NRLF LIBRARY OF THE UNIVERSITY OF CALIFORNIA. > ,, RECEIVED BY EXCHANGE \ Class be IHntversitE of Cbfcago FOUNDED BY JOHN D. ROCKEFELLER ON THE CONVERGENCY OF THE SERIES USED IN THE DETERMINATION OF THE ELE- MENTS OF PARABOLIC ORBITS, AND THE ERRORS INTRODUCED IN THE ELEMENTS BY IMPERFECTIONS OF THE OBSERVA- TIONS A DISSERTATION 6 SUBMITTED TO THE FACULTIES OF THE GRADUATE SCHOOLS OF ARTS, LITERATURE, AND SCIENCE, IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (DEPARTMENT OF ASTRONOMY) BY WILLIAM ALBERT HAMILTON CHICAGO 1903 Gbe "Gintvetsfts of Cbfcago FOUNDED BY JOHN D. ROCKEFELLER ON THE CONVERGENCY OF THE SERIES USED IN THE DETERMINATION OF THE ELE- MENTS OF PARABOLIC ORBITS, AND THE ERRORS INTRODUCED IN THE ELEMENTS BY IMPERFECTIONS OF THE OBSERVA- TIONS A DISSERTATION SUBMITTED TO THE FACULTIES OF THE GRADUATE SCHOOLS OF ARTS, LITERATURE, AND SCIENCE, IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (DEPARTMENT OF ASTRONOMY) f UNIVERSITY 1 V of / BY WILLIAM ALBERT HAMILTON CHICAGO 1903 PRINTED A T THE UNIVERSITY OF CHIC A GO PRESS PART I. i. Introductory. The elements of the orbit of a comet are usually determined by means of the data obtained at three separate, complete observations ; and it often becomes a question of importance to the astronomer as to the suitability, or perhaps we might say the suffi- ciency, of such a set of observations to determine with accuracy the required elements. He is confronted, on the one hand, with a set of formulae somewhat complex in their nature, and which are subject to limitations in their application owing to the properties of the various functions involved in their construction ; on the other hand, the data of observation are subject to limitations owing to unavoidable inac- curacies in the construction of the telescope and the multitude of items which fall under the class known as errors. It is thus both a mathematical and a physical problem with which he has to deal, and it becomes important, first, that a careful analysis be made of the proper- ties of the formulas and the conditions under which they may be applied, and, secondly, that the errors which present themselves in the observations, in spite of the greatest care and skill on the observer's part, may not be allowed to become obscured in the final results of the computation. It has been the purpose of the study of which this paper is a partial result to investigate the formulas for computing cometary orbits from each of these two standpoints. In pursuance of this plan, we have investigated, among other questions, the nature of the func- tions usually known as the " ratios of the triangles;" and have found the precise conditions under which they may be developed into con- verging power series of the time intervals between the observations. 1 This discussion is given in the first part of this paper. In another part of the investigation we have found the effects of the errors of the observations upon the computed elements of the orbit of a comet, using Obers's method as a basis of the study. To this is added the results of a computation, by use of the formulae so deduced, of the differentials of error in an actual case. We proceed to discuss the ratios i That the series converge for sufficiently small values of the time -intervals follows from CAUCHY'S existence theorems published in 1842 (Coll. Works, ist series, Vol. VII, pp. sf.). PROFESSOR PAUL HARZER has attained the same general results in the Publications of the Kiel Observatory, Vol. XI, Part 2, by direct treatment of the series. Evidently the results of both Cauchy and Harzer are of the nature of existence theorems and were not intended for practical use by computers, because the true radius of convergence was not found. 183484 4 CONVERGENCE OF SERIES USED IN DETERMINATION of the triangles and convergency of the series. We use the following notation : 2. Notation. Let / If / 2 , / 3 denote the first, second, and third times of observation, respectively. And if k denote the Gaussian constant and m the mass of the comet in terms of the mass of the sun taken as unity, then the differential equations of motion of the comet referred to the sun's center as origin of co-ordinates are, r 3 dt* r 3 d*z _ _6*(i + m)z where r is the heliocentric distance of the comet, and x, y, z are its rectangular cartesian co-ordinates. In all practical cases m will be infinitesimal in comparison with the mass of the sun, and therefore may be neglected. Furthermore, if we so change the unit of time that the new unit shall be equal to the old when the latter has been multiplied by k, and denote the time when expressed in the new units by / / , where / is any particular epoch, we may express these equations of motion very simply, thus : dt* In these equations the attractions of all the bodies of the solar system are neglected, except that of the sun. 3. Preliminary notions. Suppose now the co-ordinates of the comet at the time / to be x ot y , z ot and its velocities to be =^, -17, -77 ; at at at then, at any other time, the co-ordinates and velocities are functions of these initial conditions and / 4 ; or, as we say, , -j\ x >y< z > Tt' dt* dt' ) with similar expressions for the other co-ordinates and the velocities. f.,*v OF ELEMENTS OF PARABOLIC ORBITS 5 Now it is known from the theory of differential equations that the co-ordinates and velocities are expansible into power series in (/ f )r of the form which have finite radii of convergency, if r does not vanish for t f = o. 1 In the partial derivatives above t t is to be placed equal to zero after differentiation. Hence, as seen from (3), _8/J ~ dt ' LBr\ ~ dt" From equation (2) we obtain d*x Q _ _ #o .'::';V *"/;,' ldx ~dP"~ri~dt ~Vl~dt ' ' Equations (5) enable us to find the coefficients for the developments of the type (3), by means of which the co-ordinates and velocities of the comet at any time / are expressed as power series of the time inter- vals / / , and coefficients depending only upon the co-ordinates and velocities at the initial time / . By means of these developments of the co-ordinates the so-called ratios of the triangles are built up in the form of series which depend upon particular time intervals selected from those determined by the three observations. It is in regard to these latter series that we wish to find the conditions of convergency; and it is at once evident that their convergency will depend upon the convergency of the series of the type (3), since the ratios of the trian- gles are functions of the co-ordinates alone. 4. Convergency of series. From well-known theorems of the theory of functions of complex variables it is clear that any expansions what- ever of the ratios of the triangles into power series for given time intervals and initial conditions cannot have greater radii of conver- gency than the values which are determined by the poles and branch points of the expressions of those ratios as functions of the time inter- vals. First, however, we study the nature of the functions which express x, y, z in terms of /, and from these find the true radius of convergency. 1 JORDAN, Cours d" 1 analyse, Vol. Ill, p. 99. 6 CONVERGENCE OF SERIES USED IN DETERMINATION 5. Co-ordinates as functions of the time. From the geometrical relations of the orbit of the comet we have the relations x= r [cos (v -f- o>) cos li sin (v + w) sin li cos i] , y = r [cos (v -j- o>) sin O -j- sin (v -f- is the argument of latitude of the peri- helion, O is the longitude of the node, and / is the inclination of the orbit to the ecliptic. The last three quantities are independent of the time ; while v and r are expressible in terms of / by means of the rela- tions - l + COS V where/ is the latus rectum of the parabolic orbit of the comet and II is the time of perihelion passage. II and / are thought of as expressed in units described in section 2 above a usage which we shall con- tinue throughout this paper. 6. The solution of the cubic. By means of the equation (7) we are enabled to express x, y, and z in terms of the time intervals / II. In order to do this we introduce the auxiliaries < tan- . 2 Then the last equation of (7) becomes < 3 + 3< 2T = o . (9) This is the so-called normal form of the cubic in the quantity <. Its solutions by Cardan's formula are (10) where ^ z (r-J- j/i-f-r 2 )*, q 2 = (T i/i + T 2 )*, and i, e, 2 are cube roots of unity. 1 i See BURNSIDE AND PANTON, Theory of Equations, p. 108. OF ELEMENTS OF PARABOLIC ORBITS 1 7. Branch points. We wish to express in a power series in /, and must, therefore, find the branch points and poles of the function. At once we have the branch points r = i and T= i, where i=i/ i. Also T = oo is a branch point, as is easily seen by putting r = , , and letting T' approach zero. This is the same as putting r= oo , and we easily find that all three solutions have the same value at this point. If now we consider a Riemann surface of three sheets with branchpoints at T = /, T = /, T oo , then by the theory of functions of a complex variable we know that the quantity is a uniform function of position on this surface. 8. Connection of the sheets. In order to get a clear idea of the sur- face, it is necessary to find what sheets pass into each other at the two branch points which are in the finite part of the plane. To do this we need to follow only the purely imaginary values of T; for the two branch points in question are on the axis of pure imaginaries. Indeed, we may also consider the branch point T = 00 to be on this same axis. In order to simplify matters and at the same time render the rea- soning clearer we make the transformation T = / cos , (u) where 6 is real or complex. Then q^ and q 2 become _z* q^ = [/(cos 6 i sin 0)]* = ie 3 , 16 q z [/(cos -f i sin 0)]i = ie* . yirz _ ziti And since we may write e e 3 , **= e s , we obtain from (10) / _\ ! t\& + 3 = / \ 2 and < 3 become equal when T approaches A; but when T arrives at /4' along the path selected, this does not repeat itself; but instead we have 3 are connected at r = i; while 2 and cannot become infinite except for infinite values of 0. Moreover, owing to the periodicity of the function cannot become infinite except for infinite values of T. Hence there are no finite poles of < in the sheets of the Riemann surface. 10. Zeroes in the sheets. By use of (12) we are also enabled to find at once the zeroes of <#> in the r-surface. The general condition for the vanishing of is given by either of the conditions e cos - = o , cos These are virtually the same, since we may get the one from the other by putting 0= 2*. It is then only necessary to find the value of 6 e for which cos-=o. Now, by methods well known to the theory of trigonometry 1 it is readily proved that the only values of 6, real or com- plex, which satisfy this condition are where n is a positive or negative integer or zero. It follows that only one zero of < is to be found in each fundamental region of the 0-plane for each of the sheets on the Riemann surface. Thus, for the first sheet, it is that value of r which corresponds to the value of 6 = - which gives by use of (n) T o . We have already seen in the table following (12) that only one branch of < vanishes at this point ; and i See CHRYSTAL, Algebra, Vol. II, chap. 29. 10 CONVERGENCY OF SERIES USED IN DETERMINATION that the particular one which vanishes thus is dependent entirely upon the sheet of the Riemann surface in which T is found. ii. Resume. We note here the following summary of results as to critical points upon the Riemann surface upon which < is a function of position : ( T = o in the r-surface, Zeroes at < ,, *>,* r^/ii ) 6 = - in the fundamental region of the 0-plane ; I 2 Poles, none in the finite part of the r-plane; (13) Branch points < T = / , [ T= . 1 2. Rational functions of < # as a function of T and its branch points are at the same places? It is to be remembered, however, that this theorem does not apply to the zeroes and poles of such a rational function of < and T. These may be located otherwise than as described in (13). 13. Co-ordinates x and y as functions of r. We may write the first two equations of (6) as follows: V 2 7 T/ v\ y = [ cos v j z sin v] 1 1 -}- tan* -1 ; where/ = r(i + cos v) ; and 2C = cos o> cos II sin w sin 11 cos i , P 2S = sin sin H cos * , P 2C = cos CD sin O + sin C

as a function of T, and the branch points of x and y are r = ij r = i, and r=x> . Moreover, since c, c^ s, and s t are constants and never infinite, x and y cannot become infinite except where < becomes infinite, viz., at T= oo . Hence we have : Theorem: x and y have poles in the Riemann sheets only at T w and they have branch points at r = i, T== /, and T = . It follows from the above that x andj> are holomorphic functions of T in the sheets of the Riemann surface except at points T = I, r= i, and T = oo . Therefore they may each be expanded into power series with argument T r in the vicinity of any point T = T O . These series will be convergent inside of a circle whose center is at T O and whose radius reaches from T O to the nearest of the points T = /, or T = /. 14. Radius of convergency. If in (3) we replace / and / by their corresponding values in T by relation (8), we have just such an expan- sion as described in the last article. If we should at the same time take / o, the expansion in x becomes of the form where a , X , etc., are constants. This series will be convergent inside a circle whose center is T = o , and with radius unity reaching up to the branch points T = i and T i. Hence, the true radius of con- vergency in this case would be |r|= i; or from (8) c )=* <"> 3 Suppose in (17) we give to / any value, say i, which would corre- spond to a perihelion distance of 0.5 ; then if we make == , which is approximately its value, the case under supposition would give as the 12 CONVERGENCY OF SERIES USED IN DETERMINATION limit for the time interval for which the expansion of x into power series would be convergent, the value 20 days. The same period would hold for the corresponding expansion of y. If T O were any finite point not equal to zero, say some point on the real axis of the r-plane, then the radius of the true circle of conver- gency would be larger than that given above. In this case the radius would be 1? T = V i -f- T*, which holds for both the x and the y series. The radius of convergency of the corresponding series in / is at once deducible from the series in r through the relation (8). The relation is always Rt= XT , where the subscripts denote the argument of the series. Since in an absolutely convergent series we are at liberty to change the order of the terms at will, we may express (3) and the corresponding equation my by use of coefficients of the kind given in (5), as follows : dx where A and B for their first few terms are A- i?i'+i--+-r- i^Y+i^ 2 rl^ 2 ridt " h 2 4 U r\\dt ) ^ ri dt' B / " 3 + " 4 ^4 = / -673 + 47;*- In these series we have taken / = o. They may be written as series in t t by use of the theory of continuation of power series. Since a power series serves in every way to define the behavior of the function from which it is derived so long as we remain within its circle of convergence, we can deal with the series (19) as with quanti- ties which obey all the laws of ordinary algebra association, commu- tation, etc.; such as ordinary polynomials or rational quantities, and the resulting series will be convergent. 1 15. Ratios of the triangles. We denote the triangle between the positions of two radii vectores of the comet's orbit by the expression [>i, ^J, where / and j denote the order of any two of the three obser- vations ; also in general we denote the co-ordinates of the first, second, * See CHRVSTAL, Algebra, Vol. II, pp. 139-43. OF ELEMENTS OF PARABOLIC ORBITS 1 3 and third observations by the subscripts i, 2, 3 respectively. Now the ratios of the triangles [r it r^\ are equal to the ratios of their projections upon any plane, which fact may be expressed thus : Let now x at y a , -r^-r De taken as the zero values of the co-ordinates at at and velocities in the expansions (19) and (20). Then we get i \ dx (22) where A t , B^, A 3 , B^ are defined by : Now, the series (22) are convergent within the same circle. It follows that, since x a ,y at -j- 2 , -^ are not in general equal to zero, the series (23) are also convergent in this same circle. 1 It follows that since for such series the law of distribution holds 2 the products A lt B 3 and A 3 B^ are also convergent series. Hence, from the law of addition, we have that A^B 3 B*A 3 is also convergent. 3 We are now at liberty to substitute the values of x s , y lt x 3 ,y 3 as given by (22) in the ratios on the right of (21). We get, after the substitution indicated and by canceling the factor *rs ~-^~v7 the two members of the ratio, i Ibid., p. 178, 5. 2 ibid., pp. 142, 143. 3 Ibid, p. 141. 14 CONVERGENCY OF SERIES USED IN DETERMINATION [>.. 'J = - g 3 = ( / s~^)r I i(/ 3 -/ 2 ) 2 -(A-/ 2 ) 2 . [r f , r.] ^ (/x-/ a )L 6 ,* 4" "^ "i 2 '"]' , } [rrj = -^ = (A-/.)r , i (/ 3 -/.)"-(/.-/.) 4 ^ ^/ where ^ r , ^ x , ^4 3 , jff 3 have the meaning given by (23). Now, B^ and ^ 3 are series with arguments /, / 2 , and / 3 / 2 respectively. They hold, /. ^., are convergent, as long as / x / 2 and / 3 / 2 obey the relations (25) Also the series A 1 ^ 3 B^A^ which has two arguments, viz., / t / 2 and / 3 / 2 is convergent so long as (25) holds. 16. The zeroes of B^ and A^B^ B^A 3 . If B t should vanish, or if A^B 3 B t A 3 should vanish, then the fractions on the right of (24) evidently would no longer be legitimate. It is easily seen that the former contingency cannot occur unless / r / 2 =o. As to the latter, we state two theorems which are readily proved, but the proofs of which we will here omit. They are stated as follows : Theorem I: The expression x T y 3 y t x 3 can vanish only for the real values of v 3 v,. Theorem II: In all cases where the times of the observations are distinct and where the difference of the longitudes of the comet in its orbit is not equal to an odd multiple of TT, the expression (f,,o) n 7? cannot vanish and the expressions -W,-T-B ' p are legitimate frac- z> x A 1 f 3 > t A 3 tions which may be expressed as series, each of which is convergent for all cases where |/ 3 / 2 |< T/r 2 -f- i and | A / 2 1 < I/ T, -f i . O .7 The first terms of these series are written out in the right members of (24). In many cases, however, the computer may prefer to use the fractions, and these are always the safer formulae when doubt in any way exists as to their legitimacy. OF ELEMENTS OF PARABOLIC ORBITS 17. Computation by use of the series. The fractions on the right of (24) have both numerators and denominators in the form of p\ / series. Their radius of convergency is R t = V i -f- ^ , where O r 2 = -4 (/ 2 II). If we make &= , the following table will give cor- pt oo responding maximum intervals of time for different values of / for which the series are convergent when t 2 is taken equal to II : TABLE I. 4 2 160.0 103.4 2-5 i.2S 79-8 2. 56.0 0.75 36.2 1-25 0.62 27.4 i. o.S 20.0 14 0.25 0.12 2-5 1.8 0.6 0-44 .08 o 0.05 0.04 0.02 0.01 0.22 0.06 It is evident that for any particular value of p the time intervals should be well within the limit of values for which the series are con- vergent. This is especially true if we would have the most rapid con- vergence a thing most desirable from the standpoint of the computer. In fact, as is well known, it is imperative to have this convergence so rapid that at most but one or two terms will give sufficiently approxi- mate values of the ratios. The reason for this is at once evident when we consider that the series are transcendental in character. Thus, the quantities ^ and r 2 which enter into the terms higher than the first are essentially unknown from the start, and cannot even be guessed at with any degree of certainty until an approximate value of p has been obtained. It cannot be too strongly insisted upon, there- fore, that, in order to get the closest determination of the ratios of the triangles, the greatest care must be taken to secure a set of time inter- vals which, by their co-ordination with the parameter of the orbit in hand, will make the series rapidly convergent. It is true that this is more or less a question of trial to start with ; yet, when a value of p has been once computed by means of any set of time intervals, it will be seen at once whether the value so obtained is one for which the series are sufficiently convergent for the time intervals employed. If this is not the case, then new time intervals should be taken and the computation made over again. . - PART II. IN this part of this paper are deduced differential equations of relation which give the errors of the computed elements of the comet- ary orbit expressed in terms of the errors of the observations. At the close is also given the results of a computation in which the formulae are applied to an actual example. 1 8. Geometrical relations. Let p represent the geocentric distance of the comet, A and ft its geocentric longitude and latitude, respec- tively. Let R represent the heliocentric distance of the earth, Z and B the longitude and latitude of the sun, respectively. The subscripts i, 2, 3 are used as before to represent the order of the observation to which the corresponding co-ordinate applies. With this notation any one of the following three equations will express the relation between the geocentric distances of the comet at the first and third observations. For their derivation the reader is referred to Moulton's Introduction to Celestial Mechanics, chap. x. p 3 = m f + M' Pl ^ (i) where m ' and M' are defined by rt-iiJz' ^-zlf/^rj ' cos ft sin (\ 3 L 2 ) sin ft cos ft sin (A 2 , [V, , rj sin ft cos ft sin (A 2 Z 2 ) cos ft sin (A T Z T ) (10) - fc , rj cos ft sin (A 3 Z 2 ) sin ft cos ft sin (A 2 Z 2 ) ' Z/ = z sin (Z T Z 2 ), L 3 = 3 = sin (Z 3 Z 3 ) . where w" and M" are defined by sin ft [|> 2 , rj ^ cos (Z t - Z 2 ) + [r t , r^ R 3 cos (Z, Z 2 ) - [V, , r J /? 2 ] ^ [r x , rj [sin ft cos ft cos (A 3 Z 2 ) sin ft cos & cos (A 2 Z 2 )] jlf : 2) 3 : sin ft cos ft cos (A 3 Z 2 ) sin ft cos ft cos (\ -Z 2 ) . ^*I > ^2_ where w r// and M'" are defined by sin (A 2 - A,) cos ft _ ,, rJsin(A 3 -A 2 )cos t , rJcosftsin(A 3 -A 2 ) 16 OF ELEMENTS OF PARABOLIC ORBITS 1 7 19. Generalities. Equations (i), (2), and (3) involve the dynami- cal law that the motion of a comet is in a plane passing through the center of the sun. The quantities m' ', m n ', m'" , M ', M' ', J/'" are functions of the geocentric longitudes and latitudes A,, ft, A 2 , etc., and of the ratios of the triangles. They also involve R* , R z , R^ and the longitude and latitudes of the sun. But /?,,/?,, Z,, ^ 2 , etc., are independent of the errors of observation, since they are taken from the Ephemeris. Likewise the time intervals which are involved in the ratios are independent of A,, ft, A 2 , ft, A 3 , ft, which are derived directly from the right ascension and declination determined by the settings of the instrument. Owing to this last fact, we shall speak of A-D A. 2 , ft, etc., as observed co-ordinates. In practice some one of the three equations (i), (2), (3) is usually more advantageous than the others owing to the particular problem of computation at hand. For the method of determining the one to be used, the reader is again referred to the same treatise and chapter as in the last article. For our purpose here, the fact just stated is of interest because it necessitates the derivation of a separate set of formulae for the errors in the elements to correspond to each of the relations (i), (2), (3). In addition to the relations already given, we shall have need of the following equation, due to Euler : (r t +r 3 + ,)l - (r, + r 3 - s)*= 3 (/ s - /,) f = (* 3 - XiY + (y 3 -y^ + (z 3 - z^ = pl- 20A cos ft cos (A, - Z, ) + 2H + 2 PI 3 cos ft cos (A, - Z 3 ) (4) + P3 - 2 P3^3 COS ft COS (\ A) + 2 P3#* COS ft COS ( X 3 - Z where s is the chord connecting the first and third positions in the orbit. The quantities / 3 , t^ are given here in the units used in the first part of the paper ; and it is understood that / 3 is larger than / x , so that / 3 /! is positive. 20. Plan of procedure. From the relations already set forth, it is proposed to deduce formulas expressing the variations, in/, O, /, o> and II due to a variation in the observed co-ordinates A,, A,, A 3 , ft, ft, and ft. In this work /,, / 2 , / 3 are considered constant, as are also the quantities RI , R^ , R^ , Z t , Z 2 , Z 3 , which are obtained from the Ephemeris and depend directly on /,, / 2 , / 3 . It is evident that r lt r a , r 3 , p t , p 2 , p 3 , v i> V 2> V 3 > which occur in the relations, are functions of A,, A 2 , A 3 , ft, ft, ft, and will vary with these co-ordinates. Hence, incidentally, 1 8 CONVERGENCY OF SERIES USED IN DETERMINATION formulae are derived showing variations of these in terms of the observed co-ordinates. One thing further. The quantities j=-^- and Y~^ ^ are functions of X x , X 3 , X 3 , ft , etc. ; but when these ratios [ftt r **\ are computed by means of series, it is easily seen that the above quan- tities enter only in the higher terms of the series terms which are small and hence neglected on first approximation. It will remain to show that we may always neglect such terms in taking the partial differential coefficients of m f , M' , etc., in respect to the observed co-ordinates. This is taken up later. It resolves itself to this, then : Each of the elements p, O, o>, /, and II is a function of the six variables X x , X 2 , X 3 , ft, ft, ft ; but it is found that the work of derivation of the formulae naturally divides itself into three parts. Thus we may get expressions to determine 8/, 8O, 8o>, 8/, and 811 where these variations arise from a variation 8X, in Xj and 8ft in ft of the first position. A second set of formulae will determine 8^>, 8O, etc., where variations of like character are given to the observed co-ordinates of the second position ; finally, a similar set are obtained for the third position. We take up the work in the order just indicated. 21. The variations 8X X , 8ft. The equations (i), (2), and (3) are all of the type Hence we get for each of these , S = v (5) where 8p 3 , 8p, are changes in p 3 , p t for the variations 8ft and 8X X in the arguments ft and X x . From (i) a , (2)^, (3)^ we get the following expressions for the partial derivatives in (5), where we count the ratios of the triangles as independent of ft and X, according to the remarks of the previous article. . Q ii*M am om (6) BM' _ [> 2 , rj cos ft cos ft sin (X 2 - Z 2 ) -f sin ft sin ft sin (X t Z 2 ) "9ft" ~ [>, '.] sin ft cos ft sin (X 3 - Z 2 ) - sin ft cos ft sin (X 2 ~ Z 2 ) ( 7) %M' = |> 2 , rj sin ft cos ft cos (X, - Z 2 9^i [^t ''J sin ft cos ft sin (X Z 2 ) sin ft cos ft sin (X 2 Z 2 ) OF ELEMENTS OF PARABOLIC ORBITS 19 = |> 2 , r J cos ft cos ft cos (A 2 - Z 2 ) + sin & cos (A 2 - Z 2 ~ [r, , rj sin ft cos ft cos (A 3 - Z 3 ) - sin ft cos ft cos (A, - Z 2 ) ' = [r., rj _ cos ft sin ft sin (A, -Z ) __ ( 8 ) [r z , rj sin ft cos ft cos (A3 - Z 3 ) - sin ft cos ft cos (A, - Z 2 ) ' [>, , rj cos ft sin (A 3 - A,) ' [r r , rJcosftsin(A I -A 2 ) Using the second relation of (4), we obtain where we have BlfiiJI' | = I [ Pl - Xl cos A cos (X, - Z.) + ^? 3 cos ft cos (A, - Z 3 )] ; pi 9j i ST =-|>3-^3 cos A cos ( A 3- Z 3 ) + ^ cos ft cos '(A3- Z,)] ; Ps ^ (n) 8j i o" g^ == - [ft*; sin ft cos (A, - Z x ) - Pl ^? 3 sin ft cos (A x - Z 3 )] ; a gj- = 7 [ Pl ^ cos ft sin (A, - Z.) - Pt X 3 cos ft sin (A, - Z 3 >] ; By means of (5) we may eliminate 8p 3 from (10), and get -f- M From the first relation of (4) we get, where for brevity we put r, + r = K , ^ ( } VK+S VK- S V This equation may be written ' _ r . I /S\ ^r + ^i-y 8 ^ = ~T~ 20 CONVERGENCY OF SERIES USED IN DETERMINATION Kow, where z> 3 v t < 180, which includes all practical cases in which it is possible to make use of the expansions treated of in Part I, will always be less than unity; and hence we may write AT 3-5 2* K From this it follows that (13) may be written 2K Ti s i i/J\ 3 * 8 A = --- dJ -f I - -f- - : ~ I ~F I ~r ' ' '~ j- L 2 ^ 2 4 VAT/ The series on the right will, in all legitimate cases, be very convergent, owing to the fact that there must always be the strong inequality s < K. In general not more than three terms will be necessary (usually two are sufficient) where six-place logarithms are used. By use of (12 and 14) we are able to express 8 (r t -f- r 3 ) directly in terms of variations of ft, X lf and p x . We do not write out the result, which is very simple. It may be noticed here that in (14) a singularity enters into the coefficient of 8s when s approaches zero. In this case the value of 8J would depend on the first term on the right and would be large. The same difficulty is found in the value of 8s itself, since, as seen from (n), the partial derivatives in respect to p lf p 3 , X r , ft each become large. From this we conclude that when the observations are taken at very short intervals in the orbit, then the computed value of r, -f- r 3 will be very inaccurate, owing to the errors of the observations. From a well-known trigonometrical relation upon the triangle whose sides are p, r, and R we have the two relations for the first and third positions of the comet and earth in respect to the sun : r? = p* - 2 Pl , cos ft cos (X, - L,} + /?,- , , . r 3 * = p 3 * - 2 p 3 tf 3 cos ft cos (X 3 - Z 3 ) + R* . From (15) we obtain 8r t = -5-' Sp x + ^ 8ft -f- * 8\, , op ap, OA, , ,s *.-* OF ELEMENTS OF PARABOLIC ORBITS 2 I where we have ^ = - [ Pl - R, cos ft cos (X r - A)] , d/o x r x Jj = ^ [P, - *s ^s ft cos (X 3 - /,)] . By means of (5) equations (16) may be written fr. = r'fc +!&+. . dp, dft dA - I+P ' I+ Now from (14), when combined, as suggested above, with (12), we have Sr^SK-Sr* , / x 8^ = 8^-^ , where K involves only the differentials Sft, 8X t , 8/o t in linear expres- sions. Hence, by combining (18) and (19) we may get 8r 3 and 8r x in expressions of the form : ' 1 , ( . , , where <:, c' , d, d' are constants. By substituting these back into (18) we may also get 8^ in the form 8p 3 = .8ft + ^8X I , (21) where e and e' are constants not involving 8ft or 8X X . If we place this last in (5), we also get 8p 3 in the form X I , (22) where /, /' do not involve 8ft or 8X,. These auxiliary relations, 8r,, 8r 3 , 8/5 t , 8p 3 , are very useful for the work which follows in getting the variations of the elements/, O, o>, /', and II in terms of 8ft, 8X X . We start with the equations (7) of Part I, as follows : 22 CONVERGENCY OF SERIES USED IN DETERMINATION p = r ( i -f- cos v) = 2r cos 2 - , (23) ^ tan - 4- - tan 3 - = 2 . 2 /I . From the first of these we get for the first and third positions where the radicals are to be taken positive, if - is the first or third quadrant; and negative if in the second or fourth quadrant. By use of these relations we obtain from the second relation of (23) Eliminating n from (24) and clearing of fractions, we get 2/3 / - V 2r-p -f i/( 2 r s -/)3 - V(2r t -/) 3 ^^s-O] ( 2 5) By giving variations Srj and 8/* 3 to ^ and r 3 respectively, we obtain, after collecting and simplifying and solving for S/, -/-('i -/)!-> = Sr 3 . (26) In (26) it must be kept in mind that the radicals are positive or nega- tive according as tan - is positive or negative. This equation, when the auxiliary equations (20) are used, gives 8/ as a linear function of 8&, &,. From the relations OF ELEMENTS OF PARABOLIC ORBITS 23 we derive by differentiation *T-/ , sn = - 8r 3 r, sin v 3 8v 3 , from which o_. .x A.** 1 r* sin v s T r, sin z>, ' ^^,-i-C- (27) r 3 sin z> 3 r 3 sin z> 3 By means of (27), (26), and (20), we may get 8^, 8z 3 expressed as linear functions of 8ft, 8\ T . If we denote the argument of latitude by u .= ^-{-w, we have then the relation from which From (6) of Part I and corresponding relations expressing the car- tesian co-ordinates in terms of geocentric longitude and latitudes, we get z 8^ = 8r x -f- r z sin u t cos io/ + r^ cos u 1 sin tou t , / \ sin pdp z -f- p, cos ^d/J, , z = 8r. + r, sin w, cos 101 4- f, cos w_ sin tou, , x x r 3 (30) By means of (28), (29), and (30), we may get 8/, 8w x , 8 3 , in linear functions of 8/? x , 8X, . We also obtain x == Br t ^SO-f-z, sin O8/ /, (sin x sin O cos w x cos O cos i)^, , Sx t = cos ft cos Mp, p x sin ft cos X t 8ft p l cos ft sin X 1 8A I . By means of these equations, or the corresponding ones for the third position which may be used as a check, we may get 8O. From the relation o> = u v, we obtain 24 CONVERGENCY OF SERIES USED IN DETERMINATION Finally from v.i tv 2(/ II) tan--f -tan^- = -^ - ' , 2^32 # we obtain m= -^(A-n^-^sec^'Sv,, 42 With these we have a complete set of formulae by which the variation of any one of the five elements of the cometary orbit is expressible as a linear function of Sft and 8X Z . 22. The variations 8ft 8X 2 . If now errors are made in the observa- tion of the second place, these errors will also have an effect upon the elements. We consider the errors thus caused and give formulae for their computation. In these formulae it is to be noticed that we use the expressions 8/Oj, 8p 3 , Sr 1} etc., to designate the variation of the quantities p l} p 3 , r zt etc., only so far as ft and X 2 are concerned. They must not be confused with the same expressions used heretofore, where only X x and ft were considered to vary. The apparent ambiguity is justified by the simplicity which this usage gives to the writing of the two sets of formulae. Furthermore, we content ourselves here by simply writing down the results of the derivations. This is done because the actual work of obtaining the equations is very similar to that pursued in the last article, and where any divergence occurs the formulae themselves enable one to see the method used. Attention is again called to the approximation used in the values of the ratios of the triangles in obtaining the partial derivatives of M and m in respect to ft and X 2 . To start with we have the equation analogous to (5) : x M* */? ixi -r> 8/> 3 = M8 Pl + _ 8ft + g^- 8X 2 +* jg- Sft + /.- 8A 2 , (34) where the partial derivatives are gotten from the expressions for m' , m n ', m"', M' ,M" , M'" given in (i) a, (2) a, and (3) a; owing to their length we omit them here. It is to be noticed, however, that BM'" . here f^ is zero. = - [ P - jR, cos ft cos (X, - Z z ) + R 3 cos ft cos (X, - Z 3 )] 8 Pl (35) + - [p 3 7? 3 cos ft cos (X 3 - L 3 ) + ^ cos ft cos (X 3 - Z f )] 8p 3 . OF ELEMENTS OF PARABOLIC ORBITS 25 2K i I 3.5---(2"-3)/ s (36) where Kr^ -\- r 3 . 8r t = - [p x RI cos p l cos (X z Z,)] , (37) *3 == ~^ [Ps ~~ -^3 COS A COS (^3 :r!5~t 3) (38) (r 3 -/) V 2r. -/ - (r, -/) I/ 2r 3 -/ 8r x ^ 3 2 ^ 8r, . (39) (r, - p)V,r t -p-(r,-p) V *r t -p ' * sin v x r t sin v z (40) 8^ = Sv 3 8v t . (41) 8r T + r, sin 2/ t cos /8i + ^ cos a, sin t'8u t , . . r i (4 2 ) Bz 3 = 8r 3 -f- r 3 sin 3 cos i'S/ -(- ^ 3 cos 3 sin /8w 3 , r s (43) X 8x t = 8r t y^l-\-z l sin H8/ r x [sin u t sin li cos w, cos Q cos /] 8w t , ^ x / "k 26 CONVERGENCY OF SERIES USED IN DETERMINATION This completes the list of formulae for obtaining the variations of the elements in terms of the variations 8ft, 8X 2 . 23. The variations 8ft, 8X 3 . It is easily seen that the formulas in the case of errors in the third position will be very similar to those of the first position. This is due to the fact that in the computation of the elements the co-ordinates of the first and last positions play very similar roles. We do not give the formulas for the last position here, since it would unnecessarily prolong this discussion. 24. Dependence of the ratios of the triangles on X It ft, X 3 , ft. We come now to a question left over from a previous article. It is as to the dependence of the ratios of the triangles upon the co-ordinates of the first and third positions. The question reduces to one as to the dependence of r 2 and 2 upon the above co-ordinates. That this is true is foreshadowed by the terms of the series written out on the right of (24), Part I ; but it is also capable of derivation from the Newtonian law of motion itself that such is the case. For we have in general and by means of this relation all the higher derivations of r 2 in the development of the ratios are reducible back upon r 2 and ^ . Now, the quantities r 2 and ^ are by the manner of their computa- tion in Olbers's method each functions of ft, X,, ft, X 3 as well as of X 2 , or ft, or both of the latter two, according to the particular one of the equations (i), (2), or (3) which have been employed. But, in taking the partial derivations of M and m in respect to the co-ordinates X t , ft, X 3 , etc., we have assumed the ratios to be independent of these quantities. It is necessary, then, to justify this assumption. To begin with it can be justified only from the standpoint of its being a near approximation to the truth when the computation has been carried out in the method described in article 17, namely, when the time intervals have been so chosen as to give series (24), Part I, the proper convergency. As we have already remarked, this convergency should be so rapid that for any given case the remainders after the second term will be of the order of smallness of the lowest decimal place which is omitted in the process of computation. In order to OF ELEMENTS OF PARABOLIC ORBITS 27 verify the statement made above as to the validity of the approxima- tion in question, we prove the following theorem, which we then proceed to apply. Theorem: The variation 8r 2 arising from uncertainties in A,, A 2 , A 3 , Pi, &, ft is at least of the order of smallness of 8^, and 8Q. For, in the same manner as we obtained (33), we get, after reduction, 811 -. (48) From the relation r 2 (i -f cos v 2 )=#, we get sn whence, by use of (48), *>. =J + ^l [' 36-4 Comp.* No. Comp. aO^-o* Greenwich M. T. 1900, July 24 d I3 h 37 m 2i s August 2 d 1 2 h 9 m 56 s August 6 d i o h 26 m 1 6 s Lamb 1541 Bonn 2544 Lamb (3) 1 385 20 ; 6 20; 8 20 ; 6 16' 47^2 2' 28 ?4 2' 27?9 2o h 37 8*63 I9 h 9 m 43?63 I7 h 26 m 3?63 Elements computed by class and Dr. Laves : log q 0.006363 0=327 59' 59' 29' 32 24' 47 - II = August 3, 1900, Greenwich M. T. K =I2 24' 47" The following values were obtained for co-ordinates by Dr. Laves, which I have corrected for the notation used in this paper and used throughout in computation : A, = 49 13' 22-6, &=i3 28' 31 A 2 = 54 20' 46 fo , /> = : ^ 3 6o 4' 43fo , $, = 35- i 56", log M' 9.950021 10 , log (r t + r 3 ) = 0.308352 , log ^ = 0.007514, log ^ = 9. 278413 10. log r 3 = o.< We have numbered the equations giving the results for each step so OF ELEMENTS OF PARABOLIC ORBITS 29 as to correspond with the formulae from which they are derived in art. 21. Where the coefficients are logarithms, we have indicated it by placing the left-hand member in parenthesis, thus (8r,), (8r 3 ), etc. 8/33=1.05854 8/0, -2.266778)8, -O.I470438A, . (5') 8^ = 3.496908/3, 3.779478/3, + 0.20662 8A X . ,. (Bs) = 0.543683 8p, 0.577421 8ft + 9.315172 SA, . 8AT= 74.76678 8p z + 80.80672 8ft 4.41772 8A t . (14') 8r, = 0.236288 8p, -0.577421 8ft + 0.032823 SA, , Sr 3 = 0.2106228 8p,+ 0.022165 8/8, 0.416143 8X, . 8r 3 = 74.97740 8p x + 80.78458/3, -4.00I588A, , , 8r,= 75. 00307 8p x +81.312728)8, 4.38489 8X, . From (i8 r ) and (19') we get one and the same result, as follows : 8p T = 108080 Sj8 r 0.052766 8X, . (5") Putting this in (5') 8/o 3 = 1.12271 8)8, 0.202898 SA, . Putting these in (18') and (19'), 8r, = 0.24981 8)8, 0.42726 8A, , . g . 8r 3 = 0.25062 8ft 0.04530 SA, . The uncertainty in/ was found to be 8p = 0.058051 8ft 0.437117 8A, . (26') The following then were obtained in the order given : 8^, = -5-31986 8ft + 3.96961 8A, , 8? 3 = -5.i8 7 358ft + 4.o6573 8A X . 8# 3 8^ = 0.13251 8ft + 0.09612 8A, . (28') 8/= 15.0415 8ft 2. 4690 8A, . (3') 8 f = 1.6237 8ft + 0.18104 8A, , 8^=1.75628 8ft + 0.2772 8A X . m = 0.6799 8ft 0.0763 8A X . (31 ; ) i8n = -22i. 8ft+i 7 o. 8A, . (33 r ) ^ n (33') tne dig^ i n units place is uncertain by 3 in each coefficient. We finally get for the uncertainty in the longitude of the perihelion 8cu =6.9436 8ft 3.7886 8A, . (32') 30 CONVERGENCY OF SERIES USED IN DETERMINATION In order to get the amount of an error of an element for a given error in an observed co-ordinate, we need only substitute the values of 8f} t and &X t which are allowable from the nature of the conditions under which the settings of the instrument were made. Suppose, for instance, that 8/3 t equals to one second of arc. This, in circular measure, will correspond to 0.00000485. Then by (33') an error of one unit in the third decimal place would result; while by (27 ') the error would be confined to the seventh decimal place, and in this latter case be perfectly harmless for ordinary six-place tables. As to the amount of error actually probable, various observers differ in their estimates. LeVerrier says: 1 "Experience has proved that, for stars with feeble light, errors of 4" or 5' do not exceed the limits of the possible nor indeed of the probable." In this he was referring to the observation of the asteroids. Dr. Hussey, of the Lick Observatory, considers this estimate far too large for the case where a star and an asteriod are being compared with large modern instru- ments. However he states that "for comets the conditions vary greatly; and for those without visible nuclei, large and cloudlike ; in such cases, even under favorable atmospheric and instrumental condi- tions, " one might be doing very well indeed if he kept his errors of observation below 10." Dr. Barnard, of Yerkes Observatory, esti- mates, with Dr. Hussey, that ofi ought to be the limit of error for an asteroid of the fainter kind if the conditions are favorable; but he says: "A comet's position is far more uncertain; and discordances of several seconds of arc are not unusual in the work of good observers. Much depends, of course, on the presence or absence of a nucleus and the faintness of the comet ; but in general comet observations are distressfully discordant" The above statements of Dr. Hussey and Dr. Barnard are from personal letters to the writer in answer to inquiries in regard to the accuracy of comet observations with the best modern instruments. If we admit, then, that the errors in observation may vary from one to ten seconds of arc in the case of comets, the results (5') to (32 ') show that the computed elements will be discordant, owing to such errors in a single co-ordinate, as follows : Element/ is discordant in sixth or seventh decimal place, Element * is discordant in fourth or fifth decimal place, Element O is discordant in fifth or sixth decimal place, Element II is discordant in third or fourth decimal place, Element