs. min. o ; at a', alg. max., num. max. (-J-) ; at a", abs. min. o ; at a'" t alg. min., num. max. ( ) ; at d', abs. min. o ; at a iv , alg. max., num. max. (-{-) ; at a v , abs. min. o ; at a vi , alg. min., num. max. ( ). For the close circumpol. star d> L (Plates B and C), at low. cul. d', abs. min. o ; a', alg. min., num. max. ( ) ; d' t upper cul., abs. min. o ; a", alg. max., num. max. (-{-). For -j- d < Z, at low. cul. d", abs. min. o ; b', alg. max., num. max. (-(-) ; b", abs. min. o ; b'", abs. max. oo ; at upper cul., abs. min. o, etc. For d = o, at E and W indeterminate, and in their vicinity bad conditions. Observe at transit on the merid. For d < Z., at low. cul. abs. min. o ; <:', abs. max. oo ; c" y abs. min. o ; c'", alg. max., num. max. (-J-) ; d'", upper cul., abs. min. o ; ^ T , alg. min., num. max. ( ), etc. For d > Z,, e" t d^, and e v , abs. min. o ; while at e" f and ^ v , num. max. In any case, the choice of the horizon or the mecid. to give abs. min. points depends on their relative proximity to a point of max. error. Observations east and west may be made to eliminate the error. For -\-d L, avoid the zenith, as giving indetermination ; and its vicinity as giving a large error, and observe at low. cul. Time -Altitude -Azimuth, error in h Time -Altitude -Azimuth, error in k J3 f------ LOCUS No. 8. TlME-ALTITUDE-AziMUTH ERROR IN /. , .. dZ tan Z cos d cos t [Summary of pp. 28, 46-48, 80, 8i.J -- = = (62) dt tan t cos h cos Z Branches: I. Abs. max., the p. v. WZE. 2. Abs. min., the six-hour circle Pb'EP '. 3. Alg. max. and min., giving num. max. and min., the merid. NPS. Not considering the abs. max. and min., the alg. max. and min. occur as follows: For -|~ d > Z, at low. cul. cos t is ( ) and cos "(+); .'. dZ is ( ) and at upper cul. cos t and cos Z both (-J-); therefore an alg. min. at lower and alg. max. at upper cul. whatever the relative values of cos h. But these values make the numerical max. and min. correspond to the algebraic. For d= L the same, but at upper cul., regarding Z as changing from < 90 to > 270, not passing into the 2d or 3d quadrant, the absolute max. will be also an algebraic max. = -\- oo . Therefore at low. cul., alg. min., and at upper cul., alg. max. correspond to num. min. and max. For -f- d < Z, at both culminations the function is ( ) ; hence the value of cos h governs. For d between d L and d = o, {h is less numerically } cos h is greater numerically > than at upper cul. dZ is less numerically ) Therefore at low. cul. we find an algebraic max., and at upper cul. an algebraic min.. but numerically the names are changed. For d=.Q the error has a constant numerical value and is ( ) at each culmination ; hence there is neither max. nor min. of any kind. For d < Z at both culminations the function is ( ) ; ( h is greater numerically ) at low. cul. \ cos h is less numerically v than at upper cul. ( dZ is greater numerically ) Therefore an alg. min. at low. cul. and alg. max. at upper cul.; but numerically the names are changed. For d> Z, alg. max. at low. cul., num. max. (-f-); and alg. min. at upper cul., num. min. ( ). Considering, now, the absolute max. and min. in connection with the preceding, we have for numerical max. and min. on the meridian the same name as given by the algebraic max. and min., except for d > Z. For, these stars not crossing the p. v., at both transits occur numerical max. For-)- d > Z, at low. cul. alg. min., num. max. ( ) ; a', abs. min. o; at d' (upper), alg. max., num. max. (+) ; a", abs. min. o. For d= L, at low. cul. alg. min., num. max. ( ); on six-hour circle, abs. min. o ; at Z, alg. max. -(- oo , absolute max. oo ; on six-hour circle, abs. min. o. For -f- d < Z, at low. cul., alg. max., num. max. ( ) ; b ', abs. min. o; b", absolute max. oo ; at upper cul. alg. min., num. min. ( ) (greater, however, numerically than the numerical max, .'( ) at low. cul.: in most cases our numerical max. is numerically greater than our numerical min., distinguished from alg. max. and min. by disregarding algebraic signs ; but the intervening absolute max. and min. now interchange the names of numerical max. and min., as first assigned when comparing with only alg. max. and min.); at b'", absolute max., oo ; # v , absolute min. o. For d = o, a constant value to dZ. equal to : r -. For sin Z d < Z, at low. cul., d"", alg. min., num. min. ( ) ; c', abs. max. oo ; c", abs. min.b; d'", upper cul., alg. max., num. max. ( ) (less numerically than the num. min. at lower cul.) ; at c'", abs. min. o; c {v , abs. max. oo. For d = L, at low. cul., alg. max., -f- oo (Z changing from > 90 to < 270), absolute max. oo ; on six-hour circle, abs. min. o; at upper cul., alg. min., num. min. ( ) ; on six-hour circle, abs. min. o. For d > Z, at low. cul., alg. max., num. max. -f-J *', abs. min. o ; upper cul., alg. min., num. max. ( ) ; at e" , abs. min. o. For all -f- d's avofd observing on that side of the six-hour circle towards the p. v. For all dTs the horizon is theoretically the best place to seize the star at. ' Practically, both these conditions must yield something to the best conditions foi the altitude (a given part of the triangle entering the problem), and, therefore, a point near the meridian, in bearing, may be chosen. v The error in Z owing to error in t will not be eliminated by observations east and west. No X Time 'Altitude Azimuth, error in I Time -AltUu>de -Azimuth, error in f i Time -Altitude -Azimuth, -error in t Wo. 8 Locus No. 9. TIME-ALT i TUBE- AZIMUTH ERROR IN d, re , , -, dZ sin t sin d [Summary of pp. 28, 48, 49, 8i.J - = tan Ztan */ = r (63) act cos Zcos A Branches: I. Absolute min., the meridian NPS. 2. Absolute max., the prime-vertical WZE. 3. Algebraic max. and min., giving num. max., the curve of elongations Pa'Za" and e'Pe". Alg. max. and min. only for d > L ; alg. min. west and alg. max. east for -j- d, and conversely for d. For + , at low. cul., d", abs. min. o; a', alg. max., num. max. (-J-) ; at upper transit. d'j abs. min. o ; a", alg. min., num. max. ( ). For -f- d = L, at low. cul., abs. min. o ; at upper cul. indeterminate, but may be called abs. max. for close to the zenith the error dZ is very large. Z is the meeting-point of the an- tipathetic curves, but its vicinity, through which d=. L travels, partakes of the character of the absolute max. For -f- d < L, at d", low. cul., abs. min. o ; b', abs. max. oo ; d", upper cul., abs. min. o ; b" t abs. max. oo . The equator and d < L, on merid., abs. min.; on p. v. abs. max. For d L, the converse' of -\-d=. L ; the vicinity of the nadir the unfavorable region, upper cul., abs. min. o. For d> L, num. max. east (/) and west (e") have contrary signs to those for -f- d > L. Observations east and west on the same parallel of altitude will eliminate the error in dZ. For error in every datum, the prime-vertical gives abs. max., and points near the prime- vertical, in bearing, should be avoided. Since errors in d and in h will be eliminated by observing east and west on the same paral- lel of altitude, and since the error in /will not thus, be eliminated, but, if the body is observed on either side of the meridian, on the six-hour circle the error will reduce to o, the best con- dition will be afforded by a close circumpolar star observed east and west as near to the six- hour circle as it can be seized. Time Altitude -Azimuth, error in d A Time -Altitude -Azi/n uth, error in d M.9 JVo.9 ' -A/filuf/r Axt/ttufh, error in cl C Locus No. 10. FOR ERROR IN L. TiME^-zrrKfuTH AND ALTITUDE-AZIMUTH EQUALLY GOOD. [Summary of pp. 49-53, 81-83.] Branches: I. Identical errors in the resulting azimuths computed by both methods, the prime-vertical, WZE, and the curve of elongations, Pa"Za'" and 2. Equal resulting errors, numerical, but having contrary signs, the curve Pa'b'Ec"e"P'e'"c'" W^a^P. In the plates, the six-hour circle, instead of being drawn full above the horizon, is a dotted line throughout. The branches of the locus form the boundaries to the regions within which the one method of finding the azimuth is to be preferred to the other. The method that is the more favorable is : For -j- d > L, time-azimuth from the time of low. cul., until the star arrives at a' ; alti- tude-azimuth from a' to a"; time-azimuth from a" to a'", crossing the meridian ; altitude-azi- muth, from a'" to # lv ; time-azimuth from # iv , through lower culmination, to a' again. For -J- d = L, time-azimuth until cutting the curve at a point between a' and b'\ thence, passing the zenith to the point between # lv and # v , the altitude-azimuth ; thenceforward, to a repetition, time-azimuth. For -f- d < L, from lower culmination until b' is reached, time-azimuth ; b' to b", altitude- azimuth ; b" to b'", time-azimuth ; b"' to 6 1V , altitude-azimuth ; thenceforward time-azimuth during the course of the star to b' again. For d o, time-azimuth throughout, with the reservation that altitude-azimuth is equally good at the points E and W, the error reducing to o by each method. For d < L, from lower culmination to c', time-azimuth c' to c", altitude-azimuth ; c" to c" f , time-azimuth ; c"' to lv , altitude-azimuth ; thence to c' again, time-azimuth. For d = L, time-azimuth during the time the star is above horizon. For d> L, from lower culmination to e', time-azimuth ; e' to e", altitude-azimuth ; from e", passing d lv to e"', time-azimuth ; e'" to ^ v , altitude-azimuth ; thenceforward to e' again, time-azimuth. For all bodies having negative declinations, and for d = o, the time-azimuth is the -better during the whole time of visibility of the star. M. 10 For error in. L, lime -Azimuth an,d Altitude-Azimuth equally favorable A AfalO For erroj" in. 2,, Time-Azi?mit7i and Altitude -Azimuth e equally favorable ' in /., Time-AzimiLth rt/td Altitude -Azimuth rtf/ffl It/ ft/ r ,/IU / / \ \ / / \ \ / / \ \ / / \ / / \ \ 1. I f- cl = LJ 1 i .:.*' .'.** L.BB STAMPED - 1915 UNIVERSITY OF CALIFORNIA LIBRARY