I-NRLF C 3 Ifib 351 j-': ; '. . - \ ; -..-,. mil urn T R A N S A C T I OF THE AMERICAN PHILOSOPHICAL SOCIETY, 1L ..ajj HELD AT PHILADELPHIA, FOR PROMOTING USEFUL KNOWLEDGE, , VOLUME XIX.-NEW SERIES. PART I. * 1 . A New Method of Determining the General Perturbations of the Minor Planets. By William McKniyht Bitter, M.A. ARTICLE IT. An Essay on the Development of the Mouth Parts of Certain Insects. By John B, Smith, Xc.D. PUBLISHED BY THE SOCIETY, AND FOR SALE BY THE AMERICAN PHILOSOPHICAL SOCIETY, PHILADELPHIA N. TRUBNER & CO., 57 and 59 LUDGATE HILL, LONDON. ASTRONOMY LIBRARY TRANSACTIONS OP THE AMERICAN PHILOSOPHICAL SOCIETY AETICLE I. A NEW METHOD OF DETERMINING THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. BY WILLIAM MCKNIGHT RITTER, M.A. Read before the American Philosophical Society, February 28, 1896. PREFACE. In determining the general perturbations of the minor planets the principal diffi- culty arises from the large eccentricities and inclinations of these bodies. Methods that are applicable to the major planets fail when applied to the minor planets on account of want of convergence of the series. For a long time astronomers had to be content with finding what are called the special perturbations of these bodies. And it was not until the brilliant researches of HANSEIST on this subject that serious hopes were entertained of being able to find also the general perturbations of the minor planets. HANSEN'S mode of treatment differs entirely from those that had been pre- viously employed. Instead of determining the perturbations of the rectangular or polar coordinates, or determining the variations of the elements of the orbit, ho regards these elements as constant and finds what may be termed the perturbation of the time. The publication of his work, in which this new mode of treatment is given, entitled A.useinandersetzung einer zweckmassigen Methods zur Berechnung der absoluten A. P. s. VOL. xrx. A 6 A NEW METHOD OF DETERMINING Storungen der Ideinen Planeten, undoubtedly marks a great advance in the determina- tion of the general perturbations of the heavenly bodies. The value of the work is greatly enhanced by an application of the method to a numerical example in which are given the perturbations of Egeria produced by the action of Jupiter, Mars, and Saturn. And yet, notwithstanding the many exceptional features of the work commending it to attention, astronomers seem to have been de- terred by the refined analysis and laborious computations from anything like a general use of the method ; and they still adhere to the method of special perturbations devel- oped by LAGRANGE. HANSEN himself seems to have felt the force of the objections to his method, since in a posthumous memoir published in 1875, entitled Ueber die Stdrungen der grossen Planeten, insbesondere des Jupiter s, his former positive views relative to the convergence of series, and the proper angles to be used in the argu- ments, are greatly modified. HILL, in his work, A New Theory of Jupiter and Saturn, forming Vol. IV of the Astronomical Papers of the American Ephemeris, has employed HANSEN'S method in a modified form. In this work the author has given formulae and devel- opments of great utility when applied to calculations relating to the minor planets, and free use has been made of them in the present treatise. With respect to modifica- tions in HANSEN'S original method made by that author himself, by HILL and others, it is to be noted that they have been made mainly, if not entirely, with reference to their employment in finding the general perturbations of the major planets. The first use made of the method here given was for the purpose of comparing the values of the reciprocal of the distance and its odd powers as determined by the pro- cess of this paper, with the same quantities as derived according to HANSEN'S method. Upon comparison of the results it was found that the agreement was prac- tically complete. To illustrate the application of his formula, HANSEN used Egeria whose eccentricity is comparatively small, being about -%. The planet first chosen to test the method of this paper has an eccentricity of nearly ^. And although the eccentricity in the latter planet was considerably larger, the convergence of the series in both methods was practically the same. It was then decided to test the adaptability of the method to the remaining steps of the problem, and the result of the work has been the preparation of the present paper. HANSEN first expresses the odd powers of the reciprocal of the distance between the planets in series in which the angles employed are both eccentric anomalies. He then transforms the series into others in which one of the angles is the mean anomaly of the disturbing body. He makes still another transformation of his series so as to be able to integrate them. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 7 In the method of this paper we at first employ the mean anomaly of the dis- turbed and the eccentric anomaly of the disturbing body, and as soon as we have the expressions for the odd powers of the .reciprocal of the distance between the bodies, we make one transformation so as to have the mean anomalies of both planets in the arguments. These angles are retained unchanged throughout the subsequent work, enabling us to perform integration at any stage of the work. In the expressions for the odd powers of the reciprocal of the distance we have, in the present method, the La Place coefficients entering as factors in the coefficients of the various arguments. These coefficients have been tabulated by RUNKLE in a work published by the SMITHSONIAN INSTITUTION entitled New Tables for Determin- ing the Values of the Coefficients m the Perturbative Function of Planetary Motion j and hence the work relating to the determination of the expressions for the odd powers of the reciprocal of the distance is rendered comparatively short and simple. In the expression for A 2 , the square of the distance, the true anomaly is involved In the analysis we use the equivalent functions of the eccentric anomaly for those of the true anomaly, and when making the numerical computations we cause the eccentric anomaly of the disturbed body to disappear. This is accomplished by dividing the circumference into a certain number of equal parts relative to the mean anomaly and employing for the eccentric anomaly its numerical values corresponding to the various values of the mean anomaly. Having the expressions for the odd powers of the reciprocal of the distance in series in which the angles are the mean anomaly of the disturbed body and the eccentric anomaly of the disturbing body, we derive, in Chapter II, expressions for the J or Besselian functions needed in transforming the series found into others in which both the angles will be mean anomalies. In Chapter III expressions for the determination of the perturbing function and the perturbing forces are given. Instead of using the force involving the true anom- aly we employ the one involving the mean anomaly. The disturbing forces employed are those in the direction of the disturbed radius- vector, in the direction perpendicular to this radius-vector, and in the direction perpendicular to the plane of the orbit. Having the forces we then find the function W by integrating the expression dW A dQ . D f - = A.a r - - B . ar n . dt dg in which A, and B are factors easily determined. 8 A NEW METHOD OF DETERMINING From the value of W we derive that of W by simple mechanical processes, and then the perturbations of the mean anomaly and of the radius-vector are found from n . te = nfw.dt C be the angular distance from the ascending node of the plane of the disturbed body on the fundamental plane to its ascending 12 A ]STEW METHOD OF DETERMINING node on the plane of the disturbing body. Let ^ be the angular distance from ascend- ing node of the plane of the disturbing body on the fundamental plane to the same point. If TT, TI', are the longitudes of the perihelia, 8, &', the longitudes of the ascending nodes on the fundamental plane adopted, which is generally that of the ecliptic,, we have IT = n - - Q' - - . (3) The angles 4>, 4 1 ? & &', are the sides of a spherical triangle, lying opposite the angles t\ 180 , 7, ^ 'i', being the inclination of disturbed and disturbing body on the fundamental plane. The angles /, 4>, ^, are found from the equations sin | /sin \ (^ + <) ~ sin | (& -- 8') sin | (i + *) sin | /cos | (^ + 4>) = cos J (8 -- Q>') sin | (z -- i') cos | /sin | (^ 4>) = sin \ (Q Q') cos \ (i + i) cos J /cos I (^ - - $) =. cos | (8 -- Q>') cos | (^ -- z 7 ) In using these equations when Q is less than Q f we must take \ (360 + Q, &') instead of \ (Q - - Q'). We have a check on the values of /, , ^ by using the equations given in HAX- SEN'S posthumous memoir, p. 276. Thus we have cos p . sin q cos p . cos q cos p . sin r cos p . cos r sin jp sin /sin 4> sin / cos 4> sin / sin (^ sin /cos (^ cos 1 sin ^ . cos (Q, Q,') COS l' cos i . sin (8 Q') cos (8 &') sin ^' sin (8 Q') sin ;j cos p . sin (^ - - q) sin .;; . cos (i -- q} sin (^ q) cos jj . cos (i - - q} (5) THE GENERAL PERTURBATIONS OP THE MINOR PLANETS. 33 To develop the expression for (-V we put cos /. sin IT = k sin K, sin II' k L sin K^ ) cos n' = k cos Kj cos /cos II' = k L cos .STj, j and hence II = cos/". cos/" . & cos (n - - K) -\- cos /'. sin f . k L sin (II K^ - sin/*. cos/' . k sin (II - - ^T) -}- sin/, sin/' . A?i cos (II JSQ. Introducing the eccentric anomaly F, we have cos/" zz - (cos e), sin/ 1 = - . cos

' . fa sin (n K^ sin e' . e . cos . fc sin (II .5T) + sin .e' . cos <|> . k sin (n J5T) + sin . sin t' . cos $ . cos $' . ^ cos (II jffi). Substituting the value of r . 7 , . // in the expression for f-J we have / j\ - ( J = 1 + a 2 2e . cos F + e 1 cos 2 e 2 aee'k cos (II /iT) + 2ae'k cos (II K) cos e 2ae' cos

. k^ cos (II JSTj) sin F + 2a cos <^' . A;, sin (II Id) cos e] . sin f' a 2 6 /2 .COS V. Putting yj, /^ , y 2 , for the coefficients of cos f', sin t', cos V, respectively, and y for the term not affected b}^ cos t' or sin e', we have the abbreviated form y y] . cos F' ft . sin F -f- 7^ cos V. (7) 14 A NEW METHOD OF DETERMINING / A \ - In this expression for ( J , y c , y 1? and (3 are functions of the eccentric anomaly of the disturbed body ; y-> is a constant and of the order of the square of the eccen- tricity of the disturbing body. In the method here followed the circumference in case of the disturbed body will be divided into a certain number of equal parts with respect to the mean anomaly, g. mu f -11 4.1 u no 360 360 o 360 -, 360 The various values 01 q will then be (Jr. , z. , d. , . . . . n 1. n 7 'it ' n ' n For each numerical value of #, the corresponding value of F is found from g e sin F. Before substituting the numerical values of cos F, sin F, for the n divisions of the cir- cumference, the expressions for y^ y l7 /3 , will be put in a form most convenient for computation. Let p. sin P - 2a 2 e - 2aJc cos (n K) \ e f ($) p. cos P =: 2a cos <>' Jc L sin (II J5Q, J and fr=/'cosp- 1 (9) we find ($Q r=/sin F =. 2a . cos

'. & x cos (II JKi). sin F + p cos P. cos F ep . cos P yi =fcosF= (Z'j? * p sin PJ. cos F 2a . cos

.& sin(Il JT). sin PJ . sin F + p 2ot 2 - sin P ] . cos F ep f. cos (F P) = [2a . cos ^ . cos <' . ^ cos (11 - - K { ) . sin P 2a . cos

/ in the form = [(7- g. cos (*'-#)] [1-^. cos (E' -,)], in which the factor 1 q l . cos (V - - Qi) differs little from unity. For this purpose, if we perform the operations indicated in the second expression, and then compare the coefficients of like terms, we find y = C+q.q.s'mQ. sin Q l YL zz q . cos Q -f- q^ . C COS Q { ft = . sin Q + #1 0f = (& + )? (c) When is known, is found from (a) ; and the difference between (a) and gives >7 when f is known. The equations (a) and (c) give A. P. 8. VOL. XIX. C. 18 A NEW METHOD OF DETERMINING Deduce the values of (3 Q + , ^ Y\ from (a) and (d), substitute them in (c), we find The last equation then takes the form This equation furnishes the value off; and with f known, we find , >?, from equations already given. The three equations giving the values of the quantities sought are =OJ Finding the values of f, , >7, from these equations, and arranging with respect to preserving only the first power, we have /2 (9) Substituting these values in equations (16), they become h C sin =. n (17) noting that C = y + f . If more accurate values of f, , >?, are needed than those given by equations (#), we proceed as follows : Substitute the value of f given by (7 =/>?'. COS ^ if Substituting these in the expressions for q sin , q cos Q, they become 2 sin Q=f. .sin^ 7 (22) 20 A NEW METHOD OF DETERMINING The value of q^ is found from ?L = r * (23) The quantities q, q^ Q can be expressed in another manner. The equations (22) give / give ^-V - g (y + /) . or Further ' 2 sin *F-\- >?' 2 cos 2 F =. 1 + 2 -^ (% . sin 2 F ^' cos ~F) - and J log. (f 2 sin 2 ^+ >/ 2 cos 2 ^) = ^ (% sin 2 ^ ^' cos TT (z s^ 11 2 ^ %' cos Substituting the values of , ^', (7, given before, we find ( K sin ^- ^ cos *F) = JL + - - ? + cos THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 21 The equation y., nz q . q { gives lg- 72 = log. q + log. ^ Putting: log. ff = log. we have for 3^ log. g 1 = k>g. y. Writing s for the number of seconds in the radius, and ^ for the modulus of the common system of logarithms, we find (24) in which log. q =\og.f+y log. 0i = x = s . sn s - sn (25) cos 2^-X - -- 2 cos And for C we have from the first of (15) C^yo + ^.sin 2 ^. (26) By means of the last three equations we are enabled to find the values of ft > w ^h the greatest accuracy. The equations (17), where not sufficiently approximate, will, nevertheless, furnish a good check on the values of these quantities. (A\ 2 J are thus known; and substituting their values corresponding to the various values of g, we have the values of (~\ for the different points of the circumference. 22 A NEW METHOD OF DETERMINING Using the values of (7, + & (1) . cos + 6 (2) . cos 29 + & ( " . cos 3,9 + etc.] [1 + Z> 2 26 cos (e' + Q)]~ s - [l ^ (0) + BV . cos (e'+ Q) + B-\ cos 2 (/ + ) + etc.] where s = ^, =: e' , we are enabled to make use of coefficients already known. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 23 For 2 . cos 0, write x + - , and then we have 7 + a 2 2a cos 0]~ s = [l + a 2 a (x + ^ Expanding we have , I r+l S _+J S + 3 S + 4 55 1*2 3 ~T~ ~5~ * C> _ a~|~ s _ -j, s a . s s + 1 a 2 , s^ s + 1 s -f- 2 a 3 , s 8 + 1 s+2 g + 3 a* ^J 1 * * ~" 1 * 2 " ^ 2 " t " I ' 2 ' 3 ' ^ 3 ' 1 ' __ 2 * 3 ' ,s s+ 1 s_+2 s + 3 s + 4 a 8 . , 1" 2 3 4~ ^5~ '^r 5 " 1 And hence, for their product, we have / s a + i s + 2 v 2 8 + 3 . a * + etc.! / l " u ~2 3~; ^^ J r " * -4- f s S -^ 2 , /s \2 s + l s + 2 4 / s + l\2 s + 2 s + 3 6 " Ll * 2 " VU ' 2 3 U'~2 ;'~3"~'~~4~- tt , (s s + 1 , + 2y 8 + 3 + 4 8 -l / o, 1 u ~2 3 ; ~4~ "T^ a tc> J r -1- P 1_1 S +J /.S.L/'M 2 ?-l !+-? S + 3 5 Ll ' 2 3 VI / ' 2 3 4 fs s+l\ 2 s + 2s + 3s + 4 7| + (r T-) 4 -5 a + etc - + etc. But aj + 1 - = 2 cos 0, or + ^ = 2 . cos 20, a? 3 + \ = 2 . cos 30, etc., 1 / ' /y* ' /T 1 * * ' 24 A NEW METHOD OF DETERMINING and hence ^ = 2,a[l + 1^ .* + 1, (- +3* * + ? .Ct 1 ^ 2 ) s _)_ 2 o i s s+ls + 2s + 3 4 , a + l /s + 2\2 s + 3 S + 4 . , t "I I ' ~2" V 3" V ' ~T~ "5" s + 3 j , . s+1 S + 3 S+ 4 , (29) _0 81 , , -^T-~2~'~T~- a L - f . 4 -.a r*- 4 5 s s + 1 s + 2 s + 3 s + 4 s + 5 6 , "r-"^"-~3^'~"4"' :, 6 C '' and generally >= 2 . - . ' .... . . . . .._ Since s = ^, we find from these expressions the values of the 6 {0 coefficients for different values of n. RUNKLB has tabulated the values of 5 (?) in a paper published by the SMITHSONIAN INSTITUTION. Thus the value of [1 + 2 2a cos (s' )] ~ is obtained with great facility. _ n The value of [1 + 6 2 26 cos (e + Q)~]~ 2 is found in the same way. We now let c"> = i.^r.^.cos2^| s (i) =i.JSr. (i) .sm2.iQ) And hence have c f0) = i . N. .Z? (0) etc.= etc. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 25 Multiplying the series [| 6 (0) + & (1) . cos + Z> (2) . cos 20 + 6 (3) . cos 30 + etc.] by [J (0} + B (Y > cos (e' + Q) + ^ (2) . cos 2(e' + Q) + etc.] , noting that (& e ', and arranging the terms with respect to cos *0, sin iQ, we find = . c (0) . c (2) . c (0) etc. cos gin sin 20 cos 30 sin 30 etc. (31) c L cos / = & (i) . c (0) sn ^ Cr' . C * ' and we find (2) ( a j = A: t [cos J . cos iO + sin ^ . sin . = Tci cos (*0 JSTJ =, ki . cos (i Q K t ). Subtracting and adding the angle ig t this becomes -} i- &,cos \i(0 Q} KI + (iq i A I \ 4^ 7 / \ / ^z ^ cos * ( Q g] KI cos . i (g e') lc>i . sin i If we put (32) (33) i sin . t (g e') (34) ^ K = - ^ cos p ( Q K g K ) Ki, J (35) h i A. P. S. VOL. XIX. D. 26 A NEW METHOD OF DETERMINING n being the number of divisions, we find fa\ l c] / A (S) / , \ (Qa\ ( } A L K . cos i (g K E J AI K . sin ^ (g K e K ) (ob ) \ A) If now, for the purpose of multiplying the series together, we put (0 (0 J& ^ A^-^C^. we have . i>v ^ () (o () . t . = 2 $, . cos y# + 2 fi/,; . sin v# + 2 Cl> smvg] cosi(ge')[2 S Lv cos r#-f-2 $, sin w/] sh (#' (38) Performing the operations indicated we get (c) (c) (c) 22 cos (gr is'}. Cj-, cos vg 22 J C<, cos \_(i-\- v) 9 *V ] +22 1 (7 (> cos [(* i^) ^ te'J (s) () (s) 22 cos (ig ie'). C i>v sin vg =. 22 J C^> sin [( + v) ^ w'] 22 J C^> sin [(* v) g tV] (c) (c) (c) 22 sin (V? &') /S',,, cos v^r =. 22 J ^> sin [(t-f v) ^ *V] 22 J ^> sin [(* v) ^r ie'~] (s) (s) (s) 22 sin (ig ie f ) S i>v sin vg 22 J S liV cos [(*+ v) g tV] 22 1 /S t> cos [(*' v) g u'~\ Summing the terms we find (c) () () (c) = 22 i ( C,,, =F , J cos [( =F v) ^ &'] =F i 22 ( a> , J sin [(i =F v) g u'] (39) (c) (c) From the formula of mechanical quadrature just given, we have C it , 8 it , when (c) (c) v n ; but we know that they are J . CJ, , J ^ o? as shown by their derivation. Thus (c) (c) (c) (c) AI = J Q o + O { i cos (/ + C/ 2 cos 2g + etc. I (c) () w ( S ) > = 2 C it v cos ^ -f 2 C-, sin vg + <7 U sin gr + C it 2 . sin 2^r + etc. J (s) (c) (c) (c) At = \ BW + Ski cos ^ + Si ,2 cos 2^ + etc. ^ (c) w w ( J = 2/S z> cos r^ + 2/S' /> sin vg. + /S/ ; i sin # + /S i; 2 sin 2g + etc. J Hence where v = 0, each series is reduced to its first term. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 27 In the application of the very general formulae care must be taken to note the signification of the various terms employed. In case of (c) 2 = ~ \ - cos P (ftflO KI, J n (s) 2 = &/, siu [ (&-#*) KI, J > >i n shows the number of divisions of the circumference ; and we divide by ^ m form- ing Jc t K to save division when forming the coefficients c,,, s v . The index and multiple i shows the term in the series Ji (0) + Z> (1) cos (V Q) -\- & (2) cos 2(Y Q) -f Z> (3) . cos 3(e' - - Q) -(- etc. The double index i, x shows the term of the series of La Place's coefficients and the particular point in the circumference. The index v shows the general term of the series expressing the values of (c) (s) AI :IK , -Af,,, when we give to v values from v r= 0, to the highest value of v needed in the approximation. 2 In .&,,, ^*(* g K ) -5^ for each value of i 9 there are n values of each it/ quantity. (c) (c) () (c) () The next step is to express the w values of A , ^ , A l , u4 2 ? A > etc., respec- tively in terms of a periodic series. And since these quantities are functions of the mean anomaly #, if we designate them generally by Y, of which the special values are ~y ~y ~y ~V~ we have Y Jc + c, cos g + c 2 cos 2g + etc. ) + i sin ^ + s 2 sin 2g -\- etc. ) The values of c,, , in this series are found from the n special values of Y. 28 A NEW METHOD OF DETERMINING From (C) (8) I , or AI | c + ^! cos g + c 2 cos 2 -* 2> J- 11 When we divide the circumference into sixteen parts, each division is 22. 5. We find the values of Y 0) Y^ Y 2 , . . . . Y^ as in the case of twelve divisions. To find the values of c v and s,,, in the case of sixteen divisions, we put (0.8 )=Y a +Y s (f) =T,-T, (i.9)=r,+ r, (i) =r,-r, (2.10)= T,+ r 10 (A)= r 2 - r IO (7 . 15) = F, + F u ( A) = F 7 - F B 30 A NEW METHOD OF DETERMINING (0.4) = (0.8) +(4.12) (0.2) = (0.4) + (2.6) (1.5) = (1.9) +(5.13) (1.3) = (1.5) + (3. 7) (2. 6) ='(2. 10) + (6. 14) (3. 7) = (3. 11) + (7. 15). Then 4(c 2 + O = (0.8) -(4.12) ' 4fo- c ) = { [(1 . 9) - (5 . 13)] - [(3 . 11) - (7 . 15)] j cos 45 4(s 2 + s 6 ) - \ [(1 . 9) - (5 . 13)] + [(3 . 11) - (7. 15)] } cos 45 4(s 2 -s,) =(2. 10) -(6. 14) 8.d=(0.4) (2.6) 8. s 4 = (1.5) (3.7) 4(c, + c,) = (f) + [(A) - ( T 6 4)] cos 45 4( Cl - e,) = [(i) - (A)] cos 22 5+ [(&) - (A)] cos 67 5 4(c, + c 3 ) = (-1) - [(A) - (A)] cos 45 4(c, - c 5 ) = [(I) - (A)] sin 22 . 5 - [(-ft-) - ( T %)] sin 67 . 5 4( Sl + S7 ) = [(i) + (^) ] sin 22 . 5 + [(&) + (A)] sin 67 5 4(s, - ,) = [(T%) + (A)] cos 45 + (A) 4(s 3 + *) = [ft) + (A)] cos 22 ." 5 - [( T T ) + (A)] cos 67 . 5 cos 45- (A). When the circumference is divided into twenty-four parts, each part is 15. Let (0.12)= r o + F J2 (0.6) = (0.12) + (6. 18) (4) = (0.12) -(6. IS) (1.13)= F x + F 13 (1.7) = (1.13) + (7. 19) (*) = (1.13) -(7. 19) (2.14) = F 2 + r i4 (2.8) = (2.14) + (8.20) (|) = (2 . 14) - (8 . 20) (11.23)= F u + F 23 (5. 11) = (5. 17) + (11. 23) (^-) = (6 . 17) - (11 . 23) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 31 Then 6(c + 2 . c 12 ) = (0 . 6) + (2 . 8) + (4 . 10) 6(c 2 + c jo ) = () + [(I) - (^)] sin 30 60, - Clo ) = [(*)-- (A) ] cos 30 6(c 4 + c. ) = (0 . 6) [(2 . 8) + (4 . 10)] sin 30 6(c t c 8 ) = [(1 . 7) + (6 . 11)] sin 30 (3.9) G)s, + 10 ) = [(I) + ( T S T)] sin 30> + (f) 6(s 2 -s 10 ) = [(*) + (A)] cos 30 6(s 4 + s s ) = [(I) (^)] cos 30 Q(s,-s s ) Further, let 23 Then 6(0, + c,, ) = (A) + [(A) (M)] cos 30 + [(A) (A)] cos 60 6(c, c,,) = [(A) (H)] cos 15+ [(A) (A)] cos45 + [( 1 'V) (A)] c s75 c 6(c 3 + c, ) = (A) - (A) + (A) 6(c 3 - c, ) = j (A) - (M) -[(A) - (A)] - [(A) - (A)] I cos 45 6(cj + c,) = (A) [(A) (H)] cos 30 + [(A) (A)] cos 60 6(c 5 -c 7 ) = [(A)-(tt)] sin 15 - [(A) - (A)] sin 46 + [(A)-(A)]W 6(., + ) = [(A) + ()] sin 15 + [(A) + (A)] sin 45 + [(A) + (A)] *in 75 6( 8l ,,) = [(A) + (if)] sin 30 + [( A) + (A)] sin 60 + (A) 6(b + s, ) = { (A) + (tt) + (A) + (A) - [(A) + (A)] } cos 45 6( Si + , ) = [(A) + (H)] cos 15 - [(A) + (A)] cos 45 + [(^) + ( T V)] cos 75 6(s 5 -s 7 ) = [(A) + (if)] sin 30= - [(A) + (A)] sin 60 + (A). 32 A NEW METHOD OF DETERMINING When the circumference is divided into thirty-two parts, each part is 11. 25 Let ( 0.16)= r + Y u (0.8 ) = (0.16) + ( 8.24) (0.4) = (0.8 ) + (4 ( 1.17)= F,+ r 17 (1.9 ) = (1.17) + ( 9.25) (1.5) = (1.9 ) + (5 ( 2.18)= F 2 +r M (2. 10) = (2. 18) + (10. 26) (2.6) = (2.10) + (6.14) (15.31)= F 15 +F 31 (7. 15) = (7. 23) + (15. 31) (0.2) = (0.4 ) + (2.6 ) (1.3) = (1.5 ) + (3.7 ) ( 8.24) (f) = (0.8 ) (4. ( 9 . 25) Q) = (1 . 9 ) (5 . : (|) = (2. 10) -(6. 14) - (7 . 23) (15 . 31) (f) = (3 . 11) (7 . 15) Then 8(c +2.c 16 ) = (0.2) + (1.3) 8 (c 2 . c 16 ) = (0 . 2) (1 . 3) 8 (c 2 - Cl4 ) = [(i) - (A)J cos 22 .5 + [(A) - (A) J cos 67 . 5 -' \* / 4 I 12/ \^"/ 8 (c 4 c 12 ) = [(I) (f )] cos 45 + c 10 ) =(|)-[( T 2 o)-(T 6 4)]cos45 - Clo ) = [(i) -(A)] sin 22 .<> 5 - [(A) ~ (A)] s^ 67 . 5 16. c 8 =(0.4) -(2. 6) 8 (s 2 + s u ) = [(i) + (A)] sin 22 . 5 + [(A) + (A)] sin 67 . 5 8(s 2 -s 14 ) = [(A) -(A)] cos 45 + (A) R fa -- o ^ O ^o 4 6i 2 y 8 (s 6 s 10 ) /2\ VS/ [tt) + (A)] cos 22.5-[(A) + (A)] cos 67 .' 5 [(A) -(A)] cos 45 -(A)- TUB GENERAL PERTURBATIONS OF THE MINOR PLANETS. 33 Further, let \ 5 \ V 3TV * M And besides, let [(A) - (tf )] cos 11-.25 + [(A) [( T V) - (tf )] sin 11.25 - [(^) cos 78.75 sin 78.75 [(A) - (**)] cos 22.5 + [(JL) _ (i|)] cos 67.5 A) - tt!)] sin 22.5 - [( A ) - (if)] sin 67.5 ( A) - (tf)] cos 33.75 + [( A) - (if )] cos 56.25 (T S ) - ttf)] sin 33.75 - [(,fr) - (i|)] n 56.25 '" = (A) + [(A) - (it)] cos 45 '" = ( A) - [(A) -ttf)] cos 45 . ' = [( iV) + (if)] sin 11.25 + [(A) + (A)] ^n 78.75 = [( iV) + ()] cos 11.25 - [(A) + (A)] cos 78.75 ? = [(A) + ()] ^ 22.5 + [(^) + (fi)] sin67.5 ' = [(AM tt4)]cos22.5 -[(^) + (io)] cos 67.5 " = [(A) + (tt)] sin 33.75 + [(^) + (J|)] sin 50.25 >" = [(A) + ()] cos33.75- [(Jj-) + (W-)] cos56.25 "' = [(A) + (41)] cos 45 '" = [(A) + ()] cos 45 A. P. S. VOL. XIX. E. 34 A NEW METHOD OF DETERMINING Then S( Cl + Cu) = A" + A' 8( Cl C*) = A + A" 8 (* + ) = &" + 3' 8 fa - c 13 ) = [ A A" + B + JB"] cos 45 8 (c 3 - c u ) = [ J. - -4" - (-5 + J?" 8(c 7 + c 9 ) = A"' A S( Si 8^=C"'+C' 8 ( 3 + *!) = [D + D" ~ ( C - C")] cos 45 8 (8 5 + ) = ID + Z>" + (7 C"] cos 45 8(8 6 ) = />' --D'" 8(s 7 + s 9 ) = D i>" 8(5 7 s 9 ) = C""+ C". The expressions for the determination of the values of c v and s v , just given, are found in HANSEN'S Ausewandersetzung, Band I, Seite 159-164. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 35 CHAPTER II. Derivation of the Expressions for BESSEL'S Functions for the Transformation of Trigonometric Series. The value of Qj given thus far is found expressed in a series of terms the argu- nents of which have the eccentric anomaly of the disturbing body as one constituent. 3ut as the mean anomaly of both bodies is to be employed, it will be necessary to make me transformation ; and the next step will be to develop the necessary formulae for this mrpose. HANSEN, in his work entitled Eatwickelung des Products einer Potenz des Radius Vectors et cet., has treated the subject of transforming from one anomaly into mother very fully ; what is here given is based mainly on this work. Calling c the Naperian base, and putting ni f*y 1- fti' /iY 1 y c > y > ve have '= (cos e + V 1 sin <0 (cos e'+' V 1 sin e')> ilso , ,__ _ I gin *, =. cos (i e i' e') + V 1 sin (i e' i' e'). Denoting the cosine and sine coefficients of the angles (is i' e') by (, i' 9 c) d (i, i', s) respectively, the series F^^ (i,i, c) cos (ie i f e' ) 22 V T(^, i', s) sin (ie i'e 1 ) (1) ;an be put in the form F = i 2 2 5 (*; i', c) - V^ (*, t% s) | y 1 y' 1 ". (2) 36 A NEW METHOD OF DETERMINING In a similar manner we get F = J . 2 2 { ((i, h', c)) V T(te A', 8)) y< . a'-*', (3) where z' = c-*^. We have now to find the relation between y and z. Let g n the mean anomaly, and s = the eccentric anomaly. Then from g zz 8 e sin e, introducing V 1? we get g V l = eV 1 esine V Since 2 V l.sinerry 2T 1 , we find from 7/ r yi y we obtain V 1 =log., V l = log.y, THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 37 and (4) Thus g V 1 = log. z log. and hence z = y.c "2" . (5) From z = y . c~ ^(y-y~'\ we have and Let I be denoted by X ; then &* - y~\ (8) and c ^(y~y ) c l y . c~ il ^ . (9) But C- 7lA -V . C^ A ^rz Tl ^ . y + T-O 2/ 2 - jVs ' ^ "^" 1234 ' &* ^ e ^ Ct ) 38 A NEW METHOD OF DETERMINING and + & . y + 2 . f + J . ys Performing the operations indicated, we have 2 liars etc \ / _ 2 . O ^ + , ; ' 3p h ^ s \ f + 1^3 - I^2^4 db etc.] (y < - F Ctc , ^"^ /-i _ W _, _ A^ 4 " 1.2..m V " l.w-fl 1.2.ml. LC __ ^2 , JW . ^ 6 , ^ 8 1 2 .2 2 p 2 2 3 3 I 2 2 2 3 2 .4 2 ' (l'3)3 ,'6)6 ,757 \ / + * - i4 + TO - K^ki etc -) ( - y THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 39 As we may write li in place of i, we have, thus, also given the value of c l i(y V l ) put + 00 ( TO) + GO (m) Then, from the preceding developments, we see that ( ni) (m) m (m) m (m) (m) Again (-3) (1) (2) + 00 (m) (0) (1) (2) -oo e/AA . y m = Jti + '/M . 2/ + ^A . f (-1) (-2) (-3) '~ 3 + etc. / 3 -f etc. 3 + etc. '~ 3 + etc. + co (TO) Comparing the values of 2_ oo Jl^ . y~ m and have 7~ 7 7,-i ^ 3 W KX' f -1 ^tt :- J hx -Kk-- ^j + p-^ i^;3^r etc., tor y , (1) (1) " " (-2) (2) (2) (2) etc. r= etc. 7,3)3 1655 L7)7 ft AT | ft A /tA .. ^ -i -^ - 2 - + p- 23 j jTg^g^ dz etc., for y\ + 11 A - . ft O ,20204 ^F etc., for y \ i.i 1 . .'J Ji ,i .O.4 etc., for 2/ 2 , 1.2 P.2.3 1 2 .2 2 .3.4 etc. (10) (ii) (12) (13) 40 A NEW METHOD OF DETERMINING Comparing the values of 2_* J, lX - y m and c 7 ^ 2T 1 )' we get the same expressions for y m and y~ m . (1) (2) We see from the values of 7 AA , J hK , etc., found above, that the general term is J,WI)TO Awi+2 j^wi+2 ^jWi+4 ^[wi+4 T _ __ _ I I pf"O " A ~ 1.2...m r.2..m.m+l " I 2 .2 2 ...m.m-f l.m+2 1 1.2.m+l.m+2 Further, we have and, by putting m = 7i i t this becomes Ji J n.i /-i r\ * "h\ y (^j Let + 00 (h) ^ + " (*> Multiplying the second of these equations by z~ h . dg, we obtain + co (i) \z- h .dg = 2_ P h .dg. Integrating between the limits + n and 7t, we have THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 41 From z = c gv l = cos g + V 1 sii we have dz == ( sin g -f V 1 cos g} dg ; also z V 1 V 1 cos g sin g. Therefore dz z V 1.<<7, and (17) becomes In like manner we find '(A) i / n. 1 . j " 27r 1 A : rTJ Integrating by parts we have h h -C y^.z^.dz. (18) O / 1~ * ' \J / x-, . Comparing this value of Q t with that of P h , we obtain or (0 ,- (A) s-fc ( 19 > A. P. 8. VOL. XIX. F. 42 A NEW METHOD OF DETEEMINING Thus we have, between the mean and the eccentric anomaly, the relations In the application of these relations, since MO the expression for F is changed from F = J 2 2 { (;, f^, r c) - V- 1 ft *\ *) j y*. 2/' - into F-^^ jfttVc) V lft*V) tf. The other value of JP is ((t, ^', c)) - V- 1 ((*, ^, A comparison of these two values gives (-*') $ ((, 7^', c)J = 2 P_, t , (, t', c) z 2 . , . J"//v (*, ^', c) In transforming from the series indicated by (i, i', c) into that of ((?, A', v + l, c)) +etc. The expression ^ - l^rn ( l ' " 1^+1 + 1.2.m+l.m+2 ~" 1.2.S.m+l,m+2.i+3 ^ GtC< enables us to find the value of J hx for all values of m. A simpler method can be obtained in the following manner : ,._i\ (0) (1) (-1) (2) (-2) -* ) Putting c i~ y ~ in the form (0) = J + J . y-J y+ +J fl-^ Fl-% fl~2 - we have, for the differential coefficient relative to y, (1) (2) (-1) (2) ,+2.e7. e .y ""2" """2" If we multiply the second member of the first equation by ^|(1 + y 2 )> we have an expression equal to the second member of the second expression, and by comparing the two we find (m+l) (m-1) ^. (m) (22) 44 A NEW METHOD OF DETERMINING Let (m) h-iT -f^> - (23) then (m) (m-1) J e J e ' 2 From this general expression we find (1) (0) (2) (1) (0) etc. zz etc. z= etc. (m) From the values here given, since jjj-f- is put equal to p m , we have, by increas- ing m by unity, (m+l) " Putting -7 zz r w , equation (22) h-^ takes the form ^m Pm+l I J- t From this we find 1 I'm - THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 45 We also have i ~ - (25) a form more convenient in the applications. (m) The general expression for J is 2 (m) (0) e=J,e'Pl-p2.p3.. -Pm (26) - - where ,() I* , I 1? etc " if we put l h^. From the expression ( i') i (h'-V) ((*, h', c)) = 2P_ A / (*, tf, c) = 2 ^ J h w (i, i, c) it is evident that when &' z= 0, or when both f and h' are zero, this expression cannot be employed. To find the values for these exceptional cases let us resume the equation /I When h = we have (i) 46 A NEW METHOD OF DETERMINING The equation z = y.c- ( y~y~^ gives dz dy e /- , _ 2\ j.. /9Q\ ~ ^ (.* T~ y ) dy. / Hence d) /.i-T^-i --v 1 , 1 jo is a whole number ,-"V except when jt? = 1, when this integral is 27tV 1- Hence it follows that When i = 0, we have (i) (-i) PQ =. PQ ZZ 5 (0) Po = 1. Using the expression (-i') (-i') (-t'-l) ^ /,/ ,,\\ v p /,' ' ^,\ p / 7 * /' r "i _i_ p , / ? " /'_i_ 1 r \ z, li , cjj 2, . x'y ^, * > c ; -^ -A' I,*? * > 6 ^ n^ -L-W \^ - T^ J-j y we have ((o,o,c))_=(o,o,c)-ai'(o,i,c) for the constant term, the double value of this term being employed. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. For Ji' 0, we have ((1, 0, c)) = (1, 0, c) - X (1, 1, C ) - x (1, - 1, c) ((1, 0, *)) = (1, 0, 8) - V (1, 1, 8 ) - X (1, - 1, 8) . ((2, 0, o)) = (2, 0, c) - X' (2, 1, c) - X' (2, - 1, c) . ((2, 0, 6-)) = (2, 0, 8 ) - X (2, 1, 8 ) - X (2, - 1, 8 ) etc. etc. In what precedes we have put g r= the mean anomaly, e i= the eccentric anomaly, c the Naperian base, z cf*-\ y = tf v ~\ and obtain where c~ T& V ^ is expressed in a series, the general term of which is .m 1.2.m+l.m+2 1.2.3.m+l.m4-2.m+3 Thus l.m-j-I 1.2.m-f-l.m-f-2 1.2.3.m+l.m+2.m+3 etc. )y" We have also put and since (m) (m) 48 have found A NEW METHOD OF DETERMINING (m) (ft-O if Again supposing we have found m =. h i. (i) p -fc * * (i) (A-t) Thus we have (h-i) y (h-i) r - "i Jh\ |_cos * + sin is V 1 J 5 (A i) Equating real and imaginary terms, we have i A=00 _i (h ~ i} COS *V=:-. 2 A= _ Jh\ COS t - A=oo J (h-i) siuie=-.^ h ^J h , .sin (29) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. We notice that (i) (-i) For all other values of i (0) Po = 1. (0 o = 0. 49 If a large number of the J functions are needed they are computed by means of equations (24) to (27), as shown in the example given in Chapter V. If we wish to determine any of them independently we have from ()__ hm)m r- ^ fcp ** Z 1.2...m L " Lm+^ 1.2. fr 4 . m+l.m+2 i.2.3.m+l.w+2.m+3 d ' (0) (i) (2) (3) 1 - '- -\ - . - -J- etc. 14 4 16 36 64 ' 7,2 p i 7,4 p* 7,6 p e ~] I ' I gf Q " 2 '4 " 12 '16 " 144 '64 " 1.2 3 ' 4 ~^~ 24 ' 16 T.2.3 . .- 44 40 16 w (j, e-Y r / 2 ^ ~\ ^ = ra3 i 4[ 1 --j-f etc J (30) (m) In these expressions we have written for X its value Je. Since h has all values from &=+co to --co we find any value of J" AA by at- tributing proper values to h. From equations (29) we find the values of the functions cos is, sin ie, in terms of cos kg, sin hg, and the <7 functions just given ; always noting that when h 0, we have only for i 1, Je as the value of the function. We can employ equation (22) when only a few functions are needed, or as a check. A. P. s. VOL. xix. G. 50 A NEW METHOD OF DETERMINING It may be of value to have y* in terms of z h and the J functions. Prom the sec ond of equations (20) we Jiave (0) (1) (2) z 2 + -J 3 A.* 3 +etc. J K . z- 1 - - i<7 2A . z~ 2 iJg, . z~* etc. (0) (1) (2) - Z" 1 - Z~ 2 *~ 3 + etc. (2) (3) (4) J.z J.*. cfe. etc. (1) (0) (1) 2/ +2 = - ye/A . Z + | / 2 A 2 2 + |^3A - & + et C. (8) (4) (5) -ty.z- 1 -%J*.z-* -KsA-^" 8 -etc. (i) (0) _ 2 (i) (3) (4) (5) Then from y~ l = 2 cos y* y~ l = 2 V 1 sin ie we find the values of cos e, sin 8, cos 2e, sin 2e, etc. In case of the sine, as for example when i =. 1, we have y y~ l 2 V 1 sin e ; but in z z~ l = 2 V 1 sin g, we have the same factor, 2 V 1> in the second member of the equation. From r =. a (1 e cos e) we find f- J zr 1 2e cos e + e 2 cos 2 e ^ V = 1 + 2e cos e + 3e ? cos 2 f + ^e 3 cos 3 e + etc. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 51 For (- ) we have \a/ (r \ 2 - J = 1 -f 2 C " 2e cos + Je 2 cos 2e / But d :_/rj\ _ 2e gin e - Q _ e cos e j ^1 - 2e sin e, (Zgr Vav 7 rfy and p (0) (2)-, p (1) (3)-. p (2) (4)-. sin = U7 A + J,, sin ^ + \ \ J^ + J 2A sin fy + i I Ju + ^ A sin 3^ + etc. L _l L_ _l l_ J Multiplying by 2e . d^ we have for the integral of - (0) (2) ~i 9^> r (1) (3) n +J* J cosg--^\_J 2 , + J 2 , Jcos2#-- 3A + 3 A cosS etc. where c = 1 + f e 2 . By means of (22) this becomes /r\2 (i) (2) (3) ( a ) 1 -f f e~ f eA cos g J 2J , cos 2g ^ cos 3^ etc. ( \ 2 -J ", we have Se 2 . cos 2 e == |e 2 (1 + cos 2e), 4e 3 cos 3 a = e 3 (3 cos e -f cos 3e), 5e 4 . cos % = |e 4 (3 + 4 cos 2e + cos 4e), 6e 5 . cos 5 e z= T %e 5 (10 cos e + 5 cos 3e + cos 5e), 7e 6 cos 6 e =z ^e 6 (10 + 15 cos 2e + 6 cos 4* + etc.) and hence ~ 2 = 1 + + * 4 + ^ + etc. + [2e + 3e 3 + f|e 5 + etc.] cos e + [f ^ + ^-e 4 + -y^e 6 + etc.] cos 2e + [e 3 + f|e 5 + etc.] cos 3e M e + etc -] cos 4e 52 A NEW METHOD OF DETERMINING Attributing to i proper values in equation (29) we find the expressions for cos e, cos 2?, cos 3e, etc. We then multiply these expressions by their appropriate factors and thus have the value of ( r \~' \aj (2) , r . / r \~ c (a j ~ 2 & COS & (a) ~ 2 - Ei (2) (-2) The following are the values of R t and R L to terms of the seventh order of e. (2) J.TI - 2e -}- -%e - - - (2) -fl/ - " -& ~~ /2 (2) & ~ ~ 256 e (2) 7? _ . _2 5 fP _|_ 625 .o? ^^S 192 e l4608 e (2) -"6 -^O 60 (2) 7? _ 2401 P 1 **! 2"3040 6 ' ^o" 2 = = 1 + e 2 + f e 4 + -W + etc. (-2) r> 1 J.4/4 - 24 240^ (-2) J __ l Q 9 7 ff> 1 6 6 2 1 g7 (-2) r> 1223.06 ** 1 6 6 See HANSEN'S Fundamenta nova, pp. 172, 173. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. (2) ( 2) We add also the differential coefficients of R, , R, relative to e. l 7 Is (2) dJR a o . de 53 (2) de = - - 1* + tt* ^Wo 6 etc (2i w (2) 4875^ e de etc. (-2) dR<) (-2) de (-2) re -4- etc. etc. e + Se 3 + 2 + f e 2 + 5e + le 3 4 + ? 4 + _ 64 (-2) de (-2) 4608 ^ 54 A NEW METHOD OF DETERMINING The value of -, found by integrating d( r \ 2e . sin e . dg, is (Jj \Cu / , r i (1) (2) (3) - 2 = I + fc 2 f J K cos g ft/a* cos % t f J" 3A cos 3g etc. (2) In terms of the R t functions, r 2 (2) (2) (2) - ZT: 1 + -le 2 jRi cos ^o cos 2q R^ cos og etc. a 2 2 * y Again, since Let then dg r* we have a 2 T> ( ~ 2) - 2 = ^- cos ^^ 2 and hence ^!~ 2) = The coefficients represented by C t designate the coefficients of the equation of the centre. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 55 Using the values of the d coefficients given by LE YERRIER in the Annales de I'Observatoire Imperial de Paris, Tome Premier, p. 203, we have f-g = 4: (I) - 2 (|) 3 + | (iy + JftL (|) 7 + W (|)o] sin g I) 2 ~ V (I) 4 + V (l) c + If (I) 8 + etc. ] sin 2g (I) 4 - W (I) 6 +* ^iP (I) 8 - etc. ] si (I) 6 ^W^ (I) 8 + etc. ] sin Qg _l_ [10661993 /_ \9 "1 Q |_10080V2/ J sm y y Converting the coefficients into seconds of arc, and writing the logarithms of the numbers, we have for the equation of the centre, + [5.9164851 (I) 5.6154551 (|) 3 + 5.5362739 (|) 5 + 5.787506(|) 7 + 6. 25067 (|) 9 ] sin g + [6.0133951 (|) 2 6.1797266 (f) 4 + 6.067753 (|) 6 + 5.59571 (|) 8 ]sin2^ + [6.2522772 (|) 3 6.6468636 (|) 5 + 6.690089 (|) 7 6.22336 (|) 9 ] sin 3g + [6 5491111 (|) 4 7.093540 (|) 6 + 7.27643 (|) 8 ] sin 4# + [6.8775105 (|) 5 7.533150 (|) 7 + 7.82927 (|) 9 ] sin 5g + [7.225760 (|) 6 7.96973 + [7.587638 (|) 7 8.40484 + [7.95944 (|) 8 ] sin 8# + [8.33880 (|) 9 ] sin 9g 56 CHAPTER III. Development of the Perturbing Function and the Disturbing Forces. By means of the formulas given in the preceding chapter, the functions ^(^), ft . a2 (;!) 3 j etc., can be put in the desired form. The next step is to determine the com- plete expression for the perturbing function, and also the expressions for the disturb- ing forces. If k 2 is taken as the measure of the mass of the Sun, and m the relation between the mass of the Sun and that of a planet, the mass of the planet is represented by mk 2 . If x, y, z, be the rectangular coordinate of a body, those of the disturbing body being expressed by the same letters with accents, the perturbing function is given in the form i xx'+yy'-\-zz f ~l J " 1+m J r' 3 Now A 2 = (af x)* + (y' - 1/) 2 + O'-z) 2 , hence n =*[*_'* 1-f-m L^J r' z If a fl is regarded as expressed in seconds of arc, and if we put s = 206264". 8, p = -=-., =-, 1-J-m a " we have THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Finding the expression for (H] first by the method of HANSEN, we let h ! ' u , . k . cos (n K), h' = ** .cos \r/ cos ^ \r / cos ^ Putting cos/' = y\ . cos g' + y' 2 . cos 2g' + y' 3 . cos 3g' -f etc. = ^ . sin \ T-T A fl.A (m) (0) (1) we are enabled to find J hK if J hK , J hK are known. It must be noted that the argument of BESSEL'S table is 2 . h%, or 2 . fcl, or he. a) Thus if it is sought to find the value of J 2X , we enter the table with 2 . 2A, or 2e as the argument. When we need the functions for k from h =. 1 to It 4, we must find the (3) (2) (1) (0) e (-2) values of $J ' , J \J , \J e , -, and - - \J e . 4 2 '' T> * r ^ j r sn / out - cos / rr cos e e, and -. sin r a' a' cos S * a' 2 sin _ cos () (2)-| r (1) (3) -I cos e' = ^ + / A , J cos <;' + J LS^ 4, J cos 2gr' + etc. r (o) (2)~| r (i) (3)-i sm f' \jf v + J K , J sin^' + j[J, A , + ^ J sin "2g f + etc. From the values of cos e' and sin e' we have a' 2 r T (Q} I 2 n , r (1) (3) ~i r ( <2) (4 n ,T, COS / : |_e7 v J,, J COS #' + 2 |y 2A -- e^ J COS 20' + 3 [^ -- e/^ J COS 3(/' + etc a' 2 sin /' r (0) (2) ~i r w ( :5 n r ( 2 > rCsi = L^' +^'J sin ^ + 2 L^'+ ^ J sin %' + 3 L^'+ We now assume * (i ~ i) (ui) i A + ^ .>*_ f^* ^* (i'-i) (t'+i)-i p (i'-n (i'+i)-i - -^ J ^=r[^-.+ ^]. Comparing these expressions for y'?, S' i>9 with those found in the expression for '2 * -/*' ^2 S ~ / given above, we see that the relation between them is i' 2 . r 2 cos ( * 9 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 61 The expressions for cos F, sin F, are the same as those of cos F', sin F', if we omit the accents. Hence if we perform the operations indicated in the expression for (H), we have - i- cos' ( ig V) ^ WtfvS&r#*'} sin ( *7-*V) (2) y* and i' having all positive values. Attributing to i and i' particular values, we find, noting that $ = 0, and 8' = 0', -^ \]i . w\ + Wx ] cos ( g - - g') - - \ [U l7 \ + Z'y^'J sin ( g - - g>) 1P-71/1 *'W'i ]cos( flr-- 3r')-j-4[>Vi /VA] sin ( flf-- 5r') P-7o-/i cos ( ^') - - i Z'^'i sin( ^') 2 [A . y,/ 2 + A'. W J cos ( 5r 2gr') - - 2 PV*+ Z>^ 2 ] sin ( ^ 2g') 2 [A. yi / 2 A'. WJ cos( g 2g') 2 [Z.V 2 ^'J sin ( ? + = r* + r' 2 --2rr' H, we find dQ _ m' I" _!_ dJ _ _ ' r dH~} dv ' ' l-\-m L A* ' dv r" dv J dQ m' T 1 (r r'm _ H~\ dr^ '' : r+^i L " T 2 V J j " 7 2 J' j/-v w*' r 1 j A > dz~l dl - -- To-^A z'. /3 L l-fm L ^ r 3 J 7 _ A dA ,dH A dJ , Tr dJ 2' A rr rr , A zr r r //. - rz -- dv dv dr dz A Hence dtl _ m' dv ~ 1-fri dti _ dr ' l-f ' f 1 1 ~1 , r -T ni' r 2 ff H _ m L A 3 r'*_\ 1-fm ' A 3 / /., . T-T.X = r -. ? sm ~Ir' z sin -( /' + II') l?/i J 5 v>/ i = rr- ( Va A-,) sin 1. r sin (/+ n) l+m V J 3 rv Vt/ sm - r sm m' 3 . 97- , / a' i TT\ / ^-/ i TT/\ m' / 1 1 sm " J - rr sm ( f + n) sm (/ + n } + cos To eliminate ^T from some of these expressions we find from that _ i*+r j^ 2J " 2l The expression for r then becomes dr -> y-^ / iT~~ f*>. 9 T ^^ di2 m i r 2 r 2 1 r TT\ " -.-- ti dr 14-m I 24 s 2J r' 2 J dr From the value of A 2 we have, further, r^rr'H r' 2 r 2 2J 5 64 A NEW METHOD OF DETERMINING and hence *- _ in Bin j ^ Bin J 5 J :1 J _ 2 the latter of which, by means of the expression for , becomes ' |~ r' 2 r 2 1 ~| r / /. . T^N m' . r r . / ~ , -- sin Jr sin (f+ H) sin I -- sm (f II) m L J 5 3J :! J 1m r' 3 The expression for A 2 also gives (r^rr'HJ _ (r' 2 r 2 ) 2 _^^_ i J 4J 6 by means of which we find - m ' l"3(r /2 r 2 ) 2 _ / 2 i J_~|_ ^_ J^ rr '~ Im" L 4J^~ " J 3 " 4 J J 1+m ' r 72 ' . T dr'~ + If we put, for brevity, (J) = ,. sin J; 2 sin THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 65 the expressions which have been given for the forces, together with the perturbing function, are r" d * S + ('!") = ^(".Y \ sin 2 / r' 2 . 0/ /., . nA /o\ J -it a - sin -u + n ) - K J > a f 9 sin r / * , aa = - " sin n - drdZ' \J/ a' 2 a 2 a 2 a a \3 sin / r sin (/+ n) Vt/ i 2 /\3 sin / r . / ^ U ~^~ a Sm sin 2 / r' / /, , r-r.x r . / / , o/a\ 3 cos / ' The form given to these expressions is the one best adapted to numerical compu- tations ; and the equations are readily derived from the preceding in which the magni- tudes occur in linear form. , Thus from r __ __ _ 1 _r TJ "1 ~ " 2J " " r'* '9 9 * - . ^ y** rfr " 1+m L 2J~ A. P. S. VOL. XIX. I. 66 A NEW METHOD OF DETERMINING we have fji ra 3 r' 2 __ a 3 rH /z a j*_ ^a" 2 r rr Z 2 La** J* a''~J s J~ 2*J "a" *r"* ( 'a* where, as before, /^ /'\ 2 r TT a ' rz - . ( - t ) . . H. a=- a \rj a a . 1-fm In a similar manner all the other expressions for the forces have been derived. When we compute only perturbations of the first order with respect to the mass we need the perturbing function = p C and the forces 1 J ~1 I fa\ " ;,- a ,] " If ( J - - sin (/ + n') + (/) The other forces are only needed when we take into the account terms of the sec ond order also with respect to the mass. An inspection of the expressions for the forces shows that besides the functions we need expressions for the magnitudes '\ 2 1 r 2 sin / r f . / /. T-r,\ sin I r (r'\ 1 r 2 sin / r f a') ' *'*>-* a THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 67 When these are known we multiply the function ^a 2 (^j by F/r'V 2 1 r 2 ~| sin/ r' / /> , T-T,\ sin/r . / / , T-T\ ( /) - 2 -i U -/sinff + n'), sm (/ + n), L Va / a a 2 J a a a a the function fj.a ( a J by r n i r 2-]2 Q cin r r ' . rV'2 1 s F-?3- I > in (/+ n/ )&-i:= 3 ^ . ^ . sirf (/ + no, | *=-' r sin (/ + n) g - i, H t* ^ * t* L t/> 6t Ci J 3 ^ r sin (/ + n) ^sin (/ + H'). i /T. \/ / fj' \' sin / cos II' a 2 I' - l ~ sin I sin IT, and noting that /'\2<,iTi / 7 r <> ( <2 n r (l) (3) ~i (1) ^ = [^ + J v J sin flf + 2[y 2V + ^ J sin 2g' + etc. /a'\ 2 r (o) (2) ~i r (1) (3) ~i (-,) cos f = [Jv Jx J cos^' + 2|_/ 2V J^ J cos 2g' + etc. 68 we have A NEW METHOD OF DETERMINING r (0) (2) -I [- (0) ('2)-j (1) = I |y A , + ,/ A , J sin (- - g') + I' [y* J A , J cos (- 0') r o) ( 3 )~i r (l) (3 n + 26 [_<7 2A , + ,7 2A , J sin ( 20') + 26' |y w '^' J cos ( 2 ^') r (2) (4)~1 [- (2) (4)~1 + 36 [_J W + e/ 3V J sin ( 3g') + 36' [_ J 3A ^ J cos ( 80') + etc. + etc. The value of (/)' is found from From r' - 1 cos e we find Expanding, + (fe' 2 +|e' 4 + etc.) cos 2^' + ^e' 3 cos 30' + ^J-e' 4 cos 40' + etc. ; which, for brevity, we write, (~>) = PO + 2 pi cos 0' + 2 p 2 cos 20' + 2 p 3 cos 30' + etc. But r sin f (0) (2) (2) ~1 f (1) (3) ~1 + ^ J sin ^ + i L^A + J<* J sin % + etc. r f (0) (2)-i r (i) (3)-i ~ .cos/ = fe+ [_j; J x J cos0 + I |y 2A J ^ J cos20 + etc. (3) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 69 Putting I = ~ f . cos

) 2 - * 2 -H it is only necessary to 70 A NEW METHOD OF DETERMINING In terms of the eccentric anomaly we have, at once, : : 1 2e cos 6 er cos 2 a = 1 + |e 2 2e cos e + Je 2 cos Substituting the values of cos e, and cos 2e, we have r \2 (1) (2) -; = 1 4- fe 2 f 7 * cos | J" 2A cos 2# To find an expression for the factor ^L . ^ s i n (/ + H'), for brevity, we let O. CL , _ sin7 (1) (2) (3) " etc. _ _ cog , ?* ^i n / i /4 and from the known expressions for -, -A, -> cos/ , we get a a r (o) (2)~i r w (3) ~l I J + J* \ * sin^r' + J [^ + J, v J Cl sin 2gr' + etc. ' r (0) (2)-i r (i) (3)n |e'c 2 + L^A' J* J G, cos g' + i L^ix' J-* J C 2 cos 2^r' + etc. In the same way, if sin I sin / . ^ ft, = - . cos ct> cos n, c 4 = - - . sin IL a a we find (0) (2)-i r (i) (3) (6) By means of the expressions for the factors (.-)' THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 71 just given, we can form those for 3 -' 2 1 r 2 ~ 2 3 p-' 2 _ 1 r 2 ~| 4 La 72 a 2 a*J 3 sin 2 ~~~a 3 sin 2 7 r' 2 . 2 / /, , -r- . -.sm- (/> + n' + n) . sin (f + 72 A NEW METHOD OF DETERMINING CHAPTER IY. Derivation of the Equations for Determining the Perturbations of the Mean Anomaly, the Radius Vector, and the Latitude, together with Equations for Finding the Values of the Arbitrary Constants of Integration. HANSEN'S expressions for the general perturbations are cos* where / o) 1 + 2- h * p - [cos (/ )- In this chapter we will show how these expressions are derived from the equations of motion, and from quantities already known. The equations for the undisturbed motion of m around the Sun are dt* (1 + m} y ~ - -? + V (1 + m) *- = dt* } r* THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 73 The effect of the disturbing action of a body m' on the motion of m around the Sun is given by the expressions ,79/V - X x'\ ,79/V - V y' \ ,T>( ' Z' - Z Z' \ mKr( - ;), wifcv y -"7-), m,'Tc( ,). \ A 3 r/ J \ J 3 r' 3 /' V J 3 r' 3 J Introducing these into the equations given above we have in the case of dis turbed motion dt* (1) dP dt 2 ' r 3 The second members of equations (1) show the difference between the action of the body m' on m and on the Sun. The action of any member of bodies m', m" ', m"', etc., can be included in the second members of these equations, since the action of all will be similar to that of m'. The second members can be put in more convenient form if we make use of the function m' /I xa/-\-yy'-\-zz' \ +m Differentiating relative to x dtt _ m! (_ 1 d^ J x - - i+m \ j" ' d. x But since A 2 = (x' - xy + (y - we have the angle of eccentricity, 7t the angle between the axis of x and the % perihelion, v the angle between the axis of x and the radius- vector, /o the true anomaly, e the eccentric anomaly. These elements are constants, and give the position of the body for the epoch, or for t = 0. Let us now take a system of variable elements, functions of the time, and let them be designated as before, omitting the subscript zero, and writing # in place THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 77 of 7t . The former system may be regarded as the particular values which these elements have at the instant t = 0. In Elliptic- motion we have e e sn e r cos /' = a cos e ae r sin f =. a cos <> sin e v - f + X a?n 2 - F (1 -f- m) ISTow let n& be the mean anomaly which by means of the constant elements gives the same value for the true longitude that is given by the system of variable elements. Further, let the quantities depending on n^z be designated by a superposed dash, and let the true disturbed value of r be given by the relation r = r (1 + v). We have then ~ e e sm i r cos f rr a cos e - r sin f = cos <> sin e V = f + 7t a 77 ~~ #*"' I I 1 171 1 Q /CQ .^ A/ I JL j II v I We will now first give BRLTNNOW'S method of finding expressions for the pertur- bation of the time, and of the radius vector. Neglecting the mass ra, multiplying the first of equations (1) by y, the second by x, we have dy dx C being the constant of integration. Introducing cos /' x -* and sin f '=. -. / /i 7 * r' 78 A NEW METHOD OF DETERMINING into equations (2), neglecting the mass w, we find d*x _|_ k*. cos f -^~ dt* 7* 2 (4) (i If t L ' . Sin f -rr j ,n \ n ^^~" We have also dx v dr /. df /~>nQ T ^ i* si n r __ i, -^ l_'*JO / . ~~z~. / Olll / - -' dt * dt *. dt dy / dr /. df jr = * m f-jr + rc f-^r dt J 'dt "^ 'dt and hence or and In the undisturbed motion we have being the semi-parameter. Hence THE GE^EKAL PERTURBATIONS OF THE MESTOK PLACETS. 79 From these relations we derive (5) and also ' " i/P I _ - 1 f V'fr .Q r dt ^=r - X I =^. A}/ . Ul> //\ If we eliminate from equations (4), noting that 1 -^ = l-- p dt k p we have dz fcsin/ p neglecting the constants of integration. Since r rz r (1 + v), we have also = x The equations (7) then become . y dt dt ^/p~ J \ p From the equations = a cos F OQ^O = 0o cos > sn (7) dt dt ~/p J-\ p (8) 80 we have dx =: - a sin e ds dy = cos

o using the values of sin e, cos e, in terms of sin/, cos/^ we find dx k sin/ dy cos/-|-e e ^ i/S~' ^ "~t i/^T ft And these give k sin/' dx i/p<> l/p dz i/p k cos/ _ dy VP ke !/_p~~ rfz i/p VP* v. . i ft The equations (8) then become _ , dy T (1 , x d5 _ I/Pol = f( dt dz L V ; d< j/p j j n f the constant being included in the integral. SLU ^ # r ) ^ " dt dz L^ ' dt Vp l ~J p -to , ft r n , ^d L . . T/J.-I We will now transform equations (9), and for this purpose we multiply the first -~ by , the second by -~ , and noting that THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 81 we have dv_ dt _ **f . Sr \ dt + si" / r ( ) j \ Now multiply the first of (9) by y, the second by x, putting for J its value VP given by (6), noting that we have * ~ (11) We can write -- in the form dt dz dz __. We have r ' dt dz dt 2 Wn . - . . COS n , Of 3 cos 2 Making use of these relations we find dz 1 and for ,- given above we have ^ dz yp v ' 'dT A. P. S. VOL. XIX. K. 82 A NEW METHOD OF DETERMINING The equation (11) is thus changed into dz = f (l + 2^) Srdt- J v ) The equations (10) and (12) can be put in briefer form. Let X s = X- sin / 8r, Y c = Y+ cos/+ ^ 8r. P P Then - M 4_ 8in / f ** + i*:J jo. _ ~ (13) The values of a?, /, found in these equations we get from _ _ From the expressions for , -^ , we have also dz 7 dz sin/ __ . j ( _ (14) . ,_< + etc. (15) The quantities given by equations (14) and (15) are found in equations (13) without the integral sign. They can be put under the sign of integration and regarded THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 83 as constant if we designate all magnitudes in these factors dependent on t by a Greek letter. We thus obtain d(z l C(-\ O I Vo\ o J* 2 T/-\r TT r\ J4 ^ r= 1(1 -f 2 ' to } Sr at | ( X, . v Y e .t)dt at k |/P( J \ \''T> ' k vPo J (16) These equations include terms of the second order with respect to the mass. If we put W= we get /i/v (17) J In equations (17) 1 d~\ m/. . T - J r djj - -- sm. . dr J r dj -T^ . , rffl . 7- 1 rf^l Y = )sm/.- hcos/.-.- ^ dr J r dfj 84 A NEW METHOD OF DETERMINING neglecting the common factor Jc~ (1 + m), we have dt 19 VP*\ - cos A/i2 . >T 1 c?i2 7-\ r , fsin/" f/fi , (cos /+,) dfl r ~l 4- ~ ~ I sin / 4- cos / I r + . f 4- - . . r ^T/^o \*r f r d/ ' 7 / ? AT/P. L p d/ 1 jj d/ 1 ? J And as this becomes v p sin 0, ^ = p cos dt -.- d df 4- 2p cos Q . sin /'. - v 4- 2p . - cos o cos /' -)- 2p . r r dj p aj 2p oos.co/dfi,A P )^- If we write h*. a Q cos 2 <> in place of Jc 2 in equation (18), we have the same ex- pression for - - as that given by HANSEN. Equations (17) and (18) are fundamental in HANSEN'S method of computing the perturbations. We will now give HANSEN'S method of deriving them. Using the same notation as before, we have, since a 1-f-ecos/ r cos V also r cos 2 q a ~ ~ 1 -f e c cos / ' hence r.a l+ecos/ cosVo r.a t cos V * l-j-P cos/' ' Using f + 7t % in place of f, and developing, we get r - a . _ r-{-rcosf.ecos(x 7r )-f r sin /. e sin (/ 7r ) r.a c ~ o,cos 2 ^^ QA Let us put e sin (% 7t ) z= >7 cos 2 <^ , ecos (^ 7t ) =: since e = sin <|>, we have cos 2 cos Vo (1 2e ^ cos 2 ^> P - cos 2 ^> 7 2 )- 86 A NEW METHOD OF DETERMINING With this value of cos 2 <2>> and r cos >2 ^ e o r cos/", we find r.a _ a cosVo e .r cos f -\-rcosf (^ cos 2 y -\-e )-{- r sin f.r] cos cos a t cos Vo~H r cosy. I 1 cos Vo~l~ ? ' si 11 /- ^ cos Vo a cos Vo ( 1 2e cos Vo 2 cos Vo 1 ?' 2 j and hence 1-f-^. .CO8j^-f-^. . sin/' r.a 1 2e cos'Vo* 2 ( From d v _ ^/ _ df dz dl dt dz dt 9 and we have . COS 111 like manner we find df o 2 - =. UQ. =7 - COS d) - az r We have therefore n.a?,r 2 . cos (Z< ^fl.a 2 - **' 2 cos THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 87 If we put 1 + 6, substitute the values of a . and cos 2 <2>, we get "n * f n ' (20) (1 2e cos Vo^ 2 cos : Further, in the case of v, we have Then since and we have If we let we find 1 2e? ? cos Vo- 2 cos 'Vo- 1 ? 2 cos/. + - . sin/. = 1 e cos - - cos h . _ (1+6)* = (l + 5)ir, B 88 A NEW METHOD OF DETERMINING From the latter we have Hence __ > ^ I /i /) dt If we put f we have We have yet to express r in terms of the elements. From and from COS we have _ / \ ^ cos y> V / ' CfOS <> ' THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 89 or h _ an cos ?> A cos (p ' a c If we put cos we have 7 an cos

7, o, f,'/?, , v, respectively. Whenever we integrate, these new symbols are to be treated as constants, noting that the original symbols are used after integration. A. P. s. VOL. xix. L. 90 A NEW METHOD OF DETERMINING If in equation (21) we introduce , /> " We have also The coordinates of a body vary not only with the time but also with the variable elements. In computations where the elements are assumed constant, that part of the velocity of change in the coordinates arising from variable elements must, evidently, be put equal to zero. Coordinates which have the property of retaining for them- selves and for their first differential coefficients the same form in disturbed as in undis- turbed motion, HANSEN calls ideal coordinates. If L be a function of ideal coordinates, it can be expressed as a function of the time and of the constant elements. Thus let the time, as it enters into quantities other than the elements, be itself variable and, as before, designated by t. The function dependent on , r, and the elements we designate by A. Then dL ~dA dt ' dr or where the superposed dash shows that after differentiation r is to be changed into Let us write the equation (24) in the form THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Differentiating relative to r, we have dPf The differentiation of (23) also relative to t gives _ , C 2 ~ " ' ' ' dr />, ' 1/? 3 ' dr Eliminating - by means of (24), we have e h dr dTF 2/3 "" r JO Substituting in the expression for J- we have dW Since ^ is an ideal coordinate, we get from this ^T being the constant of integration, and the dash having the same signification as before. This expression for v is a transformation of that given in the equation 1 2e, cos Vo- 2 cos Vo- 1 ?* . sn Since 2; is also an ideal coordinate, we have from (23) 2 (26) being the constant of integration and being the mean anomaly for t = 0. 92 A NEW METHOD OF DETERMINING When we consider only terms of the first order with respect to the disturbing force, f changes into r, and we have n

7 a Q e p sin G) zz a cosji) sin >?. Also in the last two terms of WQ, - is put equal to unity. "o When terms of the order of the square and higher powers of the disturbing force are considered, f cannot be changed into r. In this case let n t Likewise let where w^J is a function of r and According to Taylor's theorem we have W= TTo . ar ' dr the value of W Q being given by (28). THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 93 We then have dW_ _ dW, ^ ^ L . ^ , dS dr dr* <0 * 2 ' d^'^ Retaining only terras of the second order, the equations (25) and (26), replacing by $z, give v z dt dr (29) The equation (26) has been put in simpler form by HILL. For this purpose from (21) and (22) we have ~ dt ~ dt Hence Developing the second member and adding W, we have = n, t + #o + o 2 dt. (30) dh . Vb OLCJJ IB IAJ CA.JJ1COO _ ^ttQ dt we find The next step is to express " an d in terms of the disturbing force. From (19) dt dt COS 94 A NEW METHOD OF DETERMINING Using these values of and vj 9 and e p cos o = cos 2 <> p, in equation (28), we find . . , ft, a cos Vo V*o cos Vo Since T _ an _ |/l-f-m " cos? " we have from the expression of h already given, dt By means of - 1 =1 e cos /. r cos.? we may transform the expressions dv a 2 = . n cos df r 2 an . /. e sm /, cos into r - 37 ^ cos (f ") he cos (# 7t co) + sin (/ o) . Ae sin (% n t dr / j? \ T / \ / /^~ \ i / = sm (j co) . he cos (% 7t o) cos (/ o) . fte sm (^ 7t THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 95 Multiplying the first of these equations by cos (f G>), the second by sin (/ o>), and adding the results, we have he cos (jc 7t G>) = (r~ h) cos (/ o>) + ~ sin (/). at ctt Substituting this value of h . e . cos (# 7i o) in the expression for TF" > noting that we have -nr^ 2.fc../ ^ cQg ,-j._ x r dv . 2h 9 .p ^ gin / -_ s dr_ Differentiating relative to the time t alone, r remaining constant, and having care that all the terms of the expressions be homogeneous, we have c^r /' civ and dh _ F(l+m) d 2 u . fcV d 2 v Substituting \ p d*r + ) r 96 A NEW METHOD OF DETERMINING we have = *, 3f- cos ( / - o) - 1 + - - [<=os ( / - ) -1] ~ 1 J - dt ( r h~a cos 1 $P O J * \ dv sn - < 30 V d W This expression for -^ is the one used by HANSEN in his Auseinandersetzung. It is given in a much simpler form in his posthumous memoir, and as the latter is the form in which we will employ it, we will now give the process employed by HANSEN to effect the transformation. Substituting first the value of h, omitting the dash placed over certain quantities, noting that in the posthumous memoir takes the place of o, and remembering that we are here concerned only with terms of the first order with respect to the mass, we have /? \ ( dQ \ o) r( } \dr ) From the relation p z= a(l e 2 ) ep cos G) we have v _ -i ep cos (u dW An inspection of the value of -^- shows that its expression consists of three parts, one independent of r, the other two multiplied by p cos o, and p sin G>, re- spectively. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 97 Put = dB [ + *Y,p _ dt dt dt \a l ) dt a V t e j and we have dS q a f [~ae cos / , e cos / , 1 ~| / cZi2 \ , ae sin / dF 9 (Faces/' , (cos/-f-e) ~| ( d& \ , a sin/ ^ ) ' Eliminating f J from the expression for - , we have dY 2 (a 2 (l e 2 ) r 2 fd&\ r sin f /dQ\ - - J - - - d ( ] _l- - ' n. f [ _ ] ndt~~le 2 \ a*e \dg ) r a T /l e 2 \dr J A (// - fi^ 1 ^ In the same way we find z - ' dQ\ r~a cos/ ^ - e sin 2 / r* sin/ /rfw\ pa cos/ ^ __ 2^ But if we employ the relation x,/1 ^>2\ I r . re cos/ fl2\ C y , , | ': -H"^ A (T a cos/ in the term, - - Vl e2 'A^ ^ ne preceding expression, the whole term becomes Crcos/ e re ~i /d&\ a (1 e 2 )? v 71 " e" a (1 e 2 )d P Vdr / Using the equation zr re cos/ r + (1 6 2 )? multiplying by e CL T I - THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. adding^ to the preceding, it becomes [- ~~ La rcosf 2e Further, we have dg rr U sm/ + ? cos f sn "Reducing this expression in the same manner as employed before, it becomes /' -i 2 r cc s)J ~ a d r~r , r 3 sin/ ~i 2 r cos/-{- 3 a e 8 ^ a 2 (l-e s ) Multiply this by cZf/, the last expression for - becomes ttdv o ;j~ the integral to be so taken that it vanishes at the same time with g. d3 dY d>F Substituting these values 01 - , , , m ndt ' ndt ' nrf^ ' dS dT f p \ ?.. =- 4- ( - cos 6) + f e ) -f - sm ndf nrf^ \a / ndta^ this expression can be made to take the simple form dW in which - a 2 (l e 2 ) r 2 2 /> sin w /-/2 99 J 1 e 2 a 1 e a 100 A NEW METHOD OF DETERMINING Since d . r 2 _ r sin / a e . tfig.de ^ cos /> we have _^L r r^_ a 2 edJ L a 8 . rfe 2(1 These expressions for A and ^ can be much simplified. Thus from r 2 e 4 - = 1 + f e 2 (2e J e 3 ) cos # ( J e 2 J- e 4 ) cos 2# J e 3 cos 3g - cos 4g etc., ^JL^J- *^ tft^VVV** ^L^'tt^ ^ ^^r-t. x{!^6^ ^. p* and a similar expression for -, we get cZ 7*^ / f > ^\ / ^^\ a , e ' d = (2 - J sin ^ + (e -J e sn e 3 sin 4^ + etc., /LoT / La. " lain r--- sin 4flr etc., a 2 . de 4e = e (2 f e 2 ) cos # ( e f e 3 ] cos 2g f e 2 cos 3^ | e 3 cos THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. From which we obtain 101 2e 2 )cos(y g) B + (e + ^)cos(y 20) -J cos y -30) - (2 -(e + e 4 )sin(y 20) cos sin (7 Sg) + 3 cos (y 40) (32) Jtjv. These are the expressions of A and B whose values are used in the numerical compu- tations. When we have the coefficients of the arguments in which y is + 1> and 1, we obtain the coefficients of the arguments in which y is i * with very little labor. d W Let us resume the expression for =- , that is, ndt dW ndt A and B having the values given before. Since can be put in the form a 2 we have a-j/1 e 2 e.dg d.- 7 o r '? *\d BW 7 sm fc\, THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 103 where de li i'g' CTand V being two functions depending alone on JL. Putting x + 1, and 1, we have and hence Thus we find 2 or putting we have de de ,0) _ (- 7T Y U j r>(i) ) r ^(i) 2^- 2- T ,d) de - de de de de '.JL M&"V ,(-i) _ W a (D + ^W a (-D (33) 104 A NEW METHOD OF DETERMINING The values of YI (K] and O (K) are readily found from ") "K; /O n 1 t$ I 1 ^5\ ^rvc A/ /I y^2 1 y>4 _j_ J_ ^o\ f*r*si *A f\/ 1 ~1^/C"""^|,C/ '" Q o (5 I Ov& Y cos We have etc., = etc. ^^ 3e de de (3 # 45 ff de ~(* e ' 64 e > (1 p* 4 V3 6 ~5 de etc. = etc. For >7 (2) we have ^(2) = a*- A ^ - - T k e 5 ) + a - A ^ 3 + 3k ^); or ^< 2 > = 46-1^-^^^ . . _ (34) For (2) we get at once THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 105 In a similar way we have In case of the third coordinate we also compute the coefficients of the arguments having no angle y from those having i y. For this purpose, putting K = in the expression for OL (K} we have 5?rr- j. ty de v de where For >7 (0) we then have Perturbation of the Third Coordinate. Let 5 the angle between the radius- vector and the fundamental plane, i the inclination of the plane of the orbit to the fundamental plane, v ) Instead of s, let us use r U = S and we have u= ~ q sin ( & ) - 1 p cos Introducing r and calling R the new function taking the place of u, we have, putting 6) + 7t for v, 7t being the longitude of the perihelion, ($ =. r cos v, and y = r sin v. Hence 3 L =: x sin sin G -}- y sin i cos cr ? THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 107 and a = sin i sin cr, (3 zz sin i cos cr. The values of p and q then are given by the equations p - - a cos /3 sin &, 2 zz - - a sin &o + /? cos & sin ^ ; from which we have dp da dp - = _co S ffi --sma -, dq da dp - = - 8 mS -+oo 8 ffi -. From the equation z l zz a a? + /2 y we have, first regarding a and /3 as constant, then regarding x and ^/ as constant, dzA dx ~ dy dt) ~~ a W"rP ^ - rf^-i da dp Differentiating the first of these, regarding all the quantities variable, we have d 2 *! _ da dx dp dy d?x d*y ~d?~ ' ' dt ~di ~~ ~dt ~dt ~ ' a dt* ' " ^ df ' ^i being the component of the disturbing force parallel to the axis z l9 and X and !Fthe other two components, we have Z a X + j3 Y+ Zcos i. "Writing for X and Y their values tfx .x d v , 70/1 \y _ + ^ 2 (l + m) ;3 , - + Ar(l + m) - , 108 A NEW METHOD OF DETERMINING and reducing by means of we have * = ;+ + * a + ) !i + or Comparing this with the other expression for --* , given above, d V we have da dx dp dy From this equation, and the value of [ _- j , since L ' / / J dy dx we find da dt dp dt = h r cos i sin v Z , = krcos icosv Z . Substituting these values in the expressions for --- and --- , we have dp dt = hr cos i sin (v dq - = hr cos i cos (v Q> ) Z . THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 109 Introducing these values into the expression for dt we have = 7i r cos * cos (v & ) p - sin (o> + 7t & ) Z at @ - h r cos i sin (v & ) -- cos (w + ^o So) a Q h r cos i -- |sin G> cos (0 So (^o So)) I Z a L - h r cos * - cos o sin (v So (^o So)) - li r cos i p - sin (o> f) . Introducing n = - - , and h zz we have ==-- sin (o / ) a 2 - cos i . (37) Let 1 ' P / s>\ ~i/r=?'o o' sm ^~~f>> then cos *.nc?^ To find an expression for (7 similar to those for J. and J5 we have, first, l rp r P r ^~1 (7 = - sin G).- cos/ coso). sin/ . /l e 2 La a a, a 110 A NEW METHOD OF DETERMINING Substituting the values of r cos/, ? sin /J given before, and similar ones for - cos a, - sin a, we find fe V afde ) \a?edg J \ay (2) fc, sin 2 y + r ^ I * THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Ill Then in case of A - we have J &i sin y - - i & 2 cos i -i) >? (2) ^ sin 2 7 - - V 2) fc 2 cos 2 y + ete. ^ "J In the second integration we call the two new constants C and JV^ and the con- stants of the results are in the forms C + & nt + &! sin # & 2 cos g + % ? (2) &i sin 2 # - - | >y (2) & 2 JV~ >y 2 cos 2 ^ >7 (2) A?! cos 2 ^ - - J >? (2) ^ 2 sin 2 , 5. In case of the latitude the constants are iven in the form sin g sn cos 7 The constants are so determined that the perturbations become zero for the epoch of the elements. Hence also the first differential coefficients of the perturbations relative to the time are zero. We substitute the values of g and g' at the epoch in ni fl the expressions for nz. v, - - , - - (nbz), etc., including in g' the long period term. cos % nat Putting the constants equal to zero, and designating the values of nfiz, v, etc., at the epoch by a subscript zero, we have the following equations for determining the values of the constants of integration: . 13; C + ^ sin gr -- 7c 2 cos ? ( cos g sn cos cos cos 2r/ -fetc. + (nbz) Q g* = sn sin g + I cos gr + >? (2) Zi sin 2^r + >y (2) ^2 cos 2^4- etc. + (~-) = l> cos 2,7 - cos j, - , sn g 112 A NEW METHOD OF DETERMINING To find ki and k>. we derive from the preceding , [cos? (2) cos 2 [sin (j + >y (2) sin 2 ! [sin gr + 2 7< 2 > sin 2 g + 3 >y (3) sin 3 g +)^etc.] - - & 2 .[cos g + 2 >7 (2) cos 2 The value of N is found further on. Having Jd we find ^ from - fc e ^ - 3 Z + 3 ~- (nte)o + 6 ( v ) = 0. We have ' 2 e 6 () e L , jY 3- & A^i 2 A) ? Av^ where Z Q is the eonstant- of W. Let us find the expressions for the constants N and K, K being the constant of integration in the expression for 5 - . "0 The equation (22) we can put in the form The differentiation of nz relative to the time gives dz - 1 + &o + &o + &i + periodic terms, Ct6 where Z Q =. - - 32".7162, in the case of Althaea, and Z^ the part to be added when terms of the second order of the disturbing force are taken into account. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 113 The expression for v is v =. N -\- periodic terms. The approximate value of being 1, the complete expression for the integral of d ~ is given by =. 1 + & 3 + periodic terms, & a being the constant of integration. Putting (3^ 2 4z^ 3 + etc.) - - 2v (- - l) = FI + periodic terms, and substi- it/ \tL / tuting this expression, together with those of v and - , in the expression for , we /I dt have, preserving only the constant terms, It is necessary now to find the value of k & in terms of the constants. If in the expression for given by equation (18) we write for p , its equivalent cos 2

/ ) dt. J V *o 2 ; \VJ ..& Introducing the true anomaly instead of the eccentric, we have, cos cos H since cos >7 = , sin YI =. , ^ 1 -4- e cos (a \- e cos iA J ^ /;2 ^ - . r '} ^ 1< /" C V * t C^a W^ Neglecting the terms having p cos o and p sin G> we have in TFo the constants and e h. h, The integral of d T is \ '\_ ^ Y^ From the expression for d 7 we find 7i h Integrating this, making use of the value of 7i , and adding the constants, we have And since the quantities under the sign of integration do not have any constant terms we can write 2 - r = l + & + ek v + periodic terms I, * 7i = 1 + & 3 + periodic terms THE GEISTEKAL PERTURBATIONS OF THE MINO^B-PLANETS. 115 Since ( - - Ij is a quantity of the order of the disturbing force we have from which we get Now putting 7 O / *-! (JT- -1) - ~ ( '- -l) rb etc. =: HI + periodic terras, substituting this expression and those for ~ h 7) /io /) h ' ft ' the preceding expression for 2 gives, preserving only constant terms, h Introducing this value of ^ 3 into the expression for JV it becomes ^= - K 4 A; 4 e^ + 3^ ) + i (3 ^ + 2/^- Preserving only the terms of the first order we have To find the value of A", the constant of integration in case of . - , we have ''o h =2 1 + K + periodic terms, 116 A NEW METHOD OF DETERMINING also ' 1 4- &3 + periodic terms. h From these we get j-i +!'-!= A- /? h Hence K = \ +H,~l (fc -f ek,,) or, neglecting the term of the second order, THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 117 CHAPTER Y. Numerical Example Giving the Principal Formula Needed in the Computation Together with Directions for their Application. ALTH^A 119. JDPITER. g - 332 48' 53".2 g' = 63 5 48.6 7i = 11 54 21.1 1 7i' = 12 36 59.4 ] I I = 203 51 51.5 } 1894.0 & = 99 22 59.9 } 1894.0 i= 5 44 4.6 J i' = 1 18 36.9 J $ = 4 36 24.9 $' = 2 45 57.2 n = 855".76428 n' - 299".12834 log n - 2.9323542 log n' - 2.4758576 log a = 0.4117683 log a' = 0.7162374 The epoch is 1894 Aug. 23.0. The elements of Jupiter are those given by HILL in his New Theory of Jupiter and Saturn, in which the epoch is 1850.0. Applying the annual motion of 57". 9032 in 7t', of 36". 36617 in &', to HILL'S value of 7t', and of &', we have the values given above. The mass of Jupiter is TOTT-ST^' ^he elements of Althsea are those given in the Berliner Astronomisches Jalirbuch for 1896. The ecliptic and mean equinox are for 1890. To reduce from 1890 to 1894 we employ the formula of WATSON in his Theoretical Astronomy, pp. 100-102. i' = i + YI cos (Q, 6) Q' = Q + (t' t) -- YI sin (Q6) cot . i' dl t t) + YI sin (& 0) A NEW METHOD OF DETERMINING B = 351 36' 10" + 39".79 (t 1750) 5".21 (f t) YI - 0".46S (f 4* = 50".246. dt These expressions for ?'', &' and 7t', can be used for the disturbed body as well as for the disturbing body by considering the unaccented quantities to be those given, and the accented quantities those whose values are to be found for the time, f. HARKNESS, in his work, The Solar Parallax and Its Related Constants, using the most recent data, gives the following expressions for 0, >?, and , when referred to (' u 1850.0: 6 = 353 34' 55" + 32".655 (t - - 1850) -- 8".79 (t t), vi = 0".46654 dl dt - [50".23622 + 0".000220 (t-- 1850)] Let an:-, n we have then (i = 0.34955 2^ = 0.69910 3^ = 1.04865 4^ - 1.39820 5^ = 1.74775 6^ = 2.09730 etc. =. etc. Hence 1_ S(i = .04865, 2_ Qu = .09730. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 119 This shows that the arguments (g 3#'), and (2g 6*7'), have coefficients in the final expressions for the perturbations greatly affected by the factors of integration. In case of the argument (g 3g'), \ve should compute the coefficients with more deci- mals ; also those of (0 3g') and (2g 3g'), since in the developments the coefficients of these affect those of (g 3) = sin J (& &') sin J (i-ht) sin i- /. cos \ (V + ) = cos (& &') sin J (i i') cos i /. sin i (*P 4>) = sin J ( cos i 7. cos J (V 4>) = cos (& where, if &' > ^, we take | (360 + Q &'), instead of i (^ ^') 5 we find // * = 116 15' 36.7 3>= 11 50 33.9 I- 6 11 35.3 An independent determination of these quantities is found from the equations cos 2) sin q sin i' cos (& Q') cos j9 cos g cos i' cos jp sin r =. cos i' sin (& ^') cosjacosf cos (S S') sin jt? sin i' sin (S Q') sin /sin O sinp sin /cos $ zr cos p sin (i q) sin /sin ( 1 P r) sinj? cos (* g) sin /cos ( 1 P r) =. sin (^ g) cos / = cos p cos ( g). 120 A NEW METHOD OF DETERMINING From n =TI --& --$> IT = n' & ip we have n = 156 11' 55".7 , H' = 156 58' 22". S. Then from k sin K r= cos 7 sin II' ~k cos .BT = cos n' & t sin TTj =. sin II' &i cos ^ = cos / cos II' p sin P = 2a 2 - 2a^ cos (H ^ cos P = 2a cos (|)' &i sin (n - - 7 u sin V = 2a cos $ ^ sin (II - - K) v cos >"= 2a cos <|) cos <|>' A^ cos (n e' tw sin TF= ;^ 2a 2 - sin P e t. w cos TF"= v cos ( F" P) Wi sin TFi = v sin ( T r - - P) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 121 we find K = 157 5' 36".6 log k = 9.999614 X l =156 51 7.4 log ki = 9.997849 P = 93 3 27.0 \ogp = 9.932748 V 359 6 2.4 log v = 0.601463 TF=266 4 39.5 log w = 0.605196 15 380 log w, = 0.601352 Then from R- 1 +a 2 2aV% y 2 = aV 2 , we have log 72 = 702855 , log y, = 7.976024. The values of the quantities from II to ^ 2 should be found by a duplicate compu- tation without reference to the former computation, since any error in these quantities will affect all that follows. We now divide the circumference into sixteen parts relative to the mean anomaly, and find the corresponding values of the eccentric anomaly E from g zz E esin E , where e is regarded as expressed in seconds of arc. Substituting the sixteen values of in the equations /sin (F P) = w sin (E W ) -- we obtain the corresponding values of /' and F. A. P. s. VOL. xix. P. 122 A NEW METHOD OF DETERMINING Then in a similar manner from \ogq-\ogf-\-y where s = 206264".8, logX = 9.63778, we find the values of Q, (7, log q , x , and y. Thus we have found all the quantities entering into the expression Instead of this, we use the transformed expression Q" = N n (1 + a 2 2acos (E'Q)}-~> (1 + V 2&cos (E' + Q))~ " , and have, for finding the values of JVj a, &-, the equations =. sm THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 123 NN _2i r (0) (1) (2) n ' O)) ' 2 := 16,, + &.. cos(J' O)+6 cos2(^' O)+etc. i^y/ i j* .**-** v t' / ' "' v \j / l L_ 7T IT -n _1 (1) (2) n cos (.#' + Q) + ^ cos2 (^' + Q) 2 2 + etc.] For finding the values of the coefficients in these expressions we use RUNKLE'S Tables for Determining the Values of the Coefficients in the Perturbatiee Function of Planetary Motion, published by the Smithsonian Institution. With the sixteen values of a as arguments we enter these tables and find at once the corresponding values of (1) (2) (3) (o) fe, 6. 6-. o* (0) a 3 (i) a 2 (2) & ! , then those of - , - 2 , -" , etc., etc. ; - . 1% ,&<>* > 5 *J ' etc '' etc> > where P 1S found Y a a a p 2 P 2 > 2 a 2 from the sixteen values of /5 2 = - - 2 . A. ~~ Cv Since & in (1 2& cos (j^ r + )) is very small it will suffice to put i^' = l ,!!"=:> 2 2 (1) (1) B, =36, ^ 5 =56. 2" 2" Then from () (i) \ -\-r T> r /~\ r, -i- N rS ^n 2 M 2" 2" (i) (0 , = t.JV B 2 2 we have, in case of ^ ( J, 124 A NEW METHOD OF DETERMINING and, for ,ua 2 f?) , (0) 3 (I) 3 (1) 3 6-3 =JVr C = .#3&cos2 * = JT 36 sin We divide by 8 to save division after quadrature. With these values values of k t , A, , from (0 (0 (*) With these values of c n9 s,,, and the values of the coefficients & M , we find the 2" 2" 2 >'-l) J'+lK (1) C M ~T~ I ^ n 2" 2" X 2" 2 ' 2 (&, \ 7, V't- 1 ; \ (.1 TT / TT _(i-l) -, (i+1) \ (1) "% ' \ For i 0, we find A; from (0) (0) (1) (1) 2 n w - n n 2" 2" 2" 2" Then in case of / f from (c) where m' is the mass of the disturbing body and s = 206264/ / 8 and from [i(Q g) (s) f> = i m' s a? Jc ( sin \i(Q g) A*] , (a\ 3 ( c ) *) ^J , we find the values of 4 t and 4 f) for the 16 different points of the circumference, and the various terms of the series. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 125 (C) (8) Again, since A lt K , A L K are given in the forms (c) (c) (s) A it K zO^v cos v g + 2 C, sin v g () (c) (s) J.^ = 2$> cos y# + 2$, sin >gr, (c) () (c) we have the following equations to find the values of the coefficients Q>, CJ; r , o<,^ w (0.8 )= F + Y s () = F -F 8 /9 1 fh V-4- V ( % \ V - . V \A.\J) _t 2 -j- _i IQ VTTT/ 2 -* 10- ^ /7ir;\ T^_j_y" ( i \ V V ( i.LO) J-T ~ ' J-15 \TS) --7 -"-M (0.4) = (0.8 ) + (4.12) (1.5) = (1.9 ) + (5.13) (2.6) = (2.10) + (6.14) ' (0.2) = (0.4) + (2.6) (3.7) = (3.11 1 ) + (7.15) (1.3) = (1.5) + (3.7) 4 (c + 2c 8 ) = (0.2) 4(c -2c 8 ) = (1.3) 4 (c 2 + <%) = (0.8) (4.12) 4 (c 2 - c 6 ) = { [(1.9) - (5.13)] [(3.11) (7.15)] } cos 45 C 4(s 2 + * 6 ) = | [(1.9) (5.13)] + [(3.11) (7.15)]| cos 45 C 4(s 2 - s 6 ) = (2.10) -(6.14) 8c 4 = (0.4) (2.6) 8s,- (1.5) (3.7) 126 A NEW METHOD OF DETERMINING 4 ( Cl + c 7 ) = (f ) + [(-&) (A)] cos 45 4 ( Cl c 7 ) = [(i) (yV)] cos 22.5 + [( TT ) (A)] cos 67.5 4 (c 3 + c 3 ) = (f) [(A) (&)] cos 45 4 (c 3 c 5 ) = [( | ) (yV )] sin 22.5 [(fV) (fV)] sin 67.5 4 Oi + s 7 ) = [( | ) + (yV)] sin 22.5 + [(^-) + ( 1 %)] sin 67.5 4 ( A 7 ) = [(A) + (A)] cos 45 + ( T %) . 4 ( S3 + 6 ) - [( i ) + ( T 7 5 )] cos 22.5 - [(yV) + (fV)] cos 67.5 4 (8 3 -. 5 ) = [(A) + (A)] cos 45 -(A) The values of c,, s,, must satisfy the equation (c) (s) A it K , or A it K = \ c + C L cos g + c 2 cos 2g + etc. + S A sin g -\- s> sin 2# -f etc. i answering to i in 6 H , and x being any one of the numbers, from to 15 inclusive, "2 into which the circumference is divided. We use c t , s v as abbreviated forms of C^ (s) (c) (c) (s) C ijV , etc. Having found the values of c,, from the 16 different values of JL , A ly A^ (C) (S) (C) (8) / a \ / a \ ^4 2 , A 27 . . . A 9 , Ac,, both for [i \ ^) and ^a 2 ^/i we have the values of these func- tions given by the equation (c) The values of the most important quantities from the eccentric anomaly E to' c t , s needed in the expansion of ^ f j and pa 2 l-J , are given in the following tables, first for ^ ( a -\ , and then for ^cr I a j , when not common to both. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 127 Values of Quantities in the Development of /*(-) and^a' 2 (-) . I 9 ^ E + W E + W, F P F ( 0) 1 If 0.0 1 II 266 4 39.5 O I II 266 15 38.0 1 II 266 21 17.2 O 1 II 359 24 44.2 ( i) 24 24 4.2 i> ( .() 28 43.7 290 39 42.2 290 8 7.8 23 11 34.8 ( *) 48 26 37.2 314 31 16.7 314 42 15.2 313 40 58.4 46 44 25.4 ( 3) 71 52 24.9 337 57 4.4 338 8 2.9 336 53 39.3 69 57 6.3 ( 4) 94 35 14.0 39 53.5 50 52.0 359 41 1.3 92 44 28.3 ( 5) 116 36 51.7 22 41 31.2 22 52 29.7 21 59 7.8 115 2 34.8 ( < ; ) 138 4 29.4 44 9 8.9 44 20 7.4 43 47 3.8 136 50 30.8 ( T) 159 8 !!>.<; 65 12 59.1 65 23 57.6 65 8 48.4 158 12 15.4 ( ) 180 0.0 86 4 39.5 86 15 38.0 86 13 41.4 179 17 8.4 ( ! >) 200 51 40.4 106 56 19.9 107 7 18.4 107 15 14.8 200 18 41.8 (10) 221 55 30.6 128 10.1 128 11 8.6 128 28 47.5 221 32 14.5 (11) 243 23 8.3 149 27 47.8 149 38 46.3 150 8 27.6 243 11 54.6 (12) 265 24 46.0 171 29 25.5 171 40 24.0 172 23 51.4 265 27 18.4 (13) 288 7 35.1 194 12 14.6 194 23 13.1 195 17 19.4 288 20 46.4 (14) 311 33 22.8 217 38 2.3 217 49 0.8 218 43 0.9 311 46 27.9 (15J_ 335 35 55.8 241 40 35.3 241 51 33.8 242 28 57.5 335 32 24.5 V 1613 47 17.9 w 1433 47 18.6 9 L g-/- y X Q Log. q. Log. a ( ) 0.612427 .001251 ii - 12.2 1 II 359 24 32.0 0.611176 01706582 ( i) 0.612078 .000860 '+431.5 23 18 46.3 0.611218 0.706349 ( 2) 0.609315 .000081 +598.0 46 54 23.4 0.609234 0.705534 ( 3) 0.605242 +.000981 +390.0 70 3 36.3 0.606233 0.704403 ( 4) 0.601312 +.001292 - 58.6 92 43 29.7 0.602604 0.703241 ( 5) 0.598569 + .000846 476.9 114 54 37.9 0.599415 0.702241 ( 6) 0.597310 +.000091 626.7 136 40 4.1 0.597401 0.701493 ( ?) 0.597194 .000956 435.1 158 5 0.3 0.596238 0.701011 ( ) 0.597621 .001322 - 15.7 179 16 52.7 0.596299 0.700788 ( '') 0.598109 .000997 +408.7 200 25 30.5 0.597112 0.700494 (10) 0.598532 .000152 +618.1 221 42 32.6 0.598380 0.700021 (11) 0.599177 +.000777 +496.6 243 20 11.2 0.599954 0.699872 (12) 0.600584 +.001278 + 96.7 265 28 55.1 0.601862 0.700504 (13) 0.603163 +.001032 363.1 288 14 43.3 0.604195 0.702020 (14) 0.606734 +.000148 600.1 311 36 27.8 0.606882 0.704038 (15) 0.610302 .000825 452.4 335 24 52.1 0.609477 0.705810 2 1 4.823835 + 3 0.5 1613 47 17.4 4.823838 5.622201 i? 4.823834 2 0.7 1433 47 17.9 4.823842 5.622200 128 A NEW METHOD OF DETERMINING Values of Quantities in the Development of and 9 X Zi Log. b. Log. a. a. Log. IT. / // i n ( ) 53 23 45.3 7 57.83 7.063818 9.701484 0.502902 9.695669 ( i) 53 26 41.3 7 57.78 7.063792 9.701945 0.503437 9.695880 ( 2 ) 53 14 15.6 7 59.97 7.065778 9.699988 0.501173 9.695892 ( 3) 52 54 33.7 8 3.30 7.068781 9.696876 0.497594 9.695837 ( 4) 52 28 55.6 8 7.35 7.072405 9.692804 0.492951 9.695616 ( 5) 52 6 31.2 8 10.95 7.075601 9.689226 0.488907 9.695421 ( 6) 51 53 41.2 8 13.23 7.077613 9.687169 0.486597 9.695400 ( 7) 51 46 50.0 8 14.55 7.078774 9.686068 0.485364 9.695430 ( 8 ) 51 49 41.2 8 14.49 7.078721 9.686526 0.485877 9.695629 ( 9) 52 52.3 8 13.57 7.077913 9.688321 0.487889 9.696120 (10) 52 18 36.9 8 12.12 7.076635 9.691160 0.491089 9.696905 (11) 52 36 21.2 8 10.34 7.075061 9.693986 0.494294 9.697532 (12) 52 49 37.5 8 8.19 7.073153 9.696093 0.496699 9.697631 (13) 52 58 10.6 8 5.58 7.070825 9.697448 0.498251 9.697141 (14) 53 5 12.5 8 2.58 7.068133 9.698559 0.499527 9.696354 (15) 53 13 54.4 7 59.70 7.065534 9.699932 0.501109 9.695743 r 77.553783 3.956815 77.569096 r 77.553803 3.956845 77.569088 (0) (i) (i) (0) (i) (2) 9 Log. ! 9.735322 9.306586 (n) 8.794442 5.94812n 6.07618 0.330914 9.738805 9.312970 (12) 8.794541 6.16466^ 5.36611 0.331246 9.741407 i 9.317738 (13) 8.794051 6.07296/1 5.94202n 0.331460 9.743073 9.320808 (14) 8.793264 5.23742n 6.16200n 0.331637 9.744461 9.323327 (15) 8.792653 5.97789 6.04134?? 0.331858 9.746165 9.326443 2 2.647715 77.912926 74.508222 I' 2.647721 77.912945 74.508268 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 129 Values of Quantities in the Development of ^ and (3) (4) (5) (6) (7) (8) (9) 9 Log. &j Log. Jj Log. &! Log. l l Log. &J Log. b l Log. &j 2 "2" 1 2" 2" 2 2 ( ) 8.954999 8.60017 8.2570 7.9215 7.5915 7.2654 6.9426 ( i) 8.956515 8.60214 8.2594 7.9244 7.5947' 7.2691 6.9468 ( 2) 8.950082 8.59373 8.2490 7.9120 7.5804 7.2528 6.9286 ( 3) 8.939865 8.58036 8.2326 7.8926 7.5578 7.2271 6.8997 ( 4) 8.926521 8.56292 8.2110 7.8668 7.5280 7.1932 6.8617 ( 5) 8.914818 8.54760 8.1921 7.8444 7.5020 7.1636 6.8285 ( 6) 8.908100 8.53882 8.1812 7.8314 7.4870 7.1466 6.8094 ( 7) 8.904506 8.53411 8.1754 7.8244 7.4789 7.1373 6.7991 ( 8) 8.906000 8.53606 8.1778 7.8273 7.4822 7.1411 6.8033 ( 9) 8.911861 8.54373 8.1872 7.8386 7.4953 7.1561 6.8201 (10) 8.921142 8.55588 8.2024 7.8565 7.5160 7.1796 6.8464 (11) 8.930392 8.56797 8.2172 7.8742 7.5367 7.2031 6.8728 (12) 8.937298 8.57701 8.2285 7.8875 7.5520 7.2205 6.8923 (13) 8.941742 8.58283 8.2355 7.8960 7.5618 7.2317 6.9048 (14) 8.945388 8.58760 8.2415 7.9030 7.5700 7.2410 6.9152 (15) 8.949898 8.59349 8.2488 7.9117 7.5800 7.2524 6.9280 2 71.449530 68.55219 65.7484 63.0060 60.3071 57.6402 54.9995 I' 71.449597 68.55223 65.7482 63.0063 60.3072 57.6404 54.9998 3 (1) (i) (0) (1) (2) (3) 9 Log.i^ Log. -i c 3 Log. | s 3 Log. % 6 3 Log. 6 3 Log. 6 8 Log. 6 3 "2" 2 "2". 2 2 2 ( o) 8.183917 5.42374 3.73837n 0.280319 0.417421 0.200612 9.961097 ( i) 8.184550 5.25307 5.29293 0.281000 0.418474 0.202090 9.963016 ( 2) 8.184586 4.24928ft 5.42550 0.278120 0.414013 0.195824 9.954877 ( 3) 8.184421 5.31430ft 5.23627 0.273612 0.406981 0.185917 9.941987 ( 4 ) 8.183758 5.43028n 4.40987ft 0.267827 0.397890 0.173060 9.925223 ( 5) 8.183173 5.24454n 5.31797?? 0.262860 0.390004 0.161858 9.910585 ( 6) 8.183110 4.20163 5.43607w 0.260054 0.385513 0.155458 9.902210 ( T) 8.183200 5.29621 5.27852n 0.258559 0.383116 0.152039 9.897732 ( 8) 8.183797 5.43847 3.83805w 0.259184 0.384116 0.153464 9.899598 ( 9) 8.185270 5.31804 5.25490 0.261621 0.388024 0.159038 9.906900 (10) 8.187625 4.49962 5.43747 0.265530 0.394254 0.167901 9.918485 (11) 8.189506 5.21681ft 5.34487 0.269488 0.400515 0.176758 9.930076 (12) 8.189803 5.43364n 4.63509 0.272484 0.405223 0.183435 9.938754 (13) 8.188333 5.34047/1 5.20953ft 0.274429 0.408267 0.187732 9.944350 (14) 8.185972 4.50257n 5.42714ft 0.276036 0.410773 0.191265 9.948948 (16) 8.184139 5.24121 5.30466ft 0.278037 0.413885 0.195644 9.954643 S 65.482568 2.159554 3.209203 1.421019 79.449192 Z' 65.482592 2.159606 3.209266 1.421076 79.449289 A. P. 8. VOL. XIX. Q. 130 A NEW METHOD OF DETERMINING Values of Quantities in the Development of l*(~) and [*a 2 ( J (4) .(5) 1 (6) (7) (8) (9) 9 Log-. &o ^D O T Log. 6 3 Log. 6 3 Log. 6 3 Log. 6 3 Log. 6 3 2 f 2 2 2 2 ( o) 9.70884 9.4484 . 9.1822 8.9118 8.6383 8.3621 ( i) 9.71121 9.4512 9.1854 8.9155 8.6423 8.3665 ( 2) 9.70116 9.4393 9.1716 8.8998 .8.6247 8.3471 ( 3) 9.68524 9.4203 9.1496 8.8747 8.5965 8.3158 ( 4) 9.66450 9.3955 9.1207 8.8418 8.5595 8.2747 ( 5) 9.64638 9.3739 9.0956 8.8131 8.5273 8.2389 ( 6) 9.63600 9.3614 9.0813 8.7968 8.5089 8.2184 (I) 9.63043 9.3549 9.0735 8.7880 8.4991 8.2077 ( 8) 9.63276 9.3576 9.0766 8.7914 8.5030 8.2119 ( 9) 9.64181 9.3684 9.0893 8.8058 8.5191 8.2298 (10) 9.65617 9.3856 9.1093 8.8287 8.5449 8.2585 (11) 9.67052 9.4028 9.1292 8.8515 8.5705 8.2868 (12) 9.68125 9.4156 9.1440 8.8684 8.5893 8.3078 (13) 9.68816 9.4237 9.1537 8.8791 8.6015 8.3213 (14) 9.69382 9.4305 9.1614 8.8882 8.6118 8.3329 (15) 9.70087 9.4389 9.1711 8.8992 8.6240 8.3464 2 77.37450 75.2339 73.0471 70.8269 68.5804 66.3134 I' 77.37462 75.2341 73.0474 70.8269 68.5803 66.3132 9 Log. k Log. lc l Log. & 2 Log. & 3 Log. Jc 4 Log. 7c 5 Log. k 6 Log. k 7 ( o) 8.824187 8.54492 8.12562 7.750420 7.39550 7.0523 6.7168 6.4105 ( i) 8.824302 8.54433 8.12588 7.751220 7.39678 7.0540 6.7190 6.4054 ( 2) 8.823605 8.53875 8.11916 7.742693 7.38634 7.0416 6.7046 6.3714 ( 3) 8.822665 8.53172 8.10982 7.730361 7.37091 7.0232 6.6832 6.3298 ( 4) 8.821701 8.52543 8.09963 7.716100 7.35261 7.0007 6.6565 6.2932 ( 5) 8.821143 8.52236 8.09246 7.705215 7.33807 6.9826 6.6349 6.2764 ( 6) 8.821183 8.52300 8.09009 7.700585 7.33130 6.9737 6.6239 6.2809 ( ?) 8.821397 8.52470 8.08981 7.699023 7.32855 6.9698 6.6187 6.2913 ( 8) 8.821810 8.52671 8.09164 7.701551 7.33151 6.9732 6.6226 6.3027 C 9) _8.822444 8.52829 8.09567 7.707159 7.33895 6.9824 6.6337 6.3093 (10) 8.823323 8.52965 8.10077 7.715298 7.35002. 6.9965 6.6506 6.3129 (11) 8.824009 8.53059 8.10550 7.723069 7.3607GI 7.0100 6.6669 6.3147 (12) 8.824233 8.53159 8.10915 7.728940 7.36874 7.0202 6.6793 6.3196 (13) 8.824055 8.53359 8.11233 7.733450 7.37462 7.0274 6.6879 6.3342 (14) 8.823809 8.53721 8.11622 7.738311 7.38053 7.0345 6.6960 6.3608 (15) 8.823826 8.54164 8.12113 7.744423 7.38795 7.0433 6.7062 6.3901 S 70.583851 68.25726 64.85258 61.793910 58.89655 56.0927 53.3503 50.6520 I' 70.583841 68.25722 64.85260 61.793920 58.89653 56.0926 53.3505 50.6512 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 131 Values of Quantities in the Development of fi(^\ and f.ia 2 ( a \'. <7 Log. fc 8 Log, & 9 JT, K> JT S K t K. JZi K, -X. ( o) 6.0606 5.7378 - 0.6 - 0.4 / - 0.3 - 0.3 - 0.3 - 0.3 0.3 - 0.3 ( ] ) 6.0636 5.7413 . +20.3 + 12.9 +11.4 + 11.1 +10.6 + 10.1 + 9.5 + 8.3 ( 2) 6.0454 5.7212 4-27.9 + 17.8 +15.6 + 15.2 + 14.6 + 14.0 +13.5 + 12.5 ( '>) 6.0178 5.6904 + 18.4 + 11.7 +10.2 + 10.0 + 9.7 - 9.4 + 9.2 + 8.8 ( 4) 5.9830 5.6515 - 2.8 - 1.8 - 1.6 - 1.5 - 1.5 - 1.5 - 1.5 - 1.5 ( 5) 5.9541 5.6191 22.7 14.5 -12.7 12.0 -11.8 -11.6 11.4 -11.0 ( 6) 5.9391 5.6019 29.8 19.0 -16.7 -15.7 -15.3 14.9 -14.5 13.7 ( *) 5.9316 5.5934 20.7 -13.2 11.6 10.9 -10.5 10.1 - 9.7 - 8.9 ( 8) 5.9364 5.5985 - 0.7 - 0.5 - 0.4 - 0.4 - 0.4 - 0.3 - 0.3 0.3 ( ) 5.9512 5.6151 + 19.3 + 12.3 + 10.9 + 10.2 + 9.8 + 9.4 + 9.0 + 8.2 (10) 5.9737 5.6405 4-29.1 + 18.6 + 16.4 +15.3 + 14.9 + 14.5 + 14.1 +13.3 (11) 5.9959 5.6656 +23.4 + 14.9 +13.1 + 12.3 + 12.1 +11.9 + 11.7 +11.3 (12) 6.0124 5.6842 - 4.5 + 2.8 + 2.5 - 2.4 + 2.4 + 2.3 + 2.3 + 2.2 (13) 6.0251 5.6968 -17.0 -10.8 9.5 8.9 - 8.8 8.7 - 8.6 - 8.4 (14) 6.0341 5.7083 28.1 17.8 -15.7 -14.7 -14.3 13.9 13.6 13.0 (15) 6.0468 5.7224 21.0 -13.4 -11.8 -11.0 -10.6 -10.2 - 9.8 - 9.0 JT 45.3439 - .5 .3 .2 + .3 .3 w 45.3441 .1 + .8 .1 9 Log. Jc Log. \ Log. k 2 Log. & 3 Log. & 4 Log. & 5 Log. Jc 6 Log. 7 ( ) 8.465272 8.60289 8.38621 8.14674 7.89481 7.6341 7.3679 7.0975 ( i) 8.466247 8.60407 8.38777 8.14874 7.89694 7.6369 7.3712 7.1013 ( 2) 8.462637 8.59849 8.38030 8.13935 7.88563 7.6238 7.3561 7.0843 ( 3) 8.457236 8.59018 8.36903 8.12505 7.86829 7.6033 7.3326 7.0577 ( 4) 8.450550 8.58006 8.35509 8.10719 7.84645 7.5774 7.3026 7.0237 ( 5) 8.445362 8.57214 8.34391 8.09259 7.82837 7.5559 7.2776 6.9950 ( 6) 8.443224 8.56872 8.33868 8.08543 7.81922 7.5446 7.2645 6.9800 ( "0 8.442508 8.56750 8.33651 8.08224 7.81495 7.5395 7.2581 6.9726 ( 8) 8.444020 8.56954 8.33902 8.08521 7.81840 7.5433 7.2623 6.9771 8.444679 8.57452 8.34564 8.09354 7.82847 7.5551 7.2760 6.9925 (10) 8.453274 8.58206 8.35573 8.10632 7.84401 7.5734 7.2971 7.0165 (11) 8.458368 8.58906 8.36522 8.11851 7.85895 7.5912 7.3176 7.0400 (12) 8.461465 8.59345 8.37153 8.12680 7.86927 7.6036 7.3320 7.0564 (13) 8.461922 8.59532 8.37468 8.13126 7.87506 7.6105 7.3405 7.0660 (14) 8.461886 8.59651 8.37704 8.13471 7.87957 7.6163 7.3472 7.0739 (15) 8.462852 8.59905 8.38088 8.13992 7.88616 7.6242 7.3564 7.0845 _y 68.69172 66.90360 64.93175 62.85706 60.7165 58.5297 56.3095 I' 68.69184 66.90364 64.93185 62.85719 60.7166 58.5300 56.3096 132 A NEW METHOD OF DETERMINING Values of Quantities in the Development of and 9 Log. & 8 Log. & y J5T, K, K, (Q-g)-^ m-9)-K* 3(Q-g)-K 3 1 / / I o / a i it ( ) 6.8240 6.5478 0.1 0.1 0.1 359 25.1 358 49.5 358 13 55.0 ( i) 6.8280 6.5522 +4.4 +4.4 +4.4 28.5 1 24.6 2 14 57.0 ( 2) 6.8092 6.5317 + 6.0 + 6.0 +(>.0 l 26.5 3 31.0 5 27 34.4 ( 3) 6.7795 6.4988 +3.9 +3.9 +3.9 2 15.2 4 55.5 7 30 33.9 ( 4) 6.7414 6.4566 0.6 0.6 0.6 2 46.3 5 28.8 8 12 2.4 ( 5) 6.7093 6.4209 47 4.7 -4.7 2 47.3 5 3.8 7 26 37.11 ( 6) 6.6921 6.4016 6.2 6.2 6.2 2 9.9 3 39.1 5 16 57.0 ( 7) 6.6837 6.3923 4.3 4.3 4.3 55.7 1 23.2 1 56 :5!i.u ( 8) 6.6887 6.3976 0.2 0.2 0.2 359 17.6 358 34.3 357 51 3.3 ( 9) 6.7058 6.4165 +4.0 _|_4.0 +4.0 357 36.2 355 38.7 353 35 39.5 (10) 6.7327 6.4463 + 6.1 + 6.1 +6.1 356 13.4 353 6.5 349 51 14.9 (11) 6.758!) 6.4752 +5.0 +5.0 +5.0 355 26.8 351 25.5 347 17 2<>.4 (12) 6.7773 6.4958 + 1.0 + 1.0 -1-1.0 355 24.4 350 55.0 346 24 12.8 (13) 6.7883 6.5081 3.5 3.5 3.5 356 1.7 351 40.2 347 23 40.2 (14) 6.7976 6.5187 6.0 6.0 6.0 357 4.6 353 30.7 350 5 3.8 (15) 6.8093 6.5317 4.5 _4.5 _4.5 358 15.9 356 3.1 353 56 22.5 V 54.0630 51.7961 .0 .0 .0 1793 47.8 1781 22 3.6 I' 54.0628 51.7957 + .3 + .3 + .3 1433 47.3 1421 21 59.1 9 42-< l) % ^(Q-9 ) -^ ; 6(e-0 )-^ '( Q 9\ K im-9\ ) A's 9( :- (s) 9 A Q A Q Aj Arj // // II II II II // // // // ( ) +.2217 .0114 +.1023 .0063 +.0505 .0036 + .0226 .0019 +.0107 .0010 ( i) .2223 +.0151 .1027 +.0085 .0498 +.0048 .0226 +.0025 .0108 +.0014 ( 2) .2138 +-0350 .0978 +.0194 .0451 +.0105 .0211 +.0057 .0099 +.0030 ( 3) .2028 +.0454 .0916 +.0249 .0401 +.0128 .0192 +.0071 .0089 +-0038 ( 4) .1916 +.0465 .0856 +.0252 .0365 +.0126 .0176 +.0070 .0080 +.0037 ( 5) .1848 +.0401 .0821 +.0215 .0356 +.0109 .0167 +.0059 .0076 +.0030 ( 6) .1832 +.0277 .0815 +.0147 .0368 +.0078 .0166 +-0040 .0076 +.0021 ( 7) .1833 +.0099 .0816 +.0052 .0384 +.0028 .0168 +.0014 .0077 +.0007 ( 8 ) .1847 .0116 .0823 .0062 .0394 .0035 .0169 .0017 .0078 .0009 ( 9) .I860 .0346 .0826 .0185 .0388 .0102 .0168 .0051 .0077 .0026 (10) .1870 .0561 .0827 .0301 .0372 .0160 .0166 .0083 .0075 .0043 (11) .1880 .0722 .0827 .0389 .0354 .0199 .0163 .0108 .0072- .0056 (12) .1904 .0793 .0837 .0429 .0350 .0216 .0163 .0120 .0072 .0062 (13) .1956 .0756 .0867 .0411 .0369 .0210 .0173 .0116 .0077 .0060 .2041 .0613 .0918 .0336 .0414 .0180 .0190 .0096 .0087 .0051 (15) .2140 .0387 .0978 .0214 .0468 .0120 .0210 .0062 .0098 .0033 S + 1.5765 .1105 +.7077 .0598 +.3219 .0318 +.1467 .0168 +.0674 .0087 i' + 1.5768 .1106 +.7078 .0598 +.3218 .0318 +.1467 .0168 +.0674 .0086 134 A NEW METHOD OF DETEKMINING In the expansion of jj. a 2 1 ^ (e) (C) (8) (c) (8) (<0 () (c) () 9 A A, A, A.^ -4 2 A.% A.% A A, it n n // // n n n n ( o) 23.3520 +32.0569 0.3301 + 19.4613 0.4009 + 11.2092 0.3464 + 6.269 0.258 ( i) 23.4045 32.1423 -j-0.41 99 19.5273 +0.5272 11.2569 +0.4603 6.300 +0.347 ( 2) 23.2107 31.7192 +1.0033 19.1618 +1.2486 10.9731 +1.0737 6.096 +0.802 ( 3) 22.9239 31.1043 +1.3503 18.6375 +1.6470 10.5748 +1.4097 5.813 +1.041 ( 4) 22.5737 30.3821 +1.4503 18.0367 +1.7240 10.1342 +1.4580 5.516 +1.063 ( 5) 22.3056 29.8387 +1.2952 17.5937 +1.5122 9.8190 +1.2644 5.310 +0.912 ( 6) 22.1960 29.6180 +0.9110 17.4156 +1.0505 9.6988 +0.8734 5.239 +0.626 ( *) 22.1595 29.5473 +0.3342 17.3564 +0.3782 9.6618 +0.3118 5.219 +0.222 ( 8) 22.2368 29.6867 0.3713 17.4552 -0.4367 9.7264 0.3654 5.259 0.204 ( 9) 22.4249 30.0100 1.1187 17.6808 1.3068 9.8617 1.0915 5.331 0.786 (10) 22.7157 30.5036 1.8033 18.0224 2.1155 10.0630 1.7762 5.436 1.285 (11) 22.9837 30.9679 2.3042 18.3471 2.7150 10.2558 2.2962 5.536 1.667 (12) 23.1482 31.2707 2.4810 18.5835 2.9616 10.4121 2.5144 5.627 1.839 (13) 23.1725 31.4193 2.3026 18.7500 2.7837 10.5580 2.3763 5.739 1.748 (14) 23.1706 31.5386 1.8212 18.9291 2.2155 10.7412 1.9027 5.895 1.409 (15) 23.2222 31.7564 1.1097 19.1791 1.3716 10.9764 1.1843 6.091 .882 r 182.6038 246.7758 3.4423 147.0656 4.1071 82.9580 3.5000 +45.337 2.564 w 182.5968 246.7862 3.4356 147.0719 4.1125 ' 82.9644 3.4985 + 45.339 2.563 (C) (8) (C) (8) (c) () (C) (8) (c) () 9 4 ^-5 A A A A ( ) +3.440 0.177 + 1.863 .115 +1.000 .072 +.532 .044 + .282 .027 ( i) 3.458 +0.240 1.874 +.157 1.005 +.098 .535 +.060 .283 +.036 ( 2) 3.318 +0.550 1.781 +.356 .944 +.221 .497 +.134 .260 +.076 ( 3) 3.130 +0.706 1.660 +.453 .868 +.279 .450 +.167 .231 +.098 ( 4) 2.937 +0.713 1.540 +.453 .797 +.276 .409 +.164 .208 +.095 ( 5) 2.812 +0.606 1.467 +.381 .756 +.232 .377 +.133 .196 +.078 ( 6) 2.772 +0.413 1.448 +.260 .748 +.157 .383 +.092 .195 +.053 ( ?) 2.766 +0.146 1.446 +.091 .750 +.055 .386 +.032 .197 +.019 ( 8) 2.789 0.175 1.459 .110 .757 .053 .389 .039 .199 .023 ( 9) 2.824 0.522 1.474 .329 .760 .199 .389 .117 .197 .067 (10) 2.870 0.855 1.491 .540 .759 .326 ^.385 .192 .193 .111 (11) 2.915 1.115 1.505 .705 .757 .425 .379 .251 .187 .144 (12) 2.963 1.235 1.528 .783 .767 .473 .382 .280 .188 .162 (13) 3.042 1.179 1.582 .753 .803 .457 .404 .272 .201 .158 (14) 3.164 0.957 1.670 .615 .867 .378 .446 .227 .227 .133 (15) 3.312 0.604 1.775 .391 .942 .243 .495 .147 .259 .087 I 24.253 1.723 12.780 1.094 + 6.639 .648 +3.423 .392 + 1.752 .232 v/ 24.259 1.722 12.783 1.095 6.641 .660 +3.415 .395 1.751 .225 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 135 . (0 (s) (c) (s) The Quantities %C ttV , JC1 >: , $> , J$ (> , arranged for Quadrature in the Expansion of u G)- <=0 i-l ,-,=2 < = 8 < = 4 ; = 5 , = 6 ,=, (c) (c) L +[209.51454] +53.571 .735" II +20.046 .553 +8.26978 .34414 // +3.566 .199 + 1.576 .110 +.707 .060 (7 tC> ^1,1 +.25653 +.548 +.382 +.22949 +.129 +.071 +.038 v=l (8) a!i 8) .25027 + 1.706 .122 + 1.273 .046 +.78997 .01129 +.456 +.002 +.253 +.005 +.138 + .006 (c) +.022 +.017 +.00807 +.003 +.001 .000 (c) +.00463 +.257 +.096 +.05847 +.038 +.024 +.013 ,2 } .170 .003 +.01835 +.017 +.007 + .004 *_2 (8) +.12279 +.128 +.080 +.04667 + .026 +.015 + .001 (c) . +.065 + .048 + .03063 +.018 +.010 +.006 (c) + .03070 +.020 +.007 +.00662 +.005 +.002 +.001 (s) .003 +.002 +.00216 +.002 +.001 +.001 v=3- (8) + .05945 +.041 +.023 +.01319 + .006 + .003 +.002 (c) . 000 .001 .00217 .002 .001 .001 '< +.00037 +.001 +.00030 <& 000 +.00052 a!f +.00055 000 +.00076 gS .001 .00103 136 A NEW METHOD OP DETERMINING (c) (s) (c) The Quantities |Q>, , \G^ V , J$> (s) r t> , arranged for Quadrature, in the Expansion of ,/a\8 (i) fcO i=i (-=2 i-3 i=4, 1=6 4 = 6 4*zz 7 Q i=9 f (0 n V<,0 n +[364.6002] +246.7810 + 147.068 + 82.9613 n +45.338 +24.256 n + 12.781 + 6.640 II +3.419 + 1.751 (c) t,0 3.4388 -4.110 3.4992 2.562 -1.722 -1.095 .654 .392 .228 - (c) a,i +4.3500 +4.6277 +3.873 +2.8862 + 1.956 + 1.253 +.771 +.461 + .270 +.154 (s) + 7.8438 + 9.373 + 7.9505 +5.816 + 3.910 +2.488 + 1.514 + .898 +.521 a!r -1.8014 -1.1511 .801 .3643 .106 +.017 +.062 + .078 + .058 + .049 (c) ; + .1015 +.104 +.0731 +.043 +.024 + .011 .008 + .003 +.001 (c) .2566 +.0899 +.294 +.3888 +.384 +.327 +.252 +.193 + .134 +.086 (8) +.1010 +.296 +.3297 +.302 +.239 +.173 + .116 +.078 +.047 < + 1.1803 + 1.1209 +.883 +.6281 +.418 + .266 +.162 + .093 + .058 +.031 (c) $.2 L +.3367 +.400 + .3459 +.255 +.170 +.106 + .065 +.034 .018 0,8 + .1113 +.1140 +.099 +.0809 + .066 +.049 +.035 +.024 + .013 + .012 Si'* .0170 .000 +.0059 + .012 +.015 + .015 +.015 + .013 +.008 ~ (s) +.5132 + .6602 +.317 + .2097 +.130 +.076 +.043 + .020 +.012 + .002 (c) .0138 .030 .0344 .032 ^.027 .020 .005 .010 .005 sin 2y + etc., noting that the tables give one-half of the values of these quantities. Thus we have i=l i 2 i\ i 2 c i n 2^1,0 +53.571 n +20.046 (c) If! 2*^1,0 - - 0.735 - 0.553 (c) CM + 1.013 + .707 #u = + 1.306 + .974 Cu .01)4 .032 #u = + .040 + .031 (c) + .363 + .135 # = .240 .004 -1,2 + .181 + .114 (c) #1,2 = + .092 + .070 (c) (s) Q, 8 + .015 + .005 #J,3 - .005 + .004 d!s + .077 + .043 # = .001 (c) (s) Ci,4 . . #1,4 - . . rf' V/^4 (c) #1,4 = 2 II +55.126 // +21.018 V + 0.458 + 0.521 P + 6.891 + 2.627 1.^ + 0.057 + 0.065 ^f + 6.893 + 2.629 = + 0.057 + 0.065 In this way we check the values of these quantities for all values of **, in case of both p(5), and po(). Applying to the coefficients of the two preceding tables the formula (a\ n () () (*) ( c ) i- 2 ) = J22( a> =F #> ) cos [( =F v)g iE'] T JS2( 0,, dh #^ ) sin [(* T v)^ 2 3 noting that J has been applied, we have the values of ^ f ^ J , ^a Q J that follow : A. P. s. VOL. xix. R. 138 A NEW METHOD OF DETERMINING [l G) 9 E' COS sin 1 COS sin // +[209.51455] // // +[364.6002] // 1 +0.25653 0.25027 +4.3500 1.8014 2 +0.00463 +0.12279 0.2566 + 1.1803 3 +0.03070 +0.05945 +0.1113 +0.5132 4 +0.00037 +0.00055 +0.0177 +0.0182 2 1 +0.023 0.041 + 0.1310 0.6464 1 1 +0.427 0.193 0.0112 -1.4577 1 -1.158 + 0.101 3.2161 + 1.0496 1 1 +53.571 +0.735 + 246.7810 +3.4388 2 1 + 2.254 0.144 + 12.4716 -1.2526 3 1 +0.087 + 0.063 +0.1909 +0.7842 4 1 +0.016 + 0.041 + 0.0970 +0.6740 1 2 +0.099 0.287 2 +0.098 0.129 0.001 -1.283 1 2 0.891 +0.029 5.500 + 0.697 2 2 +20.046 +0.553 + 147.068 + 4.110 3 2 + 1.656 0.063 + 13.246 0.905 4 2 +0.093 +0.032 +0.590 +0.483 3 +0.00446 0.01101 ' +.0750 0.1753 1 3 +0.04011 0.07730 +.0591 0.9741 2 3 0.56048 +0.00322 5.0643 +0.2912 3 3 +8.26978 +0.34414 + 82.9613 +3.4992 43 + 1.01947 0.01936 + 10.8367 0.4375 5 3 +0.07682 +0.01603 +0.7185 +0.2822 6 3 + 0.00879 +0.01536 +0.0868 +0.2441 1 4 +0.003 0.004 +0.053 0.098 2 4 +0.020 0.044 + 0.082 0.674 3 4 0.326 0.005 3.859 + 0.062 4 4 + 3.566 +0.199 +45.338 +2.562 5 4 +0.585 0.001 + 7.772 0.149 6 4 +0.055 +0.008 +0.687 +0.163 7 4 +0.078 +0.162 2 5 +0.005 +0.045 H- 0.033 0.049 3 5 +0.016 0.025 +0.088 0.095 4 5 0.182 0.007 2.657 0.041 5 5 + 1.576 +0.110 + 24.256 + 1.722 6 5 + 0.325 +0.004 + 5.163 0.006 7 5 +0.031 +0.004 + 0.567 + 0.436 4 6 +0.009 0.008 +0.079 0.269 5 6 0.100 0.006 -1.717 0.073 6 6 +0.707 +0.060 + 12.781 + 1.095 7 6 +0.176 + 0.005 + 3.260 +0.050 8 6 +0.018 0.005 +0.426 +0.057 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 139 We have next to transform the expressions for ^ f^J and fj.a 2 (*~\ just given into others in which both the angles involved are mean anomalies. From beginning with m =. 5, we find the values of r 5 for values of e f from f to e' 4 . Then we find Putting m =. 4, we find the values of r 4 as in the case of r 5 . Then we get p from (0) We proceed in this way until we finally have the values of p { . Then we find J h , ^ or from 2" J (0) 7< j i 72 i i_ /' & ^~~ I j n 2 4 where I (m) and 7^/^ from (m) (0) T , f, = 7 , /l -2 .1 The details of the computation are as follows : 140 A NEW METHOD OF DETERMINING Computation of the J functions. * = y e' 1* 2e' 2 G 3e' n f ^ e 4,6' log.? 8.38251 8.68354 8 .85963 8.98457 9 .08148 9.16066 9.22761 9.28560 log. r b 2 .31646 2.01543 1 .83934 1.71440 1 .61749 1.53831 1.47136 1.41337 tog. #3 7 .68354 7.98457 8 .16066 8.28560 8 .38251 8.46169 8.52864 8.58663 log. r 4 2 .21955 1.91852 1 .74243 1.61749 1 .52058 1.44140 1.37445 1.31646 log.n Iog.jp 5 4 .53601 3.93395 8 .58177 3.33189 3 .13807 2.97971 2.84581 2.72983 Zech - 1 5 -12 -20 -31 -45 -62 -81 2 .21954 1.91847 1 .74231 1.61729 1 .52027 1.44095 1.37383 1.31585 log. p 4 7 .78046 8.08153 8 .25769 8.38271 8.47973 8.55905 8.62617 8.68415 log. r s 2 .09461 1.79358 1 .61749 1.49255 1.39564 1.31646 1.24951 1.19152 Diff. 4 .31415 3.71205 3 .35980 3.10984 2.91591 2.75741 2.62334 2.50737 Zech 2 -9 -19 34 -52 -76 -103 - 135 2 .09459 1.79349 1 .61730 1.49221 1.39512 1.31570 1.24848 1.19017 tog. 3 ^j _M. o 7.90541 8.20651 8.38270 8.50779 8.60488 8.68430 8.75152 8.80983 log. r 2 1.91852 1.61749 1.44140 1.31646 1.21955 1.14037 1.07342 1.01543 Diff. 4.01311 3.41098 3.05870 2.80867 2.61467 2.45607 2.32190 2.20560 Zech 4 -17 38 -67 105 -152 -206 269 1.91848 1.61732 1.44102 '1.31579 1.21850 1.13885 1.07136 1.01274 log. p 2 8.08152 8.38268 8.55898 8.68421 8.78150 8.86115 8.92864 8.98726 log. r t 1.61749 1.31646 1.14037 1.01543 0.91852 0.83934 0.77239 0.71440 Diff 3.53597 2.93378 2.58139 2.33122 2.13702 1.97819 1.84375 1.72714 Zech 13 -51 -114 -202 315 454 -618 807 1.61736 1.31595 1 13923 1.01341 91537 0.83480 0.76621 0.70633 log. pi 8.38264 8.68405 8. 86077 8.98659 9.08463 9.16520 9.23379 9.29367 log. Z| 3.53004 4.73716 5: 43852 5.93828 6. 32592 6.64264 6.1)1044 7.14240 tog- 7 2.92798 4.13210 4. 83646 5.33622 5. 72386 6.04058 6.30838 6.54034 - tog. I 2 6.7650271 7.36708n 7. 7l926n 7.9691471 8. 16296n 8.3213271 8.45522n 8.5712071 Diff 3. 83704 3.23498 2. 88280 2.6321)2 2. 43910 2.28084 2.14684 2.03086 Zech -7 -25 -57 -101 -157 -227 -308 402 log.( P + -) 6. 76495n 7.3669371 .?. 7186971 7.96813n 8. 1613971 8.3190571 8.45214n 8.5671871 4 / 3. 23505 2.63307 2. 28131 2.03187 1. 83861 1.68095 1.54786 1.43282 Zech 26 101 227 -401 625 896 1213 1575 log. J" (0) 9. 99974 9.99899 9. 99773 9.99599 9. 99376 9.99104 9.98787 9.98425 log. jpx 8. 38264 8.68405 8. 86077 8.98659 9. 08463 9.16520 9.23379 9.29367 log. J w 8. 38238 8.68304 8. 85850 8.98258 9. 07838 9.15624 9.22166 9.27792 tog. ^ 8. 08152 8.38268 8. 55898 8.68421 8. 78150 8.86115 8.92864 8.98726 log. <7 (2) 6. 46390 7.06572 7. 41748 7.66679 7. 85988 8.01739 8.15030 8.26518 log. p s 7. 90541 8.20651 8. 38270 8.50779 8. 60488 8.68430 8.75152 8.80983 log.JV 4. 36931 5.27223 5. 84)018 6.17458 6. 46476 6.70169 6.90182 7.07501 lOO". 7?, 7. 78046 .8.08153 8. 25769 8.38271 8. 47973 8.55905 8.62617 8.68415 teg. J (4) 2. 14977 3.35376 4. 05787 4.55729 4. 94449 5.26074 5.52799 5.75916 THE GENERAL PERTURBATIONS OP THE MINOR PLANETS. 141 Noting that log. (e/ (0) 1) =: log. ( I 2 + ~), K ?', and I h'X, we form the following: tables : h' Log.j[>(C x 1 (1) 1 (2) ^ J3) ! (4) A /V Log.^/A/v 1 6.7649n 8.38238 6.4639 4.3693 2.1498 2 7.0658/1 8.38201 6.7647 4.9712 3.0527 3 7.2415n 8.38138 6.9404 5.3231 3.5807 4 7.3661/1 8.38052 . 7.0647 5.5725 3.9551 5 7.4624/1 8.37941 7.1610 5.7658 4.2456 6 7.5409n 8.37809 7.2392 5.9235 4.4826 7 7.6070/1 8.37656 7.3052 6.0567 4.6828 8 7.6641n 8.37483 7.3621 6.1719 4.8562 i' h'= 2 7i' 1 ~ti -J-: Value of ,-/ (h'-V) T h' h> * h' -6 A'=7 ^'=8 l* : = 2 .'=3 *=* *=5 1 4.97l2n 6.4639/t 6.76495/z 8.38201 6.9404 5.5725 4.2455 .... .... .... 2 3.3537/1 4.6703ra 8.68341n 7.36693n 8.68241 7.3657 6.0668 4.7835 .... .... 3 6.9410 8.85913/1 7.71869/r 8.85764 7.6381 6.4006 5.1598 4 4.9714n 7.36675 8.98344/1 7.96813n 8.98147 7.8413 6.6588 5.4583 5 5.6702/1 7.6393 9.07949n 8.1614/1 9.07706 8.0042 6.8709 6 6.1012n 7.8432 9.15756/1 8.3190n 9.15471 8.1402 7 For h' =0, 6.4176/1 8.0061 9.22320n 8.4521w 9.21993 8 we have 6.6689n 8.1423 9.27965/1 8.5672/1 9 8.38251n 6.8777/1 8.2594 9.32905n In computing the values of the J functions, the lines headed Zech show that addition or subtraction tables have been used. For convenience, (J" (0 - 1) is em- ployed instead of J (0 \ its values being found in the line headed log. ( V + -J. 142 A NEW METHOD OF DETERMINING From the expression h' being the multiple of g', and being constant, and i' being variable, we have (A'+l) 9 (V+2) ' etc. Now for h' = +1? we have, if we write the angle in place of the coefficient, ((ig - g')} = 1 J SS fa - &) + | + ^') - etc., (2) (3) ((ig + g')) = Jy SS (^ - -#') 2e7 v SS (ig - %E') etc. (0) (1) + /v SS (^ + JS? ; ) - 2^ ^ (ig + 2^') etc. THE GENERAL PERTURBATIONS OF TFIE MINOR PLANETS. 143 And for the particular case of i = 1, we have (0) (1) (2) ((ff - 00) = ' SS ( g - E') 2J S ( g - 2JS7') + 3J % ( g - 3JB") =F etc. <2) (3) (4) - Jv SS($r + E')-2J Sg? (# + 2^') - 3 1), as has been noted. If we put h f = + 2, we have (1) (0) (-1) ((ig 2g')) - I J w SS (ig .#') + f ,7 2V ^ (^ 2^') + f e/ w SS (^ - 3-ET') + etc. (3) (4) ' ^ + 2J0') - etc. - ' - In the table giving the values of - J K> > , we have, under h' = 2, which applies to //' the equation just given, (1) (3) for i' = 1, log. i<7 2A , = 8.38201 log. ( i J^,) = 4.9712^; (0) (4) for i - 2, log. ( %J w l) = 7.36693w log. ( f <7 2A ) = 3.3537^ ; (i) for tf = 3, log. ( f J w ) = 8.85913/i etc. = etc. etc., etc. = etc. (3) (4) ^ We find the values of J J^'? f Sw in the table under 7^' = 2. We see that i' (h'-i') these are the forms of the function J k ,^ when h 2, and i' =: 1 and i' = 2. In the expansion of the coefficient of (ig h'g') indicated above by ((tg &'ff'))t we have coefficients of angles of the form (ig + i'E'). These can readily be put into the form ( ig i' E'\ but the form employed is convenient in the transformation. 144 Arranging the functions fi ( a \ ^0? ( a -\ in this form, we have Log. n a - Log. 9 E' COS sin COS sin 1 0.0637ft 9.0043 0.5074ft 0.0210 2 8.9912 9.1106ft 7.0000ft 0.1082ft 3 7.6493 8.0418ft 8.8751 9.2437ft 1 + 1 9.6304 9.2856 8.0493 0.1637 1 1 1.72893 9.8663 2,3923 0.5364 1 2 9.9499ft 8.4624 0.7404ft 9.8432 1 3 8.6032 8.8882ft 8.7716 9.9886ft 1 4 7.4771 7.6021ft 8.7243 8.9912* 2+ 1 8.3617 8.6128 9.1173 9.8105 2 - 1 0.3530 9.1584ft 1.0959 0.0978ft 2 2 1.30203 9.7427 2.1675 0.6138 2 3 9.7486ft 7.5079 0.7045ft 9.4642 2 4 8.3010 8.6435ft 8.9138 9.8287ft 2 5 6.6990 7.6532 3-1 8.9395 8.7993 9.2808 9.8944 3-2 0.2191 8.7993ft 1.1221 9.9566n 3 3 0.91750 9.5368 1.9189 0.5440 3 4 9.5132ft 7.6990ft 0.5865ft 8.7924 3 5 8.2041 8.3979ft 8.9445 8.9777ft 4 1 8.2041 8.6128 8.9868 9.8287 4 2 8.9685 8.5051 9.7709 9.6839 4 3 0.0082 8.2869ft 1.0348 9.6410ft 4 4 0.5522 9.2989 1.6565 0.4085 4 5 9.2601ft 7.8451ft 0.4244ft 8.6128ft 4 6 7.9542 7.9093ft 8.8976 9.4298ft 5-3 8.8855 8.2049 9.8564 ' 9.4506 5 4 9.7672 7.0000ft 0.8905 0.1732ft 5 5 0.1976 9.0414 1.3848 0.2360 5 6 9.0000n 7.7782ft there may be cases when there are sensi- ble terms arising from g + 12', g + 2.E7', etc. ; if so, we use the column for li' = - - 1, and apply the proper numbers of this column to the coefficients of the angles named. Likewise in the case of (ig -f #'), there may be terms arising from the product of the numbers in the column li' = 1 and the coefficients of the angles g -f E', etc. This will be made clear by an inspection of the two expressions (0) (1) ((ig - g'}} - J (ig - - E') - 2<7 A (ig - - 2E') etc. (2) (3) ig -W)-- etc., (2) (3) ((ig + flO) = - J S (* E') 2,7 A , SS (ig - 2E') -- etc. (0) (1) + J SS (ig + -#') - 2e/v SS (^ + 2J7') etc. where ((ig #')), ((ig + ^')) represent not the angles but their coefficients. In retaining the form (ig + i'E') instead of the form ( ig -- i'E') we can per- form the operations indicated without any change of sign in case of the sine terms. Making the transformations as indicated above, we obtain the following expres- sions for the functions - and THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 147 9 9' COS sin COS sin // + 104.78521 // // +182.3777 // 1 1.04636 0.27266 1.6046 -1.9194 2 0.05031 +0.12527 0.5606 + 1.1949 3 -f 0.02860 +0.05793 + 0.1067 +0.4943 2 1 0.1274 0.6468 1 1 + 0.411 0.193 0.0830 -1.4558 1 1.162 +0.107 3.2141 + 1.1107 1 1 + 53.583 +0.734 +246.9027 +3.4023 2 1 + 1.286 0.171 + 5.3656 -1.4496 3 1 + 0.014 +0.066 0.3758 +0.8304 2 + 0.070 0.127 0.085 -1.242 1 2 + 0.399 +0.053 + 0.456 +0.848 2 2 + 20.093 +0.551 4-147.392 +4.049 8 2 + 1.056 0.086 + 7.214 1.137 4 2 + 0.027 +0.033 0.086 +0.537 0-3 + 0.00815 0.01707 + 0.0718 0.2352 1 3 + 0.04342 0.07447 + 0.0041 0.9231 2 3 0.40733 +0.03392 + 2.0442 +0.5514 3 3 -f 8.338 +0.340 + 83.537 +3.432 4 3 + 0.675 0.036 6.432 0.659 5 3 + 0.028 +0.010 + 0.079 +0.449 2 4 + 0.027 0.043 + 0.050 0.637 3 4 0.275 +0.023 + 2.174 +2.592 4 4 + 3.628 +0.197 + 46.016 +2.512 5 4 + 0.397 0.013 4.828 0.323 6 4 + 0.021 +0.008 + 0.156 +0.188 3 5 + 0.020 0.023 + 0.080 0.074 4 5 + 0.167 +0.012 + 1.762 + 0.241 5 5 + 1.623 +0.109 - 24.829 + 1.565 6 5 + 0.224 0.004 + 3.306 0.148 4 6 + 0.012 0.008 + 0.077 0.250 5 6 + 0.092 +0.007 4.535 +0.150 6 6 + 0.731 +0.059 + 13.312 + 1.085 148 A NEW METHOD OF DETERMINING The transformation should be carefully checked by being done in duplicate, or better by putting the angle ig = 0, in all the divisions of the two functions, having thus only the angles (0 E'\ ( 2E'), (0 3E'), etc., etc. ; also (0 g'\ (0 20r'), etc. Adding the coefficients in each division of the functions before and after transformation, and operating on the sums before transformation as on single members of the sums, the results should agree with the sums of the divisions of the transfor- mations given above. The transformations of these functions were checked by being done in duplicate, but we will give the check in case of another planet. We have for the logarithms of the sums before transformation, and for the sums after transformation the following : 9 y COS sin g g' o - - 1 i .85407 1.62090n 0- - 1 - - 2 1 .25778 1.51473n - 2 - - 3 9 .7024,* 1.26993n - -3 - -4 .TlOln 0.9147n 0- - 4 - -5 .6632n 0.3899n - -5 - - 6 .4387n 9.0934 - - 6 - - 7 .1222w 9.8069 - - 7 - - 8 9 .5965n 9.8865 - - 8 For the angle (0-1), (0 -2), a 0.041 -\ 0.024 + 1.722 1 .007 0. 873 H 1.578 .042 + .076 000 0016 4- .037 + 1 .346 4-71. 462 -41.774 .012 4- 18.104 32 .019 .714 + 70. 4- 70. 548 573 - - 40.188 -40.196 4- 19.809 4- 19.811 -32 -32 .318 .319 COS II 4- 70.548 4- 19.809 4- 0.906 4.540 4.707 3.059 0.623 0.071 sm - 40.188 - 32.318 - 19.352 9.263 3.313 0.330 4- 0.739 4- 0.615 3. + .062 .037 -j- .871 1.574 4- .003 4- .097 . -f- .494 4- .791 .020 .011 .504 - 18.618 4- 0.906 + 19.352 _|_ 0.902 19.355 The numbers in the last line of each case are the sums of the divisions after con- version when ig is put = 0. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 149 To have close agreement it is necessary that all sensible terms in the expansion of and ^a 2 Qj be retained. In the expressions for these functions given a large number of terms and some groups of terms have been omitted as they produce no terms in the final results of sufficient magnitude to be retained. In transforming a series it will be convenient to have the values of the (/functions on a separate slip of paper, so that by folding the slip vertically we can form the pro- ducts at once without writing the separate factors. o The numerical expressions for ^(^) and ^a- 2 HJ being known, we need next to have those designated by (H) and (Z), which represent the action of the disturbing body on the Sun. To find (H) we use two methods to serve as checks. We have first (//) = i[A 7iyi ' + fc'SA'] cos (g - 00 - J[%/ + ZyA'] sin (g - cf) < + i [ftyiyi' - $1 cos (- g - 00 i [ZV/ - ZVA'] sin (- g - g') + i fyoyl cos ( Sf) ~ i p M' sin ( 00 2 ' + ft'SAI cos (g 2g') - 2[M 1 y 2 ' + /'yAl sin (0 20') ' -- ^A'] cos (^ 2^) -h 2[Z^ 2 ' 7yA'] sin ( cj + 2 7i W ./ cos ( 2'] cos ( E 2g') + 2[fy 2 ' - - Z'^'] sin ( E 2g cos r - 2 sn cos ( 2g') + ieZ'^' sin ( 2gr') + etc. + etc. In both expressions for (H) we have =&cos(n K) h' = t cos <}> cos ' ^ cos (n JKi) zz JT) ^ sin (n K) - I where as before v ft = . 206264."8 and a = -. I -f-m a In the second expression the eccentric angle of the disturbed body appears and we must transform the expression into one in which both angles are mean anomalies. With the eccentricity, e, of the disturbed body we compute the J functions just as we did in case of e' of the disturbing body. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 151 We have in case of Althaea (0) Log.G/-!) Log. J Log. J Log. J Log. J \ Log. J (4) e f 2e 7.20740w 7.80894n 8.16025n 8.40890n 9.99930 9.99719 9.99368 9.98872 8.60344 8.90341 9.07774 9.20016 6.90632 7.5077 7.8587 8.1068 5.0329 5.9356 6.4630 6.8365 3.0347 4.2384 4.9418 5.4403 (h-i) From these values we may form a table of -- J^ as was done for the disturbing body. The values of these quantities can be checked by means of the tables found in ENGELMANN'S edition of BESSEL'S Werke, Band I, pp. 103-109. Finding the numerical value of (H) first by the second expression, we get E 9' COS a sin 1 - i +48.154 +0.651 -i - i + 0.188 0.102 o - i - 3.884 0.044 i- 2 + 4.644 +0.062 -i - 2 + 0.018 0.010 - 2 - 0.374 0.004 1 - 3 + 0.37800 +0.00510 1 - 3 + 0.00141 0.00081 3 - 0.03048 0.00036 To transform we change from (hE i'g'} into (i'g 1 TiN). Making the transfor- mation, writing also the values found from the first expression for the sake of compari- son, and the value of (I) which will next be determined, we have 152 A NEW METHOD OF DETERMINING (I) 9 9' 1 COS II - 5.826 sin a 0.066 COS // - 5.824 sin a 0.066 sin cos // // +4.799 +2.043 2 - 0.560 0.006 - 0.562 0.006 +0.463 +0.197 3 - 0.04566 0.00057 - 0.04575 +0.038 +0.016 -1 1 + 0.149 0.103 + 0.180 0.103 1 1 +48.076 +0.650 +48.079 + 0.650 1 2 + 4.631 +0.062 + 4.605 +0.062 1 -3 + 0.37740 +0.00502 + 0.37738 +0.00510 2 1 + 1.927 +0.026 + 1.927 +0.030 2 2 + 0.186 +0.002 + 0.186 +0.002 2 3 + 0.011 0.000 + 0.015 0.000 To find the numerical value of (/) needed in case of the function a 2 ( -), we have \(LZ/ (I) = &'! sin (- tf) + 6y x cos (- g') + 4 65' 2 sin ( 20r') + 4 6y a cos ( 2g f ) + 9 M's sin ( 3gr') + 9 6y s cos ( + etc. + etc. where b = ~ cos d>' sin J cos II', &' = ^ sin I sin II'. Having the values of ^ Q), ^a 2 (**-} , (^), and (/), we next find those of r\ all, j <> -r-, and a 2 , dr dz 7 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 153 from where (1) (2) (3) e 2 - - ^t7 A cos # JJk cos 2g | J" 3A cos 3g etc. o? n T T' (0 > (2 > Q ) < 3 ) -, sin (/' + IT) = [<7 A , + 7 V ] G! sin^' - - \\J W + ,7 2X ,] Cl sin 2^' etc. (0) (2) (1) (3) -f |e'c 2 [J^/ 7 A ' ] c 2 cos (/' J [t^ A / J^' ] c 2 cos 2^ etc. and C 2 being given by the equations sin/ , . T-T, = -- COS d) COS II a sin/ . T-J. 2 = smll'. a We find _I 2 ^] - [9.5769400] 2 [8.38238] cos g' 2 [6.46366]cos 2g' etc. + 2 [7.99450] cos0 + 2 [6.29667] cos2^-f- etc. _ si ^" ^ sin ( / ' + n') = [7.18046] + 2 [8.39074] sin #' + 2 [6.77809] sin 2#' CC Of 2 [8.01941] cos#' 2 [6.40668] cos 20' A. P. 8. VOL. XIX. T. 154 A NEW METHOD OF DETERMINING In multiplying two trigonometric series together, called by HANSEN mechanical multiplication, let GC A the coefficients of the angles /\.x in case of the sine, fin those of the angles fix in case of the cosine, / those of the angles vy in case of the sine, and S p those of the angles py in case of the cosine. The following cases then occur : a A sin /la? . $ p cos py rz | a^ p sin (%x -\- py) + J a A<^> sin tyx py) ft. cos fix . y v sin vy \ (3^, sin (fix + vy) i (3^ sin (px vy) ^ cos fix . $ p cos py = J j3^ p cos (fix + p?/) + i ftA cos (^x py) a A sin %x . y v sin vy = J a A / v cos (/(x + vy) 4~ 2 < K y v cos (/la? vy). In every term of the second members the factor \ occurs. Hence before multiplying we resolve the coefficients of one of the factors into two terms, one of which is 2, Performing the operations indicated, we have the values of n, ar , a 2 that follow : THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 155 ar\ dr g g' cos sin cos sin COS sin o + 104.78521 " + 16.5202 " +0.2828 " 1 1.04636 .27266 - 2.4398 - .6940 2.6311 + 6.0177 2 - .05031 +.12527 .3040 + .3928 - .059 + .239 3 - + .02860 +.05793 + .0274 + .1494 - .017 .017 1 1 .231 .090 .431 .355 .000 .129 - 1 + 4.662 +.173 - 1.166 + .481 -1.743 4.157 1 - 1 5.504 +.084 + 18.839 + .190 + .318 + .068 2 - 1 .641 .201 - 1.652 - .577 -k596 +3.580 3 - 1 + .014 +.066 .240 + .288 - .059 + .232 2 + .632 .121 + .497 - .414 - .020 .149 1 2 4.206 .009 - 9.136 + .200 2.474 6.095 2 2 + 19.907 +.549 +45.566 + 1.270 + .095 .067 3 2 + 1.056 .086 + 1.642 - .441 - .922 +2.011 4 2 + .027 +.033 .115 + .180 - .064 + .194 -3 + .05390 .01764 + .0718 .0602 .030 + .017 1 O .33396 .07957 .4443 - .3306 .045 .166 2 - 3 + .39221 +.03380 - 2.1788 + .1339 1.424 3.658 3 -3 + 8.338 +.340 +27.227 + 1.087 .064 .134 4 3 .675 .036 + 1.796 - .269 .519 + 1.099 5 o + .028 +.016 + .043 + .157 .042 + .123 2 4 + .027 .043 .054 .210 .046 .146 3 - 4 .275 +.023 .880 + .908 .784 2.078 4 4 + 3.628 +.197 + 15.430 + .882 .038 .106 5 4 .397 .013 + .883 - .137 .282 + .586 6 - 4 + .021 +.008 .013 + .063 .031 + .083 3 -5 + .020 .023 .034 .078 + .020 .130 4 5 .167 +.012 .281 + .044 .411 1.150 5 -5 1.623 +.109 + 8.605 - .543 + .024 .227 6 - 5 + .224 .004 + 1.061 + .064 .158 + .311 4 6 + 0.012 .008 0.075 0.095 5 6 .092 +.007 2.225 + .026 6 - 6 + .731 +.059 + 4.559 + .386 156 A NEW METHOD OF DETERMINING Having all we differentiate relative to ft. and obtain a c "*. dg We then form the three products, A.a "\ B . ar ( ' -- ), C . ar( - j . To this end ilg \dr / \dz / we find A, B, C y from ^ = 3 + 2[2 + e a ]cos(y- g) B = -2 [1+- f] sin (y - g) + 2 [f + f] cos (y 2g) -2 [| + f] sin (y-20) (y Sg) - 2 |e 2 sin (y - ^-2f cos (/ -j- etc. - etc. + etc. The numerical values of Ay B, G in case of Althsea are A=-3 } ) + 2 [0.302429] cos (7 g) B = -- 2 [0.001399] sin (7 - 0) + 2 [8.604489] cos (7 - - 2g) - 2 [8.604489] sin (y- - 20) - 2 [9.304508] cos 7 - 2 [8.606234] sin r + 2 [7.2076] cos (7 30) - 2 [7.3836] sin (7 - - 30) O = + 2 [9.697567] sin (7 - 0) + 2 [8.30066] sin (7 - - 20) - 2 [8.77953] sin 7 + 2 [7.08265] sin (7 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. For the three products we then have 157 A (d& A - a (j; o. '- 7 9 9' sin COS sin COS sin COS a ,, II 1 - + 2.1035 0.5371 + 1.1341 0.6804 1.3464 3.0038 1 i - .012 + .565 .4021 + .3723 + .1287 + .2411 -1 i - .2530 + .0439 32.9502 + .0549 .3877 - .4802 1 2 - .192 + .299 .0153 - .1657 - .0049 + .0228 -1 2 - + 2.079 - .597 - 1.1310 + .6821 + 1.2995 +2.9772 1 3 - + .261 + .457 .1263 - .3720 + .083 + .2404 1 . 2 I + .462 + .181 + .432 .348 .076 + .243 1 1 I .266 .015 + .453 + .461 1.881 f4.454 1 I 10.992 + .153 18.335 + .187 + .354 - .642 1 I + .462 + .181 .477 - .349 .228 + .572 1 1 1 - 3.680 - .815 + .929 - .559 - .815 -1.785 1 1 - 1 + 1.119 .013 .449 .476 + 1.906 4.470 1 2 - 1 .342 + .477 + .306 + .276 + .067 + .098 l 2 I 11.301 + .249 + 18.336 - .188 - .178 .359 1 3 1 + 2.360 .843 .929 + .559 + .785 + 1.760 1 4 - 1 .033 + .381 .264 - .276 1 1 2 + .232 .000 .232 .060 + .194 1 2 + 6.837 + .026 + 7.300 + .235 -1.230 +3.029 -1 2 .... .... .001 + .009 1 1 2 80.684 +2.195 45.412 + 1.264 + .178 .371 -1 1 2 .848 + .002 + .132 - .406 .139 + .290 1 2 - 2 + 1.633 - .735 - 3.470 - .384 .467 1.010 l 2 - 2 + 16.433 .240 - 7.317 - .235 + 1.239 3.036 1 3 2 .422 + .316 '+ .048 + .168 + .024 + .023 l 3 2 79.078 + 2.254 + 45.412 -1.264 .053 .273 1 4 2 - 7.937 .500 .213 + .384 + .454 + .981 1 5 2 .408 + .255 + .198 - .163 1 -3 + .5985 .1553 + .4644 .3261 .0482 + .157 1 1 -3 - 2.6517 + .1927 - 1.1042 + .1641 - .7083 + 1.8160 l 1 -3 .0661 + .0161 + .0541 + .0737 + .0123 + .0180 1 2 - 3 50.140 -[- 1.905 27.2994 + 1.0854 + .043 - .174 1 2 - 3 + .828 .1733 .5308 + .3287 .062 + .136 1 3 3 .380 .492 - 2.8964 - .2201 .256 .558 -1 3 3 + 3.482 .073 1.1112 .1645 + .707 1.818 1 4 - 3 + .263 + .190 .115 + .147 + .010 + .005 1 4 - 3 49.676 +2.079 +27.299 1.083 + .029 - .206 - 1 5 3 6.395 .264 + 3.899 + .217 + .257 + .534 158 dQ dg c. &\ g = (i +_ i' n ^\ nt. The differential relative to the time is ($ _j_ i' 7 -\ ndt. The preceding table is applied by subtracting the logarithms of the column headed log. (i + i' n -) 9 or by adding the logarithms of the column headed log. ( .^). \ft/ *~T~*n We will now give the values of ^, IF, and -^-., remarking that in the inte- ndt 7 cosi 7 grations the angle 7 is constant ; after the integrations it changes into g. 160 A NEW METHOD OF DETERMINING dW w ccm 7 9 9' sin COS COS sin COS sin 1 // -f- 3.2376 // 1.2175 // 1.2175n n + 3.2376n // 3.0038 id // 1.3464 nt 1 1 .3901 + .9373 .3901 .9373 - .1287 + .2411 i 1 + 32.6972 -f .0988 - 32.6972 + .0988 + .3877 .4802 1 2 .2073 - .4647 + .1036 + .2323 - .0024 + .0114 1 2 + .9480 - .0851 .4740 .0425 - .6497 - 1.4886 j 3 + .1350 + .0850 .0450 + .0283 - .028 + .0801 1 2 1 + .894 .167 + .383 + .07 .033 .10 1 -1 1 .187 + .446 .115 .330 0.62 1.60 1 1 - 29.327 - .340 - 83.900 .973 + 1.013 + 1.84 1 1 .015 + .530 .045 - 1.516 .652 - 1.64 1 1 1 + 4.609 1.374 7.087 - 2.112 + 1.264 - 2.74 -1 \ I .670 - .489 1.030 .752 1.370 3.21 1 2 1 .036 + .753 + .022 + .456 - .040 + .06 1 2 1 + 7.035 + .061 4.263 + .038 + .107 .21 1 3 1 .019 - .254 + .007 + .096 l 3 1 + 1.431 - .284 .540 .107 - .296 + .670 1 4 1 .297 + .105 + .081 + .029 1 1 2 .03 .11 1 2 + 14.145 + .261 + 20.207 .373 1.76 - 4.33 1 1 2 126.276 4-3.459 +419.660 + 11.503 - .59 - 1.23 -1 1 2 .716 - .408 2.380 - 1.356 + .46 + .96 1 2 2 1.837 -1.119 1.410 .860 + .36 .78 1 2 2 + 9.116 .475 7.008 .365 - .95 - 2.34 1 3 2 .470 + .484 .204 + .210 - .01 + .01 -1 3^-2 - 33.666 -f .990 + 14.632 .430 + .02 .12 i 4 2 .017 -f .125 .005 + .038 I 4 2 8.150 - .116 2.469 .035 - .14 + .30 l 5 2 .210 -f .092 + .050 + .021 1 3 + 1.0629 .4814 + 1.0136 + .4591 .05 .15 1 1 3 1.5475 -f- .3568 - 31.8180 7.335 -14.56 37.33 l 1 3 .0120 + .0898 .2452 - 1.847 + .25 .37 1 2 3 77.4394 +2.9904 + 81.400 + 3.139 .04 .18 -1 2 3 + .2972 + .1554 .3124 + .1631 + .06 + .14 1 3 3 3.' ,64 .7121 + 1.679 .365 + .13 .28 1 3 3 -f 2.3706 .2375 1.216 .122 .36 .91 1 4 3 .148 -f .337 .050 + .115. .00 .00 1 4 3 22.377 + .996 + 7.413 + .338-. .01 .07 1 5 3 2.496 .047 + .627 .012 .06 + .13 1 1 4 .165 .414 .096 .29 1 2 4 1.965 ',; +1.126 + 3.265 + 1.871 + .647 + 1.71 1 3 4 44.513 + 2.479 + 27.790 + 1.548 .014 .05 1 3 4 .031 .012 .019 .007 + .015 + .03 1 4 4 2.567 .385 .986 .148 + .054 .12 1 4 4 + 1.002 .963 .385 .370 .150 .40 1 5 4 .022 + .057 + .006 + .016 1 5 4 - 13.272 + .682 + 3.686 .190 .009 .04 1 6 4 3.037 - .010 + .660 .002 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 161 sin sin // // It II H " 1 3 5 1.717 + .195 + I- 374 + .156 .!<:-; .4(5 1 3 5 + .011 - .017 009 + .014 .'>0! .01 1 4 5 - 24.846 -j-1.626 + 11. 030 + .722 .015 + .02 1 4 5 .030 - .072 + 013 .032 tie .00 1 5 5 2.473 - .194 + 760 .060 .05 1 5 5 + .356 - .080 110 .024 .18 1 6 5 .089 + .160 4 '. 021 + .038 1 6 5 7.377 + .556 + 1.735 .130 _ 1 7 5 + 1.413 + .036 270 + .007 1 4 6 .964 + .124 .507 + .07 1 5 6 - 13.223 +1.090 -|- 4.555 + .38 1 5 6 .167 + .023 + .057 + .06 1 6 6 .946 - .002 .242 .00 1 6 6 2.098 - .040 .538 .01 1 7 6 3.302 + .324 + .674 + .09 i The part of W independent of y arising from the factor, 3, in the value of -4, has not yet been given. Its integral, or J 3a \^-j , is the following: X. 5 / dQ V J 3a ' V dg 9 9' cos sin 9 9' cos , ' sin 10 + 3.1392 + .8181 4 - - 3 2.74 + .14 2 + .1509 - .3757 5 - -3 - .11 _..; 3-0 .0858 - .1738 2 - - 4 .VI + .4:5 I 1 .51 + .20 3 - - 4 - 1.54 .13 j i 25.39 - .39 4 - - 4 -1(1.74 .tu 2 1 + 2.33 + .73 5 - - 4 - 1.6* r .u.3 3- 1 .04 .22 6 - - 4 ' s 1 2 +41.934 + .090 3 - -5 .14 4.16 2 2 91.80 2.53 4 - 5 - .89 .OB 3 2 - 4.13 + .34 5 - -5 - 7.49 .50 4 .10 - .V2 6 - -5 - .96 + .02 1 - 3 20 4.9099 4 - - 6 - .07 +.05 2 3 - 2,1 5 - - 6 .48 .04 3 3 B.4S 6 - * r, 8.85 .2-7 A. P. S. VOL. XIX. U. A NEW METHOD OF DETERMINING Having the values of the coefficients of ( y + ig + i'g'\ both for IK and - ~ ^3 OOS t/' we have next to find those of (vy + ig + i'g'\ and of (Oy + if) + i'g) in the case c U of .. COS i The expressions for this purpose are (2) _ _ _ 1,$ _ 1 f 384 __ For Althaia we find log. >? (2) = 8.60309 log. >7 (8) =z 7.38308 log. >7 (0) = 9.08196/* We multiply the coefficients of (dh 7 + # + *'^') by >7 (2) , and >y (3) , respectively, to find those of ( 2/ -f ig + V), (=b 3/ + ^^ + *'y ). In case of (O/ + /(/ + '^') in the expression for -^-. we add the coefficients of OOS t (+ y + ig + i'g') to those of ( y + ig + i'g) and multiply the sum by y (0) . We will give a few examples to show the formation of W, and A - 2 d r With these two we give at once also their integrals, which are n$z and v respec- tively. W ' d Y (0 - 0) u cos sin sin cos - 1 !-<> 32.6972 +.0988 - 2 2 .0190 +.0017 32.7162 32. 7162 nt + 16.3486 +.0494 + .0190 +.0017 +.0511 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. !r _id^v 163 (1- // // - 1 2 - .474 + .042 1 +3.139 + .818 2_1_0 -1.314 - .004 1 1.2175/3* +3.2376/1* -0) + .2.37 + .021 -1.314 + .004 + 1.351 1.2l75ni + .856 +3.2376n -1.077 ,6087n + .025 1.6188n + 4.59 - 1.2175n 2.07 3.2376w 0.54 +.6087w 0.58 1.6188n (-1 12 1 + .383 + .070 _1 01 - .045 -1.516 - 2 1 1 - .041 - .030 1 1 .513 + .200 -i) +.191 .035 + .022 .758 +.041 .030 0.216 -1.246 + .16 - .92 +.254 .823 +.19 +.61 (i- - 2 3 1 .022 .004 - 1 2 1 4.263 + .038 01 1 - 25.390 - .390 1 1 - 83.900 - .973 -1) + .022 .004 + 2.131 +.019 -41.950 +.486 -113.574 1.329 174.61 +2.04 39.798 +.501 + 61.19 +0.77 . j TIT In the integration we apply the proper factor to each term of W, J -p-, and obtain the values of n&z, v, except in case of the terms (ig + og'). Let us take the term (g og') or (1 0), and let ^ the integrating factor to be applied. Let c, a, d, 5, represent the cos, sin, nt cos, nt sin terms respectively. 164 A NEW METHOD OF DETERMINING Thus we have c d a a + 1.351 1.2175n< a +.856 +3.2376n; and hence +1.351 +3.2376 1.2175w or, since ^ is unity, // // +4.59 -1.2175n< [id a .856 2.07 ' I' s 1.2175 3.2376. -3.2376/iJ In case of the term (2 0), ^ is g. In the way indicated we derive the values of nbz, and v. In the case of -- COS I we have the values at once without another integration as was necessary for nbz and v. In the value of W given above the arbitrary constants of integration have not been applied. We give these constants in the form cos y Then in case of A - sin y we have cos 2y -f sn etc. yc L sin y 5& 2 cos y + >7 (2) ^ sin 2/ - - >? (2) ^2 cos 2/ i etc. Having TF from the integration of --^-, we form W from the value of W and converting y into g. We thus have from the equation dz dt + (1".351 + Jd) cos g -|- (0".856 + Jc 2 ) sin ^ 1".2175^ cos ^ + 3".2376nt sin ^ + ( ".284 + >7 (2) ^0 cos 2g + (0".589 + >7 (2) & 2 ) sin - /7 .0488w?5 cos 2g + ".1298^^ sin 2g i etc. etc. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 165 In the second integration the constants of nbz and v are designated by C and N respectively, and the complete forms are C + Ictfit + Jci sin g & 2 cos g + i>? (2) &! sin 2g J>? (2) Jc 2 cos 2g i etc. _ZV - pi cos g - - p 2 sin g - - J>/ 2) Jc L cos 2? (2) k 2 sin 2# etc. In case of the latitude the constants of integration have the form Z + h sin g + L cos g. We thus find n* = C +[1 + Aso 32".7162]/i< + [4".59 + A;,] sin g + [ 2".07 AJ cos g - l".2175nt sin g 3".2376w^ cos g + [ 0".ll + J>7 (2) k^\ sin 2^r + [ 0".31 i>7 (2) ^] cos 2g 0''.0244^ sin 2cr Q".QQ4Qnt cos 27' 2) y cos 2g + [ ".24 iV 2) & 2 ] sin 2/ etc. =b etc. u , = 1 + 0".3616 + 0".3623n* + [1".52 + y sin ^ + [ 0".68 + ZJ cos gr 1".3464^ sin g - 3".OQ38n* cos g + 0".32 sin 2g - 0"16 cos 2g 0".0539nt sin 2g 0'M204^ cos 2g -4- etc. =b etc. COS I 166 A NEW METHOD OF DETERMINING 9' sin - -0 -f- 4.59 + a - 0.11 + - 1 + 3.10 - 2 3.00 - 3 + 0.23 - 1 -174.61 -2 +263.97 - 3 + 25.15 -4 + 5.71 - 5 + 1.64 - 6 + .49 2 + 185.18 - 4 1.10 -3 +410.16 - 1 5.25 -3 - 37.24 - 2 + 6.77 -4 + .90 -3 + .92 -5 + .17 -4 + .34 - 1 + .16 expressions for i V u ' j. 1 m v in ta hi ular form are the following : *>** * J.J 1 vCv M COS* % u V cos i cos COS sin sin COS +& nt n + ^ +Z + 0.36 // // 32.71 62w* + .0511n< + .3623n _ 2.07 & 2 U.o4 -^K l _ .58 _ i& 2 + L52 + Z, .68 + 7 It " // // 3.2376n + 0.6087?i - 1.6188rtf 1.3464n 3.0038n< // // n If g| L-fiWfe + .05 ^ w kj 9 J. 1 -yi(^) l/^ 1 24 ' 77 ?y tv + .32 .16 " ff H // .0649w + .0244n .0649n .0539w* .1204w - 3.09 + 2.12 - 1.54 - 4.83 - 2.03 + 1.92 1.30 .95 + 1.30 + .61 - 1.76 + .12 + .89 .37 + .25 + 2.04 + 61.19 + .77 + 2.69 + 1.26 - 7.21 156.21 - 4.24 - 1.15 .57 - 0.81 - 18.30 .56 - 1.60 .60 - 0.35 4.68 .29 + .03 + .02 - 0.11 1.45 .09 .05 .50 .04 + 2.10 - 43.27 + .07 - 6.64 2.70 .71 + .36 .01 .47 .17 87.44 + 14.64 + 3.15 + 4.43 + 1.73 + .87 + 4.02 + .62 1.98 + -9 + 8.03 + 16.07 + 3.78 38.24 14.92 + .04 7.08 .01 .52 + .20 .86 1.05 .70 + 1.31 + .50 + .04 .69 + .05 .24 + .03 .03 .33' .04 + .28 + .10 + .01 .38 .00 .92 + .19 + .61 , 1.62 .63 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 167 The constants of integration are now to be so determined as to make the pertur- bations zero for the Epoch. The following equations fulfill this condition : C sin g - & 2 cos g -f J>? (2) ^ sin 2g J>y (2) Jc 2 cos etc. cos y + c 2 sn g + cos# - - J& 2 sin ^ - - sin g - - po cos g + cos 2# + >? (2) & 2 sin 2# -f etc. cos 2y - - ^ (2) & 2 sin 2g etc. = sn cos etc. + (^) = + Zi sin ^ + Za cos g + >y (2) ^ sin 2g -{- >? (2) Zs cos 2g + etc. + (- A = VCOS 7-y Q - i 2 sin g cos 2 ^ I sin 2r -- etc. To find ^ and k 2 we have [cos g e + V 2) cos 2# + >/ (3) cos 3g + etc.] + k 2 [sin ^ + ^(^ sin 2g + etc.] - 3Z -f 6 ( v )o + 4- (^)o = [sin ^ + 2>?( 2 ) sin 2g + 3>? (3) sin 3^ + etc.] Jc 2 [cos g + 2^ cos 2^ + etc.] where = 32".7162, being found from li. ndt 6W. SJS \ We have also / "V VQ ~~ 662. i The symbols (n^) , (v) , etc., represent the values of n$z, v, etc., at the Epoch. 168 A NEW METHOD OF DETERMINING To find the values of the angles (ig + i'g) at the Epoch we have y = 332 48' 53".2 g' - 63 5 48 .6 The long period inequality, 5 Saturn 2 Jupiter, is included in -the value of y'. From these values of g and g' we find the various arguments of the perturbations. Then forming the sine and cosine for each argument, we multiply the sine and cosine coefficients of the perturbations by their appropriate sines and cosines. In forming ~ (nbz\ etc., we can make use of the integrating factors, multiply- itdv ing by the numbers in the column (i -f *"' - ). Having their differential coefficients we proceed as in the case of (?2&z), etc. We thus find ) = + 401".7, Wo = + 180".6, (-JL) = 22". G t V _,S = 391".6, -4r Wo = + 70".5, nd v x wrf^ v " n^ And from these we have k = + 412".8, Jc. 2 = 82" .9, 7c - --26".21, 1 = 0".0 I, = - 45".2, 1 2 = + 0".4, JV= + 28".3, (7 = 332 44' 12".6. The new mean motion is found from (1 32".7162 26". 21) nt, which gives n 855".5196. "With this value of n we find the only change is in the coefficients of the argument (1 3), having + 405".29 instead of 410".16, and 86".30 instead of 87".44. The constant G now has the value C ~ 332 44' 16".3. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 1G9 Introducing the values of the constants of integration into the expressions for nz 9 r, and , we have nz - 332 44' 16".3 + 855".5196 1 + 417" A sing + 80".8 cos# 3".2376/cos# + 16".4 sin 20 + 3".0 cos 2^ 0".0244 nt sin ( 2g - 0".0649 nt cos 2g db etc. etc. = + 28".3 + - 206". 9 cos cj + 40".9sin# 8' .2 cos 2g + cosi etc. i etc. 0".4 -i 0".3623n< 1".3464 TI^ sin ^ 3".0038 nt cos I".5sin20 0".2cos2<7 0".0539 nt sin 2r/ - 0".l 204 nt cos 2g From the expressions of the perturbations that have been given, and the elements used in computing the perturbations, except that we use C in place of g and the new value of the mean motion, we will compute a position of the body for the date 1894, Sept. 19, 10 h 48 m 52 s , for which we have an observed position. From a provisional ephemeris we have an approximate value of the distance; its logarithm is 0.14878. A. P. s. YOL. XTX. v. 170 A NEW METHOD OF DETERMINING Reducing the above date to Berlin Mean Time, and applying the aberration time, we have, for the observed date, 1894, Sept. 19, 72800, g - 339 19' 38".l, g' - 65 24'.1. Forming the arguments of the perturbations with these, we find n&z - + 4' 43".2, v - + 3".6, tt . = 2".8. cost To convert v into radius as unity and in parts of the logarithm of the radius vector we multiply by the modulus whose logarithm is 9.G3778, and divide by 206264".8. Thus we have from v + 3".6, the correction, + .000008, to be applied to the loga- rithm of the radius vector. In case of z= 2 '.8, we have cost i-- _7".19. Converting into radius as unity, we have z' = - - .000035. The coordinate z' is per- pendicular to the plane of the orbit. As we will use coordinates referred to the equator we have, to find the changes in a?, y, z, due to a variation of 0', which we have designated by &?', the following expressions : fix = (sin i sin &) &?' by ( sin i cos Q, cos e cos i sin e) $z' bz =. ( sin i cos & sin e -\- cos i cos e) &&' where e is the obliquity of the ecliptic. For 1894 we find fa = ( .0404) ^', 5y = ( .3123) &', & = (+.9491) &' And for the date we have &B = + .000001 % = + .000011 * = .000033 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 171 With i - 5 44' 4".6, Q zz 203 51' 51".5, e = 23 27' 10".8, we compute the auxiliary constants for the equator from the formulae cotg A = tg cos , ty E = ^ * , cos g3 COto- B C Si COS ( -I- e) fy Q COS " COS COto- (7 ____ cos * sin (E 9 + e) ' cos sin a = cos ^, sin I = sin ^ cos ^ sin (7 = ' sn sin B sin (7 The values of sin a, sin 6, sin c are always positive, and the angle E is always less than 180. As a check we have , . __ sin b sin c sin ((7 B) y ^ ~" sin a cos A We find A = 293 45' 29".3, B = 202 59' 46".9, C - 210 45' 55".0 log sin a 9.999645, log sin I 9.977735, log sin c = 9.498012 Applying n$z + 4' 43". 2 to the value of g, we have Tie = 339 24' 21".5 By means of g or nz = E -- e sin E we find E = 337 39' 23".4 Then from I sin J v \/a(l + e) sin J E * cos $ v = ttl e cos E 172 A NEW METHOD OF DETERMINING we find v - 335 50' 12".2, log i\ = 0.378246 where v is the true anomaly. Calling u the argument of the latitude we have u v + TI-- Q = 143 52' 41".8. Hence A + u = 77 38' ll".l, 23+u = 346 52 28".7, C+u = 354 38' 36".8. And from x =. r sin a sin (A + u) y = r sin & sin (13 -j- u) z =. r sin c sin ( C + w), where log r log ?-! + 5 log r = log r, + .000008, we have x + 2.331894, y = .515433, *==.-- .070208. The equatorial coordinates of the Sun for the date of the observation are X - 1.002563 Y - + .045198 Z - + .019611. Applying the corrections &c, ^//, ^, we have x -t-fa+X= + 1.329332, y + fy + F= .470224, + & + 2T = .050630. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 173 Then from 4 y + fy + y t \ z+ dz + Z . z -f 3z + Z tfl a =r J t g d rz - sin a = - COS a, x -f- 8x -f- J' i/ -f to -f y a; -f- dx + ^ sin 8 we have, giving also the observed place for the purpose of comparison, * tt c = 340 31' II" A & c = - - 2 3' 23". I log A rz 0.149514. a - 340 33 49.1 & = - - 2 2 25.4 where the subscript c designates the computed, and the subscript o the observed place. Both observed and computed places are already referred to the mean equinox of 1894.0. If the observed position were the apparent place we should have to reduce the computed also to apparent place by means of the formula) Aa = / + y sin (G + a) ty & AS = y cos (G + a), the quantities/", y, and G being taken from the ephemeris for the year and date. If the observed position has not been corrected for parallax we refer it to the cen- tre of the Earth by means of the formulae A _ TT p cos ' sin (a 0) _i (X ' . J cos 3 tg a>' tciy =: COS (a 0) _ TT p *sin