ster HARVARD UNIVERSITY. LIBRARY OF THE MUSEUM OF COMPARATIVE ZOOLOGY. We Mays. Toythons. See \le eos Whos sae. Pabecae: oh He ie i) iy Wy Aa i! ny Yee a ay TRANSACTIONS OF THE AMERICAN PHILOSOPHICAL SOCIETY Bie iD AV EAC bese TAs, FOR PROMOTING USEFOL KNOWLEDGE. VOL. XIX—NEW SERIES. PUBLISHED BY THE SOCIETY. Philadelphia: MACCALLA & COMPANY INC., PRINTERS. 1898. CO INTIT TEL INGIES Ova WO GOS PART I. ARTICLE I. A New Method of Determining the General Perturbations of the Minor Planets. By William McKnight Ritter, M.A. . ARTICLE II. An Essay on the Development of the Mouth Parts of Certain Insects. By John B. Smith, Sc.D. (With 3 plates) PART II. ARTICLE III. Some Experiments with the Saliva of the Gila Monster (Heloderma suspectum). By John Van Denburgh, Ph.D. . ARTICLE Ivy. Results of Recent Researches on the Evolution of the Stellar Systems. By T. J. J. See, A.M., Ph.D. (Berlin). (With 2 plates) ARTICLE V. On the Glossophaginz. By Harrison Allen, M.D. (With 10 plates) ARTICLE VI. The Skull and Teeth of Ectophylla alba. By Harrison Allen, M.D. (With 1 plate) PART III.. ARTICLE VII. The Osteology of Elotherium. By W. B. Scott. (With 2 plates) ARTICLE VIII. 4 Notes on the Canidve of the White River Oligocene. By W. B. Scott. (With 2 plates) ARTICLE IX. Contributions to a Revision of the North American Beavers, Otters and Fishers. By Samuel N. Rhoads. (With 5 plates) 199 237 267 417 Aes BME ate DEC .6 1996 Pho ee eA GE TONS DEbdLrS eta AMERICAN PHILOSOPHICAL SOCIETY, HELD AT PHILADELPHIA, FOR PROMOTING USEFUL KNOWLEDGE. VOLUME XIX —NEW SERIES. PAR bak Fi ArvicheE I.—A New Method of Determining the General Perturbations of the Minor Planets. By Wiliam McKnight Ritter, M.A. ARTICLE II.—An Hssay on the Development of the Mouth Paris of Certain Insects. By John B, Smith, Sc.D. Hhiladelphin: PUBLISHED BY THE SOCIETY, AND FOR SALE BY Tue American PuHinosopHicaLt Society, PHmapELPHIA N. TRUBNER & CO., 57 and 59 LUDGATE HILL, LONDON. 1896. DEC 16 1896 TRANSACTIONS OF THE AMERICAN PHILOSOPHICAL SOCIETY. ARTICLE 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 HANSEN 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 coérdinates, or determining the variations of the elements of the orbit, he 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 Auseinandersetzung einer zweckmiissigen Methode zur Berechnung der absoluten A. P. S—VOL. XIX. A 6 A NEW METHOD OF DETERMINING Stérungen der kleinen 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 Lacraner. HaANsEN himself seems to have felt the force of the objections to his method, since in a posthumous memoir published in 1875, entitled Ueber die Stérungen der grossen Planeten, insbesondere des Jupiters, his former positive views relative to the convergence of series, and the proper angles to be used in the argu- ments, are greatly modified. Hint, in his work, A New Theory of Jupiter and Saturn, forming Vol. IV of the Astronomical Papers of the American Ephemeris, has employed HAwnsEn’s method in a modified form. In this work the author has given formule 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 HAwsen’s original method made by that author himself, by Hit 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 formule, Hansen used Egeria whose eccentricity is comparatively small, being about ;4;. 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 Smrrnson1An InstituTIon entitled New Tables for Determin- ing the Values of the Coefficients in the Perturbative Function of Planetary Motion ; 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’, the square of the distance, the true anomaly is inyolved 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 IIT 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 5. Papo aw do al dr? aE -° iG in which -A, and B are factors easily determined. 8 A NEW METHOD OF DETERMINING From the value of IW 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 pe Oe = nf W .dt y being a particular form for g. The perturbation of the latitude is given by integrating the equation C being a factor found in the same manner that A and B were. It will be noticed that in finding the value of . dz two integrations are needed ; in finding the perturbation of the latitude only one is required. The arbitrary constants introduced by these integrations are so determined that the perturbations become zero for the epoch of the elements. In all the applications of the method of this paper to different planets the cireum- ference has been divided into sixteen parts, and the convergence of the different series is all that can be desired. In computing the perturbations of those of the minor planets whose eccentricities and inclinations are quite large, it may be necessary to divide the circumference into a larger number of parts. In exceptional cases, such as for Pallas, it may be necessary to divide the circumference into thirty-two part s. In the different chapters of this paper the writer has given all that he conceives necessary for a full understanding of all the processes as they are in turn applied And he thinks there is nothing in the method here presented to deter any one with fair mathematical equipment from obtaining a clear idea of the means by which astron- omers have been enabled to attain to their present knowledge of the motions of the heavenly bodies. The object always kept in mind has been to have at hand, in conve- nient form for reference and for application, the whole subject as it has been treated by HANSEN and others. Thus in connection with HANseEwn’s derivation of the function IV, to obtain clearer conceptions of some matters presented, the method of BRuNNow for obtaining the same function has also been given. In some stages of the work where the experience of the writer has shown the need of particular care the work is TITKE GENERAL PERTURBATIONS OF TILE MINOR PLANETS. 9 given with some detail. And while the writer is fully aware that here he may have exposed himself to criticism, it will suffice to state that he has not had in mind those competent of doing better, but rather the large class of persons that seems to have been deterred thus far, by imposing and formidable-looking formulz, from becoming acquainted with the means and methods of theoretical astronomy. In the present state of the science there is greatly needed a large body of computers and investiga- tors, so as to secure a fair degree of mastery over the constantly growing material. The numerical example presented with the theory for the purpose of illustrating the new method will be found to cover a large part of the treatise. The example is designed to make evident the main steps and stages of the work, especially where these are left in any obscurity by the formule themselves. As a rule, the formule are given immediately in connection with their application and not merely by reference. It has been the wish to make this part of the treatise helpful to all who desire to exercise themselves in this field, and especially to those who desire to equip themselves for performing similar work. The time required to determine the perturbations of a planet according to the method here given is believed to be very much less than that required by the unmodi- fied method of Hansen. Nearly all the time consumed in making the transforma- tions by his mode of proceeding is here saved. The coefficients b are much more quickly and readily found by making use of the tables prepared by RuNKLE, giving the values of these quantities. Doubtless experience will suggest still shorter pro- cesses than some of those here given and thus bring the subject within narrower limits in respect to the time required. If we compare the time demanded for the computa- tion of the perturbations of the first order, with respect to the mass, produced by Jupiter, with the time needed to correct the elements after a dozen or more oppositions of the planet, computing three theoretical positions for each opposition, it is believed there will not be much difference, if any, in favor of the latter. Again, when we wish to find only the perturbations of the first order, experience will show where many abridgments may safely be made. And whenever the positions of these bodies are made to depend upon those of comparison stars whose places are often not well determined, it will be found that the quality of the observed data does not justify refinements of calculation. One of the things most needed in the theory of the motions of the minor planets is a general analytical expression for the perturbing function which may be applicable to all these small bodies. Thus if we had given the value of aQ in terms of a periodic series, with literal coefficients and with the mean anomalies of the planets as the argu- 2X5 JES Sk A\VOlke IIDG 18. 10 A NEW METHOD OF DETERMINING dQ : es : ments, we would at once have a a by differentiation. And since G2 dQ i ahi? dw only two multiplications would be needed in finding the value of aan , whose expres- sion has been given above. In the present paper we have dealt only with the perturbations of the first order with respect to the mass. The method has been employed in determining those of the second order also for two of the minor planets ; but as those of Althsea, the planet em- ployed in our example, have not yet been found, it was thought best not to give any- thing on the subject of the perturbations of the second order, until the perturbations of this order, in case of this body, are known. The writer desires here to record his obligations to Prof. Edgar Frisby, of the U.S. Naval Observatory, Washington, D. C., and to Prof. George C. Comstock, Director of the Washburne Observatory, Madison, Wis., for kindly furnishing him with observations of planets that had not recently been observed; to Mr. Cleveland Keith, Assistant in the office of the American Ephemeris, for most valuable assistance in securing copies of observed places. And to Prof. Monroe B. Snyder, Director of the Central High School Observatory, Philadelphia, he is under special obligations for the interest manifested in the publication of this work, and for continued aid and most valuable suggestions in getting the work through the press. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 11 CHAPTER I. Development of the Reciprocal of the Distance Between the Planets and its Odd Powers in Periodic Series. The action of one body on another under the influence of the law of gravitation is measured by the mass divided by the square of the distance. If then A be the dis- tance between any two bodies, this distance varying from one instant to another, it 3 : : Wes : will be necessary to find a convenient expression for (5) in terms of the time. If r and 7’ be the radii-vectores of the two bodies, the accented letter always referring to the disturbing body, we have | N= 4+ r? — 2rr A. If we introduce the semi-major axes a, a’, which are constants, and their relation a’ S a=, we obtain = (3) + () @—2 (2) (2) (1) HT being the cosine of the angle formed by the radii-vectores. Let the origin of angles be taken at. the ascending node of the plane of the dis- turbed, on the plane of the disturbing, body. Let I, Il’, be the longitudes of the peri- helia measured from this point; also let f, 7’, be the true anomalies. The angle formed by the radii-vectores is (f’ + I’) —(f + Il); and the angles f + 0, f+ WW, being in different planes, we have H = cos (f + II) cos (f’ + Tl’) + cos Zsin (f ae iN Fsimy (G7 > UI), (2) I being the mutual inclination of the two planes. To find the values of I, Il’, J, let ® be the angular distance from the ascending node of the plane of the disturbed body on the fundamental plane to its ascending 12 A NEW METHOD OF DETERMINING node on the plane of the disturbing body. Let y be the angular distance from ascend- ing node of the plane of the disturbing body on the fundamental plane to the same point. If x, 7, are the longitudes of the perihelia, , 2’, the longitudes of the ascending nodes on the fundamental plane adopted, which is generally that of the ecliptic, we have . W=a—Q-—49, Woaw—Q—r. (3) The angles ®, y, 2 — 9’, are the sides of a spherical triangle, lying opposite the angles 7’, 180 — 2, J, - 2, v, being the inclination of disturbed and disturbing body on the fundamental plane. The angles J, ®, 7, are found from the equations sin $ sin $ () + ©) = sin $(Q — 2) sin $(¢ + 7) sin 3 J cos} () + ®) = cos § (Q — Q’) sin 4 (¢ — 7) (4) cos $ Jsin $ (Y — ®) = sin $ (Q — 2) cos $ (¢ 4 72) cos § Lcos$ (J) — ®) = cos $ (Q — &) cos § (¢ — 7) In using these equations when Q is less than 9’ we must take 4 (860° + Q — 9’) instead of $ (2 — Q’). We have a check on the values of £ ®, J, by using the equations given in Han- SEN’s posthumous memoir, p. 276. Thus we have cos p. sin q = sin 2. cos (Q — &’) COS p. COS Y = Gos U cos p. sin 7 = cos 2. sin (3 — %) cos p. cos rT = cos (83 — &’) sin p = sin?’ sin (8 — 8) \ 6) sin J sin ® = sin p [ sin J cos ® = cos p. sin (? — (2 — q) sin sin (} — r) = sin p .cos (2 — q) sin J cos (Wy — r) = sin (7 ) ( ) cos I = cos p. cos (@ THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 13 To develop the expression for (): we put cos J/.sin 1’ = sin A, sin I’ = & sin Kj, ) cos Il’ = k cos K, cos Icos Il’ = k, cos K,,J and hence i= cos f.cos f’.k cos (Il — K) + cos f. sin f’.k, sin (Il — K;) — sin f.cos f’.k sin (11 — A) + sin f. sin f’.k, cos (11 — 44). Introducing the eccentric anomaly «, we have a . A a . cos f = — (cos e—é), sin f = —. cos. sing, é being the eccentricity, and ¢ the angle of eccentricity ; and find ._. H= cos «.cos &.k cos (11 — K) — cos ¢. ek cos (11 — K) — cos e.¢k cos (11 — K) + eek cos (11 — KX) + cos «.sin ¢.cos ¢’.k, sin (II — A,) —sin ¢’.é. cos ¢’. k, sin (Il — A) — sin e.cos ¢.cos @.k sin (1— A) + sin e.é.cos p.k sin (1— XK) + sine. sin e’.cos ?. cos ? .k, cos (Il — K)). = re d ‘ A A\2 Substituting the value of ~, me fT in the expression for (=) we have (=). = 1+ a’*— 2e.cos « + & cos *« — 2aeek cos (Il — I) + 2a¢k cos (11 — K) cos ¢ — 2ae’ cos @. k sin (II — FV) sin ¢ — [2a7e — 2aek cos ((I—K) + 2ak cos (Il — K) cos « — 2a cos o.k sin (I] — XX) sin €] .cos — [ — 2ae cos 9’. k, sin (11 — K,) + 2a cos > cos 9’. k, cos (11 — Aj) sin ¢ + 2a cos ¢’.k, sin (11 — 4.) cos e] . sin & + a? é€?. cos 7’. Putting 71, Go, 72, for the coefficients of cos ¢’, sin +’, cos */, respectively, and 7, for the term not affected by cos «' or sin ¢’, we have the abbreviated form 9 er = 7) — 71. Cos &’ — By. sin & + yz. cos *e’. (7) 14 A NEW METHOD OF DETERMINING : : 4 In this expression for ce 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 2 . ° ) »% Y and 9, are functions of the eccentric anomaly be divided into a certain number of equal parts with respect to the mean anomaly, g. . : 2 2 360° 360° The various values of g will then be 0°, = Bs a6 5 8b ene —1. - : nm For each numerical value of g, the corresponding value of ¢ is found from g =e&—esine. Before substituting the numerical values of cos <¢, sins, for the n divisions of the cir- cumference, the expressions for 7, 71, 9, will be put in a form most convenient for computation. Let asin la) 20 g — 2ak cos (II — XK ) (3) p.cos P = 2a cos ¢' k, sin (II — K,), and HSjeom le |) See ) yi =f.cos F; J we find By =fsin f= 2a. cos p. cos 9’. k, cos (11 — K,). sine + pcos P. cos e — ep. cos P yi =f COS i= (24? —psin P). cose — 2x. cos p.ksin (Il — K). sine + ep.sin P. And from these equations we find, since f.sin (#— P) = f.sin F'cos P —f cos FP’. sin P J .cos(#— P) = feos #’.cos P + fsin F’. sin P, f.sin(#— P) = [2a. cos ¢?. cos 9’. k, cos (11 — Kj). cos P + 2a.cos ¢.k sin(I—). sin P]. sin e + [ _ 202% sin P| . COS E—EP f. cos (#’— P) = [2a. cos. cos 9’. k, cos (Il — K,).sin P / . — ° é — 2a.cos ~.k sin ((I—FK). cos P]. sine + 2a". .cos P. Gos é. THE GENERAL PERTURBATIONS OF TITLE MINOR PLANETS. 15 [f we now put vsin V= 2a.cos¢.ksin (II— KX) vcos V= 2a.cos.cos 9’. k, cos (Il — K,) wsin W = p— 2a’. nee weos W= v.cos(V— P) w,sin W,= v.sin( V— P) w, cos W,= 2a’. a cos P, | (10) J we get J.sin(#— P)=w.sin(e + W)—ep f.cos(f#— P) = w,. cos (e+ IV,). (QU) Further, if we put R=1+0°?—2a’.e, (12) we have Yo = R— 2e.cose + €’. cose + ey, or, ¥) = R—2e.cose+e.cos*s +e .feos F. (13) We find the value of y, from The constants, *, A, hk, Ay, p, P, w, W, w,, W,, 2, are found, once for all, from the equations given above. For every value of « we have the corresponding value of Jj and F from equations (11); hence, also the values of fsin #, fcos /, which are the values of @) and 7, Equation (13) furnishes the value of y) by substituting in it the various numerical values of <, as was done for 3, and y,. ‘The value of the coefficient : ‘ A\2 : y. being constant, we thus have given the values of (“) for as many points along a the circumference as there are divisions. 16 A NEW METHOD OF DETERMINING We can put Ay? : () = 70 — 71 Cos e’ — By. sine’ + 72. Cos .e’ a in the form (2) = [C—q. cos (¢ — Q)] [1—q- cos ( — @,)], (14) in which the factor 1 — q, . cos (e’ — @,) 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-qsin Q.sin Y, v1 =¢.cosQ+q,.Ceos Q, ¥2=7-M-cos(Q + Q) Po=q-snQ+tqu-.CsinQ, 0=sin(Q+ Q,). The last of these equations is satisfied by putting Q ==. The remaining equations then take the form Ne = mee ce > ee | | a Nn | B= (¢—a-C).sing | The expressions GS OQ) = by sb 2 1) q. cos OS ee (16) Gh. Csi C) = & | Gia C..cos O77) a pen the relations expressed by the second and fourth of equations (15‘, where — /Yo als Gi We haye now to find expressions for the small quantities £, 7, ¢ found in these equations, THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 17 Equations (16) give q-q- Csin’?Q= (8+ &).-£. The equation y= C—_.qnsin’Q then becomes (Yo + S)S = (Go + &)& (a) From (16) we haye, also, PO C= (nae Qe a> Waa from which, since y,=q.q, and C= y, + ¢, we obtain Oar 6) 9% = (Gnae BE Se (aaa (b) Equations (16) give again (yi —n)& = (Go + &) n. : (¢) When ¢ is known, & is found from (a) ; and the difference between (a) and (0) Got) 62—5) =O: 5%). (d) gives 7 when ¢ is known. The equations (a) and (c) give Bo +4 (yo + 6) 5 = (Bo + 28)" Bo+2E= 71-33 and hence Bi 4 t+ )S=ye.5 A. P. S.— VOL. XIX. C. 18 A NEW METHOD OF DETERMINING Deduce the values of 8) + &, y, — 7 from (a) and (d), substitute them in (¢), we find G — (oS eo Sra Ma eas The last equation then takes the form O= 77-5 — Bo (¥2—$) —4 (70 +S) (Y2—S).S- (¢) This equation furnishes the value of £; and with ¢ known, we find &, 7, from equations already given. The three equations giving the values of the quantities sought are Sar Soa) S se lA +P ban = B70 oo) ho Or y2=0) BP Boo = (yar S)G =) (Cr) n—y1.n+ (Yo+%) (¥2—4) =) Finding the values of ¢, £, 7, from these equations, and arranging with respect to y., preserving only the first power, we have Belt bea =- Bo: ye : a (9) jae te ty i = oom hae? ies Ae? Substituting these values in equations (16), they become q7.5n0Q = Ge ee Yo q.cosQ = y,— Fie ga an q Csin Q= ao (ait rr 2 gq. Coos Q= Esty, noting that C= y, + ¢. If more accurate values of ¢, &, 7, are needed than those given by equations (gq), we proceed as follows : Substitute the value of ¢ given by (g) in the second term of the first of equa- tions (/), we find, up to terms including y,’, Pome ReMi Coats » : _ fo: (bts By 18 GS= 7 2° V2 ap abe Ge aE APP” Yo — 4. Yo: ( ) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. The last two of (7) give also ea Gals Os = Bo les C eee Difeo = (Ove r= (x ae ates) Nn Nn Introducing the values of f, F, given by (11), putting L=72+4.y2. 7p. cos °F Ua=y—4.y2 A . sin °” we have G7 eine so that C=y+ %.sin*f. Moreover, since v2— 6 = x . cos, we find from the expressions for &, 7, given above, Boek 6 jane nein m—n=f.n'.cos Ff, if g=14+2—(4) a1 (4) Substituting these in the expressions for gsin Q, gcosQ, they become gsin Q=/f.&. sm gq cos Q = f. 7’. cos F. 19 (19) (21) 20 A NEW METHOD OF DETERMINING The value of g, is found from = 23 n= (23) The quantities g, g,, Q can be expressed in another manner. The equations (22) give ig Q= a .tg # 7 Ga fia SI Bet fi ely aCOS els from which we derive Q =F ao a sin 27 =— $ G = .sin 47’ -+ ete. ql SEG ' log. = log. f + 3 log. (&". sin *#’ + 7” cos °F). Since y and y” agree up to terms of the third order, the equations for & and 7’ give ef RON ac ie) Sp i Ve vie or, rar = ue io Me 3) ra = th a ae (2%. as ) cos 2” Further £? sin °F 4+ 7” cos*H=14 2 - (yz. sin °*L— 7’ cos °F’) — (4) and 4 log. (&° sin >2’+ x” cos “PY = Fs C (x sin’ “F— yx cos “F’) = “(x sin °L’—y’ cos “F’)°—4$ (A) Substituting the values of y, y’, C, given before, we find at i Yo 12 es 9 (Xx sin “H— 7’ cos °#’) = ey ae + S cos 2 fo 7 es Ce) cos Af THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 21 The equation y, = ¢.q; gives log. y. = log. g + log. q Putting looag = loon f= y, we have for q, log. g, = log. 7 — y. Writing s for the number of seconds in the radius, and 2) for the modulus of the common system of logarithms, we find Q=F+2 log. g =log. f + y (24) log. q: = log. i —y in which ga eee + Z,,) sin 27+ s (2 — 1.) sin 4F a p= Np i iy a dn) cos 277—A, Cue = +) cos 47 And for C' we haye from the first of (15) C=y7 +y72-sin >Q. (26) By means of the last three equations we are enabled to find the values of Q, 9; Gi, C, with the greatest accuracy. The equations (17), where not sufficiently approximate, will, nevertheless, furnish a good check on the values of these quantities. All the quantities in the expression for (e) are thus known; and substituting their values corresponding to the various values of g, we have the values of ie =) for the different pomts of the circumference. 22 A NEW METHOD OF DETERMINING Using the values of C, q, @, Q, just found, Hint, in his New Theory of Jupiter and Saturn, has given another expression for ) which we shall employ. To transform (2): = (C—q. cos (¢ — Q)) (L—m. cos (¢ + Q)) into the required form we put C= 1B = sin x, = sin 7% Gi a=t93%, b=tWan (27) _ seC 2%. 8eC2 % NORE Then a) = O[1—sin x . cos (¢ — @Q) | [1—sin y.. cos (e+ @Q) | _ C[ sec? by (1—sin x . cos (e’ — @)) | [sec? $41 (1 — sin x: . eos (¢’ + Q)) | Ce, sec” by see’ 3% _ Of 1+ ty? by — 2ty 4 cos (e’ — Q) | [1 + ty? by, —2lg yn cos (e’ + Q) | ta “sec” by sec’ 2Y, Substituting the values of a, 6, NV, we get (2)" = _W* [1 + @— 2a cos (¢— Q) | * [1+ — B cos’ + Q]? (28) We compute the values of a, b, NV, corresponding to the different values of g, and check by finding the sums of the odd and the even orders, which should be nearly the same. If we put [1 + a — 2a cos (¢ — Or = [4 6° + 6. cos 0+ 6 . cos 29 + 6°. cos 39 + ete. | [1 + 0 —2b cos (e + ®) | = [3 BO + B .cos(¢+ Q)+ B”. cos 2 (e+ Q) + ete. | where s = ”, = ¢ — Q, we are enabled to make use of coefficients already known. 9? THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. For 2. cos 0, write x + = and then we have [1 + a — 2a cos 6] ° — [1 4 @—a (e+ ye = [1—ae|~ [a—“]- Expanding we have 1G So ee ee a a’a? + ete online: Ss a ss) Ia? s stl s+2 @ s stl s+2 [1 al Sea aoa gee ee ae ae 3 s 19 (aes ! 5 +* a a ae aie Peep eae a (2 ae ee Sa eee] (w +‘) el ee Ge) ae! s stl st2\2s+3 s+4 (¢ Ve =) sees Be aa . Be + ete. wr s s 1 s+2 03 ei Size 2 Sap 2 S=- 8 6 ee ge a) Se a Ag + ete. | (2° +.) | i 5 But « + ~ = 2cos6, a? + = 2.c0s 20, a + ©, = 2.cos36, etc., 24. A NEW METHOD OF DETERMINING and hence s s+2 ,.,8 stl s4+2s5+3 4 Se SUES BY a os aia Se ne ae G i? 3 Lae 2 3 ee EE OP eg ae] 3 $ ep-bi g+9 s @ts ».,8 stil sts sta ., Vee youa? st1 242 gy | ee api Sbt Shs std (29 ath @4® gt s-L4 5-46 4 eo ee eS Se, Sr +? a + ete, | oO and generally O—® 8 Sarl @=Fe=1 ss 218 Shel Saree 9se ob 4 b = 2.5 eyo ; ee ae; Pe et te | Since s = —, we find from these expressions the values of the 6 coefticients for >? different values of 7. Runxk sz has tabulated the values of 6 in a paper published by the Sm1rrHsonran Institution. Thus the value of [1 + a? — 2a cos (’e—Q)]~2 is obtained with great facility. The value of [1 + 6° — 2b cos (¢ + Q)}2 is found in the same way. We now let - aa 2 5 IN a cos *Q ) (30) sO=21 WN. B®. sin2.1Q) And hence have CU TENGE. C= NP Be cos. Ce) = eo A oO), sin AAG) iain i 15°), COS 40) ei IN Jina) etc.= ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 25 Multiplying the series [4 6 + 6. cos 6 + b®. cos 20 + 6”. cos 36 + ete. ] by [5 B® + B® cos (e+: Q) + B®. cos 2(e + Q) + ete.], noting that 6 = Q@— ¢, and arranging the terms with respect to cos7#, sin 7, we find @'\ — i KO) AO 1 D) {O (“) = 130, 6 +B, 459.6 A 4 fb. © + (B© +b) cD + (6 +4 B®) c] cos 6 a |f + (6 — 6%) s® + (6 —}®) s@] sind + [62.6 + (6M + B®) c& + (6 + 6) c®] cos 26 (31) tall + (6 — B®) s+ (6 — 6%) 8] sin 26 + [6%. 6 + (6 + B%) c + (6 + B®) e®] cos 36 FE $24) 8-4 (HY —B) 9°7] sin 30 a= ete. ete. Now let i; cos K; — £9, 6 aL (G52 = Hem) ree) + (Gee) + Be) ce) ) (32) k,sin K; + OBEY) 5 4 GH) 5 J and we find (“) = k; [cos K;. cos 20 + sin K;. sin. 76] = k,cos (#0 — K,) = k;. cos («@ — ve’ — Ki). (33) Subtracting and adding the angle 2g, this becomes (5) = k,cos|¢(Q—g)—K + (ig—t’) | — k,cos [*(Q@—9) —K.| cos.¢(g—e’)—k;. sin | i(Q-9) — K;| sin.7(g —e’) (34) If we put (©) 9 a: A, . = 7 i, C08 [(O.=9) = all | 5 : (35) Ae esi (0) eel ] A. P. S.=— VOL. XIx. D. 26 A NEW METHOD OF DETERMINING n being the number of divisions, we find R (c) 8 GC) ree () = A,,.cost(g,—eé,) —A,,.8int(g,—€&’,) (36) If now, for the purpose of multiplying the series together, we put (c) (c) A,, => C,,.cosvg + >. Co (37) (s) (c) (s) A, =2>S,,.cosvg +> S,,.sinrvg we have G) = Ps C,, cos vg +> C,, sin vg] cos 7(g—e’)—[= S.. , cos vg+> e ‘sin vg | sint (g—e’) (38) Performing the operations indicated we get - e e (¢) (¢) e e (c) . . => cos (tg —te’).C;,, cosryg = =4C,,cos[(t+vr) g—te’ ]+35 4C,, cos [(t@—v) g—ae'] ° . (s) . (s) . . . (s) . e . => cos (4g —ze’).C;,, sinvg= =4EC,, sin[(¢+v)g—ce']—s>4 3 sin [(¢—v) g—“#’ | (c) —22 sin (tg —ite') S;, cosvg =—S> 1g. ‘sin [i i+v) g—ts' |— > 1s, sin [(¢—v) g—ve' ] e fe . (s) e (8) . . . e — sin (7g—ze’) S,, sinvg= ZS, cos[(¢+v) g—te’]— S518.,¢ cos [(¢—v) g—%’] Summing the terms we find (‘)'= SE1(0,, + S,.) ) cos | (¢=F v)g g—te |F4SS(C,48,) ) sin | (Fv) g—te' | (39) (c) From the formula of mechanical quadrature just given, we have C;,o, S;,o. when (c) (c) vy =0; but we know that they are $. C,,, § S,., as shown by their derivation. Thus (c) Jal = Le. ye Ci cos g + C., .cos 2g + ete. (c) (s) ea : = >C,, cos 1g + =C,, sin vg + 6, sin g + é. . sin 2g + ete. (s) (c) (¢) (ce) A,=4 8. + S,, cos g + S,. cos 29g + ete. | io) s = SS, » GOS VG + SS. , sin 7 + S,, sing + S. sin 2g + ete. j J I: Hence where vy = 0, each series is reduced to its first term. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 27 In the application of the very general formulz care must be taken to note the signification of the various terms employed. In case of 3] no Ki. . COS [2 (Q.— 9x) — K,,. | 2 : : si A, =" h,..sin [1 (Q—9.) — Kid, oe ° O=N0 n s n shows the number of divisions of the circumference; and we divide by ; in form- ing k,, to save division when forming the coefficients ¢,, S,. The index and multiple 7 shows the term in the series 1p +4 B cos (e’ — Q) +b. cos 2(¢ — Q) + b. cos 3(¢ — @) + ete. The double index 7, x shows the term of the series of La Place’s coefficients and the particular point in the circumference. The index » shows the general term of the series expressing the values of (¢) (s) ix) When we give to » values from » = 0, to the highest value of » needed in in) the approximation. 2 & z . In ~.&,., 0(Q. — 9.) — Ki for each value of 7, there are » values of each n yk) «9 ? quantity. © GO CO 2 © The next step is to express the n values of PA eAn As As) A>, ete, respec- tively in terms of a periodic series. And since these quantities are functions of the mean anomaly g, if we designate them generally by Y, of which the special values are i Cate a MVS Ma are ee arct NC, 25 > y we have Y = he, + c cos g + © cos 2g + ete. ) (40) +s, sing +s, sin 2g + ete. ) re, The values of ¢,, s,, in this series are found from the ~ special values or 3% 28 A NEW METHOD OF DETERMINING From (s) (e) A, ,or A, =$q+ 6, cosg + © cos 2g + ete. + s, sing + s, sin 29g + ete., abe (c) (s) and similarly, for every other value of x in A;,, A;,, we have a check on the values of C,, S, In each series. Thus if in case of sixteen divisions of the cireumference we take g = 22.°5 and find the value of the series, the sum of the terms must equal the © © value of A;,, -A;,, corresponding to g = 22.°5. And this check should be employed on each series, using that value of g that gives the most values of c, and s,. If 7 ; (Onn) ‘ extends to z= 9, we have ten separate checks for the values of A; ,, A;,., respectively. In the equation Y=3e + ¢,.cos g + ¢.cos 2g + c;,. cos 3g 4 ete. + s,.sing + s.. sin 2g + s;. sin 3g + etc, if the circumference is divided into twelve parts, each division is 30°. Then for the special values of Y we have ¥, = te + & + 6 = ¢ + ete. Y, = $q + 4. cos 30° + ¢,. cos 60° + ¢ cos 90° + ete. + s, sin 80°+ s, sin 60° +s, sin 90° + ete. Y, = q+ «.cos 60° + c,. cos 120° + c, cos 180° + ete. +s, sin 60° + s,. sin 120° + s, sin 180° + ete. Y,, = q+ ¢,.330° + ,.cos 800° + ©; cos 270° + ete. + s5,.330° -+s,.sin 300°+ 8, sin 270° + ete. In the same way we proceed for any other number of divisions of the cireum- ference. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 29 Now let Qe wy | @) =m ¥ (let SS ae Ge (GA) Ge Coe) ee eC) Se (6.W=¥%+% G)=%—Pa Then 3(@ + 2c) = (0.6)+ (2.8) + (4.10) HGE—Le)= Gl B.O)4- Gy) 3(@+ e%)= (0.6)—[ (2.8) + (4.10) | sin 30° 3(@— «)=[(1.7)+ (6.11) ] sin 30°— (3.9) K(S.+ sy) = ae 7)— (5.11) )| cos 30° B(Sss— S)= (2. 8)— (4. 10) | cos 30° 314+ G)= () +[( 2) — (4s) | sin 30° 3(— 65) =| (4) — G4) | cos 30° 6.¢ = ()—)+ Go) 3(s,+ 85) =|) + Gy) ] sin 30° + (8) 3(s:— 33) =| (2) + (tp) | cos 80° 6.8, = (%)—(G) + Gy): The values of these coefficients can be easily verified by finding the values of each one from the sum for all the different values of Y’ as given in the series for DE Atle) epi (eM ne eer When we divide the circumference into sixteen parts, each division is 22.°5. We find the values of ¥, Yi, Y.,.... Yis, as in the case of twelve divisions. To find the values of ¢, and s,, in the case of sixteen divisions, we put CO gare 8) Se (Loe) ie ase 285 () = 14 16 (Beal) = Y2+ Yio (Zr = ¥,— Vu (c= SE V5 Gy = Na oe 30 A NEW METHOD OF DETERMINING .4)=(0.8) +(4.12) @©.2)=(0.4)+4 2.6) =(,9) 26.8) €.3jS0.54 6,9) (2.6) = (2.10) + (6.14) G.0) = Goll) -4 (15): eS iS © Then A(c + 2. ¢) = (0.2) Ale) —2.¢) = (1.3) 4(e,-+¢) = (0.8)—(4.12) 4(ce,—¢) = {[(1.9)—(.18)]—[(8.11)— (7.15) | cos 45° A(s, +s) = §[(1.9)—(.13)|+[(8.11)—(7.15)]} cos 45° 4(s,—s,) =(2.10)— (6.14) 8.c,= (0.4) — (2.6) 8.s,=(1.5)— (8.7) Aleit) | = (4) G4) — Gs) cos 452 4(¢q,—c,) = L@ _ (5) | cos 22 .°5 + [GD — (4°5 )| cos 67 .°5 Ale +65) = (t)—| (Ps) — Gy) | eos 45° 4(e,—e,) =| (4)—(z5) | sin 22.°5—[ (,8,) — (5) | sin 67.°5 4(, +s) = L@ + (5) | sin 22.°5 + [a rae a sin 67 .°5 A(s,— 5) =|’) + Gip)| cos 45°=- G aA A(s, +s) =| (4) + Gs) ] cos 22.°5—| (8) + Gy) | cos 67° 5 A(s,—ss) =| (5) + Gi) | cos 45°— (G4). When the circumference is divided into twenty-four parts, each part is 15°. Let (0.12)=¥%+ Y¥, (0.6)=(0.12)4+ (6.18) (2)=(0.12)—(6.18) C= 4 ale) 1) E\=O1)=—@.) (2.14)= ¥,+ Yi, Ca) 2) OR Cae) (il By= 42, GansG. neoan 23) a y= Geil) — O23) Then Further, let Then THE GENERAL PERTURBATIONS OF THE MINOR PLANETS, Bl 6(q + 2.¢.) = (0.6) 4+ (2.8) + (4.10) 6(q—2.¢) = 1.7) 4+ (8.9) + (6.11) 6( + Go) =(@)+ [@) ws Gay sin 30° 6(a—eu) =[C) — Gi) ] 208 30° 6a+4¢) =(0.6)— [(2 .8)4 (4. 10) | sin 30° 6(4—e) =| (1.7) + (6.11) | sin 30°— (3.9) 6)s+sy) = @ + (4%) | sin 30° + (4) Gos) = @iee (347) | cos 30° G(s, + ss ) A eG cos 80° 6(s,—s,) = |(2)— Gir) | cos 30° 12.6 = (¢) —@) + Go) 12.5,= (CAG) a= (Ge) (; ts) = . os Vis Go=¥ — Yu aye Yn — ¥,, 6(6, + eu) = (as) +[G) — G0] c0s 80° + [G) — Gis) ] €08 60" 6(¢,—¢,) = LG) —G 1) | cos 15° 4 [()—- Gy] cos 45° + [ Ge — (75) | cos T5° 6(¢,+¢)= 6(¢;— ca ) = 6(¢, +6) = 6(¢;—¢, ) = 6(s, + s.) = 6(s,—s,) = 6(s; + 8) = 6(s;— 8s) ) = 6(s;-+s5,) = Gp=Ge)s (y) +Gs ae AOS: —| (3) ax (2) |— |G) — (#5) |? cos 45° (qr) ae )— G8 2) | cos 80° + Gs) — Gr] cos 60° (33,) — (Gis) | sin 75° [Gs + G4) ] sin 15° + (G8) + Ge) | sin 45° + [2p + (9) ] sin 80° + (Ge) + Gy) ] sin 60° + | Gs 1G) sim 14° — (3) — Gr) sin 45° 4 [ [ Gi) + G's) | sin 75° ap GS) §(G4) + a eee cos 45° ge) = GE) + G8 [ Gs) + G4) ] cos 15° —[ ( Gh) epi cos 45° + | (3) + (ix) | cos 75° 6(s; s— 8 )=|(2 ae (Gs $) | sin 30° —| (445) + (x) | sin 60° + (4%). 32 A NEW METHOD OF DETERMINING When the circumference is divided into thirty-two parts, each part is 11°. 25 Let (OH hte, @O8j)eGOihe(s2) @.64=0.8)£6. Ch s 42%, Cosa) 69.28) C8) =C.9 46. (2.18)=¥,+¥, (2.10) =(2.18)4+ (10.26) (2.6)=(2.10)4 (6. . : @.H)=C.id)+@. G5 3l)—= v8 Ya (7.15) 2 (0-23) G5.3l) 2) = 04 hee (1-3) = 4.5 )+G: OsCij (2) Gsi0.8j)—¢. OSG Ci25) O=a@.oj)—6. (q) = (2.10)—(6. Je) = (7.23) — (15.81) (2) = (8.11) —(7. Then 8 (e+ 2.6) = (0.2) + (1.3) 8 (¢o— 2.¢5) = (0.2) — (1.3) ( ( 8(e+¢y) =(%)4 [ (2) —( f;) | cos 45° 8 (¢,— 1) = [ (4) — (qs) | cos 22.°5 + [G4 — (,5;) | cos 67 5a 8(a+ Ce) = (4) 8(G—es) = [(a- (3) | cos 45° 8(%+¢) =(%)—| Gr) — Gp | cos 45° 8(¢—eC») = 4) —(5) | sin 22.° 5 — [Ga — (5s) | sin 67.°5 16.¢; = (0.4) — (2.6) 8(s,+sy4) = [@® + (75) | sin 22.°5 + [G+ + (;°5) | sin 67 .°5 8@—su) =(G)— (8) | cos 45° + (;4;) 8(s, + 82) =[(4) + () eos 45° 8(%— sp) = (%) 8 (s+ So) =[ (4) + Gs) | cos 22.°5—| (,3-) + (435) | cos 67 .° 5 8 (s.—S») =[ (as) — (a) | cos 45° — (74). THE GENERAL PERTURBATIONS OF THE Further, let And besides, let A B A B Ow (tr) — G4) = | (2s) — GA] [G@)— G5] =| Gs) — Gi) | = [Gn — Gi). =| Gs)— Ge] = G5) + |G) — Gb] =G;) — (Gy) — G)] | sin 11°.25 4 ‘| cos 22°.5 | cos 33°.75 = |G) + G2) | (35) = Yy— V4 Gip) = 35; (5) = ¥, cos 11°.25 + [ (5) MINOR PLANETS. cos 78°.75 rin) INO [ (35) — cos 22°.5 Gis) | (5) | sin 78°.75 (a) — (2) ] e08 67°.5 sin 22°.5 — [ Gs) = (oh | sin 67°.5 cos 33°.75 + [ +) — (4) cos 56°.25 sin 33°.75 cos 45° cos 45° [ G4) — G4) | sin 56°.25 [( tee) ae (5) | sin 78°.75 ] 08 11°.25 —[ (a's) + (e's) | cos 78°.75 |sin 22°.5 + [(%) + 44) | sin 67°.5 [ + 10) | cos 67°.5 — [ Gs) + G8) | sim 88°.75 + [Gi + (4) | sin 56°.25 ( or) ar (4 4 )| cos 56°.25 cos 45° + (8;) = [ (ts) + (42 | cos 45° — (ge A. P. 8.— VOL. XIX. E. 34 A NEW METHOD OF DETERMINING Then 8(¢,+¢5) =A” + A’ 8(¢4—«¢;) =A-+ A” 8 (¢; + 3) = BY’ + B’ 8 (c¢;— ¢3) = [A— A” + B+ B’] cos 45° 8(¢,+¢.)=B”—B 8(G— en) = [4— A” — (B+ B’) | cos 45° 8(¢,+¢)= A” —A’ 8(¢,—G® ) = B—B’ 8 (s, + 55) = C+ C” 8 (s:— 55) = CO" + 8 (83 + $3) = | D + Di —(C— Cc”) | cos 45° 8 (s3 — 83) = D’ + D” 8(s;+ $1) = [D+ D” + C—C"] cos 45° 8(6¢,— sn) = Di — Dp” 8(s; +s) = D—D’ 8(s,—s )=—CO'"+ C.. The expressions for the determination of the values of c, and s,, just given, are found in HansEn’s Auseinandersetzung, Band I, Seite 159-164. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 35 CHAPTER II. Derivation of the Expressions for Bussew’s Functions for the Transformation of Trigonometric Series. Ge 20 ° . ° -The value of (5) given thus far is found expressed in a series of terms the argu- ments of which have the eccentric anomaly of the disturbing body as one constituent. But as the mean anomaly of both bodies is to be employed, it will be necessary to make one transformation ; and the next step will be to develop the necessary formule for this purpose. HAwnseEv, in his work entitled Entwickelung des Products einer Potenz des Radius Vectors ct cet., has treated the subject of transforming from one anomaly into another very fully ; what is here given is based mainly on this work. Calling ¢ the Naperian base, and putting = GUE. | = Gua". we have yy’ = (cose + /—1 sin e) (cos e+ /—1 sine’); also yy” = (eoste+ f—I1 sine’) (cosv’ e+ Y—1 sin?’ é’) = cos Ge— 7 e') | f= sin Ge —7 &’'). Denoting the cosine and sine coefficients of the angles (¢e—7 «’) by (G75) and (7, 2’, s) respectively, the series F=>(41,c) cos (¢e—7 & )—=ES V—1 (4,7, 8) sin (¢e —7e’) (1) ean be put in the form FH=1353 §(47,c) -V—1 2,8) } yy”. (2) 36 A NEW METHOD OF DETERMINING In a similar manner we get = £55 G2) = I= (Gi®))o 2 (3) where aon We have now to find the relation between y and z. Let g = the mean anomaly, and ¢ = the eccentric anomaly. Then from = e—esineg, introducing »4/ —1, we get (i SS Qala aie eal Since B= ine ==, we find gV—l=eV—1—S(y—y"). Now from =a yao" we obtain GNM = ae. 2, THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. and = (y—y7) = log. (2 O—¥). Thus g V/—1 = log. z= log. Gc2 Gq us) and hence YC (Ce) From z=y.c 29-9"), we have PG te). and yf =e ve ZU) | Let 5 be denoted by 4; then C- 7) io Aa. y . ch " ye, and ex9—-Y7) = GY cH Bey”, But chy cr a(l—m.y + yp — Pe y + oy! Fete.) Ase) 2 ie ee (1+ ha.y7 ae ol te peel! + rear y+ ete.) oT (4) (6) (8) (9) 38 A NEW METHOD OF DETERMINING and % - e ye. )2 5 5373 5 Fs 4 Oo wr! = (L+ta.y +57 +53. Sy! + etc.) a 4.0. + ‘e e- 22 5 3 8 5 y4 )A Es ? (l—ia.y “tS Yor ae Ot eed ‘+ ete.) Performing the operations indicated, we have a ee Hea es NO te roe + pose Fete.) he2s [Poe Joe ge oa (4a — ra + pes — jogs ete. ) (y =4) (+ ty —aas + peas Fete) (ety hii ) (+ ee re re a= ete.) (y— 4) Cee) + hm jm ee? nis - als : (1 ant a 1.2.m-+1.m-+2 FP ete. )y tes a2 vA -é -1 OD (yy y= 29 2 2 =1— 77+ eee EEE: -++ TapaeE fame ete. 43)8 a5 75 ve (+ Di ra ) 293 — 12234 = ete.) (y =) Se ee (jee a) (toe OS) ey) + THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 39 . . ° e é = As we may write h in place of 7, we have, thus, also given the value of c’2(¥—Y") Now put chy sy—-y) (yy) = ae J_in oO" (<2) (—m) nz +00 (m) —— ae Dane Wc Then, from the preceding developments, we see that (—m) m (m) Sinn = ( ees 1) : Tin ) (m) m (m) Jin = ( eae 1) Sin p) (—m) (m) —hr = ha Again (—my +o (0) (= ae Sin - = Jin =F J_, 1) (=2) (=3) a np OOF spin oO) “ae dino? “se ies (1) (2) (3) + qe wine y She J 1+ ye + Sn. y + ete. Sea (Ne (0) eb) O. ©). aes) ls 0 7] = Jinan =F Sin 9 Yy =F Jin 0 OF + Jin 0 y° ain ete. Cas eo). oS) . ap Cin 00) = shiny a0h Sachin e0h a Ke O +a _(—m) ne 5 Comparing the values of =_, J_, .y~™ and c-"2—y") we have (-1) (1) h378 Aes hi ts In =In =lA— aay 1 eg erga cE ete, fory”, (1) (1) hz hope hin ; cat Tin = Tin = hrA— 72.2 ate 923. «12.92.37.4 == etc., for Y> ae ©) We nis neas ‘ Fin =In = ig E23 + pose + ete, fory”, (2) (2) A222 hye AEE A Jon =In = iz vost poe t ete, fory, ete. = ete. = etc. (10) (11) (12) (13) 40 A NEW METHOD OF DETERMINING (m) Tn -y™and chxY—y™)s we get the same expressions for y” and y~”. @ (2) We see from the values of J,,, J, , ete. found +o oo) Comparing the values of > above, that the term is (m) hinym him+2_jm-+-2 Aim+4 Jm+4 hr = Sn (2 2-2 SS ete. 1.2...m 1?.2..m.m--1 V.2?...m.m—+-1.m-+-2 eee We hin ri a ~ Im+1 + 1.2.m+1.m+2 a ete.) Further, we have I; e 4 (m) ; eg —c va(Y—y diy" Shs Oma and, by putting m= h—7, this becomes sa iy 0 yf! Let + 0 (h) a ae (Oh OP | ee Y= > aatlees neo} Multiplying the second of these equations by z~". dg, we obtain +o (2) yz" .dg=> FP, .dg. Integrating between the limits + 7 and — a, we have (7) 1 +7 PS y- 2%". dg Qa —T7 general (14) (15) (16) (17) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. From z= 0"! = cosg + /—1 sing, we have dz = (—sing + /—1.cosg) dg; also z/—1= /—1 cosg —sin g. Therefore dz=2 /—1 . dq, and (17) becomes In like manner we find e . (h) e (2) e Comparing this value of @, with that of P,, we obtain (h ©) (i) (i) ee io ll Leas at De Le or A. P. S.— VOL. XIX. F. 41 (18) (19) 42 A NEW METHOD OF DETERMINING Thus we have, between the mean and the eccentric anomaly, the relations (h—1) ; gl — wh. i y | ea ae (20) y= 7 Um 2 | Tn the application of these relations, since : (7) ; Dy ape SIPs» e aie the expression for /’ is changed from F=4333 §Gt/c) -v—1 Gi, 8s) yy into = ISS Ei,e) Vl @H,a)) oS Pee The other value of /” is F=433 $3 ((4,4,c)) —v—1((Gh,s)) y'. 2. A comparison of these two values gives Oo Ry ee) aaa a UO 9) ((4, 0) == 2) IPSs (47,6) = 2.5, Su (4, 7, €) (21) In transforming from the series indicated by (7, 7’, c) into that of ((z, /’, c)), it is evident that h’ is constant in each individual case, and 7 is the variable. Thus we find, beginning with 7’ = h’, y (WW) iil (—(Wt—=1) ) : (Gl! .e)) = Me shine (G50, @) Se = 3 din (2, h’—1, c) + ete. h h eel Gi—(-+-1)) — ial ite Siu @ (fe IL, Cc) + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 43 To transform from ((4, h, c)) into (2, 7, c) we have (—i') _(W=7) (2,0,@0) =O (2, lo c)) = Dhow KG h’,¢)). Here, 7 is the constant, and /’ the variable; and for the different values of /’, begin- ning with h’ = 7, we find (0) ((’—1) —7)) (40,6) =JSyv (4,0 6)) + Serax ((47—1,¢)) + ete. ((/+1)—v)) > Fussy ((%, v +-1, ¢)) + ete. The expression (mm jm nei aii re = weet (1 soe _— : + ete.) 1. .2..m 1.m-+1 1.2.m-+_1l.m--2 1.2.3.m+1.m—+2.m-+3 (m) enables us to find the value of -/,, for all values of m. A simpler method can be obtained in the following manner : Patting c's -Y) in the form (1) (-1) (2) aie) ges © (—2) JEP? = Y—S,.-¥! +5.c-Y tJ .-y + ete. — é e€ hy hy we have, for the differential coefficient relative to y, e ae nety—y) (1) (2) (1) - be (2) A he(lt+y) c's HJ, 2.+2.5,..yrete +S ..y?— 2S, ey ete. If we multiply the second member of the first equation by h{ (1+ y~), we have an expression equal to the second member of the second expression, and by comparing the two we find (22) 44 A NEW METHOD OF DETERMINING Let (m) Je [prada ean Ts yn (23) hey then (m) (m—1) = fOr hey pee From this general expression we find pe ca = aD hs ie (2) (1) (0) I pS k pa aah ni (24) hy hy hy = Go = Ee (m) : : g hs + sates From the values here given, since —Gey 1s put equal to p,, we have, by increas- hs s ing m by unity, (m-+1) hs ae may — Pm: Pri Putting a = Tm, equation (22) Com takes the form Pm » Pm Ste 1 = Tn = Pm: From this we find 1 ), = — Pm Tn— Pm+1 z 1 Tn — — 1 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. We also have 1 — (25) Pm = Tn — Pin+19 a form more convenient in the applications. m) ( The general expression for J, . 1s Coy (m) (0) Te =e Pi- Pa Ps--- Pos (26) 2 i where (0) oat, E I! Te =1—p+ pa pm + ete, (27) if we put l= Ad. From the expression oe (ty Puta A Ww) (GUs@) = Sizan (EO) => 7 dine (GH) it is evident that when h’ = 0, or when both 7? and #’ are zero, this expression cannot be employed. To find the values for these exceptional cases let us resume the equation When hf = 0 we have 46 A NEW METHOD OF DETERMINING The equation 1 a a =O) gives ee = ons (1+ y~) dy. (28) Hence @ l grt == i—1 € ae e i-2 ah we eee oe When p is a whole number +ny—1 ie or, ay = 0; J oony=1 except when » =1, when this integral is 27./—1. Hence it follows that When 7 = 0, we have Using the expression pe (2) « s > (=e) e 8, (EF) . : (G hie c)) = SJPag(G050) = an (6058) + ap (GU 12) (—t-+-1) se Jeg (2, v— it. C), we have ((0, 0, c)) =(OF0K0) eae 2n/ (0, 1, ¢) for the constant term, the double value of this term being employed. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. For h’ = 0, we have (GOe) = LO =w de) = 27 GE abe) (LO, 9) == (G08) = 20 GG 1h 5) yy Gal) (ZO,e) = @Oo— 27 Co) — 2) CS 10) ((2, 0, s)) = (2,0, s) — (2,1, s) — a’ (2,—1, s) ete. ete. In what precedes we have put and obtain g = the mean anomaly, é = the eccentric anomaly, c = the Naperian base, Bu = Clas a yf". ols ¥—-y » ee 2G ae yi = 2. oY y’), he(y—y) + : ‘ ewe where ¢2U—Y") ig expressed in a series, the general term of which is we hig! ney hr oe (1 rae ee ) ram l.m--1 1.2.m--1.m-+2 2.3.m--1.m--2.m-43 SE Ces) Gy) Thus | I hin hers h h mam B= 0 APOE (A — ee y lm+1 1.2.m-+-1.m-+2 1.2.3.m—--1.m+2.m+3 35 ON 10) We have also put and since Offre py +00 _(—m) eh Y ) — Dan ea . Oe (m) ca YY *) =s" my 00S (—m) (m) Jin — Sin ? AT 48 A NEW METHOD OF DETERMINING have found (m) gh — Thx 2 ima : OP (i=) =In -Y; if m=h—i. Again supposing ; +o (h) ‘ — 2 = O) . Of +o (é) y' = 2s JP, ° Zz we have found Thus we have (h—t) ee ; S=dhx oh (h—2) = Sn [ cos de + sin ve »/ =, fase RA y = h Jin °& (h-~) — sdhn [ cos hg + sin hg v= : Hquating real and imaginary terms, we have z 7 h=o (h—1) cos te = =. 2,» Um - cos hg, (29) h=n (h—) sin ve = 2 Ji, « Bin hg. h=-o THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 49 We notice that x) (-1) dE ) S = — 36, (0) 0 == alk For all other values of 7 (i) We a=)! If a large number of the -/ 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 (m) _— pmjm we na i he, 28 Ne aL Tem ) L.2m-lm+2 — 1.2.38.m2lm2m+3 = ete. |, fe Ge Fe= li-F Gta a a S| ie = ae pie oe ect | ) (30) mote D Peta oe be In these expressions we have written for @ its value 3¢. (m) Since h has all values from hk = +o to— o we find any value of J, by at- tributing proper values to h. From equations (29) we find the values of the functions cos Ze, sin ze, in terms of cos hg, sin hg, and the J functions just given; always noting that when 4 = 0, we have only for 7 = +1, — $e as the value of the function. We can employ equation (22) when only a few functions are needed, or as a check. A. P. §.— VOL. XIX. G. 50 A NEW METHOD OF DETERMINING It may be of value to have y’ in terms of z" and the J functions. ond of equations (20) we have (0) (a) (2) yPs=—AaAt+t+ J, .2 +43,.2 +43,.2 + ete. (2) oO . (Or == dh oF linn ain Oe: (0) @ (2) yt=—Aat J .et+4dy.2°% 4+ 4d, 2% + ete. (2) (3) (4) 2 — J,.zg —td,.2? —t4d,.2? —ete. - (1) (0) | iO . Op —2SJ,.2 +2d,.2 + 2J,.2 + ete. @ Oates eae —2J, .2*— 2S, . 2° — 2d, .27* — ete. ze Pace bee ee) Maa. ch He ye = — id, 27 + 3dn. 27 + 2S3,.27 + ete. (3) on oO — fd, .2 — gdn.27 —2J3,.2° — ete. Then from y' + y~ = 2 cos ve =o S2 Jl. si ve we find the values of cos «, sin ¢, cos 2¢, sin 2e, ete. In case of the sine, as for example when 7 = 1, we have y—y i =2/ —I1 sine; butinze—2z1=2/ —1 sing, From the sec- we have the same factor, 2 »/—1, in the second member of the equation. From r= a(1—e cos €) we find 7 (=) = 1 — 2e cos ¢ + & cos e: = 1+ 2e cos « + 3¢ cos *e + 4¢ cos *e + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 51 2 For (") we have Gy = 1+ de?— 2¢ cos e + de cos 2e a But d [(r* : de 9 . "\ = % 5 = 22 sin a (3) 2e sin e (1—e cos « ) ay sin ¢, and “ (0) Oa - (1) 8), (2) o>. am eel wh oh sing +3[ Jy + J, | sin 29+ 4 | Jn + J;, | sin 3g + ete. Multiplying by 2e.dg we have for the integral of i (“) ” 26 (0) 2) Qe (a) (3) j Qe (2) (4) = 0 J, th cos g——| Joy. + J, | COS A J; +3, |cos3g— ete. where c= 1 + 3e’. By means of (22) this becomes r\2 (1) (2) (3) (“) =1-+ $e— 4J, cos g —4J2 cos 2g — 4; cos 3g — ete. a In case of (om we have 3é . cos «= $e (1 + cos 2e), 4¢ cos *e = ¢' (3 cos ¢ + cos Be), be’. cos *e = 3e' (3 + 4c0s 2c + cos 4e), 6e?. cose = 38° (10 cos e + 5 cos 3e + cos 5e), Té° cos °e = x5e° (10 + 15 cos 2e + 6 cos 4e + etc.) and hence (“)°H1 +438 + A + We + ote. + [2e + 3e’ + $8 + ete.] cos « + [3e + 20¢! + 195¢° + ete.] cos 2e + [é + 2% + ete.] cos 3e + [Be + 42e'+ ete.] cos 4e 52 A NEW METHOD OF DETERMINING Attributing to 7 proper values in equation (29) we find the expressions for cos ¢, cos 2:, cos 3z, ete. We then multiply these expressions by their appropriate factors and thus have the value of ¢ Ne. Wer a +o _ (—2) @\= = | ae 19, | (=) = Bue FR; cos 1q- ic) (2) (—2) The following are the values of R; and FR, to terms of the seventh order of ¢. au 3 5 7 Lig SS Bee EO ag?” sb ae (2) 5 ‘ , fy = —te + 1e— Ae © a 1333 9 81 7 Li Bae be = ae x 4 6 Ry, = —te'+ se 25 op BOR A —— Oe) f 7 Ti; = — Pa 7e ap aes’ (2) ae R, — — soe > 2401 77 fi, = — geivaye re 1 4 5 6 Rh = j= = 1+ 64 344 Le + ete. V1—2 4 8 Gee 88 1 6595 | 2675¢7 — 38 >) 9 5 pf Ei, = 2e+ Fe + FR + Ze ane ly SO Be -p Be peice 103p4__ 88 76 1 == ga GF yarn? Oe 1097¢5__ 1662147 5 192 ie 12236 6 — 160 © jhe 472787 1 = “agg See HansEn’s Lundamenta nova, pp. 172, 173. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. ) (2) (—2 We add also the differential coefficients of ;, ; , relative to e. dk, 3 = v0é de IR, a —— 8e2 __ _5 ¢t 7 6 = = 2+ 36 — 3 + ane’ F ete. ae. 2 — D8 il) A = = Oa Be eS SE e dk, — 3 p2 5 pt i; 6 es Ae ee ee de dR, 4 = —2é + 4¢ Fete. de dR, ores 4 5 6 i — _13te 4 t8hte' Fete, IR. GLb, = D7 = —2le ete. de 40 S= dk, fp eats 168076 We = — x30406 = ete. etc. = ete. ip, = e+ dé + 436 + 1956’ é aR i — 2 4 25,0 = — 9, 4 e€ + 325 ¢ + ie é é Te, a = 43 |. 6399 Zo be + 46 +- $3¢ Rae — OY 2 5 4 F 6 = — 39 125¢! + 215 1¢ é ane in NS BS = +276 ne de 4() (—2) dR, — 5485¢t__ 11634768 Ae 4608 (—2) dR, — 36696 d —: 8 é (2) aR, — 830911,6 54 A NEW METHOD OF DETERMINING 2 The value of = found by integrating a(“) = 2e.sin e.dg, is 2 (1) 2) (3) r 5 1 + 3e — 4J, cos g — 4J,, cos 2g —4-J;, cos 3g — ete. 2 (2) In terms of the #,; functions, (2) 2 (2) (2) “=1+3€—R, cosg — R, cos 2g — R, cos 3g — ete. a 2 Again, since Oj __ @ apa oe we have 2 aes Sir V1l—e* dg Let +0 ee F=J + =, Crsin2g ; then - =1+ ae 7C; C08 219, and hence (—2) 1. C; ki; a V1 Gi The coefficients represented by C; designate the coefficients of the equation of the centre. * THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 5d Using the values of the C;, coefficients given by Lz Verrizr in the Annales de V Observatoire Impérial de Paris, Tome Premier, p. 203, we have f—g =[4G) —2G)?+ $6) + WG) + PE (H)"] sing “F >Gy— (Gp) ae ag) se a ap i | sin 2g ts Gy — 4G) + SG) — 24 G)+ ete. | sin3y Gece ce eG) eect, 9 || sini4y + [1982 (ey — pet Cy + tpn (4)! | sinbg qe: it G) === G) +, ete: | sin 6g =e [432328 (Gy SS eP | sin 7g + [2H? G) ] sin 89 r [4 Lee Ge) | sin 9g Converting the coefficients into seconds of arc, and writing the logarithms of the numbers, we have for the equation of the centre, I= 4 . 4 + Mi i cs ae 4 | 5.9164851 (5) —5.6154551 (5)° + 5.5362739 (5)° + 5.787506(5)' + 6.25067 (§)° |sin g [ 6.0133951 (4)? — 6.179726 (5)! + 6.067753 (4)° + 5.59571 (5)°| sin 2g | 6.252272 (5) — 6.6468636 (5)° + 6.690089 (+)’ — 6.22336 (¢)’| sin 3g | 6 5491111 ($)'\— 7.093540 (5)'+ 7.27643 (4)*| sin 4g [ 6.875105 (¢)'— 7.533150 (4)' + 7.82927 ($)°| sin 5g [7.225760 (4)'—7.96973 (4) | sin6g | 7.587638 ($)’ — 8.40484 (5)"] sin Tg [7.95944 (4)°] sin 8g | 8.38880 (5) | sin 9g 56 A NEW METHOD OF DETERMINING CHAPTER III. Development of the Perturbing Function and the Disturbing Forces. By means of the formule given in the preceding chapter, the functions pals), acre (4)"; 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 # 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’. If x, y, 2, be the rectangular codrdinate of a body, those of the disturbing body being expressed by the same letters with accents, the perturbing function is given in the form a=.) 2a] Now IS = (Wa) =e (=) a (C= 2) =p tr? Dr’. A; hence oO= [bas 2] ItmtL4 rl? If a © is regarded as expressed in seconds of are, and if we put SST nes oa CG). (eo 1m we have GQ = Me Gi— @): =] oO THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Finding the expression for (ZZ) first by the method of Hansen, we let A= = .k.cos (I—K), hi =". COS p. Cos 9’. k,.cos (II — K) 1=*,.cosp.k.sin (I—K), U=*,. cos 9’. k,. sin (11 — K)), and have, if we make use of the eccentric anomaly, a (17) = h.cos oe cos f’ —ch(“)*.cosf’ 1. sin €. ( i)? COSN i r u/ + 1.cos aa) pins ats op) )\.. Sle +h’. sine GC) _ sin cos ¢! 7 COS ¢ 7 cos ¢! Putting \ 2 (2) cos f’ = y’,.cos gy’ + y’,. cos 29’ + y’;.cos 3g’ + ete. a’\2 sin f’ ; : Fi , : , pees , (“) oa a= Une RING) ae 0’5. sin 2g’ + 4,’. sin 3g’ + ete. - ¢ we find (1) = $ (hy’, —'8,) cos (— g'—e) + 3(ly’, —U0',) sin (—g' —e) —ehy’, cos(— gs) + el’, sin(— g's) + ally’ +884) cos (gy — 2) + 3(/'1 +15) sin( g’—e) ; (1) + 2(hy', —h'8’,) cos (— 2g’—«) + 2(ly’.— U8’,) sin (—29' — «) —4.ehy’,cos(— 2g )+ 4.eld’, sin (—29’_ ) + A(hy’s + h's'2) cos ( 2g'—e) + Aly’. + U8’,)sin( 29’ —e) + ete. + ete., where (0) (2) (0) (2) — y 04 = Jy, nN 9 DP i IN? === CAN (1) (3) (1) (3) = 3] Sa » |» 2 = 3 Jn eae: » | ete. ete A. P. S.— VOL. XIX. H. 58 A NEW METHOD OF DETERMINING ‘When the numerical value of (#7) has been found from this equation we trans- form it into another in which both the angles involved are mean anomalies. For this purpose we compute the values of the -/ functions depending on the eccentricity, ¢, of the disturbed body just as has been done for the disturbing body. The values of the (0) (1) J functions can be checked by means of the values of J,,, J,,., given in ENGEL- MAN’S edition of the Abhandlungen von Friedrich Wilhelm Bessel, Erster Band, seite 103-109, or by equations (30),. Thus by means of the equation (m-+1) a (m—1) en (un) TN hr =< hd A (m) (0) (1) we are enabled to find -J,, if J), J), are known. It must be noted that the argument of BrssEL’s table is 2.5, or 2.hd, or he. Go) Thus if it is sought to find the value of -/,,, we enter the table with 2.2 or 2e as the argument. When we need the functions for h from h =—1toh=4, we must find the 4 a 4 HO 2 (—2) yalues of a se» ad es ates the een and — td. (1) The values of 3. J. bo (0) (3) and J, we take from the table. ‘To. find J. - we have oy z e oy oe (1) 2 Fe a ne 4.5) 4g (4) 2 of 1 7° Pe ae eae le 46 1 45 | (2) For J... we have as (2) (0) 1 (1) SI og SS td eae) et 385 385 3 85 (2) And for J, we have z (2) O 7 I, — J, ale z z mo THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 59 The expression for (JZ) can be put in a form in which both the angles are mean anomalies. ‘Thus, resuming the expression for (/7), (Ca Fi cosie Gr cos f’— ch Gr cos f’—1.sin « (yr .cos f' r 7 a\2 sin f’ a\2 sin f’ 6 a\2 sin f’ +U.coseé C) F su — a (“) a= £ +th’'.sine. @) a 7 cos ¢ r cos * cos g’’ in which a hk = —.k.cos (i—K) h' = E- . ' k TI TK ee | vcos V Vv = —. cos p. cos 9’. k.cos (I— Kj) = gu. p C 5 1 v sin V (=i COS) Pr. k.sin (11 —K) = fu. aie a - U eee “yaa , k 9 (11 Pabek K,) ee | p cos Ie = ae CUS Gp : ,- Sin 1) = 3u.—=, a : a’\2 a\? sin f” we find the expressions for (<) cos f’, (5) as as follows. We put as before ¢ @) ws" = Dar COs gy aie Y's cos 29’ + Y's GOs 3g + ete. '\2 sin f’ 5 ‘ : F BPS cu eae, (*) am = 8 sin g +0: sin 29’ + 0’, sin 8g’ + ete. c 0 7 , O If we differentiate (, cos f relative to q’ we have A d(G.cos 7’) = cos fl dr’ eis Tid sin f’ Cia sin /” dg’ a ala al” fala cos v? ; dr’ ae’ sin f’ df’ @® since —— cae —_— 7 + COS dg cos g dg ip and hence a? G cos f’) a! ees SS cose 60 Similarly, in the case of is (i) , But a cos f’ = cose’ — €, Hence Now From the values of cos é’ - COS ni Pe snin i/” ; COS @ We now A NEW METHOD OF DETERMINING r’ sin f’ pon. we have cos g a (G sin i Poesia if dg” \a' cos ¢ 7? cos ¢’ r sin f’ . and * f = sine’. a’ cos ¢ a G cos f” ) 7 @.cose Sk, a 12 cos T ee 12 9 dq” 2 : dg” 2 (7 sin f’ 1? (S =) _. @? Sin 7” __ &. sine dg” > 7? cos ¢ dg COS &’ sin ¢’ 2) Bec 0) (2) [woot oh assume Me jh + Ee) cos g’ + tlio = Te cos 2g’ + ete. i (1) OA. [ee = ee sing’ + | Av + “he | sin 29’ + etc. and sine’ we have "| cos 9’ + 2 eee abt cos 2g’ + 3 [ Ja — Say_| cos Bg’ + ete. | sin g’ + 2 lez + i | sin 29’ + 3 [wee ++ Toe | sin 3g’+ ete. (i—1) pte 1 i—1) a, x ae a Jin ik 0; =i ees Si ] 1 (v’—1) (7’+1) 1 (v/—1) fh +1) (/ i [. va —Siny ii ver | Jim +m i Comparing these expressions for y’;, 6’), with those found in the expression for sin fe “cos , given above, we see that the relation between them is 7” THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 61 The expressions for cos ¢, sine, are the same as those of cos ¢’, sine’, if we omit the accents. Hence if we perform the operations indicated in the expression for (47), we have ay (2) Tr a = 30” [hy yy + Wd; Wy] cos (41g — Vg!) — 30? [By Ely 0'v] sin(Lig—’g’) (2) é and v’ having all positive values. Attributing to 7 and 7 particular values, we find, noting that 6, = 0, and 0’, = 0’, (1) = 3 [heyy + W004 Jeos( g— 9')— 3 [byi t+ lydi] sin( g— 9’) + $3 [h.nyi—h'd,0, ]cos(—g— 9’) — [By — 7184] sin (—g— g’) + gh. cos ( — J)— flys, sin( — g’) + 2 [h. yy’, +h’. 8,82] cos( g—2q’) — 2 [h.by’.4+ U'y,8'2] sin( = g — 2g’) + 2 [h.yiy’2—h’. 30’. cos (— g — 29’) — 2 [1.d,7’.— V'y,0’.] sin (— g — 29’) + 2h.yuy's Os ( — 2q') — 20. yoo sin ( — 29’) + 3 [h.yy’3+ h’.60'3] cos( g—3q’) — $[0.by7/3+U.70'3] sin( g—38g’) + ete. — ete. + $ [h.yoy'1 + W’.6,0.] cos( 2g— 9’) — 4[bb.y. 40.7281] sin ( 29 — 7’) + 3 [he yoy + h'.630',] cos (— 29 — 9’ 3 [Ld —U.y20',] sin (— 2g — 9’) + ete. — ete. The numerical value of (#7) given by (1) must first be transformed into a series in which both the angles involved are mean anomalies before it can be compared with the value given by the equation just found. If we find the value of (H) from the preceding equation, it can be checked by means of the tables in BrssEv’s Werke. The expression for « (“) is known; and with the expression for (/7) just given, we obtain the value of Ce On" i (“) —(). The next step is to obtain expressions for the disturbing forces. 62 A NEW METHOD OF DETERMINING Let v the angle between the positive axis of X and the radius-vector measured in the plane of the disturbed body, here called the plane of X Y. The differential coeffi- cient of the perturbing function © relative to the ordinate Z perpendicular to this plane is found by differentiating © relative to z and afterwards putting z = 0. Thus from m J rr’ i (Oa Le |; ae cle 1tm AO = (Gaal Pe (Qa P se (ee) r - r? aoe ory’ #, we find dQ m [- Wy 4p &: | dv 1+ m 2) ky > ihe dQ mm [— 1 Gate A dr ~~ Vem E 4 ) Pl? mm 1 9 CB dQ = aoa [— = Ua =| dA dH dA dA Z == = Hf = A— = r—?H, == = — =, dv dv’ dr 2 dz 4 Hence dQ m’ 1 1 ery du 1+m Ls a wr id dQ m’ il m’ 7 TLidh € Raye (Asem = aller A yn GQ __ mm 1 1 a iapieg ae ois Ay j @ = tole ja |sin 2.7 sin (f’ + I’) where HY = sin (f +11) cos(f’ + TI’) — cos J cos (f+ UH) sin (f’ + I’), 2 = —r’.sin Jsin (f’ +541’). THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 63 As before the origin of angles here is at the ascending node of the plane of the dis- turbed body on the plane of the disturbing body, and the plane of reference is that of the disturbed body. E ; : dQ dQ If we differentiate the expressions for ip? - yp we find 1 OZ o @2 AQ ml Sa pO aT TNS dp le ~ Bl vr Hf) eo -=o) ipa nae 1=Em \ 2 Tage ° oP r a ee : = (7? — rr’ HZ) sin Tr’ sin (f’ + 1’) a = a ( 7 =) sin Z.r sin (f+ 1) x os =— ties : = (7° —rrH) sin 7.r sin (f+) + S ae —— =e : = .sin “J. rr’ sin (f+ I) sin (f’ + I’) + aon (4-3) cos L To eliminate H from some of these expressions we find from 2 nat Al 2. ) Y= + r?— err’ . A, that : dQ The expression for Uae then becomes ar dQ a rey? 1 r i | Oise 1m 248 2A te From the value of A’ we have, further, PoaTp Tl | eee 1 i = uae DE? 64 A NEW METHOD OF DETERMINING and hence C2 3 om (pi 1 : : 3 =o Bre SG ree seit ‘ f _/ ‘ 7 IV drdZ 2 1+m [ ae al eu a su (f 1 ) HQ nv i 1 : : dQ 7 ete ea sib [a — — | sin J.rsin TI — drdZ’ 4 iba AP | (f+ ) da’ dQ becomes the latter of which, by means of the expression for 77, RD Vo SS 1 | 2 . Sab ss Po = 2 aap 7 3x| Sm Ir sin (f + I) im I va Sin (PE TD) The expression for A’ also gives (P—rr Hy (ry 7S fe i OM Sr ah 44 2H 44? by means of which we find / 5 AQ _aQ m | 3(r?—r*)? r? 1 m r ee = eal eee oe le ip Ts 1m 44 1+m 7” If we put, for brevity, (T), = = sin [ ey sin (iF? si Il’) (ry = sin (2) (2) sin (FED (= z cos 2 (2) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 65 the expressions which have been given for the forces, together with the perturbing function, are =—po(s) 2 Zain Ge ery 9 3 sin r’ . 2 + $ua2(4) S22 7 sin (f’ +11’) ¢ oe | = Buai(“)’ 2 =. = sin *( f’ + Il’) — u(“) aa (77) =0e(G) eG sin (F + 1) dQ a\SF 2? 1. 75) sin ip . Call ZV Se A) is egy ee r ae Dee 4 a”? a2 a sin Gi oe Il) — Sua? (“)" pesintd = sin (f+) —(Z)’ aa (<,) =— Buai(*)”, eo: - “sin(f + Il’ )= sin( f+ 11) + ua (4 cos FT _ (dD 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 2 m’ py? 1 - pe = [=e — an Fz] dr 1+m 24° 24 r? A. P. S.— VOL. XIX. I. 66 A NEW METHOD OF DETERMINING we have where, as before, 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 CO SS (4) — fH and the forces dQ 4 Ee "|-3 ON . are, = eS) Lam oa] — eG) — qe — — pa2(“)" ae 2 sin (f+ Il’) + (1). 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 (> wal(G) » wai) we need expressions for the magnitudes r!\? i 7 git I 7 6 , , sti JT Rs a G)o gm a gin F 41), = 7 ain (f+ ID; (BD), (i, GBe, Gay”. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 67 ‘When these are known we multiply the function ua’ (ey by r\2 ean sinI 7. , , sinir. . Le —i. eal =. osin(f +1), eT ain (f + I), (a cosl . a’ ? a b) : 5 the function ua (=) by @ |r lr]? 3 sinlr’ . ; ar \ene Ig alae pale oa “sin (f+ WW) [7 — 2 a; aie J PR ay ae 3 sinl 7 7 il © = ope tp WO), 5 “sin( f+) [7 aa pe E stale a) em (Gg? Sone We will now find the expressions for (7), (7)’, (7)”, and for the various factors just given, that are the most convenient for numerical computation. We have (1) = “sin (2). sin (f +1’). Putting, for brevity, b = — *, cos ¢’sin J cos Il’ 6.) = /sing/esinplil 3 and noting that a/\? sin f’ (0) QF. @- LOA. () ae ms = Fe + J, ] sin g’ + ole = The sin 2g’ + ete. (2) cos ie eee Tet | cos J! ++ a Fl cos 29’ + ete. - 68 A NEW METHOD OF DETERMINING we have (0) (2) (0) (2) =o (+a lene a o Loe les (a) Or). qa) (3) +26 [ Jar oly | sin (— 29’) + 28’ | J, — J _| cos (— 29) (3) (2) (2) (4) (4) + 8B [ Tye + Sou | sin (— 8g’) + 8’ [ey — Fay | 08 (— 89’) + ete. + ete. The value of (Z)’ is found from (Ty = “sin (2). 7 sin (f + 0). From _=l— 2 as, a we find a’ = , \—3 6) = (= excos ea a Expanding, ANS oil pene ; (5) = Se + (de + 22e + ete.) cos g + (%e”-+ Ze* + etc.) cos 29’ + 43e" cos3q’ + 231e cos 49’ + ete.; which, for brevity, we write, r Gy = ~ + 2p; cos g’ + 2 p, cos 2g’ + 2 p; cos 3g’ + ete. But , ‘sinof (0) (2) Es () (3) ~ r= [EA +4, |sng+4 Ba + JS, ] sin 2g + etc. (2) (1) (3) 7a (0 3 “cos f = —e+ [ee pete cosg +4 Jn. — Jn | cos 2g + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 69 Putting — = . cos @ sin J cos II, = a sin J sin, © @ 0) (2) a= oh a oy) a, Se Oh (1) (8) (0) (2 ye eos | Beer Bs cer ete. GUC, we have (Ly = — 3 be. po + l. 9.6, sing +- L,. Po. 71 COSY +1.p,.d.smn( g— g') 4 h.pieyicos( g— yg’) —l.,.6,.sin(—g— 9’) +1.—:% cos(—g— g’) — 2lep, cos ( — g') (4) +l.p..6, sn( g—29') +h.—.y, cos( g—2g9’) —l.p,.6, sin(—g—29') + 1.2.7, cos (—g— 29’) — 2he.p. cos ( — 29’) + ete. + ete. For (7 )” we have the expression Ci) Econ te( Se Putting pope = cos J, and using the p; coefficients as for (Z)’, we have (Dy = © 4 1. pcos (—g’) + hy. pr c0s(—29') + ete. (5) To obtain an expression for the factor [G)— = a | it is only necessary to a have that for (Ne 70 A NEW METHOD OF DETERMINING In terms of the eccentric anomaly we have, at once, Tr 2 9 © (“) = 1—2ecose + € cose = 1-4 $6 — 2ecose + 3’ cos 2e. Substituting the values of cos «, and cos 2¢, we have r\2 5 a) Q) (3) (2) = 1+ 3¢—4J, cos g— 4-J., cos 2g — 4-J;, cos 3g — ete. a To find an expression for the factor 8 Lis z sin (f’ + Il’), for brevity, we let sin I sin J = . cos 9’ cos IT’, — .sin Il’, a a : 7’ sin f’ Y 5 and from the known expressions for a sae A cos jf’, we get gia JT Fo ; (0) (2) s (1) (3) : = (S00 (jae IN) = [ee + Sy, | ¢, sing’ + 4 [ey =5 Ja ¢, sin 29’ + ete. a a (2) (0) (a) (3) — $2G + ee —vJ, |e,cosg’ + le ae | c, cos 2g’ + ete. we In the same way, if : ae CG = Sin T cos @ cos il, e, = 2+ sin UH, we find Shall ~P _s : : _(0) (2) : a) 6) : — poly =p) = J, +J, |essing + $| J +2 | sin 2g + ete. a ~ 6 athe (0) (2) @ _6) (6) — $ee,-+ [a —J, |c¢,cosg+3 | ahs + Jo. | c,cos 2g + ete. By means of the expressions for the factors r\2 f in I yr’. % j in I - S 5 (Z)i, — ; 5 sin (f’ + Il’), Pelee : . sin(f + I), THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. just given, we can form those for Sie 1 = 4 La? 0) a a @ Gin de HF 5 ; rl? 1 2 7 oe gl (f +11) | 5 | 1 af2 A g mt | sin’ (f +1’) 3) simi 7 Tr? iL 72 A aD la ae | sin 7p ee iF oo ; . 3 oe Sl Pe) SG Slay) 72 A NEW METHOD OF DETERMINING CHAPTER TY. Derwation 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. HLANSEN’s expressions for the general perturbations are dW, 9 _ dz + ¥* |dt Megs = at + Jo + mo f | Wo + = ae Ae: dt dk, — if = dQ her’ sin (@ AF 7) C08 aM ah} 2 (0) G0) Se [cos (f —o) —1] ‘4 + 2h, ©. sin (f— 0) r(=). 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 iF 9 Xx = PUM) = =0 “1 +e 1+m)4=0 <_+#d+m)5=0 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 2 (uv —x ce qe2(y—y y ey ae 2! tee Eg 2x) Ee — a): Introducing these into the equations given above we have in the case of dis- turbed motion ix 2 NBEO tat Shae aoe =) aa t+ (1 4 m) —; = TV Hi a dy 2 ( yo > yefy—y _ y (1) ae + k#(1+m) = =m k ee ¥) a bs 1 9 / > Zz i) 9 z'—2z tf oi + kh (1+ m) 2m ae —— =) 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’” ete., can be included in the second members of these equations, since the action of all will be similar to that of m’. b) The second members can be put in more convenient form if we make use of the function (Ove nv’ (' kei eee) tee WA Pe Differentiating relative to « CO an ( 1 dd =) dx AB ale Fas” But since we have Gia x —x dz” Ame’ A. P. S.—VOL. XIX. J. 74 A NEW METHOD OF DETERMINING and hence In the same way we derive the partial differential coefficients with respect to y and 2. The equations (1) then become d2 dx “2 4 eA+m)2 =P +m) “4A +m)4 =e (1+ m) | (2) = + (1 +m), =k (1+m) = a Let X, Y, Z, be the disturbing forces represented by the second members of equations (2), R, the disturbing force in the direction of the disturbed radius-vector, S, the disturbing force, in the plane of the orbit, perpendicular to the disturbed radius-vector, and positive in the direction of the motion. If f be the angle between the line of apsides and the radius-vector, the angle be- tween this line and the direction of S will be 90° -+ f. We then have Ka—Ssnj, v= S cos j- In case of 2, we have R= xX ee, and for S, From these we find THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 75 If we wish to use polar coérdinates we have d° = Roos f — 8 sin f daz : 2 5 = = Rsin f + S cos f, From =P COS ih, TST sm, we find dx = dr cos f — rdf sin f dy = dr sin f + rdf cos f a= dr cos f—rd7f sin f — 2dr df sin f — rdf? cos f @y = drsin f + rd°fcos f+ 2dr df cos f— rdf? sin f From the expressions for dx and dy we find dy cos f.—dxsin f= r df dx cos f.+ dysin f = dr, and hence da 1 Ge de — ™ —-— .—~ sm — GO ae peg hs te Oey do 1 ag ae —= 2,60 == ii dy pa COR ae a BID from which we see that R=VA+m) 2 saPa+m)i@ If we multiply the expression for d’w by cos f, that of dy by sin f, and add, we obtain ax cos if a @y sin f — Gp df. 76 A NEW METHOD OF DETERMINING In a similar manner we find dy cos f —d@asin f =r df + 2dr df. Operating on equations (2) in the same way, we have S 3 COS f + 5 sin f + Eee = X.cosf+ Y.snf=F a cos f — sin f = Y.cosof—X sn f=S8S Comparing the two sets of equations, we have ai ref, odr af 3 1 ap 0 Sap ar =h (1+ m), (3) ir ay” ed+tm) — 7. a aa The second members of equations (1) and (2) are small, and in a first approxi- mation to the motion of m relative to the Sun, we can neglect them. The integration of equations (2) introduces six arbitrary constants; and the integration of equations (3) introduces four. These constants are the elements which determine the undis- turbed motion of m around the Sun. Having these elements, let a) the semi-major axis, nm the mean motion, go) the mean anomaly for the instant ¢ = 0, é the eccentricity, $) the angle of eccentricity, — nm, the angle between the axis of « and the perihelion, v, the angle between the axis of x and the radius-vector, jo the true anomaly, é the eccentric anomaly. These elements are constants, and give the position of the body for the epoch, or fort=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 x in place THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 17 of m. The former system may be regarded as the particular values’ which these elements have at the instant ¢ = 0. In Elliptic motion we have nt + go= e—esine r cos f = acose—ae rsin f = acos¢sine - Dan an? = k’ (1 + m) Now let mz 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 mz be designated by a superposed dash, and let the true disturbed value of r be given by the relation r= r (1+ 7). We have then ; Ne = E€—eQ sine r COS f = A COS E — AE r sin f = a cos oy sine Ve vi + 1 ayny = kh? (1+ m). We will now first give BkRuNNow’s method of finding expressions for the pertur- bation of the time, and of the radius vector. Neglecting the mass m, multiplying the first of equations (1) by y, the second by x, we have ye = {( Ya— Xy) d+ © C being the constant of integration. Introducing _ x cS a cos f = *, and sin f = = 78 A NEW METHOD OF DETERMINING into equations (2), neglecting the mass m, we find 2, Fe} f Ore k?. cos f 2Sey a th 7 @y 4. Wsinf _ y (@) dé! pe We have also = = age a) = la joe + reosf. os and hence ee = es or eS = ( (Ye — Xy) dt + C; and ae = f Sr.dt+ ©. In the undisturbed motion we have Gin —— rT). oe =k Py po. being the semi-parameter. Hence Po r= { Sr. dt + kv/ po ON Oe THH GENERAL PERTURBATIONS OF THE MINOR PLANETS. From these relations we derive ie i oe oe in {Sr . dt, and also es oss af Sr. dt VP ky/ po) 1/p If we eliminate + from equations (4), noting that » af pee ak a fe = ih Se elt di NOE pe rae oe we have — ee =2(|[¢= a EGP | at fy _ See =/((r= ue _ Sr | dt, neglecting the constants of integration. Since r =r (1+ v), we have also a=ax(l+»), y=y(1+>). The equations (7) then become = a da ksin f _ sin f eee ia hinge =f (X—=E. 8r)at — d ») YY. _ Beosf _ cos f : Dore let er ae =J (V+. Sr) at From the equations L = My COS E— AL, Y= Ay COS Hp SIN ky 79 (5) (6) (7) (8) 80 A NEW METHOD OF DETERMINING we have dx = —d sin «de dy = d COS >. COS ede. é Then since dg = ” de, ee, “dy, “oe he as dz hr? Vv. using the values of sin «, cos ¢, in terms of sin f, cos f, we find dx ae ke sin f dy __ cos fe, Ge = 1/ Det dz ~~ V Po And these give bsinf _ _ da V Bo VP sae VP k cos f __ dy VP. key vp ae yp VP — dy YR her _ (% Sr dt dz VP VPs Pp The equations (8) then become = ao da dz “V/ Do sin f 2 ~+ =[d+% ver] = == S42 Gaya ~—dy st Ge V Po — : cos fe, : Yue dz [ato ) dt eS eye) ae (9) MEN qo : 5 the constant — Ye being included in the integral. J pe will now voor equations (9), and for this purpose we multiply the first by 2 7, the second by oe “ , and noting that THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 81 we have == sen Lt {(x- sae Sr\dt+ ae ee jens aah) Sr dt (10) Now multiply the first of (9) by y, the second by «, putting for | Ps its value V2 given by (6), noting that we have sae nar Sr)dt ( 2 (11) + poe f (H+ Eee. sryac c Czas We can write Spe the form ; a ue v) = —( +7. 24¥.% lt dt We have na df dz df a aby) = eS Ss Se LS oe (EY) ae al CE GR GG ?s df 2 AS) oD df = pe eo COS Po, UN = Ay Nh. dt r Making use of these relations we find de di (C55) Pyne ale and for 5, given above we have Zz lz y @ = Al +0). 2 — Ve + Ve, A. P. S.—VOL. XIX. K. 82 A NEW METHOD OF DETERMINING The equation (11) is thus changed into a= 1— Jo f(1 +24 H) srar— 2! J (x! sryat dt ky Po key/ Do (12) Te COE os Sir ee WA 1a ele ) a apy V Po The equations (10) and (12) can be put in briefer form. Let Rex Bs VS s: Sass A b) (Bi 5 - se 1D Then a COSTAE (5 Lv Gye ae J X,dt += J Y, dt, (18) dz 1 ) Qy > cae ee UEN. Bio gy a (Pe eee 7 dee loa mA (+ ) Seat Ed tS || Y.d The values of «, y, found in these equations we get from cA + 5% 1 ce os x= a + —"(2—t) + 4.5 (e—t) + ete. dy . Wy, \2 CS) Y=Yot Fi ee t) +4. ae (g—t) + ete. ! df From the expressions for = ; a , we have also Az Zz cos f +e, __ ue — Ki 10% : Gay LW € ee de ic t)) UC: (15) sin f 1 fds, PI fgg ; Sa a oe CO ce 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 ¢ by a Greek Jetter. We thus obtain ( = 1 aa 9 at te A eS t) — reais fa + a =) Sr dt — A J (26, DG 2) ali we ee tea) thee (16) - =| (2S i ee dt ke y/ por dt dz ke 1/ Po © dz? These equations include terms of the second order with respect to the mass. If we put vs ae (ere ve pond ce Wee sre jr we get ngz = net + gy tm | W + +r dev] at f (17) v= N—43( [9% 4+ 0 be] at In equations (17) g is the mean anomaly for £=0; JV is the constant of inte- gration in the value of ». From the value of JV given above, we have IW 1 Pi 2 es ae i ae (Lae ‘) Sr — —=(X,.v— Y,.8). dt hy/ po VP hy/ po ° Now since X = cos foe — — gin Joa ~ ¢ ) 2 = oes d ; ar < if df 12 a dr Si 1 dQ 84 A NEW METHOD OF DETERMINING neglecting the common factor k (1 + m), we have dw 1 ( VPo\ de 2 do = Ry Se esa all oe) eer meee ak dt key Do dy ky/ Do dr eee y r sin f df ; 2 dQ. q Uae F 2 sin f dQ (cos f--e,) dQ | —— ae 2 ae | SBE ee (cos eo aa r ky/ po (7 sin nf a cos f) bor ki/ Po L p wie tks p df 5 And as v =psinoa, c= 0 COS, this becomes a = [ (—1-2 a a2 2 sino. cos f. “2 +2 .psinosin f. 2 dt ky/ po Vp? df df Say. + 20 cosa. sin f. ey 2p . Fy 0080 008, f + 2p Sine sin 7 aL yp a f ae. sl . 9 COS @ a) = yy (32) ie +208 (0) oP £04 9? oa (f— oie QZ cosa (jf gf = 2e,.” cosa =| p P df But 26,9 cos a. — 2p.” i= = oe “(6 p COS 0 — py) = ps8: also = 5 k= se THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 85 Hence since k? (1 + m) is included in X, Y, R, S, we have 5 is 20. hh 5 ds — = fh, [ze cos (f —o)—1+ tee (cos (f—o) — 1) | af 2 (18) + 2hop.sin (f—o da dr If we write h,. a cos *@ in place of & 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+ecosf ; cos *¢ also r cos? g, Geeele=e\costac hence ra _ 1+ecosf COs *¢, dae cos ch lemencosias Using f + ™— x in place of f, and developing, we get r.a __ r+reos f.e cos (y—m)+r sin f.e sin (y—7) Pity A, COS “Gy Let us put esin (y— 7%) = 1 C08 "hp, (19) € cos (y—™) = £ cos*q + 43 since €é = sing, we have C08 "p = Cos) (1 — 2ey E — cos *y &* — cos “hy 7”). 86 A NEW METHOD OF DETERMINING With this value of cos*?@,and r= a, cos “py — &7 cos J, we find Pr. __ dCos*e,—e,.71 cos f --r cos f (E cos *¢,--e)--r sin -4 Cos *¢, Pi My) COS *Qy __ a cos*g,--r cos f. cos*g, +r sin f.7 Cos *¢g, , Ms COS *¢g, (1—2e,E—cos *¢, & — cos *gy7 2 and hence ai saritdh, aiken 1+&—.cos f-+-7.— . sin mh a : 7. 1— 2e,—cos’¢,€’— cos’, From GO Ui, df dz a tb @a ° de” and df __ ky/p(.--m) ae r ? we have df a di. — %- » - COs >. In like manner we find lf 2 ; — n=". cos op. dz } We have therefore dz THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 87 n . ( “a 9 If we put |» =1-+ 34, substitute the values of ——, and cos *, we get ) VT. r D U0. TONG) él ance : ooo 2 (1—2¢6,&—cos *¢,5’— cos *¢97")? 0 (20) dz qa =U + 6) Further, in the case of », we have r 1lt+v= = =~ Then since OG 2 9 n Oi = Osos 5 = (1 + 5b), 0 and cos * 2 £2 Da £ = (1— 26, £ — cos oy &? — cos 7"), COS "gy we have : 1—2e,E—cos *¢,.’—COS “gy. G7) = Panne: 26 rare (1+ —cos f+ —sin f.7)* (1--6)8 CP Ay If we let a - iP ° Al = 2 eat +—.sinf.7, a My B= 1— 2¢,& — cos *@, &? — cos *y 7”, i GeauOy hy a BS ? we find diziares: Ai ') = Fea ae = (ae) fap (Lar?) = Fae 88 A NEW METHOD OF DETERMINING From the latter we have (a) =1-20 + FS + +8. a Hence Gy oS) Ca, Fie tae If we put Wee i <2 a mn 2. “6 S cos f + a. ne sin f, we have & 14 W+h(e). dt lnc We have yet to express |, in terms of the elements. From ie SO 5 ‘ 9 cos ? Sil 3Q2 — 24 cosy. sy SS COS 79, and from iy Oe ie GB” Vi 1) S—= 5 No we have h = n y COS ¥o a “cos ¢ ’ (21) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 89 or kh an cos % ly COO Cie, If we put he Y cos @,’ we have h re an ~ ¢os g These values of h and h, being substituted in the expressions for WW, - is found expressed in terms of the elements and of », in a very simple form. To find the rela- : dz 0 tion between a and v, we use the equation 2. B ° (1 + v) —— A*(1-+0)! Z e e h and as this is also equal to eee “dt we find ie tk 1 dt sh (A+)? (22) For the purpose of keeping the formulz simple and compact, HANSEN makes use of the device of designating the time, and the functions of the time other than the elements, by different letters. Thus for é, r, «, f, 2, v, & Y, we write, T, 0, Ny @, 6, B, & v, respectively. Whenever we integrate, these new symbols are to be treated as constants, noting that the original symbols are used after integration. Ao TE: SATO, SAD, Th. 90 A NEW METHOD OF DETERMINING If in equation (21) we introduce 7 instead of ¢ we shall have dg cae j h, B 2 Sait +5), (23) where ro h hy h P = h rn ee Ul Shs — 7 sb D8 = ots m ob 2 yo ES SN. ; h, h hy ay h, Qy We have also dé at: h, (24) de ~ nh(l+p)?” The codrdinates 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 coérdinates arising from variable elements must, evidently, be put equal to zero. Coérdinates 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 codrdinates. If Z bea function of ideal codrdinates, 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 tr. The function dependent on ¢, 7, and the elements we designate by A. Then ay da Gig — Ghe 2 or dL = (--\at where the superposed dash shows that after differentiation 7 is to be changed into ¢. Let us write the equation (24) in the form + BY = 7 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 91 Differentiating relative to 7, we have at GH dv’ ; dt The differentiation of (23) also relative to t gives CGO WENAG d@ d& dt hy 2B dp aly ib” (1-Ep)8 dz : Eliminating = by means of (24), we have ae de dW 28 dg Lr ee de 1+8° dr* dt ; Substituting in the expression for ae we have LT dp dw dt ” Ge Since » is an ideal codrdinate, we get from this es 8 if (© at, (25) IV being the constant of integration, and the dash having the same signification as before. This expression for » is a transformation of that given in the equation 1 — 2e,€ — cos *g,.€’— cos *¢y.7? ITtyv= G@LEpyi@e mera coseen a “im 7) ay Since z is also an ideal coordinate, we have from (23) es is ea raj = rob + G+ mf 1 W +5) \ dt (26) g being the constant of integration and being the mean anomaly for ¢= 0. 92 A NEW METHOD OF DETERMINING When we consider only terms of the first order with respect to the disturbing force, ¢ changes into 7, and we have rie Una es tf W, dt ae (27) J yo N—}h (=) dt where Wy = 27 — #1427 £2 cos +27 .n./ sino, (28) and p and are functions of +, being found from Mt +G = xy7—EASiINY p COS @ = A COS Y — A & psIN@® = a COs sin 7. Also in the last two terms of W,, . is put equal to unity. ‘When terms of the order of the square and higher powers of the disturbing force are considered, ¢ cannot be changed into t. In this case let Nyt = MT + J + ndz. Likewise let M6 = MT + Jo + 206 where no¢ is a function of 7 and t. According to Taylor’s theorem we have W = Wi+ 2 064 4S 80 + ote. the value of W, being given by (28). THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 93 We then have GUY GN WwW, 1 dW, . dg a 3 dz: 206 SP Sa ie 06 + ete. Retaining only terms of the second order, the equations (25) and (26), replacing 6¢ by dz, give MZ = MtE+ 9+ to f [Wot . de + | dt yaaa f(s 2M sda The equation (26) has been put in simpler form by Hix. For this purpose from (21) and (22) we have (29) Mt) ere=$—a4m Hence Developing the second member and adding JW, we have M2 = Mt + Go + NH f Se dt. (30) The next step is to express sale and = in terms of the disturbing force. From (19) we find i) TY | ~~ e082, C08 (% = 7%) — COS", 4 = ——.sin (y— m). 94 A NEW METHOD OF DETERMINING Using these values of £ and z, and pcos @ = 4,cos*$)— p, in equation (28), we find 20 20 h W, = —" — -heeos (y —m—o) + —*+_ -h— 2 — 1. ” Taga, COS 7, G=a=es hqa, COS*¢, h Since k= an __ kj/ltm cos ¢ VP ? we have from the expression of / already given, Fi+m) (amar a re di By means of h _ Gar ~ cose ” we may transform the expressions du a = —.. G08 di ® dr an . —— = é sin dt cos ¢ I; into r. —h=cos (f —o) .he cos (y — 2 —) + sin(f—o) .hesin (y —™— @) + =sin (f —o). he cos (y —2%— 0) — cos (f —@) .he sin (y —™—«@) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 95 Multiplying the first of these equations by cos (f— a), the second by sin (f —o), and adding the results, we have he cos (y%—7™%—o) = (re — —h) cos (f—0) +2 sin (f—a). Substituting this value of h.¢.cos (y—7,—«) in the expression for W,, noting that 1 pe hyd, Cos*g, — #(1-+m)? we have 7 — _2hop _ pl. 2h,.p fey OT, y= Fim) Reon a) TT 7 + Baum .sin (f- 0) (eas (Ft) ST cay? J, COS? go Differentiating relative to the time ¢ alone, 7 remaining constant, and haying care that all the terms of the expressions be homogeneous, we have dW, 2hop dv 2hp a&r = —o)r ‘ ae aE ae (f—2) = + Batm in (f—o).—, a re ++ dh = cos’ ¢ 5, le aS Gh a) 14 SFR and dh 04m) @e PF de aan a) dae OED Y ha — dt Substituting 96 A NEW METHOD OF DETERMINING. we have d Wy — f 2h’ rr dQ Te = hy {2 cos (f—0) —1 + , [eos (Fa) —1]} (#2) (30) + 2h sin (f— 0) » (47) : P dW. See : This expression for 7, is the one used by Hansen in his Awseinandersetzung. 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 Ss | 22 cos Go) 1+ a [cos (f —o)—1] ew VA 1—é? an Po 7k e dQ te ass oa (f o) (4) From the relation p = al —e’)—ep cosa we have Pp von ep COS w a(1—e*) a(1—e*) © . : dw 5 : — An inspection of the value of “q_ Shows that its expression consists of three parts, one independent of 7, the other two multiplied by p cos o, and p sin a, re- spectively. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Put dW _ d& dY = dt 5 ole ( cos @ + je) oe Binkon a I dt a de a = [ cos f e cos f 1 | ( =) aesinf ( ( n.dt V1—e r a 12? =P ie df aR = of ME 5 4 | (ee a. (cos f--e) le a r( oe Vie r ae df neues DE BX DEG | eae a sin f ie = a cos f r ( jock; — /l—é 7p =e df Z : But 4 2 = s 5 a 1 a ay eee cos f ¢ eae ! =a dg i ry 1 (l—e')3 (ee dr _ ae sin f i a dia (@ 1 ) aE : ae ioe ) sin Fs Ghp ao ee cos f; hence ne dQ ndt 3a ee )s Ce 2 f dQ 1 dQ oe Ge) ye ndt é dg V ie" df oo Wale LE de] Again from Gp =a) Gal A. P. 8.— VOL. XIX. M. (a) (i) dr dg 97 98 A NEW METHOD OF DETERMINING we have (2) ay (7°) a =n (2) resin f df) Nil) Me dr/ a(1—e?) © Eliminating = from the expression for — , we have na dV 2 | a (1—e?)— 7? 7d r sin f dQ = ala) ae) jel; ~~ ae dg a/1l—eé dr In the same way we find CP 2 P 2 sin a cos f e sin ?f mile | i sys Wee ()— r vI—e+ V1= resin*f = a (1—e’)? we ) But if we employ the relation as r recos f — ad) * aie) f /1—e’, of the preceding expression, the whole term becomes r cos f e re dQ alle (1—e? yo ‘1—e? a a ail ae ) . @ COS in the term, Using the equation 0=—recosf—r+a(1—e), multiplying by 99 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS adding to the preceding, it becomes [ r cose a 2e | (“) =— Oe © a y/1—e Wg Ne dr Further, we have cos f e sin?f resin ?f m= Vl1—é a (1—e?)3 Gd TP .. r’ sin == || = Sun =-cosf /1—@ dg Lis eas a (1— re lak Oy Ay ae Reducing this expression in the same manner as employed before, it becomes 2rcos f+ sae 7? sin dg _sin Ha aia l= ale Multiply this by dg, the last expression for Fae becomes recos f+2 ae dQ = (7) ar ( a 1/ 1—e* ar] ” d¥ 2 2rcos f+ 3ae = ——— g a 1—e?’ Jj ay/1—e* J dg the integral to be so taken that it vanishes at the same time with g dY dv > ndt? ndt’ Substituting these values of <.— dW dz dY¥ /p j : ndt — ndt a ndt e GUS @ a e) +° ndt a sin 2 this expression can be made to take the simple form 7) + Bar (|) dw = adele in which ; fe 2psinw 297 * f ( cos f Qa a’ (1—e’) —1 a y/ l—e? : .4 (2% c08 o + 3¢) ae r sin f 2p sino (F cos +2¢) bo 1 2p J G cos @ + 30) ———— —— a a / 1—e* a7 \—e* (31) + 3e) dg } 100 A NEW METHOD OF DETERMINING Since @). 7 9 rsin f (POdg — @@/l= Ee” a 7 = — 2- cos a e.de J we have (oe pal Be To anal ee - San ley ae ; 8e | oP These expressions for A and B can be much simplified. Thus from "=1+}¢—(c—} 2) csg = G2 @) eran — 3 é cos 3y—“ cos 4g— eles, 2 2 tnt . p and a similar expression for —, we get - a” ad. ae.dy =(2—{) a ie d. a .de — 3¢e = —(2— 2) cosy, a = @—) sin g + (e— )sindy +26 sin 3g + 3€ din dg 4 ein, f [ 58 | dg = —(2—?e’) sing— (5 — °) sin 2g —— sin 3g — ; en 4g —ete., ae be— (2—7) cos g — (6) cos 27 —* cos 8y —“ cos 4y, —4e= —e —(2— 2 €) cos g— (e—3¢) cos 2g — 26 cos 3g — 2’ cos 4g. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. From which we obtain A =—3 +(4-+ 26’) cos (y — g) +(e+ -) cos (y — 29) —(5e + 2 WS e +5 3) 25e° ; ) cos y cos (y — 3q) cos (y — 49) +o +20) B=—(24+ &)sin(y—g) | —(e + “) sin (y—2q) 7 3 —(e + =) sin y —* sin (y—39) ( = ze sin (y —4q) eae aE 5, Sin (y + 29) 101 (32) 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 + 2, with very little labor. é dW : Let us resume the expression for a , that is, dWiee ndt ~~ ‘3 a * Since - can be put in the form a” dQ dg Aa( A and B having the values given before. dr Yr —~ = >R™ cosk a w q, — — >" Rin kq, é Tr Ose 2— cos f _ ) + Bar Se) CP d Ri) de de cos kq, 102 A NEW METHOD OF DETERMINING and SY (e ) Bef da = Sie sin tg +8 9 — Beg: But since m= 1 + ge — (2e— fe’ + sige?) cos g — (he — ge! + aige*) cos 2g a? — (4é — e’) cos 3g — ete. we have Hence the integral just given is simply ae sin kg. A and B can then be written 1 a(1—e’) —r? Q2psinw dR) , A=—3 al (26 oso 3e ) 5 — eas - +P Te aoe ie oe kg | 2 psi dR) a p OF, hoe Fh Attlee 9 | B= 5 [ (22 cosa + 8¢) Sk R sin kg ofle\ cos kg e Putting 2 {= > R™ cos x y, we have likewise Oe = Te a oe 2° sino = v 1a Ss 2 RO Sinz —COS @ Les — aR aR cos XV 5 7 oo = edy —e — 7 Y: Introducing these values of 2 ” cos o, and 2° sin @ into the expressions for A and BL, after integration relative to y we can write W in the form VSS oo ae) (xy 4+ Bt ) ; THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. where Bt = ty + ey, Uand V being two functions depending alone on ¢. Putting x = + 1, and —1, we have (1) (1) Gs ee ye ey de d RY R® 1, (1) = ae ei: Qa = de e 4 and hence aY 1 aD a® — gD C= dk® ? v= R® de Ce Thus we find [ dR) dR) de RW de RW = Oi AC eae Gari | GE — 2 oe jes de de or putting dR«) A de Rie) a 9d RO + x aR® de dR) de Rix) Ql) = 9 ER® —2.4 Ro 9 de we have a® = aM + 9M ag, 103 (33) 104. A NEW METHOD OF DETERMINING The values of 7) and 6“ are readily found from f=14+3 @—(2e—té4 74 6)cosy—(se —1 eH + gs &) cos2y —(4¢é— &) cos 3 y — ete. > ee Neosiziy. We have BOSilpee R® =—(2e—Fe 4+ ge &) )) — iL 4 6 RO=—Gé—fe+ we) R=—Ge—he ) Cy = ete. d RO Ge de d RY 2 Ba 6 5 ae (CO a a) GLO So, Ih 8 de S—(0@— 3 e+ ee) ad k® 2 4 Ge 7 (Zé ay cz Ze) GO 2 BAB de aa G € ¢ ) ete = Cte: For 7” we have (2) Crs? Car 3) G " —G=1e4%9 C SS Og ca, Oar Oe @ a ee) or (9 Se he ade, (34) For 6 we get at once THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 10 In a similar way we have (0) = BP Sa ey il = ts @ T258 > 7 Col a (35) In case of the third coordinate we also compute the coefficients of the arguments having no angle y from those having + y. For this purpose, putting x = 0 in the expression for a’ we have d PO) ad RO a -+- a) OF SS eS Oe) (—1) Cy = We apart CLE 1g = 4 (4 + o ); where dk 9 de n — B d R® de For 7) we then have (0) 72 =—(Be+ Ye + etc.). (36) Perturbation of the Third Codrdinate. Let 6 the angle between the radius-vector and the fundamental plane, @ the inclination of the plane of the orbit to the fundamental plane, v —o the angular distance from the ascending node to the radius-vector. We have then sin 6 = sin 7 sin (v—o). If we use for 7 and o their values for the epoch and call them 2 and Qo, & being the longitude of the ascending node, we have sin b = sin % sin(v— Q)) + 8; s is the perturbation. Thus we find s = sin 7sin (v—o) —sin 4%, sin (v— Qj). A. P. S.— VOL. N. XIX. 106 A NEW METHOD OF DETERMINING Putting p=sin?i sin (oc — Q)) , g =sin 7 cos (o — Q)) — sin %, we find s= qsin(v—Q)) — p cos (v—Q)). Instead of s, let us use and we have n= 7gsn(@—Q,)— £ pars (=): Q FE: a ; Introducing 7 and calling #& the new function taking the place of u, we have, putting © + 7 for v, 2 being the longitude of the perihelion, a@E dgp . dp p =a 7, 22 (© =F %)—= Qy) = =F @, COS (@ + 7 — Qo). d dp : : i s To find and = we will employ the method given by Watson in the eighth chapter of his Theoretical Astronomy. Thus « and (@ being direction cosines we have A SO =— B O3 also 2 = rsin? sin (v—o). But T = 7 COS ¥, and ¥ = Tr sin v. Hence 2 = —xsin? sins + y sin? cos o, THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 107 and a=—sinzsino, (6 =sin7zcoso. The values of p and qg then are given by the equations p= —a cos 2, — sin Qo, g = — asin 2 + 6 cos Q — sin from which we have dp da. dp a = — GCOS Qo Fh SIN {2 Fe dq 3 da dp Ros SIN {2 as + GOS a: From the equation z, = a « + @ y we have, first regarding a and ( as constant, then regarding # and y as constant, dx 5 dy Giseetes dz, da. Ap oe al = 0 FF SP I) ap SO Differentiating the first of these, regarding all the quantities variable, we have az da dx dp dy au p GY = oa dt’ dt dt dt dt dt? dt? Z, being the component of the disturbing force parallel to the axis 2, and X and Y the other two components, we have Z=aX+ 6 V+ Zcosz. Writing for X and Y their values vx di? ay +h (1+m)”, ated +m)s, 108 A NEW METHOD OF DETERMINING and reducing by means of Z=an+ By, we have ped ZL He A= “+ (+m) * + Zecosi, or V2, on ae i de = Oe + be = Z GOS 2. V2, Fae given above, Comparing this with the other expression for ~ we haye da da dp dy __ Z 2 Fie Tein Tach Gia a 0 ° dz, 0 From this equation, and the value of ak since ¢ d 2S Sep 2 yi =kv pl =e we find da. Fi =—hrcosisinv Z, dp da hrcostcosv Z. a d Substituting these values in the expressions for = and “! ae we have dp dp 7 =hrcostsin (v— 2) Z, = = hr cost cos (v— Qo) Z. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 109 a i . dR Introducing these values into the expression for = a we have dk a = Pt cost cos (v — Q) 7 8n (@ + m—2) Z —hrcostsin (v — 2) ~ Cos (@ + m)— Q)) Z \3 =—hreost [ sin @ COS (v — Qo — (m— 20) | Z Q 0 —hrcosi% [ cos @ sin (v — &— (7% — &0)) | Z QX dQ = fhpeans = sim (@— 7) ==. Hsin(o—f) 4, é ke y/1 em ke y/1-m Introducing » = mean mips = E : Vv P we have ak ] Tr O dQ 4 a p P : : = —£ im (©— Sk : ndt \/ 1—e’ a aac (o f) g AIF a (37) Let il Pp CSS = =n (@— 77) e V1—e a ae (o—f); then dk 9 dQ — — o> || == Iho cos 7.ndt C ee To find an expression for C similar to those for A and B we have, first, 1 po. i p Jape oe sO sie. conf" cos. - sin f |. \/1—e’ La, a My a 110 A NEW METHOD OF DETERMINING = e ° iP Tm e a ° . ° Substituting the values of —cosf, —sin f, given before, and similar ones for a a s ® cos o, © sin w, we find My My Ca ( d.p* ) (<7) mr. (oe) () ma: aedg ¥ \agedg/ \ade/* My le Substituting the values of these factors we obtain for C’ the expression C=(1—te) sn(y— g) — (8¢ — 3) sin y + (Le— 32 &) sin (y — 29) + § @ sin (y—8y) ee) + $e sin (y—4y) —7,¢ sin (y + 29) . 5 d du Haying found the expressions for and : nat ndt . cost we have, finally, for determining the perturbations, the following expressions : noe =n f W dt, dw n f a dt, — = (Oa (>). tos 7 dZ. | n= ° : dw Two integrations are needed to find ndz. We first find W from Fae then, form- pa Re an) td . oS ge ing W and — $ ip from JV we have néz and » by integrating these quantities. In : : dw . . F the integration of aa We Sive to the constants of integration the form ky + k, cos y + k sin y + 9 k, cos 2y + 7°) & sin 2 y + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 111 aw — we have dy Then in case of — tol + 3k, sin y — 3k, cosy + 7 k, sin 2 y — » kh, cos 2y + ete. In the second integration we call the two new constants Cand J, and the con- stants of the results are in the forms C+ kh nt + k sing —k, cosg + $7 k, sin 29 — $ 7° kh, cos 2g + ete. ol— Nf —4k,cosg— tk, sing — $7” k, cos 2g — $7 hk, sin 2 g — ete. In case of the latitude the constants are given in the form AQ +isng+l,cosg + 7 1, sin 2g + 7° L cos 2g + ete. 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 4 u d : a ; 9 the expressions for ndz, », — , oa (ndz), ete., including in g’ the long period term. cost” na Putting the constants equal to zero, and designating the values of néz, v, ete, at the epoch by a subscript zero, we have the following equations for determining the values of the constants of integration: C+ ksin g —k, cosg + $7 k, sin 2g — $ 4k, cos 29 JL ef, at (ndz)y = gy = 7 Gh, ; k + k,cosg + &sing + 7° k, cos 2g + 7° k, sin 29g + ete. + — (nbz) = 0 ndt N — 1k, cosg—ik, sin g— $7 k, cos 2g — 4 7 & sin 2g — ete. + GO); = 0 2 9 1 + dk, sin g— tk,cosg+ 7° hk, sin 2g — 7 k, cos 2g + ete. + a (7) = 0 i+ %sn g+becosg + 7” 1, sin 2g + 7° L cos 2g + ete. 4 i= Jo = (I d ni 1, cos g — l, sin Gs 7” 1, cos 2g — 7 1, sin 29 + ete. + ( : y. 0) ndt \eost 112 A NEW METHOD OF DETERMINING To find k, and &,, we derive from the preceding k, [ cos g —e-+ 7) cos2 g + 7 cos 3 g + ote. | aL fh, [ sin g + 7° sin 2g + ete. | d —3 44+ 6(v),+4 fe (ndz)) = O k, [ sin g +27 sin 29+8 7° sin 38 g + ete. | — k, [ cos g + 27° cos2g + ete. | The value of JV is found further on. Having &, we find i, from ch, 84, +3" (nbz), + 6 (rp = 0. ko We have » hb=—el, N= —2h—Zh—3Z, where Z% is the constant of W. Let us find the expressions for the constants WV and K, A being the constant of : : c ° : h integration in the expression for 3 ; 6 The equation (22) we can put in the form dz hy : a hy hy fo ee ee Rae ay + (3p 4y* + ete.) = 27 (5 The differentiation of nz relative to the time gives - =1+h+4+ Z,+ periodic terms, where Fh, = — 32.7162, in the case of Althea, and 7, 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 y is vy = IV + periodic terms. . ; h ; ; ; Ii The approximate value of ie being 1, the complete expression for the integral of d 7 U 1 is given by hy 1 = 1 + k; + periodic terms, k; being the constant of integration. 2 = B h h zs ees 3 Putting (3»* — 4r° + ete. = — 2» C — 1) = JV,-+ periodic terms, and substi- U U s A 6 F lt > 6 : dz tuting this expression, together with those of » and , in the expression for uw? U have, preserving only the constant terms, I = y, (k; —k —4— %, se Vad It is necessary now to find the value of &; in terms of the constants. If in the 7 Gai F ; : 9 9 expression for ae given by equation (18) we write for p , its equivalent a cos “o — & p cos w , we will have dW, = hy} 2° co (f—0) 1-2, ee Oe) Ty es ‘e) dt hg! di, COS *g ° hy? a, cos*y, J \df + 2h) p sin (f —o) ( \dt. We also have a =h =) dt. Selecting from the expression for dJV, the terms not containing p cos o and p sin @, we have dQ dW,=—h, (1 +27.) oe A. P. S.—VOL. XIX. O. ) dt. 114 A NEW METHOD OF DETERMINING If the eccentric anomaly is taken as the independent variable we have for the complete integral W.=kh+ hk cos 4 + k, sin 7 — hy ine de 27) (F) dt. Introducing the true anomaly instead of the eccentric, we have, cos w + e : sin w cos >» sny = —___, since cos 7 = oS a is 1+ ecosw = ky ke : h’ d2 Wo=khtek + {9 608 © -+- pepsi — hh J (142 =) (FF) dt. Neglecting the terms having cos w and p sin w we have in JV the constants h and €) hy. ho The integral of dj; is h, om dQ ait hth fi) a hg From the expression for d ; we find h Wh? 7 dQ dji=—i,l(g)& a : 5 h , Integrating this, making use of the value of ;', and adding the constants, we have h h Qo = == My It 0 h? dQ =14h)+eh—h f (1+ 2.) (ae dt. And since the quantities under the sign of integration do not have any constant terms Wwe can write yh te foe == 1 +k + ek, + periodic terms h 5 0 a = 1 eis + periodic terms THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. . hy s : : : “ Since G — 1) is a quantity of the order of the disturbing force we have es ee =i) . —— (a) 4 etc., h h h from which we get Now putting hy 2 hy 3 eee Ge — 1) = (e — 1) + ete. = H, + periodie terms, substituting this expression and those for oh fx ho Fos ape aaa i Ae the preceding expression for gives, preserving only constant terms, kj = —4(kh + ek,) +2 Af. Introducing this value of /; into the expression for WV it becomes N=—}4(4h, 4: ek, +384) +4(8V,4+ 20,—3Z,). Preserving only the terms of the first order we have N=—}(4k + ek, + 3%). ; = 6 Sats h To find the value of A, the constant of integration in case of 6 7» We have L 0 : =—1-+ K + periodic terms, "0 ie 116 A NEW METHOD OF DETERMINING also f = 1+, + periodic terms. From these we get ~— 1p ele w= Fh, Hence I= — hk, + H, = 3 (hy + ey) + 3 HL; or, neglecting the term of the second order, K=3(k +e). THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Wi hs CHAPTER YV. Numerical Kxample Giving the Principal Formule Needed in the Computation Together with Directions for their Application. ALTH HA 119. JUPITER. g = 332° 48’ 53.2 G4 = 63.9 on 48.6 a le 542) wm =12 36 59.4 | = 203" bol: 1894.0 Y =o 22 1894.0 *= 5 44 46 | (roll) @¢= 4 36 249 go = 2)7 45) 50-2 nm = 855’.76428 nm’ = 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 Hitt in his New Theory of Jupiter and Saturn, in which the epoch is 1850.0. Applying the annual motion of 57.9032 in x’, of 36’.36617 in &’, to Hriw’s value of z’, and of 9’, we have the values given above. The mass of Jupiter is ;>57/ 575. The elements of Althzea are those given in the Berliner Astronomisches Jahrbuch for 1896. The ecliptic and mean equinox are for 1890. ‘To reduce from 1890 to 1894 we employ the formule of WATSON in his Theoretical Astronomy, pp. 100-102. v=it+ 7 cos (Q— 9) Q=Q+ UU —t) “ — 7 sin (Q —8@) cot .2 dl ; 9 nm =n+ (¢—t)> +nsin (2— 6) tau $2 118 A NEW METHOD OF DETERMINING where § = 351° 36’ 10” + 39”.79 (¢ — 1750) — 5.21 (¢ —t) n= 0.468 (v — 1) These expressions for 2’, ’ and 7’, can be used for the disturbed body as well as for the disturbing body by considering the unaccented quantities to be those given, and the accented quantities those whose values are to be found for the time, /. HARKNESS, in his work, The Solar Parallax and Its Related Constants, using the ° ¢ P dl most recent data, gives the following expressions for 0, 7, and ae when referred to at 1850.0: 6 = 353° 34 55” 4- 32’.655 (t — 1850) — 8”.79 (t — f), n = 0.46654 (¢ — 1850), “= [50.2362 + 07.000220 (¢ — 1850) | (” — 4). n Letu == n! ? we have then we = 0.34955 2u = 0.69910 du = 1.04865 4u = 1.39820 Su = 1.74775 6u = 2.09730 Gio, Guec, Hence 1 —3u = — .04865 , 2—6u = — .09750. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 119 This shows that the arguments (g—3q’), and (2g — 69’), have coefficients in the final expressions for the perturbations greatly affected by the factors of integration. In case of the argument (gy — 3g’), we should compute the coefficients with more deci- mals; also those of (0 — 39’) and (2g — 3q’), since in the developments the coefficients of these affect those of (g —34q’). From sin 5 Z.sin § (¥ + ©) =sin$ (Q— Q’)sin} ¢@—7) sin I. cos 2 (¥ + ®) = cos $(Q—Q’) sin $ (¢— 7’) cos 5 J. sin 5 (¥ —®) = sin} (Q— 9’) cos 4 (7+ 7) cos 5 I.cos 4 (© — ®) = cos 4(Q — 2’) cosh (7 4 7) where, if §2’ > &, we take § (860° + 9 — 2’), instead of 4 (Q— 9’), we find vy B=116° 15’ 36.7 De Il 80 sey Shes 1 0 S583 An independent determination of these quantities is found from the equations cos p sing = sin?’ cos (Q — 8’) COS p COS g = Cos 7 cos psinr = cos?’ sin (Q — Q’) cos pcosr = cos (2 — 2’) sin p = sin?’ sin (Q — 9’) sin Jsin ® = sin p sin cos © = cos psin (¢— q) sin Zsin (¥ — r) = sini p cos (¢ — q) sin Icos (J —r) = sin (t—q) cos I = COs p COS (t— 4). 120 A NEW METHOD OF DETERMINING From Il =a —2 —® inl’ = 7! — OY — we have esl God 55/2 Ue —— OG OCeAe. ee: _ Then from k sn K = cos Jsin Il’ k cos kK = cos II’ ie, Si [kG = sin IT’ k, cos K, = cos J cos Il’ p sin P = 2a? — 2uk cos (II— EK) p cos P = 2a cos 9’ k, sin (11 — A) vsin V = 2a cos > ksin (11 — KX’) v cos = 2a cos 9 cos 9’ k, cos (1I— 45) w sin W = p— Qa? sin P weos W= v eos (V — P) w,sin W,= vsin( V — P) y w, cos W,= 2a?* eos P, é THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 12] we find IK NBO Is OAS log k = 9.999614 Kee — lbOMeol T4 log k, = 9.997849 3 Ari) log p = 9.932748 Sa) © 2.4 log v = 0.601463 W=266 4 39.5 log w = 0.605196 W,=266 15 380. logw,= 0.601352 Then from We Orn Oe Che ay apn Ole Clas we have log R = 0.702855, logy, = 7.976024. The values of the quantities from Il to y, 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 # from g= H—esn LH, where ¢ is regarded as expressed in seconds of are. Substituting the sixteen values of ¢ in the equations fsin (7 — P) =w sin (HH — W )—ep f cos (Ff — P) = w, cos(H+ W,), we obtain the corresponding values of f and /. A. P. 8S.—VOl. XIX. DP. 122 A NEW METHOD OF DETERMINING Then in a similar manner from C=y + y.sin*Q logqg= log f+ y Yo V2 a : 3 Yoo, Ee L=s (4 see ) sin 2 74 s( av — js) sin4 2 f? where s = 206264’.8, log”, = 9.63778, we find the values of Q, C, log ¢, «, and y. Thus we have found all the quantities entering into the expression 9 (-) = (C— qcos (E’ — Q)) (1—F cos ( 2 + Q)). Instead of this, we use the transformed expression a ({) =" (1 + —2acoa(H—@Q)) * (1+ —2e08(B' + Q)*, and have, for finding the values of V, a, b°, the equations Q a = sin y ? — sin val q a=tgsyz b= tg 5 Xa Va Vara G THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 125 To find the value of C, . we put (1 + a— 2a cos (Z’ — Q))~ ve +, “cos( B—Q)+ cos 2( #—Q)-+ ete. | ) 2 (1+ 6 — 26 cos (H’ + Q)? =[t2. nes cos (4 + + Q+ By cos 2 (#’ + Q) ls oO Hm “bE ete. | For finding the values of the coéfficients in these expressions we use RUNKLE’S Tables for Determining the Values of the Coefficients in the Perturbative 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 q@) (2) (3) (0) by br by Ge - (O) GP .Gb) Gea «@)) ae b 1 , then those of A OOD etc., ete. 5 Ae bs > bs ae bs , ete., ete., where 6" is found a* from the sixteen values of 0? = eaae Since 6 in (1 — 26 cos (’ + @)) is very small it will suflice to put | Ou a Nilo Then from iz) I| n (i) zN B,,cos271Q z ro) n (i) NM Be sina 2 5 a we have, in case of u (), (1) = il ee SS Va )- La, iN, nO = — qe NV bcos2 Q, 3 o4% = cd bsin2Q; 124 A NEW METHOD OF DETERMINING and, for ua? =) 3 (1) 3 iz NV 3bcos2Q, $8, = 7,wN 3bsin2Q. We divide by 8 to save division after quadrature. () (2) @) With these values of c,, s,, and the values of the coefficients 6,, we find the 7 z 7 7 Uy values of k,, ;, from 3 (7) (0) (i-1) (+1), (1) k, cos K; = 0, ¢, + (6 + 5b, \e Dy TN z B (=i )) (+41), (1) bs ) For 7 = 0, we find /) from , (0) (0) (it) CY) ko — 3 Dr Cn b,, Cn 2 2 my . a Then in ease of tu (“) from A,, = }m'sk,cos [1(Q—gq)—4\] A,, = tm'sk,sin [0 (Q—g)—&A,], where m/’ is the mass of the disturbing body and s = 206264.’8 ; and from A,, = 3m sak, cos [i (QY— g) — Ki] A,, = tm soek,sin [¢(Q—g)— A], (a) : 3 (c) (s) z Zs in case of uwa> ( “) , we find the values of A,, and A,, for the 16 different points of the circumference, and the various terms of the series. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 125 (c) (s) Again, since A;,, A;, are given in the forms (c) (s) A,, = >C,,cosvg + >C,,sinvg (c) Om A, = 2S,,cosyg + >S,,sin vg, (c) _ (8) (ce) we have the following equations to find the values of the coefficients C,,, C,,, Si, (s) i,ve (O38 )= 4% Xs (2) = Yo— ¥, dona 7.8% (i) Sie GIO) =e Ke Aj = B=, (AUS) Ss Ee G)=V%— Vs (0.4) = (0.8 ) + (4.12) (1.5) = (19 ) + (5.13) (2.6) = (2.10) + (6.14) (0.2) = (04) + (26) (3.7) = (3.11) + (7.15) (1.3) = (1.5) + (3.7) 4 (Gy + 2e,) = (0.2) A(G—2o) = Gs) AGE @) = ©@8)=Ge) 4(e— o) = §[ (1.9) — (5.13) | — (0) —@aB)]/ cos 45° 4(s + 8) = $[ (1.9) —(5.13) | + [ (8.11) — (7.15) |{ cos 45° (Ge) — (210) (G14) 8c, = (0.4) — (2.6) 8s, = (1.5) — (3.7) 126 A NEW METHOD OF DETERMINING 4 (c+ 6) = (2) + | Gr) — Gx) | cos 45° 4(¢— 7) = | (4) — Gs) ] cos 22°.5 + [ 3) — G4) | cos 67°.5 A (e+ 6) = (8) —| Gr) — Ger) | cos 45° A (¢—6) = | (4) — Gs) |sin 22°.5 — | (3) — Gs) | sim 67°.5 4 (s:+ 8) = [(4) + G's) | sin 22°.5 + [ GA) + Gs) | sin 67°.5 L(a—s) = [ (27) ap (Gh | cos 45° + (45 4 (s,+8;) = | (4) + Gs) | cos 22°.5 — | (2,) + Gs) | cos 67°.5 A(s,—s;) = |G?p) + Gér))| cos 45°— Gs) The values of ¢,, s, must satisfy the equation (c) (s) in OF A; = 4+ 6,c08g + & cos 2g + ete. + s,sing + s,sin 2g + ete. . Oo 0 (7) . 2 answering to? in b,, and x being any one of the numbers, from 0 to 15 inclusive, 23 (¢) into which the cireumference is divided. We use ¢, s, as abbreviated forms of C,,,, (s) (c) (ec) (s) C;,,, ete. Having found the values of ¢,, s, from the 16 different values of A), A,, i, (Cc) (s) (ec) (8) a > (4 Aly, Aly 5 6 9 lay 4b, lootiin ioe m (4) and war ( “)s we have the values of these func- tions given by the equation a (c) (s n (s) c = (3) = $33 (C= S,,) eos [GF ») g 1B" F £35 (C,,4 S,,) sin[ GF») gE | The values of the most important quantities from the eccentric anomaly £ to ¢,, s,, needed in the expansion of w (“) and «a> Cr are given in the following tables, first for uw (5) , and then for wa? fe 3 when not common to both. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 127 Values of Quantities in the Development of « (3) and woe(“). H H+ W H+ W, Hr — P Br SF 2 Mh Ona ® fF. MM So 7 Li ; Ol aie ( 0 0 0.0 266 4 39.5 266 15 38.0 266 21 17.2 399 24 44.9 ( 2A JA ALD, 290 28 48.7 290939) 4272) 290 8 1.8 23 11 34.8 ( 48 26 37.2 314 31 16.7 314 42 15.2 313 40 58.4 46 44 95.4 ( Tl 52) 2429 331 bT 4.4 dae) ON 2.9 Bax) Ges Be) Oe) HY (0,33 ( 94 35 14.0 0 39 53.5 0 50 52.0 309) 41 13 92 44 98.3 ( 116 36 51.7 22 41 31.2 DY 2) DS) MN BG ish 1d) 253428 ( 6 138 4 29.4 44 9 8.9 44°90 7.4 | 43 47 3.8 136 50 30.8 ( 159 8 19.6 65 12 59:1 65) 23) 57:6 65 8 48.4 158 12 15.4 ( 180 0 0.0 86 4 39.5 86 15 38.0 86 13 41.4 WG) aly Bet ( 200 51 40.4 106 56 19.9 NOT) Ska 107 15 14.8 200 18 41.8 ) 221 55 30.6 128 O 10.1 128 11 8.6 128 28 47.5 221 32 14.5 243 23 8.3 149 27 47.8 149 38 46.3 150 8 27.6 243 11 54.6 265 24 46.0 171 29 25.5 171 40 24.0 172) 23 51.4 265 27 18.4 N33 Ball 194 12 14.6 194 93 13. 195 27 19:4 288 20 46.4 dll 33 22:8 Pike ats) 233 217 49 0.8 218 43 0.9 311 46 27.9 335 35 55.8 241 40 35.3 24 di 33.8 242 28 57.5 | 330 32 24.5 | 1613 47 17.9 | 1438 47 18.6 Log. f. y x 0) Log. q. Log. C. i Oo Pm (( 0.612427 —.001251 | — 12.2 359 24 32.0 0.611176 0.706582 ( 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 ( 0.605242 —-.000981 +390.0 70 3 36.3 0.606233 0.704403 ( 0.601312 + 001292 — 58.6 Ce dish WELT 0.602604 0.703241 ( 0.598569 --.000846 —476.9 let By! So!) 0.599415 0.702241 ( 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 C& 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 ( 0.598532 —.000152 618.1 22) 42) 32.6 0.598380 0.700021 0.599177 +.000TTT + 496.6 243 20 11.2 0.599954 0.699872 0.600584 —-- 001278 + 96.7 265 28 55.1 0.601862 0.700504 0.603163 +.001032 —363.1 288 14 43.3 0.604195 0.702020 0.606734 --.000148 —600.1 3811 36 27.8 0.606882 0.704038 0.610302 —.000825 —452.4 335 24 52.1 0.609477 0.705810 4.823835 at 3 — (5 1613 47 174 4.823838 5.622201 4.823834 = 9 == (hy a yee Sy ets) 4.823842 5.622200 128 Values of Quantities in the Development of «(4) and war or A NEW METHOD OF DETERMINING g x VG Log. b. Log. a. a. Log. WN. (e) 7 // i; ie (0) 3 93 45.3 1 51.83 7.063818 9.701484 0.502902 9.695669 @ 1) 3 96 41.3 7 54.78 7.063792 9.701945 0.503437 9.695880 ( 2) 53 14 15.6 7 59.97 7.065778 9.699988 0.501173 9.695892 ( 8) 52 54 33.7 8 3.30 7.068781 9.696876 0.497594 9.695837 ( 4) 52 28 55.6 8 1.35 7.072405 9.692804 0.492951 9.695616 ( 5) 52 6 81.2 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 (Gn) 51 46 50.0 8 14.55 T.OTSTT4 9.686068 0.485364 9.695430 (8) 51 49 41.2 8 14.49 7.078721 9.686526 0.485877 9.695629 (9) 52 0 52.3 8 13.57 1.077913 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 0.499597 9.696354 (15) 53 13 54.4 7 59.70 7.065534 9.699932 0.501109 9.695743 ay TT.553183 3.956815 77.569096 x 17.553803 3.956845 77.569088 © @ @ ©) @ Q g Log. te, ILOS, 5 Oy, Log. ts, Log. b, Log. b Log. b, 0) 2 2 2 21 a ( 0) 8.792579 6.16064 4.475270 0.332110 9.748094 9.329969 () 8.792790 5.98934 6.02920 0.332186 9.748669 9.331018 @) 8.792802 4.985510 6.16173 0.331867 9.746235 9.326571 @3) 8.792731 6.05070n 5.97267 0.331369 9.742375 9.319511 ( 4) 8.792526 6.16734n 5.14693n 0.330730 9.737346 9.310298 (5) 8.792331 5.98219n 6.05562n 0.330182 9.732946 9.302224 ( 6) 8.792310 4.93934 6.173782 0.329872 9.730425 9.997590 (% 8.792340 6.03383 6.016142 0.329707 9.729076 9.295111 C8) 8.792539 6.17549 4.575070 0.329776 9.729636 9.296143 ( 9) 8.793030 6.05359 5.99045 0.320045 9.731836 9.300183 (10) 8.793815 5.23282 6.17067 0.330477 9.735329 9.306586 (11) 8.794449 5.948120 6.07618 0.330914 9.738805 9.312970 (12) 8.794541 6.16466n 5.36611 0.331246 9.741407 9.317738 (13) 8.794051 6.07296n 5.942029n 0.331460 9.743078 9.320808 (14) 8.793264 5.237420 6.16200n 0.331637 9.744461 9.323327 (15) 8.792653 5.97789 6.04134 0.331858 9.746165 |° 9.326448 } 2.647715 77.912926 74.508222 x 2.647721 77.912945 74.508268 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Values of Quantities in the Development of u(“) and wa?(“) . 129 (3) (4) (5) (6) (7) (8) (9) g | Log. b, Log. b, Log. b, Ios yOnen|) oss 0) |) Losvdy "Loe: b, 2 2 z z z 2 z ( 0) 8.954999 8.60017 8.2570 1.9215 7.5915 7.2654 6.9426 (I) 8.956515 8.60214 8.2594 7.9244 T.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 (Co) 8.904506 8.538411 8.1754 7.8244 TAT89 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 © YSIS 7.5520 7.2205 6.8923 (13) 8.941742 8.58283 8.2855 7.8960 7.5618 7.2317 6.9048 (14) 8.945388 8.58760 8.2415 7.90380 7.5700 7.2410 6.9152 (15) 8.949898 8.59349 8.2488 teSilaly 7.5800 7.2524 6.9280 Dy 71.449530 68.55219 65.7484 63.0060 60.3071 57.6402 54.9995 a! 71.449597 68.55223 65.7482 63.0063 60.3072 57.6404 54.9998 3 ‘- @) (On rari) @) (2) (3) g |Log.t N | Log. te, | Log. ts, | Log. 46, | Log. 6, | Log. 6, | Log. bg 2 2 2 2 2 2 ( 0) 8.183917 5.42374 3.1383 7n 0.280319 0.417421 0.200612 9.961097 (Ib) 8.184550 5.25307 5.29293 0.281000 0.418474 0.202090 9.963016 ( 2) 8.184586 4.24928n 5.42550 0.278120 0.414013 0.195824 9.954877 ( 3) 8.184421 5.314307 5.23627 0.273612 0.406981 0.185917 9.941987 ( 4) 8.183758 5.430287 4.409877 0.267827 0.397890 0.173060 9.925223 ( 5) 8.183173 5.244547 5.31797n 0.262860 0.390004 0.161858 9.910585 ( 6) 8.183110 4.20163 5.43607n 0.260054 0.385513 0.155458 9.902210 Ga) 8.183200 5.29621 5.278520 0.258559 0.383116 0.152039 9.897732 ( 8) 8.183797 5.43847 3.838057 0.259184 0.384116 0.153464 9.899598 @ 9) 8.185270 5.31804 5D: 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 5.21681n a 34487 0.269488 0.400515 0.176758 9.930076 (12) 8.189803 5.433640 4.63509 0.272484 0.405223 0.183435 9.938754 (13) 8.188333 5.340477 5.20953n 0.274429 0.408267 0.187732 9.944350 (14) 8.185972 4.50257n 5.42714n 0.276036 0.410773 0.191265 9.948948 (15) 8.184139 5.24121 5.304667 0.278037 0.413885 0.195644 9.954643 2) 65.482568 2.159554 | 3.209203 | 1.421019 | 79.449192 a! 65.482592 2.159606 3.209266 1.421076 79.449289 A. P. S.—VOL. XIX. Q. 130 A NEW METHOD OF DETERMINING 3 Values of Quantities in the Development of « (4) and uor(“) 5 i (4) is (5) KG) (7) (8) (9) g Log. bs Ihog.b, | og. 63 Log. bs Log. b, Log. b, P 2 2 2 2 ( 0) 9.70884 9.4484 9.1822 8.9118 8.6383 8.3621 ( 1) 9.71121 9.4512 9.1854 8.9155 8.6425 8.3665 ( 2) 9.70116 9.4393 9.1716 8.8998 8.6247 8.3471 ( 8) 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 9.0818 8.7968 | 8.5089 8.2184 ( 7) 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 (2) 9.64181 9.3684 9.08983 8.8058 8.5191 8.2298 (10) 9.65617 9.3856 9.1098 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 9.4937 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 x 77.37450 75.2339 73.0471 70.8269 68.5804 66.3134 a 717.37462 75.2341 73.0474 70.8269 68.5803 66.3132 g | Log. k, | Log. k, | Log. k,; Log. k, | Log. &,| Log. &, | Log. k, | Log. k, ( 0) 8.824187 8.54492 8.12562 7.750420 7.89550 7.0523 6.7168 6.4105 ( 1) 8.824302 8.54433 8.12588 7.151220 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 8.52360 8.09009 7.700585 7.33130 6.9737 6.6239 6.2809 ( 7) 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 ( 9) 8.822444 8.52829 8.09567 7.107159 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 7.36070 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 8.11238 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 8.121158 7.144423 7.38795 7.0433 6.7062 6.3901 2 70.583851 68.25726 64.85258 | 61.793910 | 58.89655 56.0927 53.8503 50.6520 ral 70.583841 68.25722 64.85260 | 61.793920 | 58.89653 56.0926 53.8505 50.6512 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Values of Quantities in the Development of «(4) and ua*( ate 131 g | Log.ks Log. ky || Ky JEG ISG K, K; Ky / i? / / J ( 0) 6.0606 5.7378 — 0.6 —04 —038 — 0.3 3 0.3 0.3 (1) 6.0636 5.7413 +20.3 +12.9 +-11.4 E111 10.6 aa + 9.5 + 8.3 (2) | 6.0454 5.7212 JE) ET SRR —t5T16,0) “igen exe SEO | Ss ( 3) 6.0178 5.6904 +-18.4 11.7 10.2 10.0 EO + + 92 + 8.8 ( 4) 9.9830 5.6515 —28 —18 —16 —1.5 3) 1.5 1.5 ( 5) 5.9541 5.6191 —22.7 —14.5 —12.7 —12.0 8 11.4 11.0 ( 6) 5.9391 5.6019 —29.8 —19.0 —16.7 —15.7 3 : 14.5 BBE (C®)) 5.9316 5.5934 —20.7 —13.2 —11.6 —10.9 5 —10.1 — 9.7 — 8.9 ( 8) 5.9364 5.5985 — 0.7 — 05 — 0.4 — 0.4 A —0.3 —03 -- 0.8 (9) 5.9512 6.6151 +19.5 +12.8 10.9 L 10.2 8 + 9.4 + 9.0 + 8.2 (10) 5.9737 5.6405 +29.1 +18.6 +16.4 15.3 a) : 14.1 +13.3 5.9959 5.6656 +-23.4 +14.9 +13.1 +12.3 lL +11.9 +117 +11.3 6.0124 5.6842 + 4.5 + 2.8 + 2.5 L 2.4 | 2.4 + 2. + 2.3 + 2.2 6.0251 5.6968 —17.0 —10.8 — 9.5 — 89 = 88 — 8 — 86 — 84 6.0341 5.7083 —28.1 —17.8 —15.7 —14.7 —14.3 ‘ 13.6 13.0 6.0468 5.7224 —21.0 —I13.4 —I11.8 —11.0 —10.6 9.8 9.0 45.3439 Sah) ee ee ee aS 45.3441 b = a @ ds. 8 = ofl g | Log.k& Log.k lLog.k lLog.k; Log. k, Log. k Log. k& Log. k, ( 0) 8.465272 8.60289 8.38621 8.14674 7.89481 7.6341 7.3679 7.0975 (Gp) 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 T.5TT4 7.3026 7.0237 (Gay) 8.445362 8.57214 8.34391 8.09259 7.82837 eDooO9 1.2776 6.9950 ( 6) 8.443224 8.56872 8.33868 8.08545 1.81922 7.5446 7.2645 6.9800 (C1) 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 (G9) 8.444679 8.57452 8.34564 8.09354 7.82847 7.5551 7.2760 6.9925 0 8.453274 8.58206 8.35573 8.10632 7.84401 7.5734 7.2971 7.0165 8.458368 8.58906 8.36522 8.11851 7.85895 7.5912 7.3176 7.0400 8.461465 8.59345 8.37153 8.12680 7.86927 7.6036 7.3320 7.0564 8.461922 8.59532 8.37468 8.13126 7.87506 7.6105 7.3405 7.0660 8.461886 8.59651 8.37704 8.13471 7.87957 7.6163 7.8472 7.0739 8.462852 8.59905 8.58088 8.13992 7.88616 7.6242 7.38564 7.0845 68.69172 66.90360 64.93175 62.85706 60.7165 58.5297 56.3095 68.69184 66.90364 64.93185 62.85719 60.7166 98.5300 56.3096 132 _ A NEW METHOD OF DETERMINING Values of Quantities in the Development of u (4) and uor(“). g Log. ks; Log. ky | A; K; Kk, (Q-g)-K, 2(Q-g)-k, 3(Q-9)—K; / / / | oO. i fo) y yy ( 0) 6.8240 G50 |) Ol —().1 —0.1 359 25.1 3858 49.5 358 13 55.0 (1) | 6.8280 6.5522 +4.4 +-4.4 +4.4 || 0 28.5 1 24.6 Dut B70) (2) | 6.8092 6.5317 +6.0 ++6.0 --6.0 Wo 2535) 3 31.0 By Pi ay! ( 3) 6.7795 6.4988 +3.9 +3.9 +3. 2 Way?) AL Byay5) to BO 33.9) ( 4) 6.7414 6.4566 -—0.6 —0.6 —().6 2 46.3 5 28.8 Sa eae ( 5) 6.7093 6.4209 —4.7 —4A —4.7 2 NTS Oy) ake) 7 26 37.6 ( 6) 6.6921 6.4016 —6.2 —6.2 —6.2 2 9.8) 3 se) 5 16 57.0 ( 7) 6.6837 6.3923 —4.3 —4.3 —4.3 0 55.7 1 23.2 1 5 Boo) ( 8) 6.6887 6.3976 —(0.2 —0.2 —0.2 358) TLS 358 34.3 357 Ol 3.3 @9) 6.7058 6.4165 +4.0 +4.0 +4.0 Bo 36.2 B55) BS Se) By) BI)s5) (10) | 6.7327 6.4463 +6.1 +6.1 +6.1 356 13:4 3538 6.5 349 51 14.9 (11) 6.7589 6.4752 | —-5.0 -+-5.0 -L5.0 355 26.8 301 20.5 SA 2 Oe! (12) 6.7773 GASH |) SLT) +1.0 1.0 305 «24.4 850 55.0 346. 24 12.8 (13) 6.7883 6.5081 || —3.5 —3.5 —3.5 Boe Ist B01 40.2 347 23 40.2 (14) 6.7976 6.5187 —6.0 —6.0 —6.0 351 «4.6 353 30.7 BO) By BUS (15) 6.8093 6.5317 —4.5 —4.5 —4.5 |) 358 15.9 3563.1 353 56 22.5 By 54.0630 51.7961 0 0 0 | 1793 47.8 ISI V22)> 326 a 54.0628 51.7957 + .3 + .8 + 3 || 1483 47.3 1497 2) 59: g A(Q—9)—K,5( Q—9)—K; 6( Q—9)— K, 1 09), 8( Q—9)— K3 9( @ - 9) — Ky (o) / (eo) / e) / (e) / e} y ( 0) 857 38.5 Sf 0) 356 27.5 30D) 2a 855 16.7 354 41.2 Gp) 3 Bos) 3 53.2 4 42.5 5 31.8 6 21.1 7 10.5 ( 2) 2204 S) Ina 11 12.4 UB YB Il 2) 1G AHA (3) 10 4.4 12 38.3 15 12.2 17 46.0 20 19.8 22 53.6 ( 4) 10 55.5 13 39.0 16 22.5 IQ BA) 21 49.5 24 33.0 (@5)) 9 50.6 12) 15.0) Wal Bw) ly &§) 19 28.4 21 52.8 ( 6) @ 5.8) 8 35.6 10 15.3 11 54.9 3) 3425 15 14.2 (Gig) 2 30:9 BS By) 3 40.1 AS ART 4 49.3 5) 25) ( 8) 357 ~— 8.0 396 24.9 Boe) LIL. T 354 58.6 354 15.5 353 32.4 (( $)) 301 31.8 BAG) Dit 347 23.6 845 19.5 343 15.4 341 11.3 (10) 346 34.9 343 17.8 340: 0:7 336 43.7 333 26.7 330) 956 (11) SAD mOED) 338 58.9 334 49.3 330 39.7 326 30.1 322 20.5 (12) 341 53.2 Solmeooel So Grllil 328 20.0 323 48.9 BIG) ie) (13) Bytes © otf 388 52.3 yey! | BXOLY) 330 21.5 326 66.1 321 50.7 (14) 346 40.5 342 16.6 3389 52.7 336 28.8 Baa 4h.) 329 41.1 (15) sol 50.4 349 44.9 347 39.4 345 33.8 343 28.2 B40 2207 py 174465 aft 1384 6.0 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 133 Ti i = n the expansion of u ( aE (c) (c) (s) (c) (s) (c) (s) (c) (8) 0 245 1 2 2 3 3 A, 4 7 7 7 "7 1 "7 ame. eee 7 ( 13.13109 6.9027 —.0701 =+-2'6281 —.0539 | +1.10745 —.03418 | --.4889 —.0201 ( 13.13458 6.8933 -+-.0571 2.6294 --.0647 1.10917 -++.04356 4901 —-.0262 ( 13.11352 6.8033 -+-.1712 2.5849 --.1588 1.08348 +.10356 AT51 --.0615 Cz 13.08513 6.6912 --.2633 9.5254 + .2176 1.04890 -—-.13827 4553 --.0809 ( 13.05615 6.5922 3192 9.4646 +.2364 1.01333 --.14604 43538 --.0840 ( 13.03939 6.5457 --.3187 2.4959 1.2150 0.99004 —-.12935 4994 | 0733 ( 13.04058 6.5584 —-.2479 2.4172 +1543 0.98367 --.09095 4190 —-.0509 ( 13.04700 6.5880 --.1067 2.4198 1.0585 0.98375 +.03339 4190 +-.0184 (8 13.05942 6.6190 —.0816 2.4317 —.0606 0.98937 —.03712 4218 —.0211 ( 13.07850 6.6377 —.2779 2.4464 —.1863 0.99667 —.11189 4249 —.0633 d 13.10500 6.6498 —.4389 2.4645 —.2979 1.00593 —.18002 | 4287 —.1023 el 13.12573 6.6578 — .5301 9.4816 —.3742 1.01487 —.22886 | 4322 —.1310 al 13.13248 6.6727 —.5359 2.4991 —.3995 1.02497 —.24789 4373 —.1431 (1: 13.12612 6.7090 —.4658 2.5224 —.3693 1.03984 —.23954 | 4463 —.1354 dd 13.11967 6.7727 —.3458 2.5559 —.2907 1.06142 —.18555._ | 4600 —.1090 qd 13.12018 6.8478 —.2074 2.5954 —.1791 1.08668 —.11537 | 4760 —.0683 104.75791 | 53.5708 —.7340 | +20.0460 —.5531 8.26962 —.34421 | 3.5661 —.1992 104.75663 | 53.5705 —.7354 |+20.0463 —.5531 8.26992 —.34409 | 13.5662 —.1992 (c) (s) (c) (s) (c) (s) (c) (s) (c) (s) g 5 5 6 6 7 7 8 8 9 A, Sly lad tad vr dA JI} I IH/ II vad (GO) eee een O eae aS: =" 00630. |e 20505) 003GNul) 0226) 0019) a= 0L0j 0010 (C10) 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 00TT +.0607 ( 8) 1847 —.0116 0823 —.0062 0394 —.0035 0169 —.0017 .0078 —.0009 (9) 1860 —.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 (14) 2041 —.0613 .0918 —.0336 0414 —.0180 .0190 —.0096 0087 —.0051 (15) 2140 —.0387 0978 —.0214 .0468 —.0120 0210 —.0062 0098 —.0033 z 1.5765 —11050| --.7077 —.0598 | +3219 —0318 |7421467 —0168 | 42.0674 —.0087 x 6 | 11.5768 —.1106 | +.7078 —.0598 | +.3218 —.0318 | -++.1467 —.0168 |-+.0674 —.0086 134. A NEW METHOD OF DETERMINING 3 In the expansion of wa? Ga) 3 (9) | (c) (s) (c) (s) (c) (8) (c) (s) g 0 A, A, A, A, A, A; 4 sly AA I I wah /I aA // AA aA ( 0) 23.3520 | 132.0569 —0.3301 | +-19.4613 —0.4009°| +-11.2092 —0.3464 +6.269 —0.258 ( 1) 23.4045 32.1423 0.4199 19.5273 0.5272 11.2569 +-0.4603 6.300 +.0.347 ( 2) 93.2107 31.7192 +1.0033 19.1618 +1.2486 10.9731 -+-1.0737 6.096 --0.802 ( 3) 29,9239 81.1043 -—-1.8503 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) 22,3086 29.8387 +1.2952 17.5937 1.51122 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 ( 7) 29,1595 29.5473 0.38342 | 17.3564 --0.3782 9.6618 -+-0.3118 5.219 0.292 ( 8) 22.2368 29.6867 —0.3713 17.4552 —0.4367 9.7264 —0.3654 5.259 —0.204 ( 9) 29, 4949 30.0100 —1.1187 17.6808 —1.3068 9.8617 —1.0915 5.331 —0.786 (10) Q2.T157 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 5.5386 —1.667 (12) 93.1482 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 5.139 —1.148 (14) 23.1706 31.5386 —1.8212 18.9291 —2.2155 10.7412 —1.9027 5.895 —1.409 (15) 23.2299 31.7564 —1.1097 19.1791 —1.3716 10.9764 —1.1843 6.091 — .882 yy 182.6038 246.7758 —3.44293 147.0656 —4.1071 82.9580 —3.5000 +45.337 —2.564 YY | 182.5968 246.7862 —3.4356 147.0719 —4,1125 82.9644 —3.4985 + 45.339 —2.563 (¢) (s) (c) (s) (c) (s) (ce) (s) (c) (s) g 5 5 A, 6 A, A, A, A, 9 A, Wt aA 1 iA // Wit aA I aA t/ Coy) 43440 Soi | 41863 115: |) 41.000 —l079)> | 2539 —0a4 +.282 —.027 (( Jl) 3.458 +0.240 Wasa SAL a7 1.005 +.098 -535 --.060 .283 +.036 ( 2) 3.318 --0.550 WS 55.126 +21.018 Se + O58 + 0.521 5 + 6.891 + 2.627 Ly = + 0.057 + 0.065 (c) (s) 1 + 6.893 + 2.629 ES OL05 i + 0.065 In this way we check the values of these quantities for all values of 2, in case of both w(“), and wa(“). Applying to the coefficients of the two preceding tables! the formula ($)" = 885(C, F S,,.) cos [(¢-F)g— 1B" | AabEX(Cl, + S,,) sin [(¢F2)g 6B] 2 3 noting that } has been applied, we have the values of u (G), a ) that follow : A. P. S—VOL. XIX. R. 138 A NEW METHOD OF DETERMINING GLa cos | sin cos sin v) J} y) /} 0 @ +$[209.51455 | +1/364.6002 ] 1—0 0.25653 — (0.25027 +4.3500 —1.8014 2 — (i) * 0.00463 0.12279 —0.2566 +1.1803 a) 0.03070 0.05945 0.1118 0.5132 4—0 -L0.00037 +0.00055 0.0177 +-0.0182 | —2—1 | +0.023 —().041 -+-0.1310 —().6464 == 0.427 —(.193 —().0112 SAT 0—1 —1].158 0.101 —3.2161 +-1.0496 ai | +53.571 0.735 +246.7810 +3.4388 2 — Il | 9 954. —0.144 19.4716 —1.2526 3 oo | 0.087 0.063 +0.1909 0.7842 4—] 0.016 0.041 0.0970 0.6740 —l1—2 0.099 — 0.287 0—2 --0.098 —0.129 —0.001 = 1283 1, —().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 —(.905 4— 2 0.093 0.0382 -+-0.590 -+0.483 0—3 0.00446 —0.01101 -.0750 —0.1753 1—3 0.04011 —0.07730 +.0591 —0.9741 Y—= 3 —0.56048 0.00322 —).0643 0.2912 3—3 +8.26978 --0.34414 32.9613 +3.4992 4—3 1.01947 —0.01936 10.8367 —().4375 > 0.07682 —-0.01603 -+-0.7185 +-0.2822 6—3 0.00879 0.01536 0.0868 +0.244] lo dt +0.003 —0).004 0.053 —0(.098 Y— A --0.020 — (0.044 -- 0.082 —0.674 3—4 —().326 —0).005 —3.859 0.062 4—4 ---3.566 --0.199 +-45.338 2.562 5 — 4 0.585 —0.001 BE elales —0.149 6—4 -+-0.055 +-0.008 0.687 -+-0.1638 Pell 0.078 =-0.162 2} —— 5) 0.005 0.045 0.033 —0(0.049 35 — 5 -+-0.016 —0).025 0.088 —0(0.095 4—5 —().182 —0.007 —2.657 —(.041 5—5 1.576 --0.110 24.256 1-722 6 —5 +-0.325 --0.004 5.163 —().006 7—5) 0.081 0.004 +0.567 --0.436 4—6 --0.009 —0.008 --0.079 — 6.269 5 —6 —0.100 —0.006 1.717 —0.073 6 —6 +0.707 0.060 +12.781 +J1.095 16 +0.176 0.005 -+3.260 0.050 S==6 0.018 —0.005 0.426 0.057 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 139 5 a os [MUN oO 7 We have next to transform the expressions for «(—) and ua? (= ust given into others in which both the angles involved are mean anomalies. From beginning with m = 5, we find the values of 7, for values of ¢’ from £ to e”’. Then we find Putting m= 4, we find the values of 1, as in the case of r;. Then we get ps from 1. = — Dae (0) We proceed in this way until we finally have the values of p, Then we find J, . or &: (Fi — 1) from where J=h’¢, (m) and J je from 2 The details of the computation are as follows: 140 A NEW METHOD OF DETERMINING Computation of the J functions. iiss se! é ag 2e ag 3e a) de’ log. 1 8.38251 8.68354 8.85963 8.98457 9.08148 9.16066 9.22761 9.28560 log. 7; 2.31646 2.01543 1.83934 1.71440 1.61749 1.53831 1.47136 1.41337 log. p; 7.68354 7.98457 8.16066 8.28560 8.38251 8.46169 8.52864 8.58663 log. 7, 9.91955 1.91852 1.74243 1.61749 1.52058 1.44140 1.37445 1.31646 log. r,—log. p,|| 4.53601 3.93395 3.58177 3.33189 3.13807 2.97971 2.84581 2.72983 ech —1 == 5 = 12 20 — 31 — 45 — 62 = 81 221954 1.91847 1.74931 1.61729 1.52027 1.44095 1.37383 1.31585 log. p; 7.78046 8.08153 8.25769 8.38271 8.47973 8.55905 8.62617 8.68415 log. 73 2.09461 1.79358 1.61749 1.49255 1.89564 1.31646 . 1.24951 1.19152 Diff. 4.31415 3.71205 3.35980 3.10984 9.91591 9.75741 2.62334 2.50737 Zech —2 —9 —19 — 34 — 52 = (8 —=103 = 1155 2.09459 1.79349 1.61780 1.49221 1.39512 1.31570 1.94848 1.19017 log. ps 7.90541 8.20651 8.38270 8.50779 8.60488 8.68430 8.75152 8.80983 log. Tr, 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 2.90560 Zech —4 =i =38 =6F =105 = =208 —200 1.91848 1.61732 1.44102 1.31579 1.91850 1.13885 1.07186 1.01974 log. ps 8.08152 8.38268 8.55898 8.68421 8.78150 8.86115 8.92864 8.98726 log. 7; 1.61749 1.31646 1.14037 1.01543 0.91852 0.83934 0.77239 0.71440 Ditf. 3.53597 2.93378 9.58139 9.33122 9.13702 1.97819 1.84375 1.79714 Zech = 18 —ff —=ll4 —=%2 = 44 =618 207 1.61736 1.31595 1.13923 1.01341 0.91537 0.83480 0.76621 0.70633 log. p; 8.38264 8.68405 8.86077 8.98659 9.08463 9.16520 9.23379 9.29367 log. U 3.53004 4.73716 5.43852 5.93828 6.32592 6.64264 691044 7.14240 log. 2.92798 4.13210 4.83646 5.33622 5.72386 6.04058 6.30888 6.54034 — log. F 6.76502n 7.36708n 7.71926n 7.96914n 8.16296n 8.32132n 8.45522n 8.57120 Diff. 3.83704 3.23498 2.88280 2.63292 243910 2.28084 2.14684 2.03086 Zech =i 25 =i =10l =I =27 =899 409 loge ( + =) 6.76495n 7.36693n 7.71869n 17.96813n 8.16139n 8.31905n 8.45214n 8.56718n 3.23505 2.63307 9.98131 9.03187 1.83861 1.68095 1.54786 1.43282 Zech = She pp 0 GG es log. J 9.99974 9.99899 9.99773 9.99599 9.99375 9.99104 9.98787 9.98495 log. p: 8.38264 8.68405 8.86077 8.98659 9.08463 9.16520 9.23379 9.29367 log. J™ 8.38238 8.68304 8.85850 8.98258 9.078388 9.15624 9.22166 9.27792 log. p» 8.08152 8.38268 8.55898 8.68421 8.78150 8.86115 8.92864 8.98726 log. J 6.46390 7.06572 7.41748 7.66679 7.85988 8.01739 8.15030 8.26518 log. ps 7.90541 8.20651 8.38270 8.50779 8.60488 8.68430 8.75152 8.80983 log. J®) 4.36931 5.27923 5.80018 6.17458 6.46476 6.70169 6.90182 1.07501 log. p 7.78046 8.08153 8.95769 8.38271 8.47973 8.55905 8.62617 8.68415 log. J 2.14977 3.35376 4.05787 4.55729 4.94449 5.26074 5.52799 5.75916 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 141 Noting that log. (J —1) = log. (— P+ oo A=, and t=W2', we form bo| & the following tables : 17 Te) ee) 1 2) 1 _@ he) Log. (Suv —1) Log. Fun Log. Jw Log. iin Log. 5 Fu 1 6.7649 8.38238 6.4639 4.3693 2.1498 2 7.0658 8.38201 6.7647 4.9712 3.0527 3 7.2415n 8.38138 6.9404 5.3231 3.5807 4 7.3661n 8.38052 7.0647 5.5725 3-JooL 5 7.4624n 8.37941 7.1610 5.7658 4.2456 6 7.5409n 8.37809 7.2392 5.9235 4.4826 7 7.6070n 8.37656 7.3052 6.0567 4.6828 8 7.6641n 8.37483 7.3621 6.1719 4.8562 » (Wi) rr u Value of i URN 1 —— fst se) WSs Was Wt WSs 1) 4.9712n 6.4639 6.76495n 8.38201 6.9404 5.5125 4.2455 2|3.3537n 4.6703n 8.68341n 7.366938n 8.68241 7.3657 6.0668 4.7835 3 6.9410 8.85913n 7.71869n 8.85764 7.6381 6.4006 5.1598 4 49714n 7.36675 8.98344n 17.96813n 8.98147 17.8413 6.6588 5.4583 5 5.6702n 7.6393 9.07949n 8.1614n 9.07706 8.0042 6.8709 6 6.1012n 7.8432 9.15756n 8.3190n 9.15471 8.1402 alpHlorsha—0» 6.4176n 8.0061 9.22320n 8.452In 9.21993 8 | we have 6.6689n 8.1423 9.27965n 8.5672n 9 8.38251n 6.87TTin 8.2594 9.32905n In computing the values of the J functions, the lines headed Zech show that addition or subtraction tables haye been used. For conyenience, (J“’—1) is em- ployed instead of J\, its values being found in the line headed log. (— P rt). A NEW METHOD OF DETERMINING From the expression (ni) ((¢, h’)) = So Tv (2, v), h’ being the multiple of g’, and being constant, and 7’ being variable, we have sim ay A 1 are cos /,° y 2 Gee) Cos (7 ((4, h )) = i Jy = (Ag) = 18)) =F i Tux Sn (ig — 2H’) + ete. (+2 / tee SD 9 Six sin (ag =F HH ) are Six h hi’ ) 98 (ig + 2H’) —ete. Now for h’= + 1, we have, if we write the angle in place of the coefficient, ° {) (pe 2 2) cos (7° D (ig —g')) =4dy S(ig—EB) +25. & (tg —2H’) + ete. 9 (2) (8) — ty sn (ig + BA) — Fy Sn (tg + 2B") — ete. ; and for h’ = —1, we have ‘ (2) (23) Bayes ((g + 9’)) = —tJ_y smn (19 — Ei") — 4 d_y sn (tg — 2.H") — ete. (0) (1) Le 4B Lo 8 Gye 19) 2d 8 Gy Ee DB) 4 ate. sin Since (=m) (m) (—m) (m) (m) (m J_ nh = Weo9 ) =(—1%h, de =(—IP Le , W the last two expressions give (0) (1) (9g —9')) = Jy & (ig — B’) — 2, & (ig — 2B") + ete. (2 (3) ) — JS, SF (ig + BH’) — 2S, & (tg + #’) — ete., 9) (3) (ig +9) = —Iv 8 (ig — B) — Dy & (ig — 2B’) —ete. (0 (1) + Jy % (ig + EB’) — dy 88 (ig + 2H) + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 143 And for the particular case of ¢ = 1, we have (1 (g— q)) = Iu ely — Bi) — OF, % (g — 2H) + 3J, © nin (q¢—3H’) Fete. 2 (4) De ey 0 (g sp Jo" OW Sn (g 55 2H’) an dS, Sia (g aF 3H’) — Cte (2) (3) (9 +9')) = —Iv B(g— EB) —2Jy & (g — 28 (4) Jy SS (g — 3H’) — ete. ye (@) + Jy es (g-+ B’) 20, v (9 + 2H") + 8dy sm (9 + 3H’) ete. (0) (0) Instead of J, , we use (-/,, — 1), as has been noted. If we put h’ = + 2, we have (0 0) ((ig —29")) = $ Jov 23 (ig —E") + Jy SE (ig — 2H") + BJy, 8 (ig — BE") + ete. L Joy °° (ig +B’) —2 Joy % (ig + 2B") — ete. a (h’—7’) In the table giving the values of | tux , we have, under h’ = 2, which applies to the equation just given, 1 : (3) for = 1, log. 2 Ji, = 8.38201 log. (4 Jy) = 4.97120: (0) (4) for? =2, log.( ey —1) = 7.366938n — log. (— 3, ) = 3.85370; (1) ie 0 == 63 log. (— $ Jo ) = 8.859182 efel sete: ete., ete. =a ete (3) (4) We find the values of — 3 -J,,, — 3, in the table under h’ = —2. We see that 1 (W—1) these are the forms of the function we Vn whens anded.— bh and: 24 == 2) In the expansion of the coefficient of (¢g —h’g’) indicated above by ((¢g —h’g’)), we have coefficients of angles of the form (7g + 7H’). These can readily be put into the form (— 7g — 7H’), but the form employed is convenient in the transformation. 144 Arranging the functions ju G); (ua ( A NEW METHOD OF DETERMINING Zs 4 Bi laa ) in this form, we have tos. o() Loe. 12) gy) LH cos sin cos sin —l1 0.06387n 9.0043 0.50740 0.0210 0 — 2 8.9912 9.1106n 7.0000n 0.10827 () —= 3} 7.6493 8.04187 8.8751 9.2437n 1+1 9.6304 9.2856 8.0493 0.1637 1—1 1.72893 9.8663 2.3923 0.5364 1—2 9.9499n 8.4624 0.7404n 9.8432 1—3 8.6032 8.8882n 8.7716 9.9886n 1—4 TATTL 7.6021 8.7243 8.9912n 2+ 1 8.3617 8.6128 QOS 9.8105 ®% — Il 0.8530 9.1584 1.0959 0.0978n 2—2 1.30203 9.7427 2.1675 0.6138 % — 8 9.7486n 7.5079 0.70457 9.4642 i —— 8.3010 8.6435n 8.9138 9.8287n Y — 5) 6.6990 7.6532 3—1 8.9395 8.7993 9.2808 9.8944 3 — 2 0.2191 8.7993n 1.1221 9.9566n 2 — ® 0.91750 9.5368 1.9189 0.5440. 3—4 9.5132 7.6990n 0.5865n 8.7924 3—5 8.2041 8.3979n 8.9445 8.97TIn 4—]1 8.2041 8.6128 8.9868 9.8287 4— 9 8.9685 8.5051 9.7709 9.6839 4h — 9 0.0082 8.2869 1.0348 9.6410 44 0.5522 9.2989 1.6565 0.4085 AL —— I 9.2601n 7.8451n - 0.4244n 8.6128n 4=— 6 7.9542 7.90938n 8.8976 9.4298n j= 2 8.8855 8.2049 9.8564 9.4506 i —— A 9.7672 7.0000n 0.8905 0.1732n 5 — 5 0.1976 9.0414 1.3848 0.2360 5 — 8 9.0000n T.1782n 0.2347n 8.8633n 6— 3 7.9440 8.1864 8.9385 9.3876 6 —4 8.7404 7.9031 9.8370 9.2129 68 9.5119 7.6021 6.7129 7.7782 6 — 6 9.8494 8.7782 1.1066 0.0394 Cy 0.0224 8.84517 7—6 0.5132 8.6990 [—17 0.8222 9.8156 7—8 9.7973n 8.7924 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. We will now give examples to illustrate the application of the tables for trans- forming from eccentric to mean anomaly, in case of the function u(5). For the angle 39g — 3q’. 5 i! ae «(3) ee g Jaf cos sin (i; = 8) Log. Product. 3— 1 8.9395 8.7993 6.9404 5.8799 5.13897 3 — 2 0.2191 8.7993n 8.68241 8.9015 7.6817n 3— 3 0.91750 7.5368 7.718697 8.6362n 5.2555n 3—4 9.51327 7.69907 8.98344n 8.4966 6.6824 3 — 5 8.2041 8.3979n 7.6393 5.8434 6.0372n For the angle g — og’. : (Br == 0) 1—1 1.72893 9.8663 8.388251n O.11144n 8.2488n 1+1 9.6304 9.2856 8.58251n 8.0129n 7.6681n For the angle g+q’. (h’ = 1) 1—1 1.7289 9.8663 6.46392 8.19287 6.3302n Jd5 ley S\N, BIDS (Se Product. + .00008 + .00005 + .07970 — .00308 — .04327 — .00180 + .03189 + .00048 + .00007 — .00011 +8.26978 +0.34414 +8.33775 +0.33973 —1.29259 — .01773 — .01030 — .00466 0.25653 —0.25027 —1.04636 —0.27266 ” " =. WG .000 +0427 £0,198 Qiu LOGE 146 A NEW METHOD OF DETERMINING For the angle og — og. W D1 0.0637n ee 8.3825n 8.4462 .-. + 2029794 +104.75727 4104.78521 For the angles represented by (7g — g’), there may be cases when there are sensi- ble terms arising from g + H’, g + 2H’, etc.; if so, we use the column for h’ = — 1, and apply the proper numbers of this column to the coefficients of the angles named. Likewise in the case of (¢g + g’), there may be terms arising from the product of the numbers in the column h’ = 1 and the coefficients of the angles g + ZH’, ete. This will be made clear by an inspection of the two expressions ©) a) (@g—g))= dv SG — B) — 2S, sn. (1g — 2H") ete. (2 (3) —— Jy % (ig + E’) — 2S, & (tg — 2H’) — ete, sin 2) 2 > C OS y i) 8) COSIN (Ey U (ig + J) = —dy & (ty — H') — 2Sy S&S (tg — 2H’) — ete. (0) ) + Jy % (ig + Bl) — Wy % (ig + 2B’) + ete.; where ((ag — g’)), ((¢g + g’)) represent not the angles but their coefficients. In retaining the form (¢g + 7H’) instead of the form (— 7g — 7H’) we can per- form the operations indicated without any change of signin case of the sine terms. Making the transformations as indicated above, we obtain the following expres- a\3 sions for the functions ue) and ua?(“) § THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 147 g g cos sin | cos | sin | || ah | | I 0—0 +104.78521 | " | +182.3777 " ino 04636, | — 0.27266 | — 1,6046 —1.9194 O60 = OIO5 030 eI +0.12527 — 0.5606 +1,1949 Bei9 + 0.02860 | S005 GE | + 0.1067 | +0,4943 20 | | — 0.1974 | —0.6468 il ii SrOrat —0.193 | — 0.0830 | —1,4558 (ai hee 0.107 | = Bic So ANOR a + 53.583 0.734 1246,9027 +3.4023 OA 12 1986 = oil | + 5.3656 —1.4496 Seas | + 0.014 0.066 i — 0.3758 0.8304 => + 0.070 | —0.127 == O05 —1,242 V9 + 0.399 +0.053 | + 0.456 0.848 2 9 + 20.093 0.551 | +147.392 +4049 PD + 1.056 —0.086 + 7.914 eS Jy + 0.027 +0.033 — 0.086 +0.537 v= 8 + 0.00815 —0.01707 + 0.0718 —0.2352 i=? + 0.04342 —0.07447 + 0.0041 —0.9231 8 + 0.40733 +0.03392 | + 9.0442 +0.5514 P= 3 aE) 81338 -+0.340 + §$3.537 +3,432 ba 8 SEONG) —0.036 | =5 61432 —0.659 eee + 0,028 0.010 f + 0.079 +0.449 7 | - | 9—4 + 0.027 —0.043 1 + 0.050 EBT 3—4 +5) 0.275 -+-0.023 | 41° 9144 9.592 aed | + 3.628 +0197 | + 46.016 +9512 pee Ee O39 —0.013 4 4.898 | 0,223 6—4 + 0,021 0.008 | + 0,156 0.188 S's | + 0.020 — 0.023 ! + 0,080 | —0.074 4—5 | + 0.167 0.012 sao | +0,241 5 | + 1,623 0.109 | + 24.829 | +1.565 6—5 | + 0,224 —0.004 ] + 3.306 —0.148 | I | | 4—6 EEO O1e —0.008 EEN OOT —0.250 FG + 0.092 0.007 | — 4,535 | 0.150 6 —6 Out 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 7g = 0, in all the divisions of the two functions, having thus only the angles (0 — #’), (0— 2H’), (0 —3#’), ete., ete.; also (0 — g’), (0 — 2g’), ete. 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 : g Ja! cos sin Ge af COS sin 0—1 1.85407 1.62090n 0—1 + 10.548 — 40.188 0— 2 1.25778 1.51473n 0— 2 + 19.809 — 32.318 0— 3 9.7024n 1.26993n 0 = 8 + 0.906 — 19.852 0—4 0.71012 0.9147n Q0—4 — 4.540 — 9.268 C= 5) 066327) 0:3899n 0=h = Ary = B33 0 — 6 0.4387n 9.0934 0 — 6 — 3.059 — 0.330 0—1 0.1222n 9.8069 0—7 — 0.623 + 0.739 0—8 9.5965 9.8865 0—8 -—— 0.071 + 0.615 For the angle (0 —1), (0 — 2), OQ = 8. 1) ey Ww 1) — 0.041 + 0.024 + 1.722 — 1.007 + .062 — .087 — 0.873 + 1.578 — 042 + .076 + 871 — 1.574 .000 — 0016 + .037 + 1.346 + .003 + .097 + 71.462 — 41.774 — 012 — .O19 + 494 + (791 + 70.548 — 40.188 + 18.104 — 32.714 — .020 — .O11 + 70.573 — 40.196 + 19.809 — 32.318 — 504 — 18.618 fae + 19.811 — 82.319 + 0.906 -+ 19.852 Bene hs, + 0.902 - — 19.355 The numbers in the last line of each case are the sums of the divisions after con- version when 7g 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 3 a of @ . . . . u (“) and wo?(“) 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 J 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. : ° a of @ 2 ° The numerical expressions for u(“) and wa?(“) being known, we need next to have those designated by (#7) and (J), which represent the action of the disturbing body on the Sun. To find (#7) we use two methods to serve as checks. We have first (HT) = s[hyiyy’ + Wd8] cos (g — 9) — 3 [ys + Uys] sin (g — 9) + alhyy — hoy) cos (—~g — 9) — aly. — ty,0;] sin (— 9 — 9’) + ly! cos(— yf) — Br sin (=) + 2[hyry! + 2'8:8:] cos (gy — 29) 9 — LM.’ + Uys8s] sin (gy — 29’) 4. 2[hyiy.! — W’8,5:] cos (—g — 29) — 2[ ldo’ — Uy,6,/] sin (— g — 29') + 2hyvy cos (— 29’) — 217d, sin (— 2¢’) + $[hyys! + W0,8.] cos (g — 3g) — $[ldys + U10,'] sin (g — 39’) + ete. where We Rk b= deh (1) (3) () (3) y2 = 4 [Jo — J, ] On = I [So oF J ] (4) by 2) e) ye = 4[Ja —Jn] & = 44a + Ya I, and similar expressions for 7,', 51’, 7’, 6.’, ete.; noting that y) = — de. 150 A NEW METHOD OF DETERMINING The other expression for (7 ) is (HI) = Shy’ — 8] cos (— Bg’) + My’ — V8] sin Bg) + Mhy! + Wy] cos (H— g) ~ — Sly’ + 18] ein (B— 9’) — ehy, cos (— 9’) + el’/dy sin (— g’) + 2[ hy.’ — h’d,'] cos (— #— 29’) + 2[ ly.’ — U,'] sm (— # — 29’) + 2[hy.! + h’d:'] cos (H — 29’) -- 2[ty.’ + Vd,’] sin (H — 29’) — 4ehy,! cos (— 29’) + 4el’d,’ sin (— 29’) + ete. -++ ete. In both expressions for (#7) we have h =" keos(1I1— KX) > V hi = — cos pcos q’ ky cos (Ml — KG) = Su — a a p. : > —_ 7, Osim | 1 =~-—coso¢k sin (ll— KX) = s4u—, a a pee aN, : > — 1. pecosP U = cos @’ k, sin (11 — #) = or a. where as before w= ,206264.”8 and a=“. 1-+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, ¢, 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. Pk We have in case of Althea te € 3e 2e (0) Log. (J—1) = 17.20740n 7.808947 8.160251 8.408900 (0) Log. J — 9.99930 9.99719 9.99368 9.98872 (1) Log. J — 8.60344 8.90341 9.07774 9.20016 (2) Log. J = 6.90632 7.5077 7.8587 8.1068 (3) Log. J = 5.0329 5.9356 6.4630 6.8365 (4) Log. J = F087 4.2384 4.9418 5.4403 ' - (h—7) i : : From these values we may form a table of —-/,, as was done for the disturbing h = body. The values of these quantities can be checked by means of the tables found in ENGELMANN’s edition of BrssrL’s Werke, Band I, pp. 103-109. Finding the numerical value of (£7) first by the second expression, we get H g | cos sin | / ea | 448.154 40.651 == a0 | + 0,188 —0,102 Qa i | BEG: —0.044 1D SEegigad 40.062 ie + 0.018 —0.010 O29 — 0.344 —0.004 iene} + 0.37800 1-0,00510 a3 + 0.00141 —0.00081 =e — 0.03048 — 0.00036 To transform we change from (hH—7'q’) into (7’g’ —hH). Making the transfor- mation, writing also the values found from the first expression for the sake of compari- son, and the value of (J) which will next be determined, we have 152 A NEW METHOD OF DETERMINING (1) (1) Ge gy cos sin cos sin sin cos ” " 1 " " " 0—1 — 5.826 —(0).066 — 5,824 —0.066 +-4,799 2,043 0 — 2 — 0.560 —0.006 — 0.562 —0.006 +0.463 +0.197 0—3 | — 0:04566 —0.00057 — 0.04575 +0.038 0.016 == | + @149 —0.103 + 0.180 —0.103 t=—i-| Aone 10.650 148.079 0.650 1—2 sats 4.637 +0.062 + 4.605 + 0.062 i= 3 | | 0.387740 +0.00502 —_ 0.37738 -++-0.00510 | 2—1 Se iL Oi --0.026 + 1.927 0.030 panto | 4+ 0,186 0.002 | 0.186 10,002 Y= 3 - 0.011 0.000 + 0.015 0.000 To find the numerical value of (Z) needed in case of the function a” ( ab we have (1) = where b = —“ cos¢’ sin 7 cos I’, i + 460’, sin (— 2g’) + 407’, cos (— 29’) + 9 bo’, sin (— 39’) + 9 by’, cos (— 39’) + ete. bs’; sin (— g’) + + ete. Biy'scos (— 9) j= sine acim 2 Having the values of « (4), war Gy. (77), and (J), we next find those of THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 1538 from a =u (“)—(H) UO TON Laer ee ei ON oF dr Ie (5) E ~ @ a zit (‘) (i) 5 dQ 9 s ind Mig 7 , ) iP = jl (=) = % sin (f’ + Il’) + (7) where rn Le Wig ea) 7 A a> 1+ 3@— 4J, cos g — 4d. cos 2g — 4d, cos 3g — ete. sinI 7’ (A) (3) : ; = jam (77+) =— Ge aye ¢, sing’ — s[doy + Sry | ¢ sin 2g’ — ete. (0) + 3é¢ a oe Jecug—e oor — Shy 16 Cos Qari. c, and c, being given by the equations sin I c¢, = —— cos f’ cos Il’ gma! _o = sin IT’. 2 [.- = “|= = [9.5769400] — 2[8.38238] cos g’ — 2[6.46366 |cos 2g’— ete. + 2['7.99450] cosg + 2[6.29667] cos 2g +- ete. —str x sin(/’+ II’) = [7.18046] + 2[8.39074] sing’ + 2 [6.77809] sin 2¢/ — 2[8.01941] cos g’ — 2 [6.40668] cos 29’ A. P. §.— VOL. XIX. T. ° 154 A NEW METHOD OF DETERMINING In multiplying two trigonometric series together, called by HansEN mechanical multiplication, let «a, the coefficients of the angles 2a in case of the sine, 2, those of the angles ux in case of the cosine, y, those of the angles vy in case of the sine, and 4, those of the angles py in case of the cosine. The following cases then occur : ad, Sin (Aw + py) + 30,0, sin (Ax— py) | a, sin A@ . 6, Cos py = tol 6, cos ue. y, sin vy = 4 B,y, sin (ue + vy) — $6,y, sin (ue — vy) GB, cos ua .d, cos py = & 3,0, cos (ux + py) + 3,6, Cos (ua — py) a Sin Aw. y, Sin vY = — $ ~My, Cos (Aw + vy) + Fay, cos (Ax — 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. 3 : ozo d2 »6dQ Performing the operations indicated, we have the values of aQ, ar As that * follow : THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Q (da aQ, ar(“ ) a(“ = dr dz G Gf COs sin cos sin GOS I Wy aA : /} dW O10 +104.78521 we 116.5202 1T>: 0.2828 i= — 1.04636 —.27266 — 2.4398 — .6940 —2.6311 = 0 = (03031 Sosy || = song SEES — .059 3= 0 J) 20 ORG SE ay Sec = O17 a) aE ORi —.090 = 3 E355 .000 C= + 4.662 1.173 ; == iL168 4 481 es ek + 5.504 1.084 eeis39 190 HENS 2 — il —= iil —.201 | = 1.652 —= n'a —1.596 Bye oie +" .066 | = O40 4+ 288 — .059 @—® a 2632 1 | —- 497 —= ul — 020 Wwe — 4206 —.009 | 9.136 + .200 — 9.474 ee 9 + 19.907 1.549 | 145.566 11.270 + .095 5) Seino sG —.086 | 1.642 == il — 922 Velo ee erp EOE Pes 1) Ses — .064 | = 3 -_ (OD =o | 22 ms = 0G — 00 ees = $2306 =O) | = 245 = S208 — .045 9—8 J. S999 dL EBRD |) == PANGS — -L SES —1.494 = 8 +. 8.338 +.340 ee oTeoon 1,087 — .064 A} JL Gn —.036 ATINTOG — .269 = S19 53 4. 028 +.016 a 043 aL RK = 09 eae Op —.048 = 054 —= M0 = 048 4 JL ON 023 = a + .908 ae ed ye 4 8.608 1.197 -+15.430 + .882 = 038 Heed JL) AG (O18 J 838 137 — OD —— a ee oo 4.008 = Og + 063 =O 3 —5 a020 == 023 — 034 = O78 + .020 Nees 2 GY 1.012 = si + 044 = Amd 5 = 8 +L 1,028 +109 4+ 8.605 + 543 + 024 6—5 JL Oy —.004 JL OSI + .064 — 52 AG + 0.012 =008 — 0.075 —0.095 ieeeG Ap 1.007 — 9,995 + .026 6=—6 SE pol 1.059 + 4.559 e386 156 A NEW METHOD OF DETERMINING 3 : : 0 : dQ Having a© we differentiate relative to g, and obtain a—. ag ig We then form the three products, A. a Bar =). Oxa: =. To this end we find A, B, C, from A = —3 = 22 = cos (7 — g) +215 + €] eos (7 29) — 2 [BS + 282] cos y + 2£ cos (y--39) + 2 cos (y —4g9) +- ete. dr B=—2— sein — 9) —2 [$+ 3] sin (y—29) — 2[5 + qee"] siny — 236 sin (y — 3g) — 2¢sin (y— 4g) — ete. C= 2[f—e] sn(y— 9) + 2 [$—75e'] sin (y — 29) + 2[—#e+ 46] sin y + 23 + 2 te + ete. sin (y — 39) sin (y — 49) The numerical values of A, B, C in case of Althza are A=—38 + 2 [0.802429] cos (y — g) + 2 [8.604489] cos (vy — 29) . — 2 [9.304508] cos v + 2 [7.2076] cos (y — 89) B = —2 [0.001399] sin (y — g) — 2 [8.604489] sin (y — 29) — 2, [8.606234] sin y — 2 [7.3836] sin (vy — 89) C = + 2[9.697567] sin (y — g) + 2 [8.80066] sin (y — 29) — 2[8.77953] sin y + 2['7.08265] sin (y — 39) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 157 For the three products we then have O O Al. al =) if; ar(=) Ol ch) dq dr ike I g 9 sin cos | sin cos sin cos VW VW yl VV dy ah o=0 EROS 35 yeeO 537 Ian a = IetS4iee 0.6804 —1,3464 —3.0038 h—@ = He JL 565 Eon ee srs + 1987 + .2411 i =O — (9530 +1 .0439 — 32.9502 + 0549 = Ay == ASOD 2—0 = 9p 1 299 SE oisoee 1657 (9049 + 0998 20 4 2.079 = 507 |) ee LIBI® ) SL Gea | esis | Beoee FO Je Gil 4.457 Z J068 2 B80 4+ 083 + 2404 | | =O — I ++ 462 ae GH 4. £89 = 883 | = iG +E 948 Sees = AG = ON cess + 461 ase 14 454 Oi —10.992 SE 1158 —18.335 a. Sy ly as BR eGo Ci ae GD ae ie a 4 (349 998 + 572 eae AL BRA | Se hop) 2S a0 Si aes 1a SGING) PS 0} Bon eG 41,906 —4.470 aT = Bho, aL darn 4+ 306 4 (216 4+ 1067 _ 098 oi pie 4 949 INGS58 = aS SECTS, | 859 Boi PSR) 1 — 929 4 1559 Se 5 ee nIn(G0 ey 033 aL 2G = 2 = HG 5,229 4.982 000 =— OEY — 09 2 iA Oo 4 6.837 4+ 026 ST S00 nee 235 —1.230 = + 3.029 O— 9 ee se om = 00 i 5 —80.684 19.195 — 45.412 LIGA || a Gye eS Ba 129 eis 002 iy 132 4+ 406 = 18 dk 0 79 | acqgie Louies me omen 364 — (G7 LOL 22 SENGLES, | OL) Sony See ORs D798) 81083 3 5) 4 499 aL BiG 4 LS + .168 4 024 L 023 BaD —19.078 42.954 145.412 —1.264 = 053 > = 8 es a 3 0 — SG kB af OE 5 — 2 £08 4. 1955 He OO 1 = GR | G8 SE 15985) * —-s.1553 4. Gin = SOG |) as OE Ey (ees = 7657 Tey | 9 =e IW 2b AGM | OS SET 13 = eel 9 Joanie So cin ora Eos! 20180 5) 8} —50.140 +1.905 —27.9994 11.0854 JL 8 eS tea a8 aa) 60, Le ales = song | 2 oR; iG) EE SRG 36 = S30 | 2 ep — 2.8964 — 2201 1p a ae 28 nico 2 073 — fai == 0 | ae oe Sane L—® 4 263 + 1190 See 2 010 4.005 dbede 3 49.676 +2.079 497.999 —1.083 to39 0 206 53 S95) e264 Eo oe eee em nO 4+ 534 158 A NEW METHOD OF DETERMINING Ala) /55 ar(S) C. aS) dg dr dz Me Ol OF sin cos sin COs sin cos ih dW ih II iA VW 1 Weed == J1G5 = ahd == 1038 JE ls 1 Oe 4 — 2,929 ae TY + 264 + 989 == 389 +1.029 = ee eee aL Oil +. Oli oe. rr 2008 + .014 ie eee —929.032 1.564 —15.481 + .915 L 022 33 —1 3—4 JL 058 ==, IR = 039) + 175 5 {pA + .051 1 4—4 — 1.0638 — 287 — 1.504 = (0S) — .140 — .3800 ied — 4 + 1,268 — 024 022 2938 L .390 —1.033 i he —28.751 +1.597 +15.479 = 5 | .033 9 =| G4! — 4,543 — .108 + 1.506 + .098 i B= 5 = 180 =5 156 L .002 + 088 ily ey ees — 1.654 a 132 ==" 1063 + .063 — .206 26 Bi) 78225 4. Oe + 014 = {001 — .008 _ 001 ae00S i 235 —16.185 JL 08 == 8.66! + 1544 | 034 + .038 —l 4—65 + 015 — .148 — 045 36 8 — .035 + .004 il B= 6 == IOI = .153 = IL — .036 == (0 = n68 —l 5—5 —E 294 — .017 + 062 — .063 | 206 = G3 —l 6—5 —16.0388 +1,100 + 8.661 — .b44 i 86 = im = 08 Teer een — 1.088 aL 083 | + 9.052 + .088 1 5—6 = BOL aL 08 == Aol6 —- 387 = NT 226 == 8818 ae pul + 4.516 == 38K Next from 2 dw we find the value of ——. ndt OW = 4. a(“) + B.ar (=) nat dq Ta Then we find W and —“. from Cost THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 159 We first form a table giving the integrating factors. From log. n’ = 2.4758576, log. n = 2.9823542, we have “ = 0.34954524, ° *, e °, nv ° * nv 1 ° ° * n’ ° *, nv 1 = =o = ary, L = og, = OF amar iy 4 vl a+r = Log. (‘4 a a Log-() CU) Cao Log (i+7 =) Log. (7) —2 — 1| —2.34954 0.37098n 9.62902 3 — 3} +1.95136 0.29034 9.70966 —1 — 1} —1.34954 0.130187 9.869827 4— 3} +2.95136 0.47002 9.52998 Q — 1) — .34954 9.54350n 0.45650 5 — 3] 3.95136 0.5968 9.4032 1—1} + .65045 9.813217 0.186783 Th 2) SIE TESTL 9.60008 0.39992n 2—1} +1.65045 0.21760 9.78240 2—4| + .601819 9.77946 0.22054 3 — 1} +2.65045 0.4233 9.5767 3 — 4} +1.601819 0.20461 9.79539 4—1} +3.65045 0.5624 9.4376 4— 4) +2.601819 0.41528 9.58472 2) 1.69909 0.230217 9.76979n 5— 4) +3.601819 0.5565 9.4435 0 — 2} — .69909 9.84467 0.1554n 6 — 4; +4.601819 0.6630 9.3370 1 — 2) + .30091 9.478423 0.521577 2=—5) | 2522974 9.40187 0.59818 2 — 2} +1.30091 0.11425 9.88575 3 — 5| 1.259974 0.09770 9.90230 3 — 2} +2.30091 0.86190 9.63810 4— 5) --2.952274 0.385268 9.64737 4 2} +3.30091 0.5186 9.4814 5 = |) =3.2520714 0.5122 9.4878 5 — 2) +4.30091 0.6336 9.3664 }/6 —5| +4.252274 0.6286 9.3714 0 — 31 —1.04864 0.020627 9.97938n 138 — 6] + .902729 9.9556 0.0444 1 — 3) = .04863572 8.6869553n 1.3130447n ||4— 6] -+1.902729 0.2794 9.7206 2— 3) + .95136 9.97835 0.02165 5 — 6| +2.902729 0.4628 9.5372 In regard to this table we may add that the form of the angles is (¢g + vg’) = (i 14 2) g= (7 4 ) nt. he differential relative to the time is (2 414 “) ndt. g n 7 t The preceding table is applied by subtracting the logarithms of the column headed , 1 log. (i aE “), or by adding the Jogarithms of the column headed log. (aa): nN We will now give the values of —— W, and ——, remarking that in the inte- nat cost grations the angle y is constant; after the integrations it changes into g. 160 A NEW METHOD OF DETERMINING ae W = ndt cosz : | vy OF of sin cos | cos sin | cos sin 7 ” an yy 7 Sale " ” 1 O=@ 2. BPRG 1S |) == Wei 26 BOR pi | SONS; — IAAL ia! 1 1—0 + .38901 + .9373. — .3901 ak 9857/33 == 1287 + .2411 1 1 0 32.6972 —- .0988 — 32.6972 + .0988 | SL BIST T — .4802° 1 Y— (() —~—s-—« 20738 = 4647 — 6.1036 + .23823 | + .0024 + .0114 =i 90 4 OAS =k OSL = A OS AE. LASS —l 3 — 0 + .13850 + .0850 — .0450 + .0283 — .028 + 0801 Te) eal 4. O04 == W167 BSB aL Oy — .033 = ld 1—1—1 + 187 + .446 + .115 — .330 —0.62 — 1.60 nee, es = 307 sb B40 — 83.900 = oe +1.013 + 1.84 —l 0—1 — 015. + .530 — .045 — 1.516 — .652 — 1.64 1 1—1 + 4.609 —1.374 — 1.087 — 2.112 - +1.264 — 2.74 ie Seibel Lo Bn = AS) — 1.030 — 5D —1.370 —= R21 1 4 —— Il — 036) = 153 + 022 + .45 — .040 + .06 —l 2—1 aE OR) SE OGL — 4,263 + .038 | a KO § — 21 eee al = OG) Se LOOT JE 9B | =I) Bi “LAB = O84 == 4) = Oy = 26 + .670 Teas al oT a als J OBL 4 029) 1—1— 2 — .03 — .ll 1 0—Q + 14.145 + .261 + 20.207 = oie —1.76 — 4.33 1 o¥ 12ALG ALBIN) +419.660 +11.503 59 — 128 he PBS) 116 + .408 + 2,380 + 1.356 + .46 + .96 1 2— 2 -—— 1.8387 —1.119 + 1.410 — .860 ++ .36 — .78 —Il 2—2 + 9116 — .475 — 17.008 — .365 — .95 — 2.34 la 8.2289 Ho AS — 204 42 = He Oil <1 §—9 — 33.666 -+ .990 ++ 14.632 + .430 == 02 = 19 1 te59 = Ole = 1% SE D5 SE 033 Se vee) — 8150 = 1G + 9.469 — .035 — 4 4. 20 = 9 = PIO se 4099 e050 4 {pH 1 O08 J- 1009 = 4914 | - 10ie6 +. A501 == .05 Seals 1 ls 3} — 1.5475 -—- .3568 — 31.8180 — 1.335 —14.56 — 371,33 =I ils = (012 + 0893 | = O52 = Lear 4. D5 = BY ae = 78 — 77.4394 +9.9904 | + 81.400 + 3.139 = 0 == 318 1 2 3 EL 2Ory Se ee — .3124 + 1631 + .06 + .14 Te 338 = 397G4 — jel | = ier9 = BB5 1 183 = 28 =—l 8—=8 | + 98708 = 235 | — 1916 = 192 = 28 = ol 1 Gea B | sk Avis se Bey = 050 a ealiity .00 .00 =| 48 = OA, +. $98 | — 74g JL SS = OI = Sl 5-58 Seo 196 ee 047 a eo = 12 = 08 ae 8 ih Shas = Ge } = Ane = 008 = 2 i Q—4 — 1.96537 41.126 | + 3.265 + 1.871 ae 64 (eae ole ae = Wiis LO | aL BILTOD + 1.548 = 0 = =—1 8-4 = 81 = 12 SO = 00" 4 O15 + 08 1 he = W5G7 = £5 + 986 = 149 J BA =D) =i 44 ae 002) == 7963 == Ba = Bi = 150 = to esd = (PQ sb soy SAO G + .016 —l 5+4 = 1B Ae 832 + 3.686 + .190 = 00 — =i G4 == ROBT — {00 + 660 = 2 " | THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 161 LW sae u : W —. ndt cost Lh Bor eaY Ve RAE S : = mG) of sin cos cos sin | COs sin i ” " A Tima 1 E i wi 1 38—5 =— iovily —- 21195 se Tae — 156 + .165 46 —l 3—5 + 011 + 017 — .009 SE 04 — .001 el 1 4—5 — 24.846 +1.626 + 11.080 + 722 — 015 == 02 1 4 5 030 — .072 + .013 — .032 | —- .016 00 a 5 — 5 — 2473 — 194 + .160 — .060 | —— 025 — (05) —l 5 — 5 -+ 306 — .080 — .110 — 0294 | G4: —= ,llis 1 6—5 — .089 + .160 - 021 + .038 | =I 685 = IRir = §58 Sei aL 180 | —l 5 + 1.413 + .036 == 2 + .007 | | | 1 4—6 a 964 + .124 = py + 07 | 1 5 — 6 — 13.228 1.090 Se ei f5x5)5) + .38 | —l 5 — 6 — 167 + .023 -- 057 + .06 1 6 — 6 — .946 — .002 —- 242 00 —l 6 — 6 — %.098 — .040 -- .038 — .01 1 7 6 3.302 —- .824 + 674 —— .09 The part of W independent of y arising from the factor, — 3, in the value of A, has not yet been given. Its integral, or f= oa cr is the following: dg S = 8a) GG! cos sin 1 Of GF Cos sin 1—@ | + 3.1399 aL ‘8181 lik tas 3 — 2.74 jullie == 0) SRE! Sear Say es al — 08 3=0 | = (eas = aes Qa | = 48 ee he 51 Ae OH | eA iy =13 L=—i | —39 = 0) A—f | S164 =P 91 Oey = 9.33 J Read | = 168 +.05 al = 204 — .22 || 6-4 | — .08 —.03 | p= 9 +41,934 JE 0) | 3— 5 |= Aen a6 Qs 9 Teo —2s8 | 2=—5 | = 29 —.06 2 9 |) = 48 4k Bl Rf = YA) —.50 49) — 10 = 19 6—5 | = 86 ALD 1— 3 | 20,0020 =e] 2—6@ | — 07 +.05 Q— 2 | — IAB = Md 5— 3 = 68 —.04 3123 5 8846 —1.57 || 6—6 | — 3.35 2 A. P. S.— VOL. XIX. U. 162 A NEW METHOD OF DETERMINING Having the values of the coefficients of (+ y + 7g + 7g’), both for W and —<, we have next to find those of (vy + 7 + 7g’), and of (Oy + 7g + 7q) in the case Uu cosa The expressions for this purpose 7? = 7) =— vo) = yO = For Althzea we find log. 7 = 8.60309 log e. n® = 7.388368 are aly _1 9 te — le — se Bis ILS at se 1T28@ le — (ge + =4e + etc.) lox. 1 = 9.081960 We multiply the coefficients of ( y + ig + 79’) by 7, and 7, respectively, to find those of (+ 2y + tq + 79’), (= 3y + 9 + 19’). In case of (Oy + zg + vg’) in the expression for <= we add the coefficients of (+ y + tg + 19’) to those of (— y + 2g + vg) and multiply the sum by 7. dw We will give a few examples to show the formation of WW, and — 4—. dy With these two we give at once also their integrals, which are néz and v respec- tively. = 1W Ww = i° > ihe (O—) cos sin sin cos ida W ” W ail =O —=890909 =E.0988 +16.3486 +-.0494 — 2 2— 0 0190 —-.0017 + .0190 +.0017 —32.7162 +.0511 ” wu —32.7162 -.0511nt THE GENERAL W ” —1 2—0 — .4i74 ® T= 23766 2—1—0 —1.314 1 O—O —1.2175nt st (1 — 0) ” | ; uw + .042 } ++ .237 + .818 | a — 004 —1.314 — .6087nt 3.2376 nt PERTURBATIONS OF THE d W + .004 —1.6188n¢ ia) MINOR PLANETS. a Mm ” wW dA 1.351 —1.2175nt + .856 +3.2376nt ” wu —1.07TT —.6087Tnt Mu " + .025 —1.6188nt “ UA W f? ” ” LA dh +459 —1.2175nt —2.07 —3.2376nt —0.54 +.6087Tné —0.58 —1.6188nt Cae pag) ” Ww | A " 1—2—1 + .883 + .070 +.191 —.035 = 1 0— 1 — .045 —1.516 + .022 —.158 —2 1—1 — .041 — .030 +.041 —.030 @) —= il —1l —..913 + .200 = — —0.216 —1.246 | + .254 —.823 W A | W uM SL G == oP AL --.61 (l= 1) Ww " | Mp Wy OD ores ee 99 = 2004 ! + 029 —.004 —1 Bil == 43} + .038 + 2.131 +-.019 0 1—1 — 25.390 — .390 — — 1 0—1 — 83.900 — .973 —41.950 -+-.486 —113.574 —1.329 | 39.798 +.501 MW uw | uw WwW —174.61 +2.04 | +61.19 10.77 : : = W In the integration we apply the proper factor to each term of VW, —3 ,and obtain the values of ndz, », except in case of the terms (7g + 0g’). Let us take the term (gy — 0g’) or (1 — 0), and let uw the integrating factor to be applied. Let c, a, d, b, represent the cos, sin, nt cos, nt sin terms respectively. 164 A NEW METHOD OF DETERMINING Thus we have € d a b V/ ih I // > +1.351 —1.21T5nt --.856 +3.2376nt ; and hence (“e wb ud — Lea wd —ub uw Wy} V1 1] Pet VT -+1.351 3.2376 —1.2175nt —.856 —1.2175 —3.2376nt or, since w is unity, dt a VI 1] +4.59 —1.2175nt —2.07 —3.2376. In case of the term (2 — 0), u is E. In the way indicated we derive the values of ndz, and ». In the case of sont we have the values at once without another integration as was necessary for ndz 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 It) + hk, cos y + ky sin y + 7) k, cos 2y + yk, sin Zy + ete. 1dW Then in case of —3 7, We have r $k, sin y — $k, cos y + 7) ky, sin 2y — 7 k, cos 2y 4 ete. Having W from the integration of ae we form W from the value of W and converting y into g. We thus have from the equation 9 ait W+e() , Selah, +(1”.351 + %,) cosg ~ + (0’.856 + k,) sin g -— 1”.2175nt cos g + 3/.2376nt sin g + (—'.284 + 7 k,) cos 2g + (0.589 + 74 hk.) sin 2g —'’.0488nt cos 2g + ’.1298nt sin 2g + ete. + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 165 Tn the second integration the constants of ndz and » are designated by C and VV respectively, and the complete forms are C+knt + k sin g —k, cosg + $y, sin 2g — $n hk, cos 2g + ete. N — 1k, cos g — 4k, sin g — 37 k, cos 2g — 37 k, sin 2g — ete. In case of the latitude the constants of integration have the form 1, + l, sin g + 1, cos g. We thus find nz = O+[1+ & —82".7162]nt + [4.59 + k,] sin g + [—2’.07 —k,] cos g — 1’.2175nt sin g — 3”.2376nt cos g + [—0”.11 + $y, &,] sin 2g + [—0”.31 — $7 k,] cos 27 — 0’.0244nt sin 2g — 0”.0649nt cos 2g —+ ete. + ete. vy = +0”.0511nt + NV + [—0".54 — h,] cos g + [— 0.58 — $h,] sin g + 0”.6087nt cos — 1’.6188né sin g¢ + [07.05 — 37” k,] cos 2g + [— ”.24 — 37 &] sin 29 + 0”.0244nt cos 2g — 07.0649nt sin 29 + ete. + ete. = = 10.3616 + 0.3623nt Cos2 + [1.52 +14] sin g + [—0”.68 + 1] cos 9 —1”.3464nt sing — 3’.0038nt cos g + 0.32 sin 2g — 0.16 cos 29 — 0’ .0539nt sin 2g — 0’.1204nt cos 2g + ete. + ete. 166 gg 0— 0 LW 2— 0 0—1 0— 2 0— 3 1—1 2— 2 3— 3 4—4 5—9d 6 — 6 1— 2 2—4 1—3 2—1 2— 3 3 — 2 3— 4 4—3 4—5 5 — 4 —l—1 The complete expressions for ndz, A NEW METHOD OF DETERMINING NOZ sin cos +k, nt _39,7162nt = Aah ook — 1.2175nt — 3.2376nt — 011 + 44k, — ‘31 — 37%k, — 0.0244nt — “0649nt + 3.10 — 3.09 — 3.00 + 1.92 + 0.23 — 1.76 —174.61 + 2.04 +263.97 — 7.21 + 25.15 = 8 a” Ril — 025 an 164 = Oi zee 49 = 0% 185.18 4. 9.1 — 41.10 — .71 +410.16 —87.44 — 5.25 + .87 — 31.94 + 8.03 + 6.17 + .04 + .90 — .86 i. + 04 ily — £8 = 34 Oil + .16 = WW COS sin + N +- ‘O51 1nt 0.54 — Hh, 58 — ih, tL 0.608Tnt — 1.6188nt aE 05 hey OA Ly Ke? + (0244nt — 0649nt + 2.12 — 1.54 — 1.30 — .95 1 HO 4 28 + 61.19 eT —156.21 — 4,24 — 18.30 — .56 — 4°68 — 29 — 1.45 — .09 — .50 — .04 — 43.97 + 07 -- 36 — .01 + 14.64 + 3.15 + 4,02 aL 52) | 16.07 aL SG = 208 — .Ol — 1.05 — 300 =>. 369 4c 05 — .33 — .04 -— 38 .00 1 AO 4. Gil | v, —— in tabular form are the following : Cost U COS 4 sin cos ef 0.36 ; a '3628nt SEO sph ae h == 98 IL I, — 1 3464nt _— 3.0038nt a “39 — 16 — "05390 — 204nt — 4.83 — 2.03 aes J Bil — .87 + .25 + 2.69 + 1.26 — 1.15 = OT — 1.60 — .60 4c 08 ze 02 = 6:64 — 270 = Ail = 7 4 4,43 ss — 1.98 == 299) —38.24 —14.92 = py + 20 si 11.30 SRD — 24 + .03 + .28 + .10 — 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+ k,sing— k,cosg + 47% k, sin 2g — $y” kh, cos 2g + ete. + (ndz)y = Yo us (ndz), = 0 k + k,cosg+ ksng-+ 7k, cos2g + 7°) k, sin 2g + ete ai ole NN — $k, cosg — 3k, sin g — 37 ky, cos 2g — dy k, sin 2g — ete. + (v)o = Smale + hk, sin g — 3k, cosg + 7K, sin 2g — x7 ky cos 2g + ete d+ %sing + 4 cosg + 7 1, sn 2g 4- 4%, cos 2g + ete. + (a) = 0. Ll, cosg — l,sin g + 7” 1, cos 2g — 7° 1, sin 2g + ete. 4 ( = .)) = 0 To find %, and k, we have k, [cos g — e + 7°) cos 2g + 7 cos 3g + ete.] + ky [sin g + 7 sin 2g + ete. | — 84, + 6 (r)) + 4-5 (nbz) = 0 k, [sin g + 27° sin 2g + 37° sin 3g + ete.] — k,[cos g + 27” cos 2g + ete. ] d aa ar Dae (ro == where N= —2k—ja4—44%4, 4 = — 82".7162, 6 ky being found from hey = chy + 3%, — 3 (nd2)) — 6 (0) We have also 1 = — él, The symbols (ndz),, (v)o, etc., represent the values of néz, v, etc., at the Epoch. 168 A NEW METHOD OF DETERMINING To find the values of the angles (2g + 7'g) at the Epoch we have g = 332° 48’ 53.2 g = 63 5 48 6 The long period inequality, 5 Saturn — 2 Jupiter, is included in the value of g’. 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 aan (ndz), ete., we can make use of the integrating factors, multiply- ing by the numbers in the column (« a < )e Having their differential coefficients we proceed as in the case of (ndz), ete. We thus find (néz)) = + 401”.7, (v)) = + 180”.6, (“) = —22”.6 Sy (te) = 89026, 2G) 7075) (——) = are. “ndt ndt \eost And from these we have SLOOP i OD SOP, ih = OO = — 45.2, iL =-+ 0".4, NG 28.3. C= Be Wy 1a", 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 C now has the value C’ = 332° 44’ 16”.3. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 169 Introducing the values of the constants of integration into the expressions for U nz, v, and— _, we have COS? nz = 332° 44’ 16.3 + 855’.5196 ¢ + 417.4 sing + 80.8 cosg — 1'.2175sing — 387.2376 cosg sip O72 sim2g ee o.0) | cos 2g; 0".0244 nt sin 2g — 0'.0649 nt cos 2g + ete. + ete. i ==) 284.3 + 07.0511 né — 206.9 cos g a 4079 sing + 0”.6087 ntcosg — 1.6188 ntsing — 8'.2cos2g + 17.3 sin 2g + 0.0244 ntcos2g— 0.0649 nt sin 2¢ + ete. + ete. See 104 4+: 07.3623 nt Cost — 44’ 2sing — 0.7 cosg — 1'.5464ntsing — 3.0038 nt cos g — 1’ 5sin 2g — 0.2 cos 29 — 07.0539 nt sin2g— 0.1204 nt cos 29 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" 48™ 52%, 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.—VOL. 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, OG = sas? UB) sel, g = 60° 24’.1. Forming the arguments of the perturbations with these, we find noz = + 4 437.2, ps LBB, ee OH (Sy To convert » into radius as unity and in parts of the logarithm of the radius vector we multiply by the modulus whose logarithm is 9.63778, and divide by 206264’.8. Thus we have from » = + 3”.6, the correction, + .000008, to be applied to the loga- rithm of the radius vector. ; U 4 In case of —— = — 2’.8, we have cost (2a — 28 S ke => 2, plane of momentum, | 1 : ! Pye pe at AA ee = y¥ + kz — —, surface of energy, r Me ed (4). 9 9 a . 878 1 oy = laws = il, quae OF mechiny. J These equations represent all possible interactions of the system, but in their present form are very difficult to interpret. The general problem to which they give rise seems to be insoluble, but we can solve and interpret them fully for one particular case which is in close accord with the conditions existing in nature; and it is possible to show by analogy that all other cases will be essentially similar to the one of which we shall treat. By taking the case of two equal stars rotating in the same direction with equal angular velocities, or substituting (3) of (4) in (1) of (4), we reduce the plaiie of momentum to a particular line of that plane : ae — He + (14 k) = a — Ae’ + 2 = 0, since & = 1. The equation of the energy surface passes into the form fo) see al Go The curve of rigidity becomes n = 24, wheren = Vy 4 #2 ve Every point in the plane of momentum represents one configuration of the system, 2. é., one distance apart, one velocity of axial rotation, one moment of momentum of orbital motion. This point therefore determines the dynamic condition of the system, and by the motion of this point we may discover the changes which are taking place in any case that may be imagined. As we have restricted the plane of momentum to one line, the guiding point representing the configuration of the system will simply glide back and forth along this line. In the same manner the surface of energy is now restricted to a curve formed by cutting that surface by a certain plane; the guiding point that would slide along the energy surface is thus restricted to one line of the surface given by the transformed equation. [The reader who may desire to examine this question exhaustively must be referred to my Inaugural Dissertation, Die Hntwickelung des Doppelsternsysteme, Berlin, 1893, R. Friedliinder & Sohn. | As the tides raised in the stars are subjected to frictional resistance, energy 1s EVOLUTION OF THE STELLAR SYSTEMS. 229 thereby converted into heat, and lost by radiation into surrounding space; thus the total energy of the system must decrease with the time. Hence it follows that, however the system be started, the guiding point representing the configuration of the system must slide down a slope of the energy curve. In the accompanying illustration the curves are drawn for the value of /Z = 4. If the guiding point is set at @ it may move either of two ways: it may slide down the slope ac, im which case the stars fall together ; or it may slide down the long slope ab, in which ease the stars recede from each other under the influence of tidal friction. This latter case is the one of chief interest in respect to systems actually existing in space, and the several other ideal cases need not be discussed in this paper. The con- dition at ais dynamically unstable, and corresponds to that of the system at the instant when the stars are first separated. At this juncture they rotate as a rigid system, but as each is losing energy by radiation, the axial velocities will soon surpass the velocity of orbital motion, and then the tides will begin to lag, and the mutual reaction of the stars will drive them asunder. Thus the guiding point in general slides down the slope ad. This means that as the stars recede from each other, the period of revolution for a long time surpasses that of axial rotation, but that in time the two periods again become synchronous when the guiding point has reached the minimum of energy at 4, where the bodies once more revolve as if rigidly connected. The question now arises with respect to the changes of the eccentricity. The differential equation for the change of the eccentricity is shown to be ff IT ; = HB 8a np = (@ @) ge Ese uU— — arcan —— ical 1a) sone \ aidan debates (5) J | I ’ ih eee —$—__——__— = i eapea Roe where B is an arbitrary constant; a, 0, a + (, are the roots of the biquadratic equation. a’ — He + 2=0. Equation (5) is illustrated in the lower part of the preceding figure, the origin being shifted downward to 0’ to prevent confusion of too many curves. in one diagram. Now as the guiding point on the energy curve slides down the slope ab, the eccentricity at first very slightly decreases, then increases slowly, finally much more rapidly, until a high maximum is reached, after which it again diminishes, owing to the libratory motion in the system. Thus it is clear that as the stars recede from each other, the orbit becomes highly eccentric, but will ultimately become circular when 230 RESULTS OF RECENT RESEARCHES ON THE the system revolves as a rigid body. This last condition cannot come about while the stars are still contracting and shining by their own light, and hence all visible systems are characterized by highly eccentric orbits. To leave no doubt that tidal friction is a sufficient cause to account for the elongation of the orbits of the double stars, I applied the theory to a special case, in which the masses, distances and velocities are known. Taking two spheroidal fluid masses each three times as large as the sun, expanded to fill the orbit of Jupiter, and set revolving in an orbit of 0.1 eccentricity at a mean distance of 30 astronomical units, I find that by tidal friction the major axis of the orbit will be increased to 48 astronomical units, while the eccentricity will rise to 0.57. In this problem the masses are set rotating at such a rate as will produce an oblateness of about 2, so that the equilibrium is stable. Different conditions will produce different results, but it is easy to see by this numerical example that tidal friction is a sufficient cause to account for the observed elongation of the orbits of double stars. Though it may be supposed that there could be little doubt of the generality of the law of the eccentricity which I inferred in 1888, yet the importance of this fundamental fact of the universe is so great that I did not feel satisfied till all the obser- vations of double stars had been examined anew and this conclusion touching the eccentricity established upon the most unshakable foundation. At length I have been enabled to show by the most exhaustive investigation of stellar orbits ever attempted, that the most probable eccentricity is 0.48; while on the other hand extremely eccentric and extremely circular orbits are equally rare, and must be referred to some unusual cireum- stances. Thus of the 40 orbits now well-known, it turns out that none lie between the eccentricities 0.0 and 0.1; two between 0.1 and 0.2; four between 0.2 and 0.3; eight between 0.5 and 0.4; nine between 0.4 and 0.5; nine between 0.5 and 0.6; two between 0.6 and 0.7; four between 0.7 and 0.8; two between 0.8 and 0.9, and none between 0.9 and 1.0. It follows therefore that by whatever process the stars developed, their orbits assumed a form which is about a mean between the nearly circular orbits of the planets and the extremely elongated orbits of the periodic comets. Now a double star can originate by but one of two processes: either such a system is the outgrowth of the breaking up of a common nebula, or it is made up of separate stars brought together in a manner analogous to that involved in the capture of a comet. That these systems are not the outgrowth of accidental approach of separate stars we may at once affirm; for if we suppose them to be so produced, there being no third disturbing body which acts like the sun in the capture of comets, the captured star would recede to a distance equal to that from whence it came. In that eyent we should observe stars moving in paths of very immense extent, and consequently EVOLUTION OF THE STELLAR SYSTEMS. 231 revolving at the quickest in some hundreds of thousands of years. If the paths be elliptical, the major axes of these ellipses would be of the same order of magnitude as the distance which separates us from a Centauri; while if the paths be parabolic or hyperbolic, the two objects would pass and then separate forever. On the other hand we can conceive of nothing which could diminish the dimensions of a very long ellipse, unless it be something analogous to a resisting medium. Such a medium to be effective in reducing the size of the orbits would have to act for a great period of time, and besides would probably be visible in space as diffused nebulosity. No nebulosity is observed about revolving double stars, nor is there any evidence of a sensible resisting medium either among the stars or in our own solar system. We may therefore reject the idea that the dimensions of the orbits were originally very large, and have since been diminished. As the orbits are now of the size of those of our greater planets, and there- fore comparatively small, it follows that the stellar systems have originated by some process other than by the union of separate stars. As a nebula is a very rare and expanded mass, and is yet held in equilibrium by the pressure and attraction of its parts, it necessarily rotates very slowly ; and hence when it divides into two parts under the acceleration of rotation due to secular condensa- tion, the orbit pursued by the detached mass must be of small eccentricity. For even if the forces producing separation could be exerted suddenly to produce a violent rupture, the detached mass in pursuing its eccentric orbit would again come to periastron, where it would encounter resistance in its orbital motion, and the result of the grazing collision would be a diminution of the size of the orbit, and consequently an exaggeration of the resistance at the next periastron passage; in this way the system would very soon degenerate into one mass. On the other hand were the initial eccentricity small, the newly-divided masses would pass freely, and when the orbit eventually became highly eccentric the secular contraction in the size of the masses would prevent disturbance at periastron. Subsequent collision could not possibly occur, because the periastron distance would steadily though perhaps only slowly increase as the stars are pushed asunder and the orbit is rendered constantly more and more eccentric. It follows therefore that in the beginning the orbits are only slightly eccentric, and that the eccentricity is developed gradually as the result of secular tidal friction working through immense ages. Accordingly in the elongation of the orbits now observed we see the trace of a cause which has been working for millions of years. The existence of this cause and its effects on stellar cosmogony could probably never be inferred except in the manner by which I approached the problem. On the one hand it appears that we have inferred the true cause of the expansion and elongation of the stellar orbits, while on the other the trace left by this cause has enabled us to detect the existence of A, BsS:—-VOW. XUX. 2D: 232 RESULTS OF RECENT RESEARCHES ON THE unseen tides in every part of the heavens. In a fluid universe tides necessarily result from gravitation, and are as universal as this great law of nature. In my later researches I have therefore been much concerned to show from the discussion of reliable observations that gravitation is really universal* and consequently that the tides we have assumed actually exist in the bodies of the stars. It is thus made certain that the foundation upon which our cosmogonie speculation rests is as enduring as the Newtonian theory itself. We now come to the second part of the problem: By what process did the stars separate ? In college lectures I had heard the annular theory of Laplace expounded for the solar system, and yet I failed to see how this theory could account for the separation of equal or comparable masses, such as we observe among the stars. Realizing that the double stars are in fact made up of two bodies of comparable mass, I reached the conclusion while still at the Missouri University that there must exist some process by which a nebula divides into equal or comparable parts, in a manner analogous to that of fission among the protozoa. About November, 1889, very soon after I entered upon my studies at the University of Berlin, I found that Darwin had recently published an important mathematical paper on the figures of equilibrium of rotating masses of fluid, and had referred therein to the profound work of Poincaré. published about a year before. When I beheld the figures of equilibrium which these mathematicians had com- puted, I recognized at once the cosmical process I had already assumed to exist; it was indeed a great satisfaction to see a demonstration that under gravitational contrac- tion homogeneous incompressible fluid masses may divide into equal or comparable parts. The next question was: Are there nebulee of this form in the actual universe? In searching over the paper of Sir John Herschel in the Philosophical Transactions for 1833, I found some drawings of double nebule almost exactly like the figures mathematically determined by Darwin and Poincaré. It was no longer possible to doubt that the real process of double-star genesis had been discovered. Further investigation and reflection haye confirmed this inference, and I believe we may now accept with entire confidence the result reached at Berlin in November, 1889. In the first investigation Poincaré begins with the Jacobian ellipsoid of three unequal axes, and imagines it shrinking in such a way as to remain homogeneous, and yet gain constantly in velocity of axial rotation. When the oblateness has become about 2 he finds that the equilibrium in this form becomes unstable, and the mass tends to become a dumb-bell with unequal bulbs—an unsymmetrical pear-shaped figure which I have called the Anioid. As the contraction continues the whole evidently ruptures into two comparable masses, and the smaller will then revolve orbitally about the larger. If * RESEARCHES ON THE EVOLUTION OF THE STELLAR SystmMs, Vol. I: On the Universality of the Law of Grav- tution and on the Orbits and General Characteristics of Binary Stars (Tue Nichols Press, Lynn, Mass., 1896). EVOLUTION OF THE STELLAR SYSTEMS. LUBY we suppose either mass to contract still further, it is evident that the rotation will begin to exceed the orbital motion ; and the tides raised in either mass by the attraction of the other will lag, and tidal friction will henceforth play just the part we have already described. Starting from a different point of view, Darwin was already at work on essentially the same problem when Poincaré’s paper appeared, and he held his results back for nearly a year longer, hoping to make application of the principle Poincaré had announced. In this second method of treatment two masses of homogeneous fluid were brought so close together that the tidal distortions of their figures caused them to coalesce into one mass; set in motion asa rigid system, the problem was to find the resulting figure of equilibrium. It turned out to be a dumb-bell with equal or unequal bulbs according to the relations of the primitive masses. Thus we see it proved from two The Apoid of Poincaré, showing how a rotating mass of fluid separates into two unequal parts. independent points of view that a division such as I assumed in 1888 can theoretically take place ; and among actual nebule of space such division seems to be a general law. During the years of 1896 and 1897, I have examined a number of such objects in the southern hemisphere, and find them substantially as drawn by Herschel many years ago. Burnham and Barnard had previously assured me that the interpretation of the figures of double nebulse based on the drawings of Herschel was in accord with the phenomena of nature, but the studies more recently made with the great Lowell telescope supple- ments their large experience in a very happy manner, and may be said to remove the last doubt that could attach to the division of nebule by the process of fission. Before concluding these remarks it ought to be pointed out that in space we have to deal with masses which are not homogeneous, nor are the nebule by any means incompressible ; yet many considerations lead us to believe that in most cases the density of 23: RESULTS OF RECENT RESEARCHES ON THE a nebula is not very heterogeneous, and hence in general the foregoing conclusions would not be greatly modified. In this reasoning I have assumed nothing but that the nebule are figures of equilibrium under the action of gravitation. That these masses are fluid is certain, for the bright lines of their spectra indicate that they are self-luminous gas ; on the other hand the same force which controls the motions of the stars must operate among the particles of the nebule, and thus determine the figures of the masses in accord- ance with the laws of mechanics. As the conditions here assumed certainly exist in the heavens, we need only add that when the masses separate they are probably revolving as a rigid system. When they contract under the influence of gravitation, they must by a well-known mechanical law gain in velocity of axial rotation, and tidal friction then begins expanding and elongating the orbits; in the course of some millions of years we have a double star like a Centauri or 70 Ophiuchi. The stellar cosmogony here suggested may be regarded as a very general theory. Our solar system is so remarkable, that it is uncertain whether a theory which explains the formation of double stars could assign also the cosmogonic processes which have given birth to the planets and satellites. The masses of the planets are very small compared to that of the sun, and the masses of the satellites are equally insignificant compared to those of the planets about which they revolve. Moreover the orbits are very circular, and these various circumstances make our system absolutely unique in the known crea- tion. Yet so far as our researches on the double stars may illuminate the problem of planetary cosmogony, they indicate that the separation took place in the form of lumpy or globular masses—not in rings or broad zones of vapor such as Laplace supposed. From the survey thus hastily made of a very large subject, it appears that we have taken a step in the generalization of the theory of tides and of tidal friction, and have indicated the probable mode of formation of the stellar systems. Little or nothing 1s known of the development or even of the mechanism of star clusters; the problem of explaining the more complicated systems must ultimately occupy the attention of astronomers if we are ever to trace the development of the visible universe. As a step in the direction of accounting for the origin of multiple systems, it may be said that observations on triple and quadruple stars have shown that they, too, developed by repeti- tion of the fission process. One or both components of a binary have again subdivided, just as I inferred was the case when still at the Missouri State University in 1888. While the views here expressed are the results at which I have arrived after a partial investiga- tion of the theory of tides and of the figures of equilibrium of rotating masses of fluid and a comparison of these theories with the phenomena observed in the heavens, I reserve the right to modify any opinion or conclusion which future research may show EVOLUTION OF THE STELLAR SYSTEMS. 235 to be unsound or incomplete. That tidal oscillations which were first noticed by the navigators of our seas are at length seen to be but special phenomena of a general law operating throughout the universe is alike honorable and gratifying to the human mind. It is equally inspiring to recall that by the known laws of these phenomena we are enabled to trace existing systems through immeasurable time, and thus disclose cosmical history which mortal eye could never witness. In our time it is no longer sufficient to maintain the traditions of the past, to trace the planets, satellites and comets through centuries, and explain observed anomalies in their figures, attractions and orbital motions by the law of gravitation. We-must essay to discover the cosmical processes by which the existing order of things has come about. Though it seems probable that a fair begin- ning on this problem has already been made, a much greater work remains to be done during this and the coming century. What is needed is a more thorough exploration of the face of the heavens, by astronomers who are familiar with the laws of mechanics; and a far-reaching investiga- tion of the general theory of tides in viscous liquid and gaseous masses such as the stars and nebule of remote space. Even if the full extent of the hopes here expressed can be realized only after the lapse of several centuries, I venture to believe that the achievement will not be unworthy of the past history of Physical Astronomy. ARTICLE V. ON THE GLOSSOPHAGIN (Plates VI-XV_) BY HARRISON ALLEN, M.D. Read before the American Philosophical Society, January 21, 1898. Having an impression that the genera of bats are best defined by minute characters in the skull, teeth and wing membranes, I am led to review the Glossophaginee—a sub- family of the Phyllostomidide, concerning which unsatisfactory accounts exist both as to structure and relationship. The bats embraced in the group are characterized by a slender protrusile tongue, an elongated jaw and a deeply cleft lower lip.* The temporal impression is faintly marked and the sagitta is absent or confined to the frontal bone. The thumb and forearm are ‘long. The olecranon lies on the upper side of the wing membrane. The canine teeth are long and the upper molars without hypocone. ‘The incisors are so diminutive as to permit the tongue to be freely projected without wide separation of the jaws. According to P. Osborne (Proc. Zodl. Soc., 1865, 82) the thumb aids in the seizure of small fruits, the teeth tear through the skin and the long tongue extracts the semi-fluid contents. As in the Edentata, the elongation of the jaws and tongue has led to the sim- plification of the teeth. But reduction in number of the teeth has gone on scarcely at all; indeed, the most highly specialized forms are those having the largest number of teeth. The genera are arranged in three alliances—the glossophagine, the chcernycterine and the phyllonyeterine. The first is composed of G'lossophaga, Leptonycteris and probably Monophyllus ; they certainly relate closely to the Vampyri. The second of the highly specialized and more doubtfully placed group of Chernycteris, Lonchoglossa and Anura, oe asin as are indebted to Prof. W. Peters (If. B. Akad., Berlin, 1868), for a revision of the group of the glos- sophagine bats. The diagnoses are unfortunately sometimes inadequate and without critical analyses of synonymy. The confusion arising from the circumstance last named is to be acknowledged ; as a result, the task of identification when not aided by inspection of type specimens is difficult. Dobson in his well-known catalogue of the Chiroptera in the British Museum, 1878, follows Peters closely—often indeed merely translating or paraphrasing his language—and on the whole shows less acumen than characterizes his admirable work elsewhere. 238 ON THE GLOSSOPHAGIN &. is probably also of Vampyrine origin. The third division contains but a single genus, viz., Phyllonycteris. It is so near Brachyphylla that it would be easy to effect the transition and remove the genus to the alliance expressed by the term brachyphylline. It is akin, therefore, if not annectant, to the subfamily Stenodermine.* The material available for the study just completed was not large, and two genera, namely, Monophyllus and Glossonycteris, 1 have not seen. I have concluded from the published descriptions of G'lossonycteris that doubts can be frankly expressed concerning the validity of this genus. Perhaps not enough stress has been laid upon the effects of age in attempting to separate it from Anura. Reliable characters are found in the lower molars. The extension forward of the ridge (anterior commissure) between the protoconid and the paraconid is more marked than in any other group, and is in consonance with the compression of the crowns. The ridge is not spinose, and is scarcely raised. In Glossophaga the ridge is constantly as in the Vampyri, but in the other genera it is an extension forward from the protoconid. No trace of hypocone is seen in the upper molars. The row of glands lying to the outer side of the nostril is discernible in all genera except Phyllonycteris. Minute distinctions are found in the degree of development of these glands. They are best developed in the glossophagine group, and least so in the cheernycterine. In Phyllonycteris the ecto-nareal gland-row is occupied by a flattened fold of skin which becomes incorporated with the nose leaf.+ The proportions of the width of the third and fourth digital interspaces taken at the distal ends of the metacarpal bones when the wing is extended is found to be as valuable an aid in determining affinities as elsewhere in the order. In like manner the shapes of the terminal cartilages of the fourth and fifth digits, the arrangements of muscles and nerve markings of the wing membrane are noted as furnishing excellent characters. The following scheme of interdigital diameters is given : Second Third Fourth Second Third Fourth Interspace. Tnterspace. Interspace. Interspace. Interspace. Interspace. Glossophaga soricina .....- Q 12 17 = Lonchoglossa.. cog 16 23 Glossophaga truei........-.. 2 11 15 JAGR acconscqcon0nc0ss9008002 3 15 30 Eeptony Ctr is ...-.eseeeeeee 3 15 25 Phyllonycterts ....+++...... 3 13 25 Cher ony cteris........-000+ 2 11 20 Enough can be gleaned in the way of inductions from the shapes of the anterior * In a paper by myself, entitled ‘‘On Ametrida minor” (Proc. Bost. N. Hist. Soc., 1892), I used inadvertently the term Stenodermatide for this subfamily. + The genera of the remote megaderminine genera are in like manner distinguished by characters in rows of glands as contrasted to folds of skin, though the structures are here not ectonareal, but infranareal. In Megaderma the glands are distinct, while in Lyroderma and Lavia they ave supplanted by a skin-fold which becomes an integral part of the nose leaf, ON THE GLOSSOPHAGIN &. 209 extremities and the details in the phalanges and terminal cartilages to warrant the intro- duction at this place of a few remarks on the subject of flight. Leptonycteris. The greatest restriction in the movements of the digits is found in Leptonycteris. The sharp flexure of the second row of the phalanges on the first impede rapidity of flight, while the axially disposed, terete terminal cartilages show absence of strain. The second and third metacarpals always maintain an acute angle to the forearm. Glossophaga and Chernycteris. These genera resemble Leptonycteris, differing therefrom in degree only in the greater degree of interphalangeal flexure and in the angulation of the second and third digits to the forearm. Anura shows scarcely any tendency to flexure or angulation of the parts above named while the terminal cartilages of the third and fourth digits are markedly deyiated from the axial positions and thus appear to correlate with increase of wing strain. Lonchoglossa is intermediate between Anwra and the preceding group. Phyllonycteris shows an isolated position from the foregoing group as a whole, on account of the terminal cartilage of the fifth digit being entirely embraced by the wing membrane. It is a curious circumstance that the remote Leptonycteris exhibits a similar peculiarity. It cannot escape notice in studying the group that the extraction of soft pulp from a fruit is not unlike the lapping of blood. Acquirements apparently so diverse as fruit-eating and blood-taking are not so improbable as they might appear to be at first sight. Geoffroy, who established Glossophaga, yet who had no knowledge of the habits of the species, concluded from the structure of the tongue that the animal was a blood-sucker.* In adapting the head so as to create a blood-lapping from a pulp- extracting form the greatly elongated jaws are shortened, the face flattened, and the teeth become knife-like. In this manner we may trace the transitions which have taken place in the Vampyri in creating on one hand the Glossophagine and on the other hand the Desmodine. In Glossophaga the Flexor carpi radialis passes along the upper border of the radius as far as the distal third, at which point it crosses the curved radius to reach the carpus. In Chernycteris and Lonchoglossa the tendon of this muscle lies to the lower border of the nearly straight radius. The Flexor sublima digitorum has the weakest development in Chernycteris, which form it supplies the first and fourth digits only. In Phyllonycteris it omits only the second, while in Lonchoglossa and Glossophaga it supplies all the digits. * The stomach in the Glossophaga villosa Rengger (Naturgesch. der Sdugcthiere von Paraguay, Basel, 1830, 80) was found to contain blood with remains of insects. Jt is not known what forms would now be included under this title. See remarks on Ania. Ay B &=VOm, SUK, YD. 240, ON THE GLOSSOPHAGINE. The origin of the Glossophagine is easily traceable to the group denominated by Peters the Vampyri. But the division between the genera composing the Vampyri is of a character to suggest two groupings at least, and the term Vampyri is best used in a restricted sense. Indeed, it is a small cluster of four genera only ( Vampyrus, Macrotus, Schizostoma and the aberrant Hemiderma), which possess a large, triangular, first upper premolar and an inflated, weak periotic region. . Of the second group (Phyllostomi), of which Phyllostoma is the type, I have imper- fect knowledge—haying studied besides this form the genera Lonchorhina and Lophostoma. But they agree in having the first upper premolar small and acicular, a peculiarity I find figured in Geryais (Hxp. du Sud.) as characteristic of Tylostoma and Monophyllum (Dolichophyllum). 1 infer that Trachyops, Phylloderma and Mimon are members of this group from Dobson’s statement (Br. Cat. Chir.) that they resemble Phyllostoma.. I have no satisfactory knowledge of the periotic region in this group, but can say that it is boldly defined, concave, and not inflated in Phyllostoma, Lonchorhina and Lophostoma. Now it has been seen that the Glossophagine yield two groups—that of the Glosso- phagi and that of the Lonchoglossi. In my judgment these do not haye a common origin. The Glossophagi agree with the Vampyri as above restricted in the shape of the first upper premolar and the inflated periotic region, while the Lonchoglossi are much nearer the Phyllostomi. Chenycteris possesses a triangular premolar (with large denticles) and a moderately truncate concaye periotic region, but its other characters, taken as a whole, connect the form intimately with the Glossophagi. The taxonomic value of the terminal cartilage can be determined only by the examination of extended series. At first I had inferred that the shapes of the cartilages of the fourth and fifth digits were of considerable value. But inspection of the largest number of individuals of the most common species—namely, Glossophaga soricina—gave me an impression that they were really variable structures ; thus in one individual from Costa Rica they were both spatulate ; in another from Bahama Islands they were both aciculate ; and yet in a third specimen from the last-named locality the fourth digit was spatulate and the fifth aciculate. Nevertheless the variability itself is of interest and I have, therefore, figured the cartilages, believing that after extended observation they may assist In more firmly defining the minor groups of species than is now the case. GLOSSOPHAGA. Upper incisors in a continuous row. Length of forearm not exceeding 56 mm.; thumb, 8 mm.; calear present; the tail is short with free tip on the dorsum of the interfemoral membrane. Proencephalon creates an eminence on brain case ; fronto-maxillary inflation conspicuous ; mastoid process small. Dental formula: 1. +— ¢. +— prm. 4—m., $= 21. ON THE GLOSSOPHAGIN®. ZAI The Mexor profundus digitorum supplies second and third digits oniy. The Semimembranosus and Biceps femoris are absent. The tendons of the Gracilis and Semi- tendinosis closely approximate and give the appearance of being fused, but by gentle traction they can be shown to be distinct. Pallas first described Glossophaga soricina as haying no tail (Mise. Zodlog., 1766, 48), the type beimg a female. He subsequently described and measured a second speci- men (Spicil. Zool, III, 1767, 24), a male, which he dissected. He now noted the presence of a short tail and figured the skeleton in which the tail is plainly seen. Geoffroy accepted the first description as final, and proposed a separate name (G. amplexicaudata) for the assumed new species possessing a tail. Gray (Ann. and Mag., N.58., 1858, IT, 490) acting on these erroneous premises proposed the name Phyllophora for Glossophaga amplexicaudata. Geryais (Expn. Amerique du Sud., 1855, 11, mem., 40) sustains Gray’s position without comment. Peters set the matter to rights in 1868, over ~a hundred years after Pallas’ first simple error of observation. Of the elaborate measurements of Pallas those taken of the male are the most accu- rate and include those of the skeleton as well. The figure of the head by Geoffroy also conforths in vertical measurement. The width of the basal part of the nose leaf is less than in our figure. Pallas, Geoffroy and Spix all accurately figure the interfemoral membrane as approaching the ankle, certainly reaching a point below the level of the middle of the tibia, which is the distance given by Dobson. The fact that the two forms of Glossophaga differ so widely makes it desirable that the characters of the first recorded species be carefully noted. A review of the original description of Pallas is of restricted value, other than the anatomy of the soft parts, notwithstanding the praise Geoffroy and Dobson award it. Geoffroy states he had dis- sected an alcoholic specimen and confirmed Pallas’ observations. But Pallas did not note so conspicuous a fact that in the first digit the metacarpal bone is much shorter than the combined lengths of the phalanges. The cranial and dental outlines are worthless ;* but one cannot gainsay the value of the figure of the fimbriated and elongated tongue. evokes tats @ Synoptical Table of Genera. Palatal portion of premaxilla forming a rostrum in advance of median incisive foramen; gland mass confined to sides of nose leaf; occipito-squamosal suture without foramen; tympanic bulla separated from postglenoid process by a conspicuous interval; ethmoid ; } bone convex in brain case; no ectopterygoid lamina; in third to fifth digits first | phalanx smaller than second; fimbriz: not confined to tip, but extending well back L along the tongue. | Glossophagina vera. * Gervais (1. c.) believes the form is not Glossophaga at all, but Hemidernu. 242 ON THE GLOSSOPHAGIN®. a. Median upper incisors larger than lateral; premolars 2; crown of lower canine with base lying inside position of lateral incisor; median incisor foramen barely in advance of paired foramina; upper incisors inclined; pit over proximal third of face vertex. b. Upper incisors in continuous row; molars 3; thumb one-fourth the length of forearm (31-34 mM. )....<.- 2.0... .2..cee-seescusecucccnescensscneneeans Glossophaga. b. Upper incisors with wide interval between centrals; molars 2; thumb one-sixth the length of forearm (45 mm.)..-.--+-+sseeeeeeeeeeeees Leptonycteris. 3. £; crown of lower canine a’. Median upper incisors smaller than lateral; premolars with base not lying inside position of lateral incisor; median incisor fora- men well in advance of paired foramina ; upper incisors vertical. e. Lower canine compressed, with cingulum; metacarpal bone of thumb exceeds length of phalanges. d. No phalanx to second digit of manus; premolars #; tail present; thumb one-seventh the length of forearm (AD aa, )) coscocmos ooocoasenocotosacoSeconUncoodISCoROASEnoSoACoSADCSOSS Charnycteris. c’. Lower canine rotund, no cingulum; metacarpal of thumb equal length of phalanges. d'. Phalanx to second digit of manus; tail present; thumb one-eighth the length of forearm (38 mm.).......-....-...-+5 Lonchoglossa. d''. No phalanx to second digit of manus; no tail; thumb one-sixth the length of forearm....-..--.-+.-.:01.eseeeeseeseen Armura. Palatal portion of premaxilla not rostrum-like; gland mass crosses muzzle back of nose leat ; tympanic bulla almost touches postglenoid process; occipito-squamosal suture with large Tul. foramen; ethmoid bone not convex in brain case; an ectopterygoid lamina. In third to fifth manal digits first and second phalanges equal; premolars 3; molars }; fim- brize of tongue at tip only. | | 4 | L Glossophagina aberrantia. Tail present; exceeding short interfemoral membrane; thumb one-fourth the length of forearm (45 mm.)..Phyllonycteris. Glossophaga soricina Pallas. Auricle emarginate at upper half of the outer border ; internal basal lobe free from head and indications of basal ridge. Lappet in side of the external basal lobe stout, pointed. Wing membrane from ankle. Terminal cartilage, fourth digit spatulate. Rudiment of an ascending process from the zygoma. Auricle subrounded, internal basal lobe with suggestion of vertical ridge, outer margin of auricle sinuate ; external basal lobe large, obtuse, retroverted, internal lappet a mere projecting nodule. Tragus straight on inner, convex or obscurely serrate on outer, margin. The nose leaf hairy and small, midrib confined to the pedicle. The leaf proper projecting nearly one-half its length above the conspicuous gland mass. The upper lip as well as the borders of the groove in the upper lip furnished with four to nine minute warts. Above, the fur is dark, sooty gray, at the tip the remainder of the hair being lighter but nowhere white. Beneath paler, unicolored. Interfemoral membrane almost ON THE GLOSSOPHAGIN”. 243 as long as tibia. The calear is one-half the length of the tibia. The interfemoral mem- brane is often incised rather than semicircular.* The tip of the tail projects from the free margin of the interfemoral membrane. Tongue on dorsum free from retrose papill. The first phalanx of the first digit is as long as the metacarpal. Entire digit one- fourth or nearly one-fourth the length of the forearm (10 to 40, or 8 to 36). The first phalanx of the second digit is one-thirtieth the length of the metacarpal; the entire digit is not as long as the third metacarpal. The first phalanx of the third digit is smaller than the second; the third is flexible; the separation from cartilage tip is indeterminate. Metatarsi equal. The row of first phalanges of toes equal. The Skull—The brain case papyraceous; the position of the body and hemispheres of the cerebellum—the mesencephalon and prosencephalon—hbeing clearly outlined on the periphery. Pretemporal crests scarcely defined and not continuous with the orbital margin; mesotemporal not seen ; posttemporal not distinct from the occipital. The face vertex is flat with shallow median depression over the ethmoid bone. The convex nasal bones are outlined by grooves, of which the median is the widest and deepest. Each nasal bone is incised on its free margin at the anterior nasal aperture. The sides of the face are convex, with a conspicuous, though small fronto-maxillary inflation. The infraorbital foramen answers in position to the junction of the premolars. The lateral border of the anterior nasal aperture is produced ; between it and the promi- nence over the canine tooth a groove is defined. The height of the alveolus is one-third the width of the neck of the canine, and one-seyenth the vertical diameter of the anterior nasal aperture. The posterior border of the hard palate near the zygomatic root is spinose. The palatal notch at the mesopterygoid fossa is acutely incised, carried back to a line answering to the glenoid notch and is without median spine. It reaches a point opposite the posterior third of the zygomatic arch. The tip of the pterygoid process lies opposite the oval foramen. The ascending process of the zygoma is inconspicuous and rounded. Base of cranium with prominent, median, vomerine ridge. The lateral depres- sions on the basioccipital are conspicuous, the mastoid process is ebtuse. The tympanic bone is separated from the postglenoid process by an interval. The coronoid process of the lower jaw is carried above the level of the condyle and is subacuminate. The angle is hamular and deflected outward with a notch between it and the lower border of the masseteric impression and projects backwards slightly beyond the condyloid process. Symphysis not carinate. The junction of the ethmoid and sphenoid bones in brain case convex. The Teeth—TVhe teeth of Glossophaga are the best defined of any of the group. The cusps are sharp, the incisors and premolars are adapted for cutting, and the molars * Geoffroy expressed it thus, ‘‘ coupée en angle rentrant,’’ but this shape is often absent. 2A4 ON THE GLOSSOPHAGINA. for grinding. In the upper jaw, with the exception of an interval on either side of the canine, all the teeth are contiguous.* In the lower jaw there is no interval on either side of the canine, for the lateral incisor and the first premolar are in contact with it. The upper incisors are arranged in a small are, which is smaller than the space between the canines. The central incisor is hatchet-shaped, the outer margin concave. The lateral incisor is smaller than central, with inner border twice the length of the outer. The canine is coneave on the palatal surface. The premolars are triangular subequal, yet the heel of the second tooth is twice the size of the first. The cingules are scarcely discernible. The first molar is subtriangular with W-shaped crown reduced, the fiuting on the para- conid, rudimental ; the metacone is united to protocone by a ridge. The second molar is subquadrate, W-pattern scarcely reduced; the fluting on the paracone marked; the ridge from the metacone not reaching the protocone, but a distinet though narrow valley intervening. The third molar is one-half the size of the second, the second V being rudimental. The longitudinal axis of both second and third molar is oblique to axis of the alyeolar processes. The third molar slightly oyerlaps the second at the buccal border. The lower incisors are proyided with flat smooth edges to the crowns and are adapted to crushing rather than to cutting food. The canine is directed slightly back- ward and is provided with a small heel. The premolars are triangular, equal, the bases increasing in thickness from before backward. The molars exhibit marked commissural extension in advance of protoconid and paraconid. The hypoconid is cuspidate and as high as metaconid ; all the teeth are much alike, but become progressively smaller and narrower from the first to the third, while the extension in front of the paraconid and protoconid become Jess and less marked. The third tooth is not more than two-thirds the length of the first. In a skull of an embryo which measured 8mm. long, the lower jaw projected well in front of the upper and. bore the deciduous canines. The shapes of the incisors and pre- molars could be discerned, while the upper jaw was edentulous. Tn an adult which retained the right upper lateral incisor only and the molars were much worn, the only teeth in the upper jaw that were in contact were the second and third molars. In the lower jaw the third molar was separated from the tooth both the first and third. The lower incisors were much worn and placed slightly in adyance of the lateral teeth. I am inclined to believe these are variations due to advanced age. * The upper incisors as represented by Leche (Studier ofver Mjolkdentionen och Tindernas Homologier hos Chiroptera, 1876, Tab. If, VIL) do not touch. ON THE GLOSSOPHAGIN &. 245 Glossophaga true, n. s. In the Proc. U.S. Nat. Mus., XVIII, No. 1100, 1896, 779, I described a new species of Glossophaga under the name G. villosa. Since Rengger (J/. ¢., p. 80) described in 1830 a species under this name I have concluded to rename the form, notwithstanding that the species is quite different from the genus Glossophaga as now restricted. See remarks under Anura. I take pleasure in dedicating this species to the accomplished Curator of Mammals of the National Museum, Mr. F. W. True. I herewith reproduce the description, which now has the advantage of appearing with appropriate figures of the head, skull and teeth. It is a remarkable circumstance that the genus Glossophaga, while the most common of any of the forms embraced in the group of Glossophagi, and has been collected from he widest range of any of its race, should haye presented degrees of variations so low as neyer to have permitted the recognition of more than a single species. The complicated synonymy successfully unraveled by Peters, it is true, contains a number of names of species, but these were proposed through misapprehension of assumed generic values and bear no relation to questions of specific distinction. A eareful study of two specimens (Nos. 9522 and 9525) belonging to the United coo) States National Museum has conyinced me of the necessity of recognizing two species of Glossophaga—namely, Glossophaga soricina and the one which I here name Glossophaga truei. Auricle entire on outer border or slightly emarginate. Internal basal lobe bound down to head without -trace of ridge. Excepting in length of head and trunk eyery- where smaller than G@. soricina. The ascending process of the zygoma twice the size of the same part in that species. Wing membrane from distal fourth of tibia. The termi- nal cartilage of the fourth digit terete. The auricle is without ridge at base of the internal: basal lobe, which is scarcely defined and closely bound down to head ; outer margin almost entire; external basal lobe and nodule inconspicuous. Tragus with trace of serration on outer margin, basal lobe large, quadrate. The nose leaf, hairy, without midrib at internarial pedicle, projecting scarcely at all above the simple gland mass of the upper lip, which it almost entirely occupies. Thumb one-fourth the length of the forearm namely, nine to thirty-two. The tail had evidently occupied a position similar to that seen in G. soricina, It had been remoyed in preparing the skin, 246 - ON THE GLOSSOPHAGIN A. Based on skins of two adults: No. 9525, U.S. N. M., La Guayra, Venezuela ;* and No. 9522, U.S. N. M., co-types. No. 9525, U.S. N. M., fur soft, shrew-lke; dull ash at basal two-thirds, sooty at apical third; it extends along the entire length of the dorsifacial region. No. 9522, U.S. N. M., quite the same, but is dark brown instead of sooty. The skull + closely resembles that of G. soricina, but is smaller and thinner walled. The ascending process of the-zygoma is longer and more pointed than in the species just named; the palatal notch is less acute. The fronto-maxillary inflation is conspicuous. The symphysis menti is carinate. The angle of the lower jaw projects backward slightly beyond the line of the condyloid process. The brain case is 12 mm. and the face 7 mm. long. The upper central incisors broad with slightly concave cutting edges; the lateral incisors are narrow with oblique cutting edges. The premolars are slightly separated from one another and the second premolar from the first molar; they are compressed, subequal, and triangular ; the second premolar is thickened posteriorly. The other teeth closely resemble those of G. soricina. ‘The first upper molar is longer than the second and the second longer than the third; there are no ridges extending from the paracone to the metacone. The third upper molar does not overlap the second molar at the buccal border. The muscle fascicles and nerve markings of the endopatagium disposed as in G. soricina. This system is the weakest of any of the group of the Glossophagi. The terminal cartilages are throughout terete. — On the whole the descriptions of Pallas and of Geoffroy agree well with Glossophaga soricina of Peters’ revision, and exclude those specimens here embraced under G. true. In Geoffroy’s figure { the measurements of the nose leaf agree with those of G’. soricina, but the shape of the tragus and internal basal lobe of the auricle are like those of the form under consideration. But the figure is evidently based upon a dried specimen. The isolation of the premolars in G. true: answer fairly well to the arrangement of the teeth in an old example of G. soricina. This is an interesting fact, inasmuch as it suggests that senile characters in one species may be the same as those found in young adult life of another. The following proportions are noteworthy: The first phalanx of the third digit is longer than the second. The third metacarpal bone is as long as the forearm. The *Tt is not certain that the locality here given is the correct one. The record in the National Museum catalogue is imperfect. + In addition to the skull in the type specimens, I possess a skull from Brazil presented by the late Mr. Harte, which answers to the above description. { Ann. du Mus., 1810, XV, Pl, XI. ON THE GLOSSOPHAGIN ®. 247 forearm is 1.15 mm., the smallest in the group. The calcar is one-third the length of the tibia. The first phalanx of the first toe extends slightly beyond the first phalangeal joint of the second toe. The first row of phalanges decreases progressively from the second to the fifth toe. Type.—No. 9522, U.S. N. M.* Measurements of G'lossophaga truet. Millimeters. Head and body (from crown of head to base of tail) ..........::cccceessessseeeeeeeeeseeee seseeeaasenenes 45 Eecadvand et rears sense sees sects eere sete eeneae sce ence sae nase Son oe senor ce wate sce soos cocoueee Soaoa0ic09000 32 First digit : Wengthyof firstimetacarpall DONes...sacce0stsseteoceeecncvssosseee-rccesecncnoceecececeseceecesasessecece 4 en op hvonehastap all axeasme erect eemtcentensrscostreenscecccetessseeeeseeeaceseresceeeceteeeeneeeteres 4 Second digit : Length of second metacarpal bone coo SS ILGNYE AN Oi Tah 79) 11 ENGb-<: ceaponosadccoscoooScooccaqdboSHo4Ro5o0e saps sbngcIDEsaee0HoNGekeRD0IINNINITEbOOIHHSOOS 2 Third digit : Length of third metacarpal DOne.............000.scsse-seseccesescceeccceecccesesceeeteetesecseteneesens 30 Length of first phalanx........ccccsccceecccseseeeeeesceseeceeseenseeee coo» dal Length of second phalanx 14 JURA ERIN Mie NTE: TAR ETE < ococnconbopcesqoodsasacnoaconoqecddacoasq¢aRcocOsaondanpLoondGUaSUDDOBUOHGoOHObOND 6 Fourth digit : Length of fourth metacarpal bone 27 Heng thnotsirstgphalanxeeeeteccsemonsecracscasasencserscosen so seeites ect eeteeeteceecsceeceeteiisetenas 9 eng thotsecond yphalanxgeseesnsceeccesoseseassseecsedeecesatee cen oeeeeeeeeecee etree eseerstntes 9 Fifth digit : idens thy of fiithtmetacarpall ones--c-casn-casccesencacsesccorseecasesscesesmccecceseersseteeereecereee Q7 Length of first phalanx ....... 50060600000000000000 6000000) -BondoNOdan9uDS BeSoDSqSUEHESodeosanNDSSEHAGCBGOD 8 ILEMEHT Ot Ses ~MnG! TATE NRIoOSE9ONNDoOSOOEDOSARdRaEDSECCHHODATSbONEsCS od 25 TSIEN DE EBPs sc0s093c9 0090009005 a09n0 qoNoNDaDED ANON O saDDESOOAODACAIDIANEHOSEDaCHONSHOAaDDSHOSDDERISSSRSoBEoRDSADORODER 10 Te (Gr Ne OE HEV FS)590080000005 900065000d90n04000000000000000000 2ooa0 00D ShDSaGRASSHeDGoNUsaDbOadboDERDAAGORAG peteseeeseeee 3 TLS OP MIEN, cooosac00s0900 000008 c00300000 soDdDDODN cOSODDBUDSSSECONDOCHODEHDSCSHODOBRADAADOEGOSE aoocnoDODOIRODORCOONN 11 IL SN GE THlOMocoscoocsa0e aaponno sods 980709903 0cdoNsCDID0009N000 ganoano9nOHONGE DoDOSeDHDScDSEODE. CUasHOURESSSSNBNOEBESCC 15 TURIN CL? TO Foon occ onoss ono ann conDab0Roo nDOSTEODOREDOGOCACHOSSD ConSOoDSODOSSEAbOSizaNSdOBON pabODOnAREAEOOHOBSsEDDeCRN! 11 Length of interfemoral Membrane. .........2..22cceceecccseeceeeceecseeseseccnessnesccecsaneceseeeecseeceeneeeees 4 Length of tail LEPTONYCTERIS. Upper central incisors separated by wide interval. Proencephalon not forming an eminence on the brain case. No spine at upper margin of the anterior nasal aperture caused by union of the free margins of the nasal bones. Tail none. Second phalanges of third, fourth and fifth digits sharply flexed on the first. Dental formula: i. 4—c. +— prm. 3—m. ?= 18. Leptonycteris nivalis Saussure. Auricle small, nearly one-half the length of the face, slightly emarginate at basal half outer border. Internal basal lobe scarcely free; external basal lobe convex, inner lappet crescentic. Tragus straight on inner, convex on outer side; basal lobe conspicuous. Nose leaf projects far beyond non-ribbed pedicle. The latter forms a wart-like contour inferiorly. The upper lip is narrow and provided with two inconspicuous nodules. Car- tilages at the end of digits are as in Glossophaga. Calcar rudimental, scarcely one-fifth the length of the tibia. Tongue furnished on sides and dorsum with minute, hair-like papille. The side of ON THE GLOSSOPHAGIN#. 251 the mental groove furnished with an obscure row of minute warts and the chin beyond the groove thickened with gland clumps. Fur short, villose, longer on neck, above deep ash verging to gray, base white, below paler. On neck, basal part tawny, but abdomen almost unicolored. The hair is slightly whiter at pubis. Distal half of humerus (above and below) hairy—the rest of the limbs, except the base of thumb, second digit and all of dorsum of foot, covered with a sparse growth of short hair. The muscle fascicles on wing membrane are much the same as in Phyllonycteris. They are wide apart generally, but do not extend over so large a field. The reticulated arrangement of fibres near the forearm is conspicuous. The longitudinal lines in the third and fourth interspaces distinct. The nerve markings are characteristic. Both arise from the digits far above the joint, the anterior being at distal third of the fourth meta- carpal bones. The terminal cartilage of the fourth digit scarcely spatulate ; that of the fifth digit is terete and not free. In this respect Leptonycteris resembles the remote Phyllonycteris. The skin in the second interspace is not pigmented. The Skull—Skull not papyraceous ; proscencephalon not defined. The pretemporal crests subtrenchant and form a short, faint conjoined line with its fellow at the sagitta ; the scarcely discernible mesotemporal depressed, not reaching sagitta; pasttemporal reaching occipital crest. Face vertex with depression over ethmoid, but the nasal bones are scarcely defined in median line and not separated at all laterally from the concave sides of the face. Fronto-maxillary inflation barely discernible and crossed by the orbital ridge. Alveolar process in height equals one-seventh the width of the neck of the upper canine and one-twenty-second the vertical diameter of the anterior nasal aperture. The depression between the lateral margin of the anterior nasal aperture and the root of the canine tooth much deeper than in Glossophaga soricina. Ascending process of zygoma rudimentary. The premaxilla weak in advance of the large incisive foramina ; posterior border near the zygoma root not spinose. The rounded notch at the mesopterygoid fossa midway between zygoma root and glenoid cavity. Scarcely any difference observed between the level of the basioccipital and the basisphenoid. The mastoid process acuminate. The tip of the pterygoid process in advance of the oval foramen. ‘The nasals are incised at the anterior nasal aperture. The angle of the lower jaw acute, not hamular; it is on the same plane with the masseteric impression, not separated therefrom inferiorly by a notch, and projects backward beyond the condyloid process. Symphysis not carinate. The lower border of the masseteric impression carried in a semi-circular line beyond the horizontal ramus. The Teeth—Teeth crowded for the most part. Upper incisors as in G'lossophaga soricina ; the central hatchet-shaped, separated by an interval. The lateral incisors as 252 ON THE GLOSSOPHAGINA. large or larger than centrals. Canine concave on palatal surface. The first premolar with- out basal cusp and separated from the canine and the second premolar. The second pre- molar with basal cusp and in contact with the first premolar. The first molar much larger than the second, the paracone subtriangular, the outer surface of the paracone and mesacone are scarcely at all fluted, hence the W-pattern not evident. The second molar without fluting on the rudimental mesocone, hence the posterior limb of the second V is absent. The single lower incisor which is seen in the two examples lies in close contact. with the canine. The canines are large and divergent, projecting to the inner side of the lateral incisor. The three premolars are triangular with conspicuous cingules; lingual aspect of the first premolar concave and in contact with the canine; the second free from the first and the third premolar. The protoconid with a long anterior extension which has the value of a second functionalized cusp. The paraconid is small and placed slightly back of the protoconid. The mesoconid is higher than either of the other elements, and together with. the hypoconid form a low, broad heel. Molars slightly overlapping at buccal borders ; the metaconid and hypoconid are of great size with wide yalley. Metatarsi equal ; first row of phalanges decrease progressively from the second to the fifth. The measurements of Dobson do not agree in some respects with the three specimens examined. The thumb is smaller, while the first phalanx of the third finger is much larger. He states the “tail none or exceedingly short.” In the cheernycterine alliance the genera Chernycteris, Lonchoglossa and Anura are placed. They have in common three premolars and three molars in each jaw.* CH@RNYCTERIS. Naked skin fold defining nostril laterally. Pterygoid process in contact with tym- panic bone. No phalanx to second digit. Length of forearm, 42 mm.; thumb, 7 mm. Dental formula: 1. 4 — c. + — prm. 3 — m. 3 = 22. Chernycteris mexicana Tschudi. Auricle subelliptical, emarginate on posterior border; internal basal lobe large, entirely free from the head and hairy; external basal lobe small, acute; internal lappet conspicuous. Tragus elliptical; basal lobe simple, deflected backward.+ Interfemoral membrane longer than tibia, semicircular. Calcar half the length of the * The only other forms possessing the same armament are the remote genera Vespertilio, Cerivoula, Natalus and Thyroptera. + In one specimen the tragus exhibited near the tip two papille seen on both the anterior and posterior borders and an additional cluster of three on the posterior surface. ON THE GLOSSOPHAGIN®. 253 tibia; the tip projects slightly beyond the interfemoral membrane ; wing membrane attached at a point midway on metatarsus. Nose leaf acuminate, sparsely hairy. Inter- nareal pedicle with midrib ; below two warts at median line in the short lip; outer flange at the nostril broad, tumid and gland-bearing. The gland mass proper well defined, but not across the face back of the nose leaf. Tail two-thirds the length of the femur and appearing free above the interfemoral membrane. Vibrissee on muzzle very long. Fur everywhere silky. Above, tips dark brown, the remainder of hair lighter brown. Beneath, lighter in shade, light brown, unicolored. No. 399, Acad. Nat. Sci., is smaller than the specimen named. The length of forearm is 33 mm. (about 1/50), and shorter than that assigned Chernyc- teris minor Peters. ‘The calcaneum, however, is not as long as the foot. The central incisors are absent in the upper jaw. In other respects the specimen resembles C. mezi- cana. I do not identify this specimen with C. minor, but regard it as a variation of C. mexicana. The Skull.—Skull papyraceous ; the divisions of the cerebellum and cerebrum discern- ible through the periphery. Temporal ridge almost n7/, not forming union at any part of the sagitta. Fronto-maxillary inflation absent, but the inner wall of the orbit and the fronto-nasal depression unite to form a ridge which bears a foramen. Face vertex without median fronto-nasal pit, but in its place a flat surface which bears a median ridge. No groove indicating positions of the nasal bones, but the outlines are seen through the translucent periphery. The sides of the face uniformly convex. The upper border of the anterior nasal aperture incised. The lateral margins of the anterior nasal aperture scarcely produced; the groove between them and the eminence over the canine teeth rudi- mental. The simple infraorbital foramen over the first premolar tooth. Alveolar process in height one-thirty-first the width of the neck of the canine and one-thirteenth the vertical diameter of the anterior nasal aperture. Six inconspicuous ruge. Zygoma incomplete. The infraorbital foramen on same vertical line between the second and third premolars. Hard palate acutely arched in molar range. The posterior border near root of zygoma with slightly convex margin; oval foramen well in advance of the pterygoid free tip which reaches the tympanic bone. The tympanic bone not reaching the postglenoid process. The palatal bone extends to the anterior lacerated foramen before forming the large subacuminate notch. Pterygoid process convex out- ward, forming bulla-like recesses. The mesopterygoid fossa with a faint vomerine ridge which is continuous with the conspicuous basioccipital ridge. The coracoid process acute, deflected outward, the angle produced beyond the condyloid process, and con- tinuous with the depressed lower border of the masseteric impression. Symphysis with pronounced carination. Brain case, 16 mm, long; face, 14 mm. long; or the face almost as long as the brain case, © 254 ON THE GLOSSOPHAGINA. The Teeth—Wide interval between upper incisors. The central as described by Dobson, is smaller than the lateral. But in two specimens examined by me the centrals were larger than the laterals. Both teeth are inconspicuous and scarcely raised above the gum line. The palatal surface of the slender canine flat. Of the two premolars present, the first possesses both anterior and posterior cingules and without increase of width back of the cusp. The second is without posterior cingule, but is widened back of the cusp. The first molar with paracone extending the entire length of the tooth, but sloping from before backward. Protocone and’ mesocone without buccal fluting or palatal ridges. The second molar as the first, but the protocone ends at the beginning of the mesocone. The third molar as the second much smaller and all parts rudimental. The lower incisors deciduous. The slender canine with rudimental lingual cingule which does not extend beyond the level of the lateral incisor. The first premolar close to canine with cingule subequal to the cusp. The second and third premolars with cusp much larger than the prominent cingules. The first molar with protocone and paracone almost coalesced ; the protocone well advanced. The posterior border of the tooth is furnished with a prominent cingule apparently developed from the hypocone. The first molar is separate from the third premolar and the second and third from one another. Chernycteris exhibits vertical muscle fibres in the endopatagium, the nerve markings of the interdigital spaces and the shapes of the terminal cartilage of the fourth digit in a manner quite the same as in G'lossophaga, though the structure last named is less spatu- late than in that genus. Measurements.—The first phalanx of the first digit shorter than the metacarpal; no phalanx is present in the second digit. The metatarsi and the first row of phalanges equal. Tongue attached to floor of mouth at the level of the space between the second and the third molars, or 12 mm. from the symphysis. Penis not pendulous. ANURA. Interfemoral membrane hairy ; tail absent; wing membrane attached to midtarsus ; calcar absent ; no phalanx to second digit; two warts on upper lip; groove in lower lip wide with many warts. First premolar large remote from canine. Dental formula: i. + — ¢. + — prm. 3 — m. 3 = 22. Resemblance to Lonchoglossa very close. The general appearance the same even to the shape of the terminal cartilages of the phalanges. Skull and number of the teeth the same. But it is held that the tail, calear and phalanx to the second digit all being absent, separate Anura from the genus just named, ON THE GLOSSOPHAGIN®. P55 The first lower premolar possesses a small, anterior, basal cusp and is, therefore, almost as large as the other premolars. The main cusp throughout scarcely higher than the basal cusp. Anura wiedii Peters. Auricle much the same as in Lonchoglossa. The tip of the tragus is pointed. Nose leaf simple, acuminate, no depression above nostrils. The gland mass at the side of the nostril continuous with that extending up to the side of the nose leaf. Upper lip with two equidistant warts. Fur everywhere long and silky. Above, apical third dark brown, basal two-thirds Isabella brown. Below, apical third Isabella brown ; basal two-thirds dark gray. Thus the arrangement of color is boldly contrasted with that of other forms in the group. Fleshy mass of forearm, the interfemoral membrane, the thigh and the feet covered with short hair. On the ventral aspect the forearm is covered with fur which extends thence a short distance on the interfemoral membrane. The proportions of the wing of Anuwra are those of a larger animal than Loncho- glossa, though the thumb is of the same size. The lower extremities are almost identi- cally the same in size, the calcar alone being larger in Lonchoglossa. The absence of the phalanx has already been noted in Chernycteris. Alliance with this genus is suggested in the great width of the cleft in the lower lip and in the possession of warts on the upper lip. The muscle fascicles and membrane markings are as in Glossophaga, but the terminal cartilages of the fourth digital interspace while spatulate exhibit the limb on the somad side greatly prolonged. This character is not seen elsewhere in the group. The cartilage of the fifth digit while terete is also greatly prolonged on the free margin of the endopatagium. These characters indicate that there is more strain on the wing during flight than in any other genus. . The Skull—Vhe skull is almost identical with that of Lonchoglossa. The alveolar height is one-third the width of the neck of the canine and one-seventh the vertical dia- meter of the anterior nasal aperture. The zygoma by careful maceration is shown to be cartilaginous. A specimen of Lonchoglossa shows the same structure. The skull is 24 mm. long. The brain case is 60 mm. long, and the face 40 mm. The lower border of the masseteric impression is not produced. Dobson’s figure, Pl. XX VII, Fig. 4, does not agree in all respects with our example. In 1830, Rengger (Naturgesch. der Sdugeth. von Paraguay, 80) described a species of bat under the name Glossophaga villosa. Since Wagner (Suppl. Schreb. Siugeth.) assigns this forma place under Chernycteris, it is well to state that while G. villosa Rengger retains three premolars in both jaws, that the tail is absent, the interfemoral] A. P. S— VOL. XIX. 2G 256 ON THE GLOSSOPHAGIN®. membrane is but half an inch deep at the rump, and the lateral upper incisors are smaller than the centrals. The interfemoral membrane is hairy. This species is nearer Anura in most of its characters than any other genus in the group. LoNncHOGLOSSA. Tail short ; wing membrane attached to ankle; calcar present but small, about one- third the length of the tibia; a phalanx to second digit; groove in lower lip narrow with a few inconspicuous warts; no warts on upper lip; basal part of nose leaf rudimental ; apical third of tongue filamentose ; interfemoral membrane not hairy. Dental formula: 1. 4#— ec. +— p. 3 — m. 3 = 22. The first lower premolar small and without anterior, basal cusp; the main cusps of the entire series twice the height of the basal cusps. The presence of the tail and a phalanx to the second digit are sufficient grounds to separate Lonchoglossa from Anura. Lonchoglossa caudifera Geoff. Auricle pointed, internal basal lobe bound down to head. External border faintly sinuate scarcely ; any external basal lobe; the inner lappet large. Tragus blunt at tip. Nose leaf simple, without pedicle ; lateral gland mass of base rudimental ; wpper lip short, without warts. Large numerous vibrissee from face, especially from mentum. Filaments on tongue large, not meeting in middle line of dorsum. Wing membrane reaches to calcar. Seven rugee on the hard palate, the last two alone divided. The tail not quite as long as the short interfemoral membrane, the tip not free. The hair of the dorsum exhibits apical third brown, basal two-thirds pallid. Beneath paler, prevailing hue brown (but with scarcely a contrasted shade toward base), tending to become grayer, almost unicolored on loin. Limbs naked. The wing markings both in the nerves and muscle fascicles are as in G'lossophaga, but the terminal cartilage of the fourth digit is terete, and that of the fifth digit is small and scarcely deflected. The Skull—The bones yery.thin, permitting the subdivisions both of cerebellum and cerebrum to be seen through the periphery. The pretemporal ridge unites with its fellow at the anterior fourth to form a faint, linear crest; the mesotemporal and _post- temporal ridges not separately defined, scarcely discernible. Fronto-maxillary inflation small. Face vertex without pit at the fronto-nasal region ; outlines of nasal bones not defined. Side of face conyex. The lateral borders of the anterior nasal aperture mod- erately produced, The foramina between the two premaxille near the incisor margin large, ON THE GLOSSOPHAGIN A. 207 The alveolar process so slender that it cannot be measured. The parts as viewed from in front embrace the floor of the nasal chambers at the premaxillary part and permit the median foramen to be seen. The zygoma without a trace of ascending process. The posterior palatal margin near the root of zygoma spinose ; the posterior palatal notch with conspicuous spines. Pterygoid process almost reaching tympanic bone and extends beyond the oval foramen. Mastoid process aciculate. Mesopterygoid fossa with incon- spicuous yomerine spine. Basioccipital depressions shallow. |The coronoid process scarcely raised above the level of the condyloid process. The deflected hamular angle projects in a marked degree beyond the condyloid. The lower border of the masseteric impression is produced conspicuously beyond the border of the ramus. Symphysis with large keel. One skull 21 mm. long; face 8 mm. long; brain case 15 mm. long. Upper Teeth_—The small central incisors separated by wide interval, and each tooth in close contact with the large lateral. The central incisor with ovoid crown scarcely wider than neck ; the lateral incisor projecting below the level of the central with crown wider than neck and conspicuously oblique outer border. The interval between lateral incisor and the canine no greater than in other genera. Canine with inner surface flat. First premolar one-half the size of the others; separated from the canine and the second pre- molar, but nearer the last-named tooth. The second and third premolar triangular, with large basal cingules. The W-pattern of the molars discernible. In one specimen the long, sloping proto- cone with suggestion of hypocone, recalling the parts as in Macrotus ; in the second the teeth were without hypocone. Canine with rudimental heel. First premolar separate from the canine and second premolar. Second premolar separate from the first and third ; third premolar separate from the second, but contiguous to the first molar. First molar with cingule of the protocone extended forward, scarcely deflected inward and overlap- ping third premolar; protocone and paracone approximate, united at base. Lower Teeth.—First lower premolar without anterior basal cusp, and is, therefore, much smaller than the other premolars. In the entire series of premolars the main cusp is twice as high as the height of the basal cusps. The first and second molars of the same plan with the foregoing, the third being slightly the smaller. The lower teeth with jaw are figured by Leche (/.¢., Taf. I, Fig. 8). The first pre- molar is represented as being exactly like others of the series. This character would prevent the Lonchoglossa of Leche’s identification being received under Lonchoglossa caudifera of this essay. ; Variations.—The above description is based on two specimens, which were subject to some variation. In one the pretemporal crests did not unite. In one the cusps of the teeth were much worn. 258 ON THE GLOSSOPHAGINZ. Notes on the Skeleton.—Ribs thirteen ; first costal cartilage not wider than the rib. Humerus with pectoral crest relatively high, one-half the diameter of distal end of bone. The sternal crest after careful removal of the pectorals is very high and apparently with- out notch, but the greater part of the interpectoral septum is membranous. The phalanx of the second digit about as in Vespertilio. The metatarsi and first row of phalanges of toes equal. . Measurements.—Forearm, 36 mm.; foot and thumb of same length, viz., 8 mm.; fore- arm, 1.35 mm. BRACHYPHYLLINA. I propose to establish the Brachyphyllina to include the genera Brachyphylla, and Phyllonycteris,* forms which have hitherto been assigned separate groups in the Phyl- lostomidie, the first named to the Stenodermata and the second to the Glossophagina. Brachyphyllina. Leaf-nosed bats with tip of tongue retaining clump of papille extending across dorsum. In the Glossophagina the papille are arranged not only at the tip but the sides for great lengths. The minute first upper premolar wedged in between the canine and large second premolar; coronoid process acute, raised high above the level of the condyloid process. Mesopterygoid fossa deep, apex answers to the junction of the anterior and middle third of the zygoma. Nasal bones high, arched, defining a depres- sion between them and the maxilla. Sagitta entire with well-defined pretemporal crests. The glands of muzzle continuous behind nose leaf. Thumb large, one-fourth the length of the forearm, nearly. Auricle narrow, oval with pointed tip. Tragus coarsely serrate entire length of outer border. Upper lip hairy, without warts. Lower lp with shallow median groove, margined with large warts. Lips not fringed internally. BrAacHYPHYLLA. Upper central incisors very much larger than the laterals. Length of forearm, 65 mm.; that of thumb, 16 mm., this being about one-fourth the length of the forearm as in Phyllonycteris. Grinding surfaces of molars with numerous large mammillations, cuspi- dation distinct. Angle of lower jaw quadrate, massive ; nostril entire, the wide outer margin and the side of the rudimental nose leaf continuous. Tragus entire on inner border. The tail rudimental, one-fourth the length of tibia, and concealed im the inter- femoral membrane. Dental formula: i. ¢— ce. +— prm. #— m. 3 = 20. * T have not studied Riinophylla, but the conclusions arrived at after reading the accounts of Peters and Dobson induce me to place the genus in the same alliance with genera just named. But in the absence of material I am com- pelled to confine my comparisons to Grachyphylla and Phyllonycteris. ON THE GLOSSOPHAGIN #. 259 Brachyphylla cavernarum Gray. The auricle lanceolate with shghtly convea margins, basal lobes rudimental. The tragus pointed, one-half the length of the inner margin of the auricle ; convex on thickened inner, and coarsely serrate on outer, margin. Nose leaf with entire nostrils and wide ectonareal flange ; erect portion of nose leaf rudimental—concaye and often minutely crenulate on midmargin. Supranarial margin concaye on either side of an obscure median ridge. Infranarial margin wide, continuous with upper lip and faintly incised. The basal gland-clump continuous across face—vertex back of nose leaf. The upper and outer parts are thick and bear a few coarse bristles, while the lower are thin and lost on the upper lip. ‘Twelve warts are arranged in pairs on the side of a mental V-shaped group, the median groove being shallow. Two median warts may be said to haye slight morphological significance. The fur above is yellowish white except the tip, which is brown. Below the tints are the same, but the shaft is more tawny and the tips much lighter. The distal third of the arm above and below is covered with hair. The distal half of the thigh is similarly covered. A sparse growth of hair is limited to the upper half of the dorsal surface of the interfemoral membrane. The calear is rudimental. The terminal cartilages of the fourth and fifth digits are uniform, elongated and scarcely wider at free margin than on the sides. The second interdigital space is almost devoid of pigment. The third space retains a vertical line for nearly its entire length, while the fourth exhibits one for about an inch near the free margin, the rest of the space being areolated. The endopatagium is furnished with numerous thick muscle fascicles ; near the tibia it is thick and leathery. Second interspace, Third interspace, — Fourth interspace, Pteral formula : 3 mm. 19 mm. 35 min. The Skull—vThe walls of the skull are thin and permit the divisions of the brain to be discerned. The sagittal, pretemporal and occipital crests are well defined and tren- chant. The fronto-maxillary inflation is conspicuous and bears the pretemporal crest. The inner orbital wall is moderately conyex, and is marked by a conspicuous foramen. The infraorbital foramen is placed well in adyance of the orbit in line of the second premolar. The zygoma with a rudimental ascending process at the posterior third, but none anteriorly to contribute to the limitation of the orbit. Lower Teeth.—The incisors are stout, in continuous row. The palatal basal cusp is on level with the crown, which thus presents a broad, quadrate surface, marked in the middle from before backward by a ridge. Canine without conspicuous basal cusp. Pre- molars subequal, the first the smaller and triangular, the second with large basal cusp. 260 ON THE GLOSSOPHAGINA. First and second molars with quadritubercular cusps well defined, a large mammillation on the anterior commissure of the second molar; the third molar triangular, tri- tubercular. Upper Teeth—The central incisors are very large, triangular, nearly fillmg the interval between the canines. The lateral incisors are minute, not over one-fourth the size of the centrals. The anterior surface is concave; the crown is blunt and quadrate, with basal cusp and cutting edge equal. The canine with anterior and posterior denticles, the posterior of the two being enormous and presenting the aspect of being an outshoot from the side of the crown. The first premolar minute and of the same form as the lateral incisor. The second premolar large, triangular and projecting beyond the molars. The basal cusp (denterocone) conspicuous. Molars tritubercular, without W-shaped pattern. Several mammillations are present on the grinding surfaces. Third molar is one-half the size of the second. Measurements of Brachyphylla cavernarum. Millimeters. Head and body (from crown of head to base Of tail)......ss0.cssseeeecseeeeeeeeseeeeeeeeeneeeetee essere eeees 66 Thength Of Arm -...22....cccccesense-eeccserecanscenescessnrsssserecnnesnsens pedsso95ES00000900989 oAODIOCeH d500800 0 eo0300 40 Length of forearm, ...-.0.....ccesessseesssnececcoreceansecsocteccanesscceonevsretestnacerenssecnnsosseacsscoateccneess 65 First digit: Length of first metacarpal DOnE.......-.:1:esssceceeceeeeensscceereseceeceesseseereceesrsscscceeenansense 4 Thength Of phalanges.......00..--0.sesesserssensersrcenesnccnesenenseoessuoersressvseecesnrecreuscennaesesrnes 12 Second digit: Length of second metacarpal bone.. Length of first phalanx.......::scccccsesceeececneenseseccueneeeeeseceeessececsceeccssrsaenscsseseanecererens Third digit: Length of third phalanx Fourth digit: Length of fourth metacarpal bone.........-.:seccecceeeeteerenneteenceseceeteeses rescue sasteecsueeoers 51 Length of first phalanx...:.....0....0ccssscssecssecccnssossaacseusscesccocrereroness secneersascemuercersnnce 15 Length of second phalanXx..........::sc:ossccecssessscssecencteecceterceenercnerscsunceesene sodoenonpgaoo5000 17 Fifth digit: Length of fifth metacarpal DOme..........sccseecsecerneeceseneetsencenneceeseseenesesereessueeeaecuserens 55 Length Of first phalamx......-01seccssesssscssrnnscescnasennrsiecacnneseserreesecccusetrressscecsseranevesers 15 Length of second phalanx.......0..ssssssssecersecsasccsescecececcenneasnesesecerevenccesesseeueccssteeeeners 14 Length of head TCI SG) OF (CAaL:conssrcnccnvclsvons-seqnocesncerisscsiusnacseancsseecm=ran= ns -écQnan0oDGNSo3soNnGEDoSONNDOTHCOOGBOSLOD +212 Height of tragus.........csese-.seeee en0ece000019990000005R000 so0D00BDoBNaRaRDqHOoNS, EoGAGRdAHGEDs9Hdudacaaq905056000R3 9 Length of thigh Length of tibia..... Length Of £00b...c.0sccccccecceecenecreetsccnecseeraeeanscensscesasanecuseuseesarseceoascaussssoscesesenessesareceassses Length of interfemoral membrane Length of tail ON THE GLOSSOPHAGIN®. 261 PHYLLONYCTERIS. Upper incisors separated from the laterals by wide intervals; naked skin-fold defining nostrils laterally ; nose leaf not reaching aboye the level of approximate club- shaped gland masses. Thumb the largest in the group nearly one-fourth the length of the forearm. Length of forearm, 45 mm. Teeth with cusps nearly obliterated, no W- pattern on molars. Large vacuity between occipital bone and pars-squamosal of the temporal. Fimbriz not arranged in rows, but form a uniform coyering to the tip of the tongue. The first and fifth metatarsal bones longest. The first row of phalanges of third to fifth digit of manus, same length as the second row. Calcar wanting. Zygomatic arches fibro-cartilaginous. Dental formula: i. 4 — ¢. + — prm. 3? — m.3 = 21 Phyllonycteris was described by Gundlach, but published under the care of Peters, who does not appear to have known the form. Gundlach correctly compares the genus to Brachyphylla. Dobson follows Gundlach closely, his description being little more than a translation of the original article. When he departs from the text he makes statements which do not agree with the specimen on which the present essay is based. Thus he says, “the incisors are as in Glossophaga; the molars like those of Carollia 3 (Hemiderma), but the W-shaped cusps scarcely developed ;” whereas the upper lateral incisor is twice the size of the central and the zygoma may be complete. With the exception of the skulls, Dobson did not study Phyllonycteris at first hand. Phyllonycteris sezecorm Gundl. Auricle simple, ovate, with rounded pointed tip. External outline without subdivision or inner lappet near the base. Internal basal lobe scarcely free. Tragus convex on inner side, straight on outer. Both sides marked by three, coarse, teeth-like processes. Basal point scarcely longer. Nose leat simple, obtuse with internarial pedicle. The perinarial flange is lamillar and distinct from gland mass. The structure last named well defined, apparently crossing muzzle back of the nose leat, but two club-shaped masses are nearly approximate. Upper lip high without warts. Interfemoral membrane deeply incised, extending from distal third of the tail to the caleaneum. The tail is short, scarcely projecting beyond the interfemoral membrane. The fur long and silky above, light gray tipped, subtip sooty, the rest of the hair pale verging to white. Beneath much paler, nearly uniform gray. The tip of hair tawny, the rest of the hair of a somewhat lighter shade. Almost the entire field of the endopatagium filled with widely separated nearly equidistant vertical muscle fascicles, There is no reticulated arrangement of fibres, The 262 ON THE GLOSSOPHAGINA. nerve markings in the fourth interspace as in Glossophaga except that from the fourth digit there are three instead of one nerye. The terminal cartilage of the fourth digit is obscurely spatulate. The Skull—The skull not papyraceous, the division of the cerebellum, but not of the cerebrum, discernible on periphery. The pretemporal crest distinct. It begins over the moderate fronto-maxillary inflation to form a delicate crest by union with the fellow of the opposite side at the anterior third of the sagitta. _Mesotemporal and posttemporal crests not discerned. The orbital ridge is rudimental, but the frontonasal pit conspicuous at proximal end of the slightly convex nasal bones. The large infraorbital foramen lies over interval between second premolar and first molar and is thatched by a ridge. The alveolus (7. e., the distance from the central incisor to the anterior nasal aperture) equals in height one-fifth of the base of the upper canine and one-eighteenth of the ver- tical diameter of the large, anterior, nasal aperture. The zygoma often complete.* The maxilla at root of zygoma with a very small ascending process. The premaxilla at the side of the anterior nasal aperture salient. Neither the grooye between the nasal bones or the depression on the maxilla at the side of the nasal bones are conspicuous. The depres- sion between the aperture last named and the eminence over the canine is shallow. The hard palate just back of the last molar is sharply defined by a double crescentic trans- verse ridge; the palatal notch is acute and deep, the apex reaching the level of the - anterior third of the zygomatic arch, the pterygoid process corresponding in position to the oval foramen. The tympanic bone touches the postglenoid process. The junction of the ethmoid and sphenoid bones in the brain case not convex. A vacuity is found in the line of junction of occipital and squamosal bones. The basioccipital bone with scarcely any pit-like depressions ; the vomerine ridge scarcely discernible in the mesopterygoid fossa. The mastoid process small, conical. The proportion of the face to the brain case is as 9 to 15 mm. Lower Jaw.—Coronoid process acuminate. The hamular angle not deflected or pro- jected beyond the condyloid process ; lower border of the masseteric impression not dis- tinguished from the corresponding border of the horizontal ramus. Back of the molars and at base of coronoid process a tubercle for insertion of temporal muscle is seen. Symphysis-menti broad, non-carinate, the surface near the incisors marked by coarse venous foramina. The Teeth—The upper central incisors hatchet-shaped, contiguous ; laterals much smaller, not half the size of centrals and separate therefrom. The incisors not entirely occupying space between the canines. Canine broad at base, robust, convex entire length * Dobson ( Cat. Chirop. Br. Mus.) in text states that they are incomplete, but acknowledges the fibro-cartilagium arch in a footnote, ON THE GLOSSOPHAGIN#. 263 of palatal surface. First premolar very small, nodular, about one-fourth the size of the second and not much larger than the lateral incisor. Second premolar triangular, with- out basal cusp; posterior half of palatal surface concave. Molars without well-defined cusps and decrease in size gradually from before backward. The third molar one-half the size of the second. ‘The protocone, paracone and metacone scarcely indicated ; no W- shaped pattern.* Lower lateral incisors twice the size of the centrals; all are non-contiguous and nodular. Canine with conspicuous concave heel; all other parts convex; cingulum extends inward so as to lie back of the lateral incisor. The premolars thick and robust, subequal ; the first smaller. The molars decreasing in size from before backward without details. Of the measurements it is noted that the first phalanx of the first digit is scarcely longer than the metacarpal bone. In the second digit the single phalanx is one-tenth the length of the corresponding metacarpal bone. The entire second digit is as long as the third metacarpal bone. In the third digit the first and second phalanges are equal— the third phalanx is nearly one-half the length of the second. The terminal cartilage of the fourth digit is moderately spatulate, and that of the fifth digit is deflected toward the body. The wing membrane attached to the tibia at the distal seventh or to the ankle. Interfemoral membrane attached to tip of the small calcaneum. The Skeleton —The sternum is boldly keeled over the presternum and metasternum. The ribs are twelve in number. The first costal cartilage is discoidal. The humeral pectoral crest is relatively low and not half the diameter of the proximal end of the bone. The fifth metatarsal bone is much the largest of the series. Palatal rug eight, last three to four interrupted in centre. The first and fifth metatarsals are longer than the others. The bones of the first row of phalanges of the toes are equal. * Peters and writers following him give all glossophagine genera W-shaped pattern of molars. I have had no oppor- tunity of examining the type of Phyllonycteris in the Berlin Museum, but I have received through the kind offices of Mr. Paul Matschie a photograph of the skull which I find conforms to the account above given. A. P! S.— VOL, Xix. 24, 264 ON THE GLOSSOPHAGIN®. Table of Measurements (in millimeters). &: | & 3 | fmee Wey es =a 5 2 S | 8 aes Saf ree lush Head and body (from crown of head to base of tail) 45 45 57 | 55 40 49 | 32 TL SIAN GP PE ease os cnocesanoecocas onnaosonnecoosnoce aoe oon sb ansstoaoansoGN: ssbarcossoaee 19 2 | 20 20 20 | 25 Teng ih Of fOrea tM accass--ceseeseresseneraaanne=aneeeece ner oeceo= = neanencccssce=acennean iar 36 39 50 | 42 | 35 38 45 First digit : } | Length of first metacarpal bone 4 4 4 3 5 Length of first phalanx .-------.----<-2-22----0--2-2c-ceeeenennneneesnanennee one 4 4 4 3 3 3 7 Second digit : : Length of second metacarpal bone......---.+--:+++----s0eeeesseeeeeeeee sees 30 25 40 40 29--) 33 33 Length of first phalanx.......- Toaoencose 1 2 3 0 yet ONC 3 Third digit : a Length of third metacarpal bone........-..---+:+s++0eeeeeeeeereeeeeeseeeeeees 34 30 | 47 45 37 38 | 38 TeeMStH Oe Tash [PRA on sec conse taco nosceoscnacaceesspeosessoneoscesccccoccees 13 sal 14 17 42 | 13 | 44 Length of second phalanx....00:.0...::22seeceeseeeeece see ce ese c es neeeeeeeeeeees 16 12 | 923 21 18) |) 21 S14 Length of third phalanx......--..--.+0:cseseececsseeeeeesceeeeeeeeeeeeneeeseeees 7 Gj 9 Quslestt eels Fourth digit : | | Length of fourth metacarpal bone Length of first phalanx ...---.+--2:::ceeseseseeeees eee eneneeesceesessseeeeeeees 10 9 11 12 | 10 | 13 Length of second phalanx...-----.---:2:::sseeseseeeessseeeeeeesecceeeseeeeeees eat) 9 16 15 12 13 11 Fifth digit : . Length of fifth metacarpal bone 30 20 40 35 30 30 35 Length of first phalanx........-..... pasagia swt ae eetectiewcet wos seescnesteen weet oee 9 8 10 10 7 8 1d Length of second phalanxX..--.-++..:ssssssscceceeeeceeseneesssseeeeceeeeeseeenees 9 8 10 13 To es | 10 Length of head.......e..ceeeeeceeceeeeeeeceecesseeseseeeecneeeererenteeceeeereeececeseasanes 23 21 o7 32 25 29 25 Height of ear 14 ) 11 12 13 13 14 11 TS erTey SNE re ea Scene celareecrner ees Ree Beason a eeeaR occas eceeocec acto 6 ASS |e} 4 5 eae 4] 5 Length of thigh......-....--:s:e++- PR ee eae abe Hees soa oaSe 10 2 15 A. jel Sella STS Tareas reer eae ee es MR ne SE, 14 [ee top |e Penta te etree Length of foot 8 8 12 10 vw 7 13 Length of interfemoral membrane in median line........---+--++111s12++eeeeeeees hal 9 20 | 4 6 7 5 2 Length of tail....----..:::eceeeeeeeeeeeeeecee cee eeeeee cece eneeeseeeaaee cesses eeeeeeseneaeees Norre.—The Secretaries deem it proper to state that this, as well as the succeeding paper, was presented to the Society after the author’s death, which lamented event occurred on November 14, 1897, and that, therefore, it has not had the benefit of his revision in its passage through the press. ! Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga truet. Glossophaga truet. Glossophaga truet. Glossophaga truet. Glossophaga truet. Glossophaga truet. Glossophaga truet. Monophyllus redmani. Monophyllus redmani. Monophytlus redmani. Monophyllus redmani. Monophyllus redmani. Monophyllus redmani. Brachyphylla cavernarum. Brachyphylla cawernarum. Brachyphylla cavernarum. Brachyphylla cavernarum. Brachyphylla cavernarum. Brachyphylla cavernarum. to 39. Brachyphylla cavernarum. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. ON THE GLOSSOPHAGIN ji. EXPLANATION OF THE PLATES. PLATE VI. Head seen from in front. Skull vertex. xX 3. Skull profile. xX 3. Skull base. X 3. Jaws with incisors and canines seen from in front. x 10. Lower teeth seen from above. Xx 2. Upper teeth. x 10. Left lower molars seen in profile from lingual aspect. Puate VIL. Head seen from in front. X 2. Skull vertex. xX 3. Skull profile. X 3. Skull base. xX 3. Upper teeth. xX 8. Lower teeth seen from above. x 8. Left lower molars seen in profile from lingual aspect. PLATE VIII. View of head from in front, showing ear and nose leaf. Skull of same. Norma verticalis. x 3. Skull of same. Norma lateralis. x 3. Skull of same. Norma basilaris. X 3. Upper and lower jaws seen from in front. xX & Teeth of the same as seen from the surfaces of crowns. PLATE IX. View of head showing ears and nose leaf. Skull of same. Norma verticalis. x 3. Skull of same. Norma lateralis. x 3. Skull of same. Norma basilaris. x 3. Upper and lower jaws seen from infront. X 8. PLATE X. Teeth of same seen from the surfaces of crowns. X PLATE XI. Head seen from in front. xX 2. Skull vertex. xX 3. Skull profile. xX 3. Skull base. xX 3. Jaws with incisors and canines seen from in front. x 8. x 8. Upper teeth. IGD Xx 8. The first molar is to the The first molar is to the X 2. oS (eb Sh Terminal cartilages of the fourth and fifth digits. 266 ON THE GLOSSOPHAGINE. Fig. 46. Leptonycteris nivalis. Lower teeth. X 8. Fig. 47. Leptonycteris nivalis. Left lower molars seen in profile from lingual aspect. The first molar is to the right. X 10. PLATE XII. Fig. 48. Chernycteris mexicana. Head seen from in front. X 2. Fig. 49. Charnycteris mexicana. Skull vertex. X 3. Fig. 50. Charnycteris mexicana. Skull profile. X 3. Fig. 51. Charnycteris mexicana. Skullbase. X 3. Fig. 52. Chernycteris mexicana. Jaws with incisors and canines seen from in front. X 5. Fig. 53. Chernycteris mexicana. Upper teeth. X 10. Fig. 54 Chernycteris mexicana. Lower teeth. X 10. Fig. 55, Chernycteris mexicana. Left lower molars seen in profile from lingual aspect. The first molar is to the right. 10. PLATE XIII. Fig. 56. Lonchoglossa caudifera. Head seen from in front. X 2. Fig. 57. Lonchoglossa caudifera. Skull vertex. X 3. Fig. 58. Lonchoglossa caudifera. Skull profile. x 3. Fig. 59. Lonchoglossu caudiféra. Skull base. XX 3. Fig. 60. Lonchoglossa caudifera. Jaws with incisors and canines seen from in front. X 8. Fig. 61. Lonchoglossa caudifera. Upper teeth. X 8. Fig. 62. Lonchoglossa caudifera. Lower teeth. X 8. Fig. 63. Lonchoglossa caudifera. First and second right lower molars seen from lingual aspect. The first tooth is to the right. x 10. PLATE XIV. Fig. 64. Anuwra wiedii. Head seen from in front. X 2. Fig. 65. Anura wiedii. Skull vertex. X 3. Fig. 66. Anura wiedii. Skull profile. X 3. Fig. 67. Anurawiedti. Skull base. X 3. Fig. 68. Anwra wiedii. Jaws seen from in front showing incisors and canines. > 8. Fig. 69. Anura wiedit. Upper teeth. X 8. Fig. 70. Anwra wiedii. Lower teeth. X 8. Fig. 71. Anuwra wiedii. Left lower molars seen from lingual aspect. The first tooth is to the right. X 10. PLATE XY. Fig. 72. Phyllonycteris sezecorni. Head from in front. X 2. Fig. 73. Phyllonycteris sezecorni. Skull vertex. X 3. Fig. 74. Phyllonycteris sezecornt. Skull profile. X 3. Fig. 75. Phyllonycteris sezecornt. Skull base. X 3. Fig. 76. Phyllonycteris sczecorni. Upper teeth. X 10. Fig. 77. Phyllonycteris sezecorni. Lower teeth. X 10. Fig. 78. Phyllonycteris sezecornt. Jaws seen from in front showing incisors and canines. X 8. Fig. 79. Phyllonycteris sezecorni. Left lower molars seen from lingual aspect. The first tooth is to the right. xX 10. ARTICLE VI. THE SKULL AND TEETH OF ECTOPHYLLA ALBA. (Plate XVI.) BY HARRISON ALLEN, M.D. Read before the American Philosophical Society, January 21, 1898. In 1892 (Proc. U. S. Nat. Mus., 1892, No. 913, 441), I described a bat from Honduras under the name of Lctophylla alba. The single specimen was without skull. I have been permitted through the courtesy of Mr. Oldfield Thomas, of the British Museum, to inspect a second example of the genus. The material consisted of a dried skin and a skull of a male individual which was mutilated by shot in the ptery- goid and orbital regions. The specimen was collected at San Emilio, Lake Nic-Nae, Nicaragua.* The norma verticals shows faint fronto-temporal lines which barely approximate near the bregma, but recede from that point posteriorly so that no trace of a temporal crest exists. The fronto-maxillary inflation is conspicuous and makes a swollen border for the upper and anterior orbital margins. The nasal bones are sharply elevated above the plane of the maxilla. Sufficient of the norma dasilaris remains intact to show that the hard palate is elongated and the palatal bones are produced, thus separating the genus sharply from Stenoderma and its allies and allying it to Vampyrops (see Synoptical Key). The basioccipital bone is deeply pitted for muscular impressions. In this respect it presents a marked contrast with Vampyrops, in which this bone is nearly fiat. The tympanic bone is small, leaying the greater part of the cochlea exposed. The norma occipitalis shows a weak occipital ridge. The junction of the ectopetrosal + surface of the pars-petrosa with the occipital bone is complete, while in Vampyrops a vacuity exists. The lower jaw retains a curved aciculate angle relatively twice the size of the same * The skin was badly mutilated by shot and the nose leaf and chin plates so distorted that no attempt is made to compare the parts with the original description. The second interdigital space is without pigment, head and neck both above and below are pure white. The lower third of the body both on dorsum and ventre is tipped with ash-gray. + I propose naming that part of the pars-petrosa lying in the brain case the endopetrosal, and that lying exposed back of the pars-squamosa the ectopetrosal part (Journ. Acad. Nat. Sci., 1896, Philadelphia). 268 THE SKULL AND TEETH OF ECTOPHYLLA ALBA. part in Vampyrops. ~ The masseteric muscle extends to the lower margin of the ascending ramus. The coronoid process is one-third smaller than in the genus last named. Dental formula: 1. 2 — c. + — prm. 2 — m. 2 & 2 = 28. The Teeth—Upper incisors conical; the centrals larger than the laterals with rela- tively broader bases. The centrals are separated from each other by a smaller interval than exists between these teeth and the laterals, or between the teeth last named and the canines. The canines are slender and slightly longer than the second premolar. The first premolar is pointed, root much exposed and is about one-third the size of the second. The first upper molar is quadrate with trenchant marginal cusps in position of proto- cone, paracone and metacone; the crown defined by these elements is concaye. ‘The second molar is pyriform, the base being toward the palate. A pointed marginal cusp is seen in the position of the paracone and a second in that of the metacone. The crown is concave and simple, save for a longitudinal ridge. The premolars and molars are separate from one another ; the greatest interval being between the premolars. The lower incisors are blunt cones, contiguous, filling space between canines; the teeth last named are deeply excavate posteriorly. Premolars are aciculate, the first tooth almost touching the canine and is smaller than second. The second tooth is deeply con- cave posteriorly with a conspicuous heel and cusp. The molars are subequal, without W- pattern. The first molar is obscurely quadrate, slightly narrowed in front with enormous sharply pointed paraconid; other cusps are absent; the lingual border is not raised. The second molar is subrounded, no trace of cusps being present other than a longitudinal ridge in the middle of the deeply excavate crown. The front and lingual borders of the tooth are greatly elevated, the former furnished with two sharp processes, the latter crenulate. The teeth are all separated from one another beyond the canine, the smallest interval being that between the canine and the first premolar and the widest between the premolars. : Letophylla is in alliance with Vampyrops. It resembles this genus in the upper incisors and first upper premolar being conical and in the prolongation of the palatal bones. The shape of the lower first molar possesses a large paraconid, but is without protoconid. In the dental characters last named ctophylla is like all other Steno- dermine, excepting Brachyphylla, Artibeus, Dermanura and Sturnira. The forms exhibiting the stunted, first, lower molar are again divided into two groups by the palate andthe lower jaw. In Chiroderma, Vampyrops and Ectophylla the palate is oblong ; the palate bone extends to a point answering to the anterior root of the zygoma, or eyen the posterior third of the arch, and the lower jaw has a well-defined posterior border to the ascending ramus, with no deflected angle. In Pygoderma, Stenoderma and THE SKULL AND TEETH OF ECTOPHYLLA ALBA. 269 Trichocorys, the palate is rounded, as a rule excavated and rarely reaches a point answering to the anterior root of the zygoma; the lower jaw has no well-defined posterior border, the boldly deflected angle almost reaching the condyloid process. The position of Eetophylla in the Stenodermine is shown in the synoptical natural key. Brachyphylla is an annectant genus to the Glossophagina through Phyllonycteris Artibeus, Dermanura and Sturnira apparently relate to the Vampyri, but while the structure of the molars is essentially that of this group, no annectant form is known. Sturnira in the simplicity of the tooth structure recalls Hemiderma. The relation between the remaining genera of the table is intimate. The Stenodermine constitute, with the exception of the Heamatophilhia, the most aberrant group of the Phyllostomidide. I recognize, therefore, the following natural arrangement of the genera : Subfamily STENODERMATIN#. eae linam ny llinie ne Seen aes. cste ha hreiad. see Brachyphylla. ( Artibeus. P Nor tillye lines eis eee eh re Mice Bare | Crake Dermanura. | Sturnira. ( Chiroderma. *, | 4 Ghimocleriminiiee eee ehh ae = ca haan ee : Vampyrops. | Eetophylla. ( Stenoderma. | Pygoderma. aie 1 Centurio. Stemod ermine cy eet kaa ks saan : Trichocorys. Ametrida. y . | Spheronycteris. A Natural Synoptical Key of the Stenodermide, Based on Characters Derived from the Skull and Teeth. I. First lower molar elongate with paraconid distinct. [ a. Angle of lower jaw broad, scarcely pointed, concave above, not deflected, ascending \ ramus defined. Hard palate oblong, palatal bones produced. Upper incisors coni- | cal, molars $ ; crowns coarsely ridged ; all cusps of the first lower molar subequal... l Brachyphylla. Group Brachyphyllini.... * Ohiroderma is not as near Vampyrops and Ectophylla as the members of other groups are to each other, 270 THE SKULL AND TEETH OF ECTOPHYLLA ALBA. ( a. Angle of lower jaw narrow, aciculate, not deflected ; posterior border of ascending ramus defined ; hard palate oblong ; palate produced. b. Palatal bones extend to point answering to the middle of zygoma. Upper incisors flat ; first upper premolar broadly lanceolate ; crowns of molars rugose ; proto- conid and paraconid of first lower molar prominent, subequal, the others rudi- ee mental. CEO pe OUCI NS eres Gy WIDE #roccsccecocsscontcooscs0ssdoonnoos.osnao2esenscasnseosoososcaerocnBNNONOADNODSONN Artibeus. Gl. WIGS 2 -cococonencoso op eanbacoadapo Scand cnoSesndoorocosDosHONENDOAboSSHHoooSSANNIG Dermanura. b’. Palatal bones extend to point answering to the anterior third of the zygoma. Upper incisors conical, contiguous ; first premolar narrow lanceolate ; crowns of molars smooth ; all cusps of first lower molar subequal, anterior commissure Ip @msjaneeine 8 NOMEN 2 oocossocoasos ope cocogasQnecdencencocosoeponHSSconBeGREoSioconnCoD4 Sturnira. Il. First lower molar subquadrate without paraconid. if d. Hard palate oblong, palatal bones produced. Upper incisors conical. e. Angle of lower jaw quadrate, not deflected, posterior border defined. Nasal bones absent in adult ; palate bones produced nearly to the line of glenoid cavity. First upper premolar acicular ; first — ae lower molar with protoconid and mesaconid subequal. Molars 2... Group Vampyropini....-.- Cee e/. Angle of lower jaw acuminate, not deflected. Protoconid of first lower molar aciculate, enormous. Jf. Hypoconid first lower molar rudimental ; molars 2... Vampyrops. jf’. Hypoconid first lower molar none ; molars 3.........-.. Ectophylla. ad’, Hard palate round, palatal bones scarcely, if at all,* produced. | e/’. Angle of lower jaw rounded, deflected, posterior border ascending ramus not defined. g. Frontal bone in orbit greatly inflated ; palatal bones extend to a point answering to the anterior root of the zygoma ; | pterygoids produced, inflated and nearly teaching the panic bones; upper incisors conical ; protoconid of first lower molar searcely larger than other cusps ; hypo- | conid of the same tooth marginal, rudimental molars 2... | Pygoderma. Cremicrenodencie | g'. Frontal bone in orbit not inflated ; palate bone produced to anterior third of zygoma; upper incisors conical ; protoconid first lower molar enormous; hypoconid of same tooth marginal ; molars }...-.....-.---..-.-+ Ametrida. g'’. Frontal bone in orbit scarcely inflated ; hard palate with posterior margin excised; pterygoids not produced. | | | | | Upper incisors flat; protoconid of first lower molar | enormous. h. Palate excised to first molar ; hypoconid of first lower | molar inside contour. Molars .........-. Stenoderma. | h’'. Palate excised to middle of first molar ; hypoconid of | first lower molar marginal. Molars 2 ... Trichocorys. * Mr. O. Thomas (Ann. and Mag. Nat. Hist., 1889, p. 70) first employed this character to separate this group from the foregoing. THE SKULL AND TEETH OF ECTOPHYLLA ALBA. 271 Measurements of Ectophylla alba (in millimeters). Head and body (from crown of head to base of tail)... 36 36 Men ot hpoisanmirenn cerca terrane scar mere atcctcine seameeiooecisciie ceccacioe tition caste celecs vce vena shade svcetestersevsiics 17 IL@IGIN Ot TOREAI TT qeaosadsoanseda0.an9odbsedb0adg6n0s8094Geqshdd90s00N60d0K000 000 Jodo bco0desHSsodDnECHE AG DoS He SEnBBEDEAOOSOEe 25 26 First digit : ent hvolsnirshame tacan pala On eseerspheecsenteecsanceimasce ses sscsteneeresreeetimesaes see seceeeessesseemececees : 3 3 Length of first phalanx...........-- 3 3 Second digit : Wencthvotsecondentetacanpalliib one ys eeeerteeer creer steerer cece enlaces eee sesereseiseereseneaseeeteeceere 21 20 TLS ON THRE, THAT TAS oc soog3n50000s900030N= INS cao SoocONbonSSaaTEG2Goqsabsoco9qsaROveS>9aHNGNDBAD9S9GBe0000 HOBHaDAea 3 Third digit : Length of third metacarpal bone.........- 25 25 ILE GE aS TONE FATED 0053090000000 ssaDDaDoADDOAGOCODBEHEDHODSEdoONdoHSHODoSENcHOADScROBOeSHASEAaEHAGIS 9 ILemaHn @ii S@COMEl TRAN, soocec cov e3c000n0009s900099c3000R00080800 neagdaNDSSoODDoGONSOD oADeooAoeSAANGACES HoaHSo90a 12 13 ILEWSUN Oi WantREl TNA eva CeccosoHoaconsdoooono5coaeINaESDDOGNbSONoROnOSCASDDKDDN|AGOGG soNEEHoSDHEOLOdesADoaEeDeGouAS 6 6 Fourth digit : Length of fourth metacarpal bone 25 25 Meno thyotuirstaphalamxerscyeccades veces ose de eeicishae cinch anced ene mare ueeseunaec ce acca sene ae soecee? sant 7h 8 ILEMAWA Oi Seeded! [MAMI .coccnacsooocoGseDoGASEABOOBEOCHESoDUA SoBNSooogonnScoRaSDoDNEgOIBAALONATeREDEOgOSUDNeS 8 7 Fifth digit : | ‘Length of fifth TEASE EN! INO, paovoboodcongkosossoass0nGenbon 7OSG9nONEGERO anEDosONH/-aagodaDRoOEsCORONanDOGaeOD 25 ? Length of first phalanx.. 6 6 ILemaWn OF SecouGl Tp ogee, coosopon20096590Racoa9Hoas nonDobsbopadeooosnHocHosHoo;endDoRSopSsOGASHADBORSoEHOODE Anae 7 ri ILE HD OF VEAL ceosenodasccsen600399c00905000 099 D500N00090000d0s NoNDOS NSESEES OED gBOsOSAOIcCoGNOoOOoRGNOSHOS PoaDbosaneHooAeaDAG 14 14 JEIGREING @E © JProsccsaqnedancoconesecoq das pss bos0Hbo0000¢bac0000000 00500 aod ObUDDUsHAIGHE teEoRAODODDAESHOSeOUSbOARsODHsoGovOHONONS 10 10 TEIGTEINE OF WREGWE .000900900060 on00000c0 0900900990000d000000000000950020000-,eaqD0D60 DonooDOINOTODOAGENNSNNOROHO SoncvasoROseDABHeE 5 2 Length of t. igh.... 8 i ILEMAIN OP HDI, coocococcecas oscosonon donc coseaa0cnqqCnObESURCBSCONDABROSaqOCaD0 EqDNSIDDboaENcHDLAGOHDOOOD ScHoCODONDOTAGUOnEOTE » 10 10 Length of foot ........ poo soaseso99eecbapcaDoDUaR ADDED DODHOUDEESORCO EDU DOONoDEANOCOORD.SoDAcHOsedSedonSEDonaaEEuDaEEeoSRaCRENcaS 8 8 Length of interfemoral membrane........... SCORE GUE SNHSGABEAdoOSGnED. cosdacHaonbacoonaDoode Load bHebsaenEoooecanereones 4 4 In concluding the account of this interesting specimen, I will call attention to the molar teeth of Cephalotes, a member of the remote group of the Pteropodidee. The two genera, however, resemble one another in being frugivorous, in retaining few or no tubercles to the molars and, probably on this account, in exhibiting elongated crests in the centre of deeply excavate crowns. A tenable hypothesis for the origin of this central cusp may be expressed as follows. The grinding away of the crowns has gone on to a degree that brings the enamel cap down near to the division in the alveolus, between the sockets for the roots of the teeth, so that this ridge acts as a point of resistance to further wear and leads to a reassertion of the principle of cuspidation at this point, A. PB. S.——Vow. XIX. 21. DD THE SKULL AND TEETH OF ECTOPHYLLA ALBA. One of the most marked characteristics of the teeth of fruit-eating bats is the dis- position for the loss of cusps in the molar teeth. This takes place without intermediate grades so far as is known. In two of the three subdivisions of the Phyllostomide it occurs as exceptions to the rule—Hemiderma in the Vampyri and Phyllonycteris in the Glossophaginee, but is the rule rather than the exception in the Stenodermine. In the Pteropodide the tendency to the loss of cuspidation is the rule, the genus Pteralopex being the only exception. Such abrupt variation within the limits of small groups indicates that the tendency to external specialization has weakened the type and exposes it under the influence of environment, ordinarily acknowledged as active in modifying forms, to gross modification always on the side of deterioration. EXPLANATION OF PLATE XVI. . Eclophylla alba—norma verticalis. . Eetophylla atba—norma lateralis. . Ectophylla alba—upper and lower teeth. = (>) mw wm ee . Ectophytla alba—lower molar (profile). . Ectophylla alba—ramus of lower jaw. . Cephalotes peroni—first right upper molar. leo) =! asI OD . Cephalotes peroni—first and second right lower molars. Ee 7 - i £4, ae Lig NOTIGE. Preceding Volumes of the New Series can be obtained from the Librarian at the Hall of the Society. Price, five dollars each. A Volume consists of three Parts; but separate Parts will not be disposed of. A few complete sets can be obtained of the ‘Transactions, New ates Ve I—XVII. Price, ninety dollars. Acdress, THE LIBRARIAN. OCT 4@ lave fens ACTIONS OF THE AMERICAN PHILOSOPHICAL SOCIETY, HELD AT PHILADELPHIA, FOR PROMOTING USEFUL KNOWLEDGE. VOLUME XIX.—NEW SERIES. PART TTT. . ARTICLE VII.—The Osteology of Elotherium. By W. B. Scott. Articue VIIT.—Notes on the Canide of the White River Oligocene. By W. B. Scott. ARTICLE IX—Contributions to a Revision of the North American Beavers, Otters and Fishers. By Samuel N. Rhoads. a, Philadelphia: 4 PUBLISHED BY THE SOCIETY, ¥ AND FOR SALE BY ae: Tue American PuirosopHicat Society, PHILaDELPHta, N. TRUBNER & CO., 57 and 59 LUDGATE HILL, LONDON. : 1898, OCT 4& 1898 ARTICLE VII. (Plates XVII and XVIII.) THE OSTEOLOGY OF ELOTHERIUM. BY W. B. SCOTT. (INVESTIGATION MADE UNDER A GRANT FROM THE ELIZABETH THOMPSON FUND OF THE A. A. A. Ss.) Read before the American Philosophical Society, February 4, 1898. Elotherium is one of the many genera of fossil mammals concerning which the growth of our knowledge has been exceedingly slow, and only of late has it become prac- ticable to give a complete account of its bony structure. The genus was named in 1847 by Pomel (47 a, 6) and shortly afterward renamed Entelodon by Aymard (’48) from a better specimen, but for several years only the dentition was known and that imperfectly. In 1850, Leidy (50, p. 90) described the first American species, but, not suspecting its generic identity with the European forms, he at first referred it to a new genus, Archio- therium. Leidy’s material enabled him to give a fairly complete account of the skull. Kowaleysky, in 1876, described an imperfect skull found in France and he further showed that the feet were didactyl, a very unexpected fact in view of the pig-lke char- acter of the dentition. In this country Profs. Marsh and Cope have added materially to our knowledge of this remarkable animal (Marsh, ’73, 793, 94; Cope, ’79) and the former has published a restoration of one of the species. In spite, however, of this list of workers who have, from time to time, occupied themselves with the study of Hlothe- rium, much still remains to be learned regarding its structure, and its phylogenetic rela- tionships are even more obscure. ; In the summer of 1894, Mr. H. F. Wells discovered in the White River Bad Lands of South Dakota certain bones, which, with the expenditure of infinite pains and skill, were excayated from the rock by Mr. J. B. Hatcher, and which proved to be a most remarkably complete skeleton of EHlotherium. This beautiful specimen (Princeton Mu- seum, No. 10885,) formed the subject of a preliminary communication which I made to the third International Zodlogical Congress, at Leyden (Scott, ’96), and will be more fully described in the following pages. Except for a single thoracic vertebra (and perhaps a QA THE OSTEOLOGY OF ELOTHERIUM. few caudals) and part of the hyoid apparatus, the skeleton is complete; it is represented in Pl. XVII, which will enable the reader to judge of its unusual state of preservation. Additional material, belonging to several species, will also be made use of for purposes of of comparison, but the description will deal almost exclusively with the White River forms. The Artiodactyla may almost be designated as the despair of the morphologist. So manifold are the forms which this puzzling group has assumed, and so variously are the characteristics of its minor groups combined, that the confusion seems hopeless. ‘The only way in which this tangled skein can be unrayeled and its many threads separated and made straight, is by the slow but sure method of tracing. the phylogenetic develop- ment of each family step by step from its incipient stages. Many years must pass before sufficient paleeontological material has been gathered to make this possible, but already some progress has been made in the work. Each successive form in a series, as soon as it is recovered, should be fully described and illustrated for the benefit of other workers, a necessity which must excuse the minuteness of detail into which the following deserip- tion enters. For the sake of conyenience the entire bony structure of the animal will be described, including those parts which are already well known, in order that the reader may be spared the trouble of searching through many scattered papers, written in several languages. I. Tue Deyririon. The teeth of Hlotherium are already familiarly known and require but a brief account here. The dental formula is I 3, C 4, P 4, M 3. A. Upper Jaw.—The incisors, three in number, increase regularly in size from the first to the third, the latter being much the largest of the series; it has a conical or some- what trihedral crown and resembles a canine in shape and appearance. In some individ- uals the crown of this tooth is worn in a peculiar manner, a deep groove or notch being formed on its posterior side, in a place where it cannot haye been made by the attrition of any of the lower teeth. The other incisors have spatulate crowns, with blunted tips, the attrition of use wearing down the apices as well as the posterior faces of these teeth. This description applies more particularly to the larger White River species, such as Lf. ingens and E. imperator ; in E. mortoni the upper incisors are of more nearly equal size and more conical shape. In all, the median incisors are separated from each other by a considerable notch, and the whole series is much more extended antero-posteriorly than transversely, the external incisor standing behind the second one. I 3 is separated by a short diastema from the canine and at this point the premaxillary border is quite deeply notched to receive the lower canine. The canine is a very large and powerful tusk, with a swollen, gibbous fang; the d c=) I ? co) fo) THE OSTEOLOGY OF ELOTHERIUM. 275 crown is long, massive, recurved, and bluntly pointed; it is oval in section, and has a prominent posterior ridge. The premolars are very simple in construction. The first three are well spaced apart and have compressed, but thick, conical crowns, without accessory cusps of any kind, and each is implanted by two fangs. In size, they increase posteriorly and p ® has a decidedly higher crown than any other premolar. P 4 is smaller than p 2 in every dimension except the transyerse, this diameter being increased by the addition of a large internal cusp (the deuterocone) and the crown is carried upon three fangs. In the smaller species of the genus, such as H. mortoni, p 2 and p 4 are placed close together, while in the larger forms these teeth are separated by a short space, and the diastemata between the other premolars and between p + and the canine are relatively somewhat greater, the enlargement of these teeth hardly keeping pace with the elongation of the muzzle. In the European species, #. magnum, the arrangement of the premolars is somewhat different, p 2, ® and 4 forming a continuous series, while p + and 2 are quite widely separated. The molars are relatively quite small; m 2 is the largest and m ? the smallest of the series. The crowns are low and bunodont, bearing six tubercles arranged in two trans- verse rows. ‘The hypocone, though functionally important, is decidedly smaller than the protocone, and structurally is still a part of the cingulum. Schlosser is, however, mis- taken in supposing that there is any important difference between the American and the European species of Hlotherium with regard to the position of the protocone. In m 3, which has a more oval crown than the other molars, the sexitubercular pattern is obscured by the development of numerous small tubercles upon the hinder half of the tooth. The cingulum of the molars is quite strongly marked, especially upon the ante- rior and posterior faces. B. Lower Jaw.—The incisors resemble those of the upper jaw, except that they are of more nearly equal size and somewhat more spatulate shape ; 1 y is little enlarged and is much smaller than the corresponding tooth in the upper jaw. The canine is a very large, recurved tusk, like the upper one in size and shape; it bites between the upper canine and enlarged external incisor, the three teeth together making up a very formidable lacerating apparatus. An interesting hint as to the habits of this animal is given by a peculiar mode of wear of the lower canine which occurs in some well-preserved specimens. In these we find a deep groove on the posterior face of the tooth, beneath the enamel cap and close to the level of the gum. No other tooth can reach this point to cause such a mode of attrition, and the groove is doubtless due to the habit of digging up roots with the lower tusks; the pull of the roots, especially when covered with sand or other gritty material, would naturally wear such a groove.* The * This ingenious and highly probable explanation of a somewhat puzzling fact was suggested to me by my colleague, Prof. C. F. Brackett. 276 THE OSTEOLOGY OF ELOTHERIUM. same explanation applies to the curious notches sometimes worn in the external upper incisor. ‘The numerous specimens examined do not indicate that there was any difference between the males and the females in the size of the canines, the tusks being invariably large and powerful. If, as here suggested, the canines served other purposes than those of weapons, the lack of any such sexual difference would be intelligible enough. The premolars are very simple and quite like those of the upper series in shape; their crowns are massive, compressed cones, without additional cusps. The cingulum is usually prominent, but varies in the different species. P is much the highest of the series, especially in #. imperator, where it rises to the full height of the canine, and gives a very characteristic appearance to the lower dentition. Pz has its posterior face flat- tened, forming an incipient fossa with a number of small tubercles in it. P sand ; stand quite close together, and p ; is separated by a short space from the canine, while p; is isolated by considerable diastemata both in front of and behind it. The lower molars are small in proportion to the size of the jaw and to the space occupied by the premolar series. In size they increase posteriorly, and they have a simple, quadritubercular pattern, the crowns surrounded by a strong cingulum. There is much variation in the development of the fifth or posterior unpaired cusp (hypoconu- lid); it is frequently absent and represented only by a strong cingulum, though some- times it is present as a distinct cusp on m > or mz. It is less commonly found on m 3 and only in the very large /. leidyanum is it well developed. The Milk Dentition—The temporary canines and incisors differ from the permanent ones only in size. It is uncertain whether the first premolar, in either jaw, has a prede- cessor in the deciduous series, none of the specimens distinctly showing such a predecessor. In one individual, howeyer, the tip of p 1 is just visible in the centre of a large alveolus, from which a milk-tooth has apparently been shed. If this change does actually occur, it must take place at an early stage, and, on the whole, it seems probable that, at least in the upper jaw, the number of deciduous premolars is four. Dp 2 has a compressed, elongate, conical crown, without accessory cusps of any kind; it is carried on two widely separated fangs, and is isolated by diastemata both in front of and behind it. Dp 2 consists of three principal cusps. The antero-external cusp (protocone) is an acutely pointed pyra- mid, while the postero-external cusp (tritocone) is lower and smaller. The internal cusp (tetartocone) is posterior in position and placed on the same transverse line as the trito- cone, while between the two is a small conule. The cingulum is distinct on the front and hind faces, obscure on the outer and absent from the inner face of the crown. Dp ¢ is molariform, but differs somewhat from the molar pattern in the fact that the postero- internal cusp is even more distinctly an elevation of the cingulum and that the posterior conule is double. THE OSTEOLOGY OF ELOTHERIUM. 277 The lower milk-premolars are even simpler than the upper; dp 5 and y are com- pressed and conical, without accessory cusps, but with serrate edges and sharply-pointed summit. Each of these teeth is supported upon two fangs. Dp < is of the usual artio- dactyl type, consisting of three transverse pairs of cusps, of which the median pair is the largest, and the anterior pair the smallest. A small talon is formed by the elevation of the cingulum in the median line, behind the posterior pair of cusps. This account of the milk dentition applies only to £. mortoni; I have not seen these teeth in the larger species. Measurements. | No. 11156 | No. 10885 | No. 11009 No. 11440 | Upper dentition, length I1to M3...... Ramee ve te pho nar acti Seti | | 20.270 Ce 2 TON Games, IOMGWISc coc onevcdoos Goce benodo cose oeuE 118 104 | .064 .065 (C joReMOlayr SOmies, NeMADN. -s2corcccsoa0cancacnu0ds Ste kiacsens .238* 175 .124 .113 CRMTOE, HIN DOM, GHANA, .16 congeadosaocosssedscaud | .048* .046 | .082 GG ee Mans Versen diam ecleneeemeniceia te emcee nee .03885* .042 .022 Come Pale mlen thie sanss acne ease tron oe aac er Toes ian Gaee> .080* | 024 .019 CEE TED De WRG File OUR elo oA Seo te i | 088% | .088 02 een e O23 EID Be lye KBs Meee ee RS Me ee mR AL Rtgs | 041 028 | .028 CMe AR pmeaL ier haar teniara® ME Se Ra ae RAS Mae | . .085* | 081 0195 .018 cS > TST TL iepayetitn d ea 0 ire i a ge emecOS52. le 038 | - .020 E09 OOD SATIN pe nee ee eer aA Gai oe alee eget | .086 | .019 05. INL Bh TTC dle Soe RR RR ea | 04 | 085 025 | .028 cy 9 "007 SRR g Oar | 039 0235 | 024 6G) NEG) Tan iatllie gas tison Raab O Une suaap Ore cone et SE eaneeree ne OR 2 ORE I ae a ies ces) 6 0G" RENT ten rN rascal ae ga eI | 088 022 | «0215 Lower dentition, length IT1toM3............................ | -432* | | 261 GG AOA SOMES, IEMA cc copcocs oss Goeoooe Guns Sosscnngen 121% | -108 | | 070 Re == premolar series: lengthy. o.% s. cnc os 0s se0ees tanec one } Bile? |) SRD 126 a 1B il Nena noes deino.ce ceeds Coen CD ODS Oe aE eaGEn coon. | .028* | .026 | 017 s cc INGA! Oi, GiOneas opbecbocossEnesbes sooo LeRsBEEGSS -026* | -019 CRaPEPEOM enh sa sais oe eres teas te SNE Sie BME lipeeeO3tcans | Giar.0asit ste] .020 Si, Bae STy@TCANIP Seas Senet ee See eer .038* | .023 “i 1D B, IGMEHI -oocoousonas oo pedoegaesouedbeeadoDanL 00000 .046* .043 027 RRC Mee pli beet me Tae Ade teh et Pee o61* | 031 Gs “TBA. Thana de Se pe Uae Sle dese ao Spec Soa Meee eb bras 046 | 037 | 025 3) 1G Teele Nice sane seer ce Reena rare Maen, 0445 | 020 Gl TAT Te Ten geoceecs ook acu OI eee enn 037 | 081 0215 65-7) (GS SIN Ha oe pes o> (geen eR REE MEE RSE ERIE EBC aese 029% 027 | 018 CINE DBT Miggooneee dosece aed: Ae aNROOU Ce Oar oR esEee .0895* | .035 | 0225 Ob. 08 ORT St Hea a: cob bE ROBO REAR eRe Re meee aee .036* .030 > neorG ci) SUL S) Deming ope’ canbe Otte: Eee nae mm nat .043* 039 | 0245 a Scam UL eecya ee a te ore eA weioe sis lemons 037% | 028 | .016 | *No. 11161. 278 THE OSTEOLOGY OF ELOTHERIUM. Il. Tue SKULL. The skull of Hlotherium is one of the most remarkable features of this very curious animal. It is characterized by great length and slenderness, with the supraoccipital and nasal bones lying in the same horizontal plane. The muzzle is exceedingly long and narrow, and tapers somewhat anteriorly, though expanded by the sockets of the great tusks; the orbit has been shifted far back, its anterior border being, in some species, over m 2, and in others above m 8. The cranium is short and of absurdly small capacity, which, with the great temporal openings, gives an almost reptilian appearance to the skull when viewed from above or below. The sagittal crest is very high and thin, and the zygomatic arches, though rather short, are enormously developed. One of the most peculiar features of the skull is the great, compressed plate which is given off from the ventral surface of the jugal and descends below the level of the lower jaw, and this gro- tesque appearance is further increased by two pairs of knob-like processes on the ventral borders of the mandible. The occiput (Pl. X VIII, Figs. 1,2) is high and very broad at - the base, but narrowing rapidly to the summit; above the foramen magnum it forms a broad, flat projection of almost uniform breadth, with a very deep fossa on each side of it. The basioccipital is stout and rather short, keeled in the median ventral line and slightly contracted to receive the auditory bulle ; at its junction with the basisphenoid it forms a pair of small, roughened tubercles. The exoccipitals are yery large bones, espe- cially in the transverse direction along the base of the occiput, dorsally they narrow fast. Above the foramen magnum they form the very broad, prominent and nearly square pro- jection which has already been mentioned ; this is thick and is filled with cancellous bone, the fossa for the vermis of the cerebellum making but a slight depression upon its internal face. On each side of the projection is a large and deep triangular fossa, which, how- ever, is not confined to the exoccipital, the periotic and squamosal both being concerned in its formation. The inferior part of the exoccipital extends widely outward, reaching to the line of the glenoid cavity, and ending in the large, prominent and massive, but not elongate paroccipital process. In this region the exoccipital is brought very close to the zygoma, but, ventrally at least, does not quite touch it, a narrow band of the tympanic inter- vening between them. The foramen magnum is strikingly small and of a transversely oval shape. The occipital condyles are relatively rather small, especially in the vertical dimen- sion, laterally they are well extended, and they are widely separated both aboye and below. In the very large /. imperator the external angles of the condyles are abruptly truncated in a curious way, and bear flat articular surfaces, though in some individuals this trunca- tion is found only on one side; while in the smaller species the condyles are of the usual form. The supraoccipital is a large bone, widest at the base (7. ¢., the suture with the exoccipitals) and narrowing dorsally. Superiorly it is drawn out into two posterior wing- THE OSTEOLOGY OF ELOTHERIUM. DATES) like processes, such as are found in Oveodon and other White River ungulates. Between these wings the hinder face of the bone is concaye and at the bottom of this concavity are two small, but profound pits. The supraoccipital is continued over upon the roof of the cranium and forms a part of the sagittal crest. A considerable part of the periotic is exposed on the surface of the skull, at the bot- tom of the lateral occipital fossa, where it is enclosed between the exoccipital and the squamosal ; it does not give rise to any distinct mastoid process. The oceiput of the European species, Z. magnum, as figured by Kowalevsky (76, Taf. XVII, Fig. 5), is different in many details from that which characterizes the Amer- ican species. It has more of an hour-glass shape, not so wide at the base, more contracted in the middle and more expanded at the top, but with much less conspicuous wing-like processes, and it has no such projection above the foramen magnum, nor such deep lateral fosse. The condyles are larger and of an entirely different shape, haying their principal diameter vertical, instead of transverse. The paroccipital processes are longer, more com- pressed and not so widely extended laterally. -The foramen magnum is large and of more nearly circular outline. The basisphenoid is narrower than the basioccipital and is not keeled on the ventral surface, but is otherwise like that bone. So much of its course is concealed by the union of the palatines and pterygoids along the median line that its length cannot be deter- mined, while the presphenoid is nowhere exposed to view. The tympanic is very extensively developed (Pl. X VIII, Fig. 1). Part of it is inflated into an oyal, somewhat flattened and rather small auditory bulla, which differs from that of Hippopotamus and of all existing suillines in being hollow and not filled up with spongy tissue. On the outer side of the bulla the tympanic is extended as a narrow strip, which broadens considerably between the squamosal and the exoccipital, with both of which it articulates suturally, as well as with the alisphenoid in front. The bulla itself terminates anteriorly in a blunt spine. ; The alisphenoid is small and forms yery little of the side of the cranium. It is most elongate antero-posteriorly along the ventral line, but has hardly any distinctly developed pterygoid process. At the line of the sphenoidal fissure, which notches but does not per- forate the bone, the alisphenoid is narrowed, to expand again at its suture with the parie- tal and frontal. The orbitosphenoid is relatively rather large, but is low in the vertical dimension, and does not extend upward into the orbit proper. Two sharp ridges on the external face of the bone enclose a V-shaped grooye, in which lie the optic foramen and foramen lacerum anterius. The parietals are very large proportionately to the size of the cranium, but quite small as compared with the entire length of the skull; they roof in most of the cerebral Ne By VO, WiIDK, DY A 280 THE OSTEOLOGY OF ELOTHERIUM. chamber, but toward the yentral side they rapidly contract, forming narrow strips between the squamosal and frontal. Throughout their length the parietals unite to form the very high, thin and plate-like sagittal crest, which is one of the most characteristic features of the skull. In the European species, #. magnum, this crest has a remarkably straight and horizontal course, but in the known American species it is gently arched from before backward. Large sinuses are developed in the parietals, so that the cerebral chamber is eyen smaller than it appears to be, when viewed from the outer side. These sinuses extend over the entire roof of the cerebral fossa, even invading the supraoccipital ; they appear to be traversed by numerous small trabeculee, the ends of which are seen, in the sagittal section, embedded in the matrix which fills the sinuses. The frontals are much larger than the parietals. In the postorbital region they are — very narrow, in conformity with the very small size of the brain, but at the orbits they expand widely to form the broad, lozenge-shaped forehead, which is convex from side to side, though slightly depressed, or “dished” in the middle; the supraciliary ridges are very inconspicuous. Anteriorly the frontals diverge to receive the nasals between them, sending forward long, pointed nasal processes, which, owing to the great elongation of the muzzle, are widely separated from the premaxillaries. The orbit is large and projects prominently outward ; it is completely encircled by bone, the long and massive postorbital process of the frontal uniting suturally with the shorter process of the jugal. The orbits do not rise above the leyel of the forehead, as they do in Hippopotamus, and present more anteriorly, less directly outward, than in that animal. Mention has already been made of a groove on the orbitosphenoid, which terminates below and behind in the fora- men lacerum anterius; this groove is continued upward and forward upon the frontal, steadily widening as it advances. The postero-superior ridge bounding the groove is the more prominent ; it extends almost to the postorbital process, from which it is separated by a distinct notch, while the antero-inferior ridge dies away within the orbit. In most of the American species the forehead rises yery gradually and gently behind to the sag- ittal crest, but in F. imgens the rise is much more sudden and steep. The frontal sinuses are large, giving the conyex shape to the forehead which has been described; these sinuses appear to communicate with those formed in the parietals. Except posteriorly, the sywamosal forms but little of the side-wall of the cranium, its suture with the parietal curving abruptly downward and forward; its compressed and prominent hinder margin forms nearly the whole of the lambdoidal crest, though a con- tinuation of it extends upward upon the supraoccipital, ending in the wing-like processes of that bone. The zygomatie process is enormously developed ; it extends widely out- ward from the side of the skull as a massive, vertical plate, which is shaped much as in Hippopotamus, and is not continued forward as a broad, horizontal shelf, such as is found THE OSTEOLOGY OF ELOTHERIUM. i 281 in Sus. The superior border curves. upward into a great, hook-shaped process, which resembles that seen in Merycochawrus, and gives a highly characteristic appearance to this region of the skull. That portion of the zygomatic process which is directed anteriorly is short and, though massive, is much less so than that which extends out laterally ; in front it is received into a notch of the jugal. The glenoid cavity is large, transversely directed and quite deeply concave, though the postglenoid process is not strongly deyel- oped and is hardly more conspicuous than the preglenoid ridge. This disposition is unusual among the ungulates, but it occurs also in the Eocene genus Achanodon and in the modern Dicotyles. The glenoid cavities of the two sides are very widely separated, their inner margins lying external to the line of the paroccipital processes. The posttym- panic process of the squamosal is small, and is closely applied to the paroccipital process. The shape of the zygomatic arches, together with the extreme narrowness of the cranium proper, causes the temporal openings to be very large and to appear widely open when the skull is viewed from above. ‘These openings are, however, less extended transversely and more antero-posteriorly than in Hippopotamus, while in Sus they are hardly visible from above. The jugal isa very remarkable bone and constitutes one of the most extraordinary features of the Hlotheriwm skull. Posteriorly it is notched to receive the zygoma, and sends out a process along the ventral face of that bone, extending to the preglenoid ridge. The jugal forms the inferior half of the nearly circular orbit, and for this purpose its dorsal border is made deeply concave, giving off a stout postorbital process to meet that of the frontal, while anteriorly it is moderately expanded upon the face in front of the orbit, where it is wedged in between the lachrymal and the maxillary. The most pecu- har feature of the jugal, however, is the immensely developed vertical plate, which descends from beneath the orbit downward and outward to below the level of the ven- tral border of the mandible, recalling the similar, but much Jess massive processes found in certain edentates, e. g., Megatherium. ‘These plates are laterally compressed, but quite thick, and when the skull is viewed from the front, they are seen to diverge quite strongly downward ; their shape varies in the different species. In the very large forms from the Protoceras beds, such as #. imperator, the process retains its plate-like form throughout, its free end being only moderately thickened. This appears to be true also of £. mortoni, though my material is not sufficient to allow me to make this statement positively, but in the large species from the Titanotherium and Oreodon beds (£. ingens) it forms a club-like thickening at the tip, which in /. ingens is coarsely crenulate on the posterior border (see Pl. XVII). These processes are, so far as is yet known, quite unique among the hoofed mammals, and it is difficult to form eyen a conjecture as to what their functional significance may have been. Some misunderstanding has arisen as to the spe- 282 THE OSTEOLOGY OF ELOTHERIUM. cies in which these jugal plates are found. Nothing is known concerning their presence or absence in the European representatives of the genus. Leidy’s material gave him no reason to suspect their occurrence in the species described by him, and he consequently restored the zygomatic arches without them (769, Pl. XVI). Marsh first discovered the processes in a skull of the species named by him /. crasswm, and it has sometimes been assumed that they were more particularly characteristic of that form. As a matter of fact, they have been observed in all of the American species of which well-preserved skulls are known, viz., 2. mortoni, EL. ingens, and E. imperator, and, in all probability, all the American forms, at least, possessed them. The achrymal is a rather large bone and forms nearly half of the anterior boundary of the orbit. On the face it is expanded into quite a large plate, which articulates below with the jugal, in front with the maxillary, and above with the frontal, the long anterior process of which prevents any contact between the lachrymal and nasal. In Aippopota- mus the very short, broad frontal has no anterior process, and so the nasal and lachrymal are connected, as they are also in Sus. Within the orbit the lachrymal is but little extended ; the foramen is single, very small, and placed inside the orbital margin. The lachrymal spine is very low. The nasals are narrow, slender and very much elongated. Their greatest width is at the anterior end of the nasal processes of the frontal, and here is also their greatest transverse convexity ; ‘from this point they narrow and flatten, both in front and behind. Anteriorly they contract very gradually and terminate in sharp points, with their free ends quite deeply notched. In /. wmgens the nasals appear to be relatively shorter than in the other species. In /Hippopotamus these bones have much the same shape as in Hlotherium, but they narrow more abruptly behind the poit of greatest width, and their free ends are not notched. In Sus the nasals are truncated posteriorly and in front their free tips project far beyond the borders of the premaxillaries. The premaxillaries ave very large and heavy bones, the horizontal or alveolar portion especially so. Posteriorly, this portion is constricted, forming a groove for the reception of the lower canine, expanding again in front to carry the large incisors. The palatine processes are not much developed, the very large incisive foramina leaving but little space for them; the spines are long and slender, extending behind the canine alveolus. The ascending ramus of the premaxillary is low and rises gradually behind, and though broad at first, it rapidly becomes very slender, terminating behind in a fine point. Though these bones in Hlotheriwm haye a very different appearance from the immensely enlarged premaxillaries of Hippopotamus, yet both may have been formed by divergent modifications of a common plan. The maxillary is greatly extended antero-posteriorly, in correspondence with the THE OSTEOLOGY OF ELOTHERIUM. 283 elongation of the whole muzzle ; its facial portion is low, gradually diminishing in height forward, where its suture with the premaxillary forms a very gentle, sweeping curve. The longest suture of the maxillary is that with the nasal, the connection with the frontal being very short, owing to the. extension of the lachrymal. Posteriorly, this bone pro- jects but little beneath the orbit, which has an imperfectly developed floor, and the pro- jection which it sends out to the jugal is much less massive than in Hippopotamus. The face gradually narrows forward, until it reaches the infraorbital foramen, expanding again in front of the foramen and swelling out into the prominent canine alveolus. The palatine processes of the maxillaries are long and narrow, and as the molar-premolar series of the two sides form almost straight and parallel lines, the bony palate is of nearly uniform width, slightly concave transversely, but almost plane antero-posteriorly. In front, these palatine processes are deeply emarginated by the large incisive foramina, and in the median line are still further notched to receive the long premaxillary spines. The palatines make up but very little of the bony palate, forming only a narrow strip in front of the posterior nares, and narrow bands along the sides. The palatal notches are small and shallow. The pterygoids are elongate, but quite low; there are no hamular processes or pterygoid fossee; the two bones meet suturally along the median dorsal line, completely concealing the presphenoid from view. The posterior nares are long, narrow and low, extending forward to the middle of m 2; the opening gradually contracts posteriorly, where it becomes very narrow, while the side-walls slope upward and die away upon the alisphenoids. Anteriorly the nares are divided by the very large vomer, Which is distinctly visible, and which at its hinder termination expands into a transverse plate, articulating with the palatines. The meeting of the two pterygoids forms a small canal, which appears to overlie the whole length of the posterior nares and to open forward into the nasal chamber on each side of the vomer. This is a very excep- tional arrangement, and I am unable to suggest what its functional meaning may be (see Pl. XVIII, Fig. 1, c). The cranial foramina are, in some respects, quite peculiar. The condylar foramen is large and conspicuous, being placed well in front of the condyle ; it is, however, smaller than in the specimen of /. magnum which Kowaleysky has figured. The close approxi- mation of the paroccipital and stylomastoid processes, and the outward extension of the tympanic between them, have given a somewhat unusual position to the postglenoid and stylomastoid foramina; they are crowded close together at the postero-external angle of the auditory bulla, and both of them perforate the enlarged tympanic bone. The fora- men lacerum posterius forms a long, narrow and curved slit at the postero-internal angle of the bulla, while the foramen lacerum medium and the opening of the eustachian canal occupy their ordinary position at the front end of the bulla. No distinct carotid canal is visible externally. 284 THE OSTEOLOGY OF ELOTHERIUM. Kowaleysky inferred from the study of his specimen that the foramen ovale “ nicht als selbstiindiges Foramen existirte, wie z. B. bei den Ruminanten, sondern mit dem For. lac. med. verschmolzen war, wie bei den heutigen Suiden und bei Hippopotamus” (’76, p- 483). This is probably a mistake; at all events, it is not true of the American species, in which the foramen ovale is a long, conspicuous opening, of oval shape, perfo- rating the alisphenoid. As in the ungulates generally, there is no separate foramen rotun- dum, that opening being fused with the foramen lacerum anterius. The latter is a large and somewhat irregular opening, which notches the anterior border of the alisphenoid, passing between that bone and the orbitosphenoid. The optic foramen is small and well separated from the foramen lacerum anterius, lying in front of and at a slightly higher level than the sphenoidal fissure ; it does not open so far forward as in L. magnum, and, in consequence, it does not form such a remarkably elongated canal as in the European species (see Kowalevsky, 76, Taf. XVI, Figs. 1 and 3, dd), but, on the other hand, it is far from being a simple perforation of the orbitosphenoid, such as occurs in the recent ungulates. This elongation of the optic canal should probably be correlated with the very small size of the brain, which would seem to have been relatively smaller than in the ancestors of the genus. Though the orbits are far behind their primitive position, the backward shifting of the optic tract would seem to have kept pace with the change in the position of the orbits. The posterior palatine foramina are large and conspicuous openings, placed at the maxillo-palatine suture, and separating the two bones at these points; the palatine plates of the maxillaries are deeply grooved for some distance in front of the foramina. The incisive foramina are likewise large, invading both the maxillaries and the premaxilla- ries; indeed, their size preyents the development of any considerable palatine processes on the latter bones. These foramina are in very marked contrast to those of ippopota- mus, in which the enormously expanded and massive premaxillaries are perforated by two small and widely separated openings; in Sus also the incisive foramina are propor- tionately much smaller than in Hlotheriwm. The infraorbital foramen is large and is separated from the orbit by a considerable interval, opening above the anterior border of p 2. In front of the foramen a deep groove channels the outer face of the maxillary for a short distance. The canal itself is much elongated, in correspondence with the great length of the jaws, and its posterior orifice, within the orbit, is very large. The lachry- mal foramen, which is single, is quite small and is placed inside of the orbit. The supraorbital foramen is subject to some variation in the different species. In E. ingens, from the Titanotherium beds, these openings are of good size, are placed quite near to the median line, and have well-marked vascular channels running forward from them. In specimens of #, mortoni from the Oreodon beds, and in the very large species THE OSTEOLOGY OF ELOTHERIUM. 285 (£. imperator) from the Protoceras beds, the openings haye become minute; they are shifted laterally and haye no anterior grooves leading from them. The mandible is not the least curious part of this remarkable skull. The horizontal ramus is extremely long and nearly straight, with an almost horizontal inferior border. The depth and thickness of the ramus yary considerably; even in skulls of the same length the mandible is decidedly more slender in some specimens than in others. The materials are, however, not yet sufficient to determine whether this difference is of a spe- cific, sexual, or merely individual character. -A remarkable knob-like process is given off from the ventral border of the mandible, beneath p z, which is subject to much -yari- ation in shape and elongation, in accordance with the age and size of the animal. In young indiyiduals still retaining the milk-dentition, the process is a mere rugose eleva- tion, and in the adults of the smaller species it is hardly more than a knob, while in the large forms it becomes greatly elongated and club-shaped. No marked difference in this regard is observable between the species from the upper and those from the lower hori- zons of the White River formation, the process being relatively quite as long and promi- nent in Z. ingens from the Titanotherium beds, as in /. imperator from the Protoceras beds, but in the huge John Day species it has become particularly long and heavy. The symphysis is quite long and very thick and massive; the two rami are indis- tinguishably fused together and laterally expanded, so as to somewhat resemble the sym- physis of Hippopotamus, though not attaining any such extreme degree of massiveness as in the modern genus. The chin is abruptly truncated and flattened, and rises very steeply from below; on each side, beneath or a little behind the canine alveolus, there arises from the ventral border a second club-shaped process, similar to, but much heavier and more prominent than the posterior process already described. These two pairs of knobs give to the jaw a highly peculiar and characteristic appearance ; they form another of the enigmatical features of the Elotherium skull, for it is difficult to imagine what part they can have played in the economy of the animal. The two inferior dental series pursue a nearly parallel course, diverging backward but little, but behind the molars the two rami turn outward and diverge rapidly, so that posteriorly they are very widely separated, in correspondence with the great interval between the glenoid cavities of the two squamosals. The angle of the mandible is prom- inent and descends below the ventral border of the horizontal ramus, much as in Hippo- potamus, though not to the same extent. The ascending ramus is not high, but of con- siderable antero-posterior extent. The masseteric fossa is quite small, but very deeply impressed, and is situated quite high upon the side of the jaw. The condyle is relatively little raised above the level of the molar teeth, and it is sessile, hence inconspicuous, though it is large, transversely expanded, and strongly convex. The coronoid process 286 THE OSTEOLOGY OF ELOTHERIUM. is strikingly low and small; it is of triangular shape, erect and not at all recurved, and is separated from the condyle by a yery wide sigmoid notch. The mental foramen is small, single, and placed below p 3. Several of the hyoid elements are preserved in connection with the skeleton of L. ingens which forms the principal subject of this description. The stylohyal is quite long and slender ; its proximal portion is laterally compressed and yery thin, but moderately broadened in the fore and aft direction. For the distal two-thirds of its length the bone is thicker and of a compressed oval section, expanding into a club-shaped thickening at the lower end, which is excavated for the connecting cartilage. The ceratohyal is con- siderably shorter than the stylohyal, but of quite similar shape; its proximal end bears a cup-shaped expansion, beneath which it becomes yery thin and much compressed, but broadened antero-posteriorly ; the inferior part of the shaft is slender and oyal in section, with another cup-shaped expansion at the distal end. The epihyal and basihyal haye not been preserved. The thyrohyal is of remarkable length and slenderness, and obyi- ously was not codssified with the basihyal; the bone is of subcylindrical shape, with expansions at the proximal and distal ends. This hyoid apparatus does not resemble that of any artiodactyl with which I have been able to compare it. The elements of the anterior arch somewhat resemble those of Fippopotamus, but are more slender and elongate. In the modern genus, on the other hand, the thyrohyals are very short, and are ankylosed with the basihyal, a totally differ- ent arrangement from that which characterizes Hlotherium. From the foregoing description and accompanying figures it will be obyious that the skull of Hlotherium is an extremely peculiar one. Among recent animals that of Hippo- potamus approximates it most closely, and displays, with many striking differences, sey- eral decided and, it may be, significant resemblances. Some of these resemblances, such as the straight cranio-facial axis and the long sagittal crest, are of no particular import- ance, because they occur so yery generally among the primitive ungulates of all groups. Other similarities, again, are not of this nature. The proportions of the cranial and facial regions, the degree of backward shifting of the orbits, the relations of the zygo- matic and paroccipital processes, the broadening of the muzzle, and the general plan of skull construction, are all similar in the two genera. On the other hand, each genus has certain peculiarities correlated with its manner of life. Thus, the elevation of the orbits and the backward displacement of the posterior nares in Hippopotamus are adaptations to its aquatic habits. Doubtless the extraordinary peculiarities of Hlotherium, such as the dependent processes of the jugals and the great knobs on the mandible, are of a sim- ilar nature, though, in the absence of the soft parts, it is difficult even to conjecture what their use may have been. THE OSTEOLOGY OF ELOTHERIUM. 287 Measurements. No, 11156. No. 10885. No. 11009. No. 11440. Skull, extreme length on basal line........................... 0.803 20,648 20.460 «« width across zygomatic arches (behind jugal process).. . 2.500 443 2297 -264 66 AINA 1) 546 po sgeodocamounalds Goodeabecad elsjelelejetelalele.ls 133 140 :089 -082 Cranium, length to anterior border of orbit.................... .282 -288 | .198 193 Face, length to anterior border of orbit.-..................... | .518 2.378 270 Occiput breadthwote base reer ee terre eee | 281 252 .160 158 Fe on AGHA TNE Uke eoeodd b GOSAEOE ies oS aU ate eee OR Ia Oe San | .158 .120 Bony palate, length in median line................-.......... 2.376 | 247 AP SOMANE OREM, WOMB, > o choco peoces cob oso ons acosudEeogo0NH 279 Sarat | 146 146 Descending process of jugal, length................-......06- | 830 256 126 Mandible wlenothencyacea eters tele csh cits aerate ever | .659* 608 Es heightaticoronoidsprocessies-eee sees bieas | .253* 171 107 co GI NUN 1) sos omc a ssaustcosdos soc laed anne aoamiacetre tc .133* 091 052 * No. 11161. Ill. Tue Bra. Attention has been repeatedly called, in the foregoing description of the skull, to the extraordinarily small size of the brain-cavity. Even on viewing the skull externally, this smallness of the cranium proper strikes the observer immediately, and, in connection with the long, slender muzzle, gives the skull something of a reptilian aspect. When the cranium is sawn open in longitudinal section, it becomes apparent that the brain is even smaller than would be inferred from the external view alone, much of the space being, so to speak, wasted in the great frontal and parietal sinuses which overlie the whole cerebral chamber. In a large, full-grown skull this chamber will hardly contain an ordinary human fist. The olfactory lobes are very large and are connected with the cerebrum by short thick olfactory tracts. The lobes are not at all overlapped by the hemispheres, but are entirely exposed for their whole length. The cerebral hemispheres are relatively small, though they are, of course, much larger than the other segments of the brain; so short are they that they do not extend over the olfactory lobes in front, or the cerebellum behind. In shape, they are low and wide, narrowing gradually forward, but with blunt anterior termination. The frontal lobe is yery small, for the frontals take but little share in the roof of the cerebral chamber. The parietal lobe, on the other hand, is relatively large and forms the greater part of the hemisphere, for there is, properly speaking, no occipital lobe, the occipital bones not tak- ing any part in the formation of the cerebral fossa. The temporo-sphenoidal lobe is also quite large and prominent, but is short antero-posteriorly. The brain-cast shows that the As 1 Se AVOWs BIDS 1 988 THE OSTEOLOGY OF ELOTHERIUM. hemispheres were conyoluted, but the conyolutions are so feebly marked that they are hardly worth description. It is obvious, howeyer, that the gyri were fewer and simpler than in any of the modern ungulates. The cerebellum is rather small, though the cerebellar fossa has a vertical diameter not much less than that of the cerebral fossa. Antero-posteriorly the former is quite short and its transverse breadth is not great. This breadth is still further reduced by the relatively very large size of the periotic bones which extend freely into the fossa. IV. THe VERTEBRAL CoLumy. The vertebral formula is: C 7, Th ? 138, L. 6,S 2, Cd 15 -+ The atlas (Pl. X VIII, Fig. 3) is very wide transversely, and at the same time it is of considerable antero-posterior extent, a shape which recalls that of Anoplotherium, rather than that of the recent ruminants or suillines. The anterior cavities for the occipital condyles are deep and wide, but low and depressed. Dorsally, these cotyles are widely separated by a broad, but not very deep emargination of the neural arch, nor do they approximate each other very closely on the ventral side, a notch of considerable width intervening between them at this point. The neural arch is thick and heavy, but short from before backward and quite narrow transversely ; it is also low, not arching strongly toward the dorsal side, and nearly smooth, being free from any but the most obscurely marked ridges. The foramina perforating the arch for the first pair of spinal nerves are unusually large. The neural spine is rudimentary and forms only an inconspicuous tubercle. The neural canal is low and broad, forming a transversely directed ellipse. The inferior arch is considerably more elongated antero-posteriorly than the neural, and has but little transverse curvature, except laterally, where it rises to form the sides of the neural canal. The hypapophysis is represented by a small, backwardly directed tubercle, which arises from the hinder margin of the ventral arch, and occupies the same position as in the pigs, but is much less strongly developed. The articular surfaces for the axis are low and broad, and haye a very oblique position, presenting inward toward the median line, almost as much as backward; they have also a slight dorsal presenta- tion. In shape, they are yery slightly concave and are surrounded by prominent borders. The facet for the odontoid is wide, and deeply conecaye in the transverse direc- tion, but quite short antero-posteriorly. This facet is connected at the sides with those for the centrum of the axis, but distinct ridges are formed along the line of junction. The transverse processes of the atlas extend out widely from the sides of the arch, attaining their greatest transverse breadth along the posterior line; they are also very long in the fore-and-aft direction, reaching far behind the surfaces for the axis. For most of their course the transverse processes have thin borders, but posteriorly the THE OSTEOLOGY OF ELOTHERIUM. 289 margin becomes much thicker and more rugose. The vertebrarterial canal, which is notably small, occupies much the same position as in Sus, opening posteriorly upon the dorsal side of the hinder border. The anterior extension of the transverse processes has conyerted into foramina (atlanteo-diapophysial) the notches for the inferior branches of the first pair of spinal nerves. On the ventral face of each process is a large fossa, enclosed between the side of the inferior arch and the greatly thickened posterior border of the process. The resemblance in shape to the atlas of Anoplotherium, to which atten- tion has already been called, affects more particularly the form of the transverse processes but they are more extended transversely than in that genus and are not so pointed at the postero-external angles. The avis (Pl. XVIII, Fig. 4) is a short, but very massively constructed bone, which in general shape and appearance resembles that of Lippopotamus. The centrum is short, anteriorly yery broad and depressed, but thickening posteriorly, and with a nearly circular and slightly concave hinder face. A strong and prominent keel runs along the ventral face of the centrum, enlarging backward, and terminating behind in a trifid hypapophysis. The odontoid process is short, heavy and conical, with no tendency what- ever to assume the depressed and flattened shape which occurs in so many White River ungulates. The yentral articular surface of the odontoid seems like something super- added to the process itself, for it is clearly demarcated by a groove running all around it, and projects slightly in front of the body of the process. On-the dorsal side of the centrum a broad and well-defined ridge runs backward from the odontoid along the floor of the neural canal. The atlanteal articular surfaces are yery broad and low, not rising so as to enclose any part of the neural canal. They are very oblique with reference to the median line of the centrum, with which they form angles of about 45°. These surfaces are slightly convex in both directions, and ventrally they project much below the level of the centrum. The transverse processes are short, thin and compressed, much less massive and widely extended than in Hippopotamus ; they are perforated by very large foramina for the vertebral arteries. The pedicels of the neural arch are low and short, but very heavy ; they are not pierced for the passage of the second pair of spinal nerves, as they are in Hippopotamus and in some of the pigs. The neural canal is decidedly small, especially its anterior opening; behind, it enlarges somewhat, particularly in the dorso- ventral dimension, the posterior opening being high and narrow, while in Hippopotamus it is low and broad. The neural spine is a large plate which is very thin in front, but becomes thick and massive behind, ending in a broad rugosity. This spine resembles that of Hippopotamus, but is not produced so far backward and does not overhang ‘the third cervical, The pestzygapophyses are large, slightly concaye, and present obliquely 290 THE OSTEOLOGY OF ELOTHERIUM. outward, as well as downward; their bases are separated by a broad and deep grooye, which is continued upward upon the posterior side of the neural spine. The third cervical vertebra also bears a considerable resemblance to that of Aippopo- famus, differing only in some points of detail. The centrum is short, heavy and moder- ately opisthoccelous, depressed, but increasing posteriorly in vertical thickness. It bears -a strong yentral keel, which terminates behind, as in the axis, in a trifid hypapophysis. The pedicels of the neural arch are not, as in the pigs, pierced by foramina for the spinal neryes; they are low and short, but very thick, and the neural canal is strikingly small. The dorsal side of the arch is short, broad and nearly flat. The neural spine is remarkably well-developed (when the anterior position of the vertebra is taken into account), rising as high as that of the axis. It is rather thin and compressed, although its base occupies the whole fore-and-aft length of the arch. From the base, however, it rapidly tapers upward and terminates ina small, rough tubercle. In Hippopotamus the third cervical has an even better developed neural spine, not higher, but broader and less tapering than in Llotherium. The prezygapophyses are large, oblique and somewhat conyex ; they are placed very low, so that their inferior margins are separated from the centrum only by narrow notches. The posterior zygapophyses are much larger and more prominent than the anterior pair; they are also less oblique in position and are raised higher above the centrum, corresponding to the posterior elevation of the neural arch. The transverse. process is a compressed plate, which has no great vertical height, but is well extended from before backward, exceeding the centrum in length; the pos- terior portion of the process is thickened and recurved, ending in a rugose hook. The absence of any distinctly marked diapophysial element distinguishes this vertebra from the corresponding one of Hippopotamus and Sus, and in the latter genus the inferior lamella is more slender and rod-like, while the spinal nerves make their exit through foramina in the pedicels of the neural arch. The fourth cervical vertebra is different, in many respects, from the third. The centrum is somewhat shorter and is less distinctly carinate on the ventral side, but is more decidedly opisthoccelous. The neural arch is remarkably short in the antero-posterior dimension, so that the articular faces of the postzygapophyses actually extend forward beneath those of the anterior pair, which gives to the pedicel of the neural arch, when seen from the side, a curiously notched appearance. The neural spine is higher, but more slender and recurved than that of the third cervical. The transverse process is altogether different in shape from that of the latter. It has, in the first place, a very prominent diapophysial element, which projects outward as a heavy, depressed bar, thickened, rugose, and slightly upcurved at the distal end. In the second place, the inferior lamella is much higher vertically, but decidedly shorter from before backward, THE OSTEOLOGY OF ELOTHERIUM. 291 In Hippopotamus and in Sus this vertebra is very similar to that of Elotherium, but the neural spine is notably heavier. The fifth cervical vertebra las an even shorter neural arch than the fourth-and a much higher neural spine. ‘The spine tapers rapidly from the base upward and becomes very slender, but it is nearly straight and only slightly recurved. The neural canal is somewhat larger than in the fourth vertebra, but, as in all the cervicals, it is strikingly small as compared with the size of the vertebra as a whole. The diapophysis is strong and prominent, but more slender than on the preceding yertebra, while the inferior lamella, though relatively short from before backward, has attained great vertical height and is strongly everted. In Elotheriwm the fifth vertebra is of the same type as the sixth, whereas in Aippopotamus it more nearly resembles the fourth. The sixth cervical is very like the fifth, but displays certain obyious differences. Thus, the neural arch is even shorter antero-posteriorly, and the neural spine is higher, heavier and much more strongly recurved. The postzygapophyses are decidedly smaller and are very characteristic in their markedly oblique position, for they rise steeply back- ward in a way that occurs in none of the other yertebree. The diapophysis is shorter but heavier than that of the fifth, while the inferior lamella is of similar shape, but larger, higher and with the free margin more thickened. In Hippopotamus this vertebra has much the same construction as in Lotherium, but the spine is shorter and more massive and the inferior lamella is much larger. In Sus the sixth cervical bears considerable resemblance to that of the White River genus. The seventh cervical is characterized by the height and thickness of the spine, which in these respects much exceeds that of the sixth. This spine tapers superiorly, but expands again at the tip into a rough tubercle. The posterior zygapophyses stand at a higher level than the anterior pair and are unusually concave. The peculiarities seen in the postzygapophyses of the sixth and seyenth vertebree are to provide for the curvature of the neck, which changes its direction at this point. From the occiput to the sixth cervical the neck is nearly straight and inclines downward and backward, while the seventh vertebra begins the rise which culminates in the anterior thoracic region. This change in direction requires greater freedom of motion, which is supplied by the modifi- cation of the zygapophyses upon the vertebree mentioned. The transverse process is, as usual, not perforated by the vertebrarterial canal; it is rather short, but heavy and much expanded at the distal end. On the posterior face of the centrum are large facets for the heads of the first pair of ribs. In Hippopotamus the neural spine of the seventh cervical is relatively much longer and heavier than in Hlotheriwm or in Sus. As a whole, the neck of Hlotherium is short and massive, with very strongly developed processes for muscular and ligamentous attachments, as are indeed necessitated 292 THE OSTEOLOGY OF ELOTHERIUM. by the immense weight and length of the head. Among recent artiodactyls Hippopotamus has cervical vertebrae most like those of Hlotheriwm, though there are many differences in the details of construction. The most apparent of these differences lies in the greater and more uniform height and thickness of the neural spines in the modern genus. Doubtless the even more exaggerated massiyeness of the skull in the latter is the occasion of this increased development of the ceryical spines. In Sus the perforation of the neural arches for the passage of the spinal nerves constitutes an important difference from Hlothervum. The thoracic vertebra would appear to haye numbered thirteen, though this point cannot, as yet, be determined with entire certainty, and while the thoraco-lumbar vertebree were, in all probability, nineteen in number, as is well-nigh universal among the artio- dactyls, yet there were doubtless variations in the number of ribs, as is very frequently the case among existing animals. The first thoracic has a rather small centrum, with decidedly convex anterior and nearly flat posterior face; the facets for the rib-heads are very large and deeply concave. ‘The transverse process is rather short, but very large, heavy and rugose, and bears an unusually large, concave facet for the tubercle of the first rib. The prezyga- pophyses are of the cervical type, but present more obliquely inward than in the vertebree of the neck, while the postzygapophyses are, as in the other thoracics, placed upon the ventral side of the neural arch. The neural canal is high and narrow and its anterior opening has assumed a cordate outline. The neural spine is inclined strongly backward, much more so than that of the seventh ceryical, and though laterally compressed it is extremely high, broad and massive, greatly exceeding in all its dimensions that of the last neck vertebra. The anterior six thoracic yertebree (see Pl. XVIII, Fig. 5) are very much alike in appearance. The first three have broader and more depressed centra, which in the others become deeper vertically and more trihedral in section. The transverse processes are very large and prominent and carry large, deeply concave facets for the rib tubercles. The neural spines are very high, thick and heavy, and are strongly inclined backward, with club-shaped thickenings at the tips. At the seventh thoracic begins a rapid reduc- tion in the length and weight of the spines, a process which reaches its culmination on the eleventh vertebra, which has a remarkably short, weak and slender spine. This arrangement results in a great hump at the shoulders, somewhat as in Zitanotherium, though in a less exaggerated form. In both genera, the length of the anterior thoracic spines should be correlated with the great elongation and weight of the skull which requires immense muscular strength in the neck and shoulders. Hippopotamus has no such hump, but this is probably explaimed by its largely aquatic habits, THE OSTEOLOGY OF ELOTHERIUM. 293 A change in the character of the facets for the rib tubercles occurs simultaneously with the shortening of the neural spines ; they suddenly become much reduced in size and are plane instead of concave. The transverse processes, however, remain very large and prominent as far back as the eleventh thoracic. In no case are these processes per- forated by vertical canals, such as occur in Sus. The twelfth thoracic is the anticlinal vertebra and has a nearly erect spine of lumbar type, though somewhat more slender than in the true lumbars. On the thirteenth the spine is quite lke that of the lumbars and inclines slightly forward. Transverse processes are absent from the last two thoracic yertebree, which display the feature, very unusual in an ungulate, of large and conspicuous anapophyses. As far back as the eleyenth vertebra the zygapophyses are of the ordinary thoracic type; they are small, oval facets, the anterior pair on the front of the neural arch and presenting upward, the posterior pair on the hinder part of the arch and presenting downward. On the eleventh thoracic a change takes place ; the anterior zygapophyses are as before, but the posterior processes are flat and present obliquely outward, rather than downward, the two together forming a prominent, wedge-shaped mass. The prezygapophyses of the twelfth vertebra are correspondingly modified; they present obliquely inward and together constitute a cayity which receives the wedge-like projec- tion from the eleyenth. Prominent metapophyses also make. their appearance on the twelfth thoracic. The posterior zygapophyses of the latter and both pairs of the thirteenth are of the cylindrical, interlocking type characteristic of the lumbars. These processes are remarkably complex and in a fashion that does not occur in Hippopotamus, but is found in Sus and many of the Pecora. The complexity is occasioned by the development of large episphenial processes, which give an additional articular surface above the zy gapophyses proper ; in section these processes have an S-like outline, and they constitute a joint of great strength. The lumbar vertebra (Pl. XVIII, Fig. 6), almost certainly six in number, have rather short, but massive centra. In the anterior part of the region the centra are some- what cylindrical in shape, but they become more and more depressed and flattened as we approach the sacrum. The neural canal is broad and very low, especially in the pos- terior part of the region. The neural spines are inclined forward and are of moderate height ; they are broad antero-posteriorly, but thin and laterally compressed, except at the tips, where they are thickened. The spine of the last lumbar is a little different from the others in being more erect and slender. Episphenial processes are present on the first, second and sixth vertebre, but not on the third, fourth or fifth. These processes are apt to be somewhat asymmetrical and better developed on one side than on the other, and it is probable that more extensive material would show them to be subject 294 THE OSTEOLOGY OF ELOTHERIUM. to much individual variation. Metapophyses are prominent only on the first and second lumbars, rudimentary on the third and absent from the others. The transverse processes are very feebly developed in proportion to the size of the vertebrae. On the first lumbar they are short and straight, and gradually increase in length up to the fifth, but in all they are strikingly thin and slender. The last lumbar has transverse processes of unusual length, space for them being obtained by the sudden eyersion of the anterior ends of the ilia, but even here they are weak. The trunk-vertebre of Hippopotamus are much more massively constructed than those of Hlotheriwm, the decrease in length of the thoracic spines posteriorly is more gradual, while the neural spines and transverse processes of the lumbars are much longer and in every way heavier. The thoraco-lumbar series of Sus bears considerable resem- blance to that of Hlotheriwm, but in the former the transverse processes of the thoracic vertebree are perforated by vertical canals, and those of the lumbars are much longer and stouter. The sacrum consists of two vertebree only. The first has a broad, depressed centrum and yery large pleurapophyses, which carry most of the weight of the ila, though the second sacral has also a limited contact with the pelvis. On the first vertebra the prezygapophyses are very well-developed and haye large episphenial processes to receive those of the last lumbar. The two neural spines are co(ssified into a high but short ridge. The second sacral has a yery much smaller and especially a narrower centrum than the first, and retains moderately complete postzygapophyses. In Hippopotamus and in Sus the sacrum is relatively much larger than in Hlotheriwm, and consists of at least four vertebrae, sometimes eyen as many as six. Hyven in aged individuals of the White River genus I haye not seen more than two vertebre in the sacrum. ; ; The caudal vertebre (Pl. XVIII, Figs. 7, 8, 9), of which fifteen are preserved in association with one individual, indicate a tail of only moderate length, and present a number of peculiarities. The first caudal has somewhat the appearance of a miniature lumbar ; its centrum is short, broad and depressed, with quite strongly conyex faces; the neural canal is relatively large and a distinct, though small, neural spine is present. The zygapophyses, especially the anterior pair, are large and prominent and project much in front of and behind the centrum. The transyerse processes are quite long and heavy, and are directed outward and backward. A pair of tubercles on the ventral side of the centrum represent rudimentary hemapophyses. The succeeding caudal vertebrae resemble the first in a general way, but passing backward, the centra become more and more slender and elongate, while the neural canal diminishes in size, and the various processes are reduced. The hemapophyses, on the THE OSTEOLOGY OF ELOTHERIUM. 295 other hand, increase in size and on the (?) fifth vertebra they curye toward each other, almost meeting and enclosing a canal, which continues as far back as the (?) eighth vertebra, behind which the hemapophyses are again reduced. The middle portion of the tail is composed of very long, cylindrical vertebree, which in shape strikingly resemble those of the great cats, and which are proportionately much longer, though apparently less numerous than those of Anoplotherium. At the anterior end of each vertebra are six prominent, nodular processes, the zygapophyses, transverse processes and hemapophyses respectively. Posteriorly the centra become more and more slender, but are not much diminished in length, for what appears to be the penultimate vertebra is nearly as long as those in the middle region. The various processes are, however, reduced to yery insignificant proportions. The last vertebra has its anterior portion shaped like that of its predecessor, but it rapidly tapers behind to a smooth, slender, compressed and subeylindrical rod, with a club-shaped thickening at the end. As I have seen but a single specimen of this curious vertebra, I cannot feel quite confident that its shape is a normal one and not due to some injury or morbid process. The tail of Hippopotamus is of about the same relative length as that of H/othe- rium, but the individual vertebree are very different, being all shorter and heavier, and diminishing in size more gradually to the end. In Sus the caudal vertebrae are somewhat more like those of Hlotherium, but none of them have such slender elongate centra. Little is known concerning the caudals of Anthracotherium. Kowaleysky says of them: ‘“ Von den Schwanzwirbeln liegt mir nur ein einziges yor. Obwohl seine Erhaltung sehr mangelhaft erscheint, kann man doch aus diesem kleinen Sttick den Schluss ziehen, dass der Schwanz bei den Anthracotherien kurz war und somit gar keine Aehnlichkeit mit dem sonderbar langen Schwanze der Anoplotherien hatte” (73, p. 333; Taf. x, Fig. 36). The vertebra described by Kowaleysky is an anterior caudal and is much smaller and in every way more reduced than the corresponding ones of Elotherium. Among existing artiodactyls, it is the giraffe which most resembles the White River genus in the peculiar character of its caudal vertebre. Measurements. ENE Sp TORE, copsascs coocconan soxsco Spo euESgONDNSSESOSOCOOIS OOSCOIUSOHODOSOND SenocasoosnosecONSeDcCeN CONOR boxeCes 0.160 JNTES, ERGRTIESE TAGLUTR. .. cononeoconscecoenabaosbsanSodosesuSeCCESoONdAN=dos poSSoRHNSsHeanccOOSSCEGHSOAENSSUECHSAOOH J 270 Axis, length GT GST TIT aces AOR Gn ECBO CCD NEC OOOH ERE EEE Boa nicsaaa2 Saccon OO pao Dodo SEER TSC IaA EAC ICRU eee Se 085 ANSE), TGTIRUE ©? OOM OTT ococaconsosscospesgoSoSHESORCoceTRESeoO NcbSoOSHNSSSKOeOOIES SoEeoSSoecanOAbTSIoDIEONS! GOSH. 6 026 Ata Sm anite nl OmMbread uhisascessonnesessncdnecct sec natacacsncscisercanceusenterte se netes seneaeoesemareesnecacaceeaaes -109 JASSS,, POOSHSHIOW | TENN hens concd ccqsonaoadosSuaNsscn5 005095 coscDONSOnoHOSOSURsaSno SondES SsancosEpSRaNSccODONDSSoOOS 054 Third cervical, length Seventinice nyaea lee en pstihveseetssesecenta reese neenanedee ca elene a eeeenertereneee eee nominee anne ea ter eenenee eens .056 TRY TIRGREO KE, LIGLEVE Nace coccescosocteoncdecbe BO ACBOSBECEDABREDSECOATIASE C90: SOSCREEIE HONE EIRE TOSI COSEADOQUOOSES! & 051 KD SSO, cade Wve 296 THE OSTEOLOGY OF ELOTHERIUM.- Measurements. Fifth thorecic, height of neural spine. ..-.....-ecesecseeeeeegeceeteceaeeetscecersceseesensessecssescreenesses esi First lumbar, IGG MoosocsosconesanscosoracznencoABAS.EeESENNCERADNa HoDOUDBONNE> sonSECOOOBDOONDOSEEEDEEODODDSEOSOD .050 SIDR REM ORNR, NE Ixa (EO jccocsocsoxdc000c0s03 29000000 90009n200299R00 D9Sdos09NDC0N9r joNHSaDEDGEgHDODOCHONBeOUE9INSIcGGN 048 Sixth lumbar, breadth across tramsverse PYOCESSES.. .+--.+02cseceecesecs ete ecn sec seeeceseeeeseceserseees 176 fSHYoTUITAS, IETDVR A Blo ccpcqoa0oomcoodocooan00se0cbon.cdoooboSdoAaSant non HDSEdoReOydOBDUDSEDCHaLCaSDOGe gaqNOKOOTaCeUAEADODOOdES c 098 Furstisacrals-waidthoots cent iimissceccsesccsscasc ccssclscGarecivencssrestincsn apices stirasleetns cascelomeensicecensne .068 Second sacral, width of centrum..... ANTEETTOR CAC, HET 29000 cosG0000Kc0 000 cbnan00009NqS00;300 p>noNIOoASHoDOnGNOAOHAEDOCIGHUHDSRDOIHCKOGBARHOEO 3 032 Wicca Gamal, WER ccopsscoaccoonnnponannnoanoscoonocH.CHDaNNSAN}Guo9qo4 AHo.oAHonDOoHOSOnUONDONEY HoBooDNNAHOOET .063 VY. THe Riss and STERNUM. The ribs of Elotherium are decidedly smaller and lighter and indicate a less capacious thorax than we should expect to find in such a large animal, a fact which adds to the apparent height of the skeleton, because of the long interval between the thorax and the ground. The first rib is short, subeylindrical proximally, but broadening considerably at the distal end; it has only a slight lateral curvature, appearing nearly straight when viewed from the front, but it arches moderately backward. The head is large and compressed, and is separated by a deep and narrow notch from the yery large and conspicuous tubercle, which is also compressed laterally. The ribs increase gradually in length up to the seventh or eighth of the series, and the posterior five, though successively shortening, retain a considerable relative length throughout. The first five or six ribs are laterally compressed and of moderate breadth, but the posterior part of the thorax is composed of very slender and subcylindrical ribs, very different from those which we find in most ungulates, except in the more primitive groups. The tubercle reaches its maximum of size and prominence on the third rib, behind which it gradually diminishes in size and becomes more and more widely separated from the head, and more sessile in position. On the twelfth and thirteenth pairs the tubercles are absent, corresponding to the lack of transverse processes on the twelfth and thirteenth thoracic vertebre. In Hippopotamus the ribs are relatively yery much longer, broader and heayier than those of Mlotherium, and grow broader toward the hinder end of the thorax, where the great bony slabs are in the sharpest possible contrast to the slender and subcylindrical rods of the extinct genus. In Sus the ribs are more like those of Hlotheriwm, but they have not such a regular and symmetrical curvature as in the latter. The sternum of Elotherium is a yery remarkable structure, and although it is of distinctly suilline type, it is, nevertheless, not altogether like the sternum of any known genus, recent or fossil. The presternum, or manubrium, forms a very large, thin, com- pressed and keel-shaped plate, which is especially remarkable for its great vertical depth, THE OSTEOLOGY OF ELOTHERIUM. 297 this dimension exceeding the antero-posterior length, and is proportionately much greater than in Hippopotamus or the modern suillines. The body of this segment is extremely thin, but the anterior border, and to some extent the ventral border also, is thickened and rugose. ‘The facets for the first pair of sternal ribs form prominences, which are situated near together and close to the postero-superior angles of the segment, so that nearly the entire length of the latter projects in front of the first pair of ribs. Of the mesosternum four segments and a part of the fifth are preserved. The first segment somewhat resembles the presternum in shape, being short, narrow and yery deep ; the dorsal border is much thicker and wider than any other part of the segment, and the ventral border is also thickened, though in a less marked degree. Posteriorly, this element becomes somewhat wider and shallower. The second segment of the mesosternum is decidedly broader and shallower than the first, but still retains a very unusual degree of vertical depth. Both the dorsal and ventral surfaces are much broadened, while the body of the bone is a thin, vertical plate, which connects the horizontally directed dorsal and ventral borders, giving a cross-section somewhat like that of an I-beam. In the third segment these progressive changes are carried still farther, and the bone becomes very distinctly broader and lower than the second segment. The dorsal and yentral borders still project much beyond the vertical connecting plate ; this plate, however, is much thicker transversely than in the preceding segment. The ventral surface is rendered quite strongly concave by- the elevation of its lateral borders. In part, this concavity may be due to the pressure which has somewhat distorted the entire sternum, but the ventral groove is so symmetrical that it can hardly be altogether due to distortion. The fourth and fifth segments exhibit similar changes, each one being broader and lower than the one in front of it; the vertical plate becomes very much thicker and the ventral groove more widely open. Though the specimen is of an animal past maturity, yet the last three segments distinctly show the median. suture, along which their lateral halves united. In Hippopotamus the breast-bone is quite like that of H/otherium, but the presternum is longer and not of such exaggerated depth, and the rib-facets are placed much nearer to the anterior end, while the mesosternum consists of fewer, broader and shallower segments. In Sus the sternum is still more like that of Hlotheriwm, but has a decidedly longer and lower presternum. VI. Tue Fore Limes. The fore limb of Elotheriwm is quite elongate and, in connection with the shallow thorax, and very long neural spines of the anterior thoracic yertebre, it gives to the skeleton a somewhat stilted appearance. 298 THE OSTEOLOGY OF ELOTHERIUM. The scapula is remarkably high, narrow and slender, at least in the White River species, while in the John Day forms there is reason to believe that its proportions are quite different. The glenoid cayity forms a narrow, elongate oval, with its long axis directed antero-posteriorly, and is not very deeply concave. The coracoid is a large, but not very conspicuous rugosity, which sends off from its inner side a compressed, hook-lhke process ; when the shoulder-blade is seen from the external side, this process is concealed from yiew. ‘The neck of the scapula is broad and rather thick, and there is no distinct coraco-scapular notch. The coracoid border in its upward course inclines forward but little, and for the upper one-third of its height curves gently backward, to join the suprascapular border, which is exceedingly short. The glenoid border is more oblique, and inclines backward and upward at a moderate angle. The spine is shifted far forward, dividing the blade very unequally, so that the prescapular fossa is very much smaller than the postscapular. Indeed, the distal one-third of the shoulder-blade can hardly be said to have any prescapular fossa at all. The spine itself is rather low, and for much of its course its free border is curved backward and thickened to form a massive meta- cromion. ‘The acromion is very short and inconspicuous, ending considerably above the level of the glenoid cavity. . The scapula associated with the large species of Elotheriwm from the John Day beds, which Cope has described under the name of Bodcherus (79, p. 59), is very different in shape from that of /. ingens from the White River, to which the description in the preceding paragraph more particularly applies. The blade is very much broader, both fossee widening rapidly toward the dorsal end; these fossee are of nearly equal width and the spine is placed almost in the middle of the blade. There can be little doubt that this scapula is properly referred to the incomplete skeleton with which it was found associated. Aside from its similarity in color and texture to the rest of the skeleton, there is no other amimal known from the John Day horizon to which so large a scapula could belong. The shoulder-blade of Hippopotamus is much broader, in proportion to its height, than that of /. ingens ; the coracoid is more prominent and the coraco-scapular notch is distinctly marked; the postscapular fossa is somewhat larger than the prescapular, but the difference is much less extreme than in the White River species, the spine occupying a more median position ; the acromion is much the same in the two forms, but the meta- cromion is larger in the fossil. In Sus also the scapula is relatively broader than in E. ingens, and, in particular, it has a wider prescapular fossa, but is without any distinct coraco-scapular notch. The spine rises from the suprascapular border yery steeply to the high (but much smaller) metacromion, and then descends gradually to the neck, without forming an acromion. In spite of these differences, the resemblance in the character of the scapula between Sus and Hlotherium is unmistakable. THE OSTEOLOGY OF ELOTHERIUM. 299 The humerus is relatively long, but is, at the same time, a massively constructed bone. The head is large and very strongly convex, especially from above downward, although it is not set upon a very distinct neck, nor does it project far behind the plane of the shaft. The external tuberosity is very large, forming a massive and roughened ridge, which runs across the whole anterior face of the head and rises toward the internal side, where it terminates in a high, thick and recuryed hook, overhanging the bicipital groove. The internal tuberosity is very much smaller, but is, nevertheless, quite promi- nent; it likewise projects over the bicipital groove, which is very broad and deeply incised into the bone. The great transverse breadth of the external tuberosity displaces the groove far toward the internal side of the humerus. The shaft is long and heavy ; its proximal portion has a great antero-posterior diameter, and its transverse thickness, though less, is still very considerable. The fore-and-aft diameter gradually diminishes downward, until the shaft assumes an almost cylindrical shape, below which point it begins to expand transversely. The deltoid ridge is rugose and prominent, and runs far down upon the shaft, but forms no deltoid hook. The distal end of the shaft is very heavy, being both broad and thick. The supratrochlear fossa is low, wide and shallow, while the anconeal fossa is very high, narrow and deep, its depth being much increased by the great production of the posterior angles of the distal end. The supinator ridge is rough, heayy and prominent. The trochlea, which is very completely modernized, in correspondence with the advanced differentiation of the ulna and radius, is somewhat obliquely placed with reference to the long axis of the shaft, descending toward the ulnar side. The trochlea differs very markedly from that of such primitive artiodactyls as Oreodon and Anoplotherium ; it is high, full and rounded and is divided into two unequal radial facets, of which the inner one is decidedly the larger. The intercondylar ridge, which, in most primitive artiodactyls, forms a broad and rounded protuberance, is, in Klotherium, compressed into a sharp and prominent ridge, and shifted well toward the external side. The internal epicondyle, which is so largely developed in Oreodon and other early artiodactyls, has practically disappeared. The humerus of Hippopotamus is relatively much shorter and more massive than that of Hlotherium ; the external tuberosity is not extended so far across the anterior face of the bone and the bicipital groove is, in consequence, not shifted so far toward the inner side; the deltoid ridge is much better developed and gives rise to a prominent deltoid hook. In the existing species of Hippopotamus the intercondylar ridge is narrower and less conspicuous, but in a Pliocene species from the Val d’Arno it has quite the same appearance as in H/otherium (see de Blainyille, Ostéographie, Hippopot- amus, Pl. V). The epicondyles are much more prominent than in the latter, and the postero-internal border of the anconeal fossa projects much more than does the 300 THE OSTEOLOGY OF ELOTHERIUM. : external border, while in Klotherium this difference is decidedly less marked. In Sus the humerus resembles that of the White River genus in form, but is proportionately very much shorter; the deltoid ridge is shorter and less prominent, while the supinator ridge and the epicondyles are more so. The radius and ulna (Pl. XVIII, Fig. 10) are firmly coéssified in all the known species of Klotherium, though the suture between them is clearly marked, eyen in old animals. ‘The radius is relatively very long, but rather slender; the head is quite thick, but of only moderate breadth, projecting most toward the external side. The humeral surface is composed of three connected facets, of which the internal one is much the largest and bears an elevated ridge for the corresponding depression on the humeral trochlea. The groove for the intercondylar ridge of the latter is quite broad and notches the anterior border of the radius. The shaft is rather narrow transversely, but quite thick and heavy, and arches forward but moderately ; the distal portion is broadened and thickened and bears upon its dorsal face a deep tendinal sulcus, bounded by very promi- nent ridges. The distal face is quite broad, but without much dorso-palmar extension, and carries two well-distinguished carpal facets, which pursue an oblique course, from before backward and inward. The scaphoidal facet, which is the smaller of the two, is concave in front, saddle-shaped behind, and is reflected up upon the posterior face of the bone. The facet for the lunar is much larger than that for the scaphoid, and has a somewhat similar shape, but the anterior concayity is not so deep, and the articular surface is carried much farther up upon the palmar side of the radius. The radius has no contact with the pyramidal. In Hippopotamus the forearm bones are ankylosed, though somewhat less intimately than in Elotherium. The radius is very short, broad and thick, and is almost straight. The external facet for the humerus is larger and more concaye and the carpal facets are of more nearly equal size, while that for the lunar rises much more steeply toward the ulnar side. In Sus the two bones are separate, and the radius is short, very heavy and arched forward ; its distal end is much more thickened than in Elotherium, the facet for the scaphoid is relatively larger, while that for the lunar is smaller and is extensively reflected upon the palmar face of the radius. In Dicotyles the ulna and radius have coalesced even more completely than in Elotheriwm. The w/na has a very long, thick and prominent olecranon, which projects far behind the plane of the shaft. The process is conyex on the outer side and concave on the inner, thickened and club-shaped at the free end, which displays a broad, shallow sulcus for the extensor tendons. The sigmoid notch is deep and the coronoid process prominent, as is required by the great depth of the anconeal fossa on the humerus. ‘The articulation of the ulna with the latter is confined to the posterior and superior aspects of the humeral THE OSTEOLOGY OF ELOTHERIUM. 301 trochlea, no part of the articular surface on the ulna presenting proximally, for the radius occupies the entire distal aspect of the humerus. Only the proximal portion of the facet for the humerus extends across the entire breadth of the ulna; for the rest of its course this facet is confined to the inner side. The shaft of the ulna is somewhat reduced, but is not interrupted at any point and, indeed, it is quite stout for its entire length ; its prin- cipal diameter is the transverse, the antero-posterior thickness being decidedly dimin- ished. Below the head it narrows and then expands to its maximum breadth, from which point it narrows gradually to the distal end. On its external side the shaft is quite deeply channeled. The distal end is small and bears a saddle-shaped facet for the pyramidal, which is concave transversely and conyex in the dorso-palmar direction ; its external border is compressed and extends as a sharp edge behind the body of the bone, forming a concavity on the palmar face. The pisiform facet is continuous with that for the pyramidal. The ulna extends distally below the level of the radius and thus arises the very exceptional condition of an articulation between the ulna and the lunar. The facet for this carpal element is small and is entirely confined to the radial side of the ulna, the distal end of the latter not extending at all upon the proximal face of the lunar. In most artiodactyls in which the functional digits haye been reduced to two, the radius tends to encroach more or less extensively upon the proximal face of the pyramidal, for which extension the diminution of the ulna makes a way. - In Elotherium the arrangement is different, the ulna occupying the entire proximal surface of the pyramidal, and by extending below the level of the radius securing a lateral contact with the lunar. Indeed, this arrangement quite precludes the attamment of the more usual radial-pyramidal articulation. ; The ulna of Hippopotamus is proportionately much shorter and in eyery way more massive than that of Hotherium ; it also has a very much larger and more prominent olecranon, as would naturally follow from the immensely greater weight of body which requires support upon the limbs. There appears to be a slight disto-lateral contact between the ulna and the lunar; at all events, the radius does not extend over upon the pyramidal. In Sus the ulna is free throughout and its shaft is relatively much shorter and heavier than in Elotherium ; the ulna and lunar do not come into contact. The ulna of Dicotyles is more reduced than that of the White River genus and the connections of the carpals with one another and with the metacarpus are upon quite a different plan. 4 Measurements. Scapula, height. ...-...--.seeesceeceeseeeeeesnereneeneecesesesceaceces, cteeecseccrecscesnececesssennseeccensceessnsees 0.430 Scapula, greatest Widthh...........-.c0s--sssececsneceeceeteceeeeecncecerenee cuneecccueseccceccccnseseceeererscerene 245 Scapula, breadth Of NECK -..--+.-..-s0scescnceeeeccececcnaeevteeouanasssscancavestceccccattsencteacncesessccccrese= 065 Scapula, glenoid cavity, ant.-post. Ciameter........:1.cesecseeeeeseceeessseeeseecceceneereesssesaneeereeees .068 302 THE OSTEOLOGY OF ELOTHERIUM. Measurem ents, Scapula, glenoid cavity, transverse CiaMeter..........ccsccccesseceseeneceecceeecanccessececeesecssesseses -050 Pimerus, Lem eth <-ee-- eee em nine nen ennnieccaneisse ste cieciens\asieseerisiosensriss>ssaseass-er/ssensenseaseee saan. rssnns .405 Humerus, width of proximal end,.........06. consoscresssereerssecssecseeeseeeecssssevsreeseesteveresanresers 132 Humerus, thickness of proximal end....-...-:.-secsesseeeeeeeeeeeeeeeecee ee .128* Etumeruss wiadtihvordistaltemdccs-ceateeessccrsecceesscesciccceneasessecheeskerneeeerenteseee cee rerceeeeesecee .095 TREGITS, JIGME H Nooo ocoons00an sno noHSoDSADNoTOMADSGOdNODD soqDoOONoECHSgOFODSEC nocoD DHS SACHCOoDONboBHOSHOSODSHODHSONNSCO .300 DEH USS yy Le fy pox ena ea Cl ep ta ett ee ele ele tel le le ele lee leet see se alee eee eee eee 074 Radaus; widbhuot Gistalliendsaneccscscscssnc sce sece casement siseseseiteaeiec seateneecbenilsshiveasisndseessce=ncteieecce .062 UME, TSE Mescrscae ceeednses2c902009d000 BooNoD oaaSoaDdsSCEDDTONOGNA DLioBeOSOUNGoOSNOSOSHoDOOCUS OCS SBOoOASoooRSOBOCRON 443 Ulna, length of olecranon fr. CoronOid ProCeSs....-.--.--.eseceeceeseccnecceccereeentsasscecenseeceessseners 103 Wika, spadlidn Ge Chie iElscc6on: sonecnacond copoonoGaBEC DOG ADCoOATAgSDoCoOHOOONUNEHSosaDEDSeONS SoeeencSoEEeNOoIOCNCON .037 VII. Tse Manus (Pl. XVII, Fig. 11). co} The principal facts of the structure of the fore foot have already been determined by Kowalevsky, but the material now at command permits a more complete account to be given. Certain differences also which obtain between the European and American repre- sentatives of the genus should not be passed over without mention. The carpus of Elotheriwm is a curious one in many ways, and while modified to suit the didactyl condition of the foot, by the reduction of the lateral and enlargement of the median elements, it has yet retained many of its primitive characteristics. The scaphoid is high and thick in the dorso-palmar direction, but very narrow trans- versely. The dorsal and internal (7. ¢., radial) surfaces of the bone are very rugose, and on the palmar border, which is the narrowest part of the scaphoid, is a blunt and massive mammillary process. The articuiar surface for the radius is of unusual shape. It is divided into two parts, an antero-external and a postero-internal ; the latter is much the larger and is saddle-shaped, conyex transversely and concave in the dorso-palmar direction, while the former is convex and descends steeply toward the ulnar side. These two parts of the articular surface are continuous, but they meet at nearly a right angle, and their junction forms a ridge, which is the highest point of the scaphoid. On the ulnar side are three facets for the lunar ; the largest one is proximal and dorsal, and is continuous with the surface for the radius, which it meets at almost a right angle ; this facet is very oblique and presents distally as well as laterally, the scaphoid here forming a projection which extends over the lunar. The second lunar facet is dorsal and distal in position ; it is small, nearly plane, and not very distinctly separated from the facet for the magnum. The third lunar facet is distal and palmar, and is placed upon the ulnar side of the mammillary process already mentioned ; it is of oval shape and nearly flat. The contact between the scaphoid and the lunar is confined to these three points, and as the * Somewhat reduced by crushing, THE OSTEOLOGY OF ELOTHERIUM. 303 facets on both bones are more or less prominent, they are elsewhere separated by con- siderable interspaces. The distal side of the scaphoid is much narrower than the proximal and is occupied by facets for the trapezoid and magnum, no articular surface for the trapezium being apparent. The trapezoidal facet is considerably the smaller of the two, and is simply concave. The magnum facet is in two parts, a very slightly concave distal portion, and a somewhat smaller lateral portion on the ulnar face of the scaphoid. In the European species figured by Kowalevsky (’76, Taf. XX VI) the scaphoid is somewhat broader than in the American forms, In both groups a remarkable resem- blance to the scaphoid of Anthracotherium is observable, which extends to even the details of structure (see Kowalevsky, ’75, Taf. XI, Fig. 38). As Anthracotherium is, however, a tetradactyl form, the scaphoid is somewhat broader in proportion to its height than that of Elotherium, though hardly so much so as would be expected. In Hippopotamus and Sus the scaphoid is of quite a different shape from that of the fossils, being distinctly shorter and wider. The /unar is a very large and complex carpal, which exceeds the scaphoid in all of its dimensions, and especially in breadth. The radial facet is in two parts, continuing those which occur on the scaphoid ; the anterior or dorsal part extends across the width of the bone and is very convex antero-posteriorly, while the palmar portion is very much larger and is concave in the same direction. The dorsal border rises steeply toward the ulnar side, where the lunar is drawn out into a blunt, projecting, hook-like process, which extends over the pyramidal, as the scaphoid does over'the lunar. On the radial side are three facets for the scaphoid, corresponding to those on the latter, which have already been described. The palmar face is greatly extended transversely, and, though lower, is much broader than the dorsal surface. On the ulnar side are two facets for the pyramidal, which constitute an interlocking joint of unusual firmness and strength. One of these facets is proximal and dorsal and overlaps the pyramidal; the second, which is very much larger, is palmar and distal in position, and has a saddle-like shape ; it interlocks closely with a similar facet upon the pyramidal. When seen from the front, the contact between the lunar and the magnum appears to be entirely lateral, but as it passes toward the palmar side, the magnum facet broadens, becomes very concave, and assumes a distal position. The unciform facet is aiso oblique and the beak between the two is not in the median, but shifted far toward the radial side. Dorsally the unciform facet is consider- ably wider than that for the magnum, but on the palmar side these proportions are reversed. The lunar of L. magnum figured by Kowalevsky resembles that of EZ. ingens, except that its proximal surface does not rise so steeply toward the ulnar side and does not A, P, S.—VOL. XIX. 2M, 304 THE OSTEOLOGY OF ELOTHERIUM. project over the pyramidal. The lunar of Anthracotherium (see Kowalevsky, ’73, Taf. XI, Fig. 37) is like that of Hlotheriwm, but is narrower, especially its palmar face, and much thicker, and the distal beak is more nearly in the median line. In Hippopotamus the lunar is broad and rests almost equally upon the magnum and the unciform, as it does also in Sus. The pyramidal quite resembles the scaphoid in shape, but is much broader, not so thick antero-posteriorly, and generally of a more rugose and massive appearance. In view of the reduced lateral digits and the codssified radius and ulna, the relatively large size of the pyramidal is somewhat surprising. The proximal end is occupied by the ulnar facet, which is convex transversely and deeply concave antero-posteriorly. On the palmar side is a narrow, plane facet for the pisiform, which is very oblique in position. This facet is carried upon a compressed and slightly recurved, hook-like ridge, which runs for nearly the full vertical height of the bone, though not quite reaching to the distal end. On the radial side are two facets for the lunar, separated by a wide and deep suleus; the palmo-distal one is larger than the corresponding surface on the lunar, and its curvatures are, of course, in opposite directions to those of the latter, being concaye in the vertical, and conyex in the dorso-palmar diameter. The distal end of the pyramidal is taken up by a large, but shehtly concave facet for the unciform. In the material described by Kowalevsky the pyramidal of Hlotherium is not repre- sented, while that of Anthracotherium is so badly preserved and of such uncertain reference, that any comparison founded upon it would be valueless. The pyramidal of Hippopotamus is broad, square and heavy, as is also that of Sus, on a smaller scale. The pisiform is quite small and slender, though of considerable length ; it is strongly recurved toward the median side of the carpus, presenting the conyexity externally ; the distal end is thickened and club-shaped, though but little expanded in the vertical dimension. The pyramidal facet is nearly plane and oblique in position, broadest exter- nally and narrowing to a point on the radial side. The ulnar facet is very much smaller and somewhat concave; the two meet at almost a right angle. The pisiform of H. magnum (Kowalevsky, ’76, Taf. XX VI, Fig. 27) is not unlike that of H. ingens, but is of a more irregular shape, which looks as though it might be due to disease, that of Anthracotherium (XKowalevsky, ’73, Tat. XI, Fig. 58) is of quite similar shape, though much larger. In Sus the pisiform is of an entirely different shape from that of either of the extinct genera, being much deeper vertically, more compressed and plate-like, and less strongly recurved. That of Hippopotamus is more like that of the fossil forms. The trapezium is not associated with any of the specimens which I have seen, nor is any facet for it distinctly visible on either the scaphoid or the trapezoid. If present at THE OSTEOLOGY OF ELOTHERIUM. 305 all, it must have been in a very reduced and rudimentary condition, haying lost all functional importance. The trapezoid is high, narrow and thin ; it is closely interlocked with the magnum, lying in a depression on the radial side of that bone. The facet for the scaphoid is simple and strongly convex. ‘Three facets for the magnum occur on the ulnar side, one proximal and two distal; the former is much the largest of the three, but is confined to the dorsal part of the ulnar side. Of the two distal facets, one is dorsal and one palmar ; they are separated by a narrow space and are situated in different planes, almost at right angles to each other. On the radial side, near the distal end, is a shallow depression, which may haye lodged a rudimentary trapezium, though there is no facet for such a bone. The distal side of the trapezoid bears a small, plane facet, of triangular shape, for the rudimentary second metacarpal. The trapezoid is not yet known in connection with the European species of /lothe- rium, or with Anthracotherium. In Hippopotamus it is lower and broader and of more functional importance than in Elotherium, as it also is in Sus, and in the latter, differing from all of the other genera mentioned, it articulates extensively with the third meta- carpal. The magnum is a relatively large and massive bone, the three diameters of which are nearly equal, though the dorso-palmar dimension somewhat exceeds the other two. The dorsal moiety of the bone is the lower, quite a prominent head rising proximally from the palmar portion. The palmar hook is represented by a short, but broad, rough and massive ridge. The proximal end is unequally divided between the facets for the seaphoid and lunar ; dorsally the former is much the wider and occupies almost the entire breadth of the bone, but it does not extend so far posteriorly and on the head is con- fined to the antero-internal aspect of that elevation. The lunar facet is very narrow on the dorsal side, and lateral rather than proximal in position, but posteriorly it widens and coyers nearly the entire head. When yiewed from the ulnar side, the lunar facet appears to be of a horseshoe-shape, narrow arms extending far down upon the dorsal and palmar borders, and separated below by a very large sulcus. These two arms of the lunar facet are obscurely demarcated from the two small facets for the unciform, in which they may be said to terminate distally. The distal end of the magnum is covered by the large, saddle-shaped surface for the third metacarpal, which is convex transversely and concave antero-posteriorly ; and proximal to this, on the radial side, is a small facet for the second metacarpal. On the radial side also is a depression, running almost the full vertical height of the magnum, for the reception of the trapezoid. The depression contains a larger proximal and two smaller distal facets for the trapezoid, corresponding to those already described on the latter. 306 THE OSTEOLOGY OF ELOTHERIUM. The magnum figured by Kowaleysky (76, Taf. XX VI, Figs. 21, 32) is of the same general type as in the American species, but with some differences of detail. Thus, the bone is of relatively greater antero-posterior thickness; the palmar face is narrower and the palmar hook very much more prominent; the sulcus which, on the ulnar side, separates the two arms of the lunar facet is much narrower, and, in consequence, the arms themselyes are broader; the head of the magnum rises less abruptly toward the palmar side. The magnum of Anthracotherium is not sufficiently well known for com- parison. That of Hippopotamus is low and broad, and differs from the magnum of Llotherium in that the dorsal portion of the lunar facet is proximal in position. In Sus also the magnum is low and wide; its lunar facet is relatively larger than in Hippopota- mus, and it has no articulation with the second metacarpal, from which it is excluded by the contact of the third metacarpal with the trapezoid; the head is low. The wnerform is the largest and most massive bone of the carpus; in shape it is low, broad and thick, with its principal diameter directed transversely, and has on the palmar side a hook-shaped process, which is not very prominent, but broad and heavy. The proximal end is occupied by the facets for the lunar and pyramidal, of which the latter is much the wider; the junction of the two forms a prominent ridge which curyes across the proximal end, from the dorsal to the palmar side. These two facets are both slightly concave transversely, but very strongly convex antero-posteriorly, being reflected far down upon the palmar face. On the radial side are two vertical articular bands, separated by a wide and deep sulcus. The dorsal band, which is much the wider of the two, is composed of two very obscurely separated facets, a minute proximal one for the magnum anda very large distal one for the unciform process of the third metacarpal. The palmar band is a high and narrow facet for the magnum only, and is much more extended vertically than the corresponding surface on that bone. The distal end carries a large facet for the head of the fourth metacarpal, and on the ulnar side is a minute facet for the rudimentary fifth metacarpal. The unciform of Kowaleysky’s specimen does not differ in any significant way from that of the American species. In Anthracotherium this bone is much wider and lower than in Klotheriwm and the facet for the fifth metacarpal is more distal than lateral. In Hippopotamus the unciform is exceedingly large, and its dorsal face is of a low, wide, rectangular outline, and its great breadth corresponds to the large size and functional importance of the fifth metacarpal. The proximal end is divided almost equally between the lunar and pyramidal facets, and the absence of a distal beak on the lunar allows a larger contact between the unciform and magnum. In Sus, which has much reduced lateral digits, the unciform is narrower than in Hippopotamus, but broader than in Hlotherium, and the facet for the fifth metacarpal is not so completely displaced toward the ulnar side as in the latter. THE OSTEOLOGY OF ELOTHERIUM. 307 The metacarpus consists of four members, two functional, the third and fourth, and two mere rudimentary nodules, the second and fifth. Metacarpal IT is not preserved in any of the specimens which I have seen, though it is figured by Marsh (93, Pl. VIII, Fig. 4), but the facets on the neighboring bones show that it was carried by the trapezoid and retained a lateral connection with the magnum, excluding me. i from any contact with the trapezoid. The manus of Elotheriwm is thus -a typical example of what Kowaleysky has called the “inadaptive mode” of digital reduction. Metacarpal IT is long and massive. The head is heavy, enlarged in both dimensions, and has a stout prominence upon the palmar side; it bears a broad, saddle-shaped surface for the magnum. On the radial side is a depression for me. ii, at the proximal end of which are two small facets for that bone. The unciform process is very large, prominent and heavy, and projects far over the head of me. iy, but is, as usual, confined to the dorsal half of the head. On the distal side of this process and on the ulnar side of the shaft is a continuous, concave facet for the head of me. iv. A second facet for the same metacarpal is borne upon the palmar projection from the head. The shaft of mc. ii is broad, but much compressed and flattened antero-posteriorly ; both width and thickness are nearly uniform throughout, but increase slightly toward the distal end. The distal trochlea is broad and rather low, but is reflected well up upon the palmar face; on the dorsal side it is demarcated from the shaft only by an obscure ridge, with no deep depression aboye it. The carina is yery prominent, but is confined entirely to the palmar face. The lateral pit on the ulnar side is large and deep, but that on the radial side is faintly marked. 7 In Kowalevsky’s specimen (’76, Taf. XX VI, Fig. 21) the third metacarpal does not differ in any important way from that of the American species, though the magnum facet is somewhat more concave transversely and the shaft is rather more slender. In Anthracotherium (Kowalevsky, ’73, Taf. XIII, Fig. 80) me. iii is very similar to that of Elotherium, but is relatively heavier; at the proximal end the tubercle for the insertion of the extensor carpi radialis muscle is more conspicuous, and the palmar projection of the head more prominent. Metacarpal IV is a little shorter and narrower than me. iii, with which it articulates by two large facets, separated by a wide and deep groove ; of these facets the dorsal one, which is overlapped by the unciform process of me. iii, is strongly convex, while the palmar facet is flat and borne upon the palmar projection. The ulnar side has a shallow groove, in which lies the nodular me. vy; the articulation with the latter is by means of a single, small, triangular facet. The shaft is somewhat narrower transversely than that of me. ili, but is otherwise like it, as is also the distal trochlea. 308 THE OSTEOLOGY OF ELOTHERIUM. In £. magnum, Kowaleysky’s figure shows a somewhat differently shaped proximal end (76, Taf. XX VI, Figs. 21, 24), the head is somewhat more extended transversely, especially toward the ulnar side, while the palmar projection is narrower and_ less prominent. In Anthracotheriwm the head of me. iii has no such transyerse extension. Metacarpal V is an almond-shaped nodule, almost exactly like the specimen figured by Kowalevsky (Taf. XX VI, Fig. 25), though of a rather more regular outline. Proxi- mally the nodule has quite a large, subquadrate, and slightly concave facet for the unci- form, which presents more laterally than superiorly, and forming a very obtuse angle with this surface, is a smaller, triangular facet for me. iy. The metacarpus of Hippopotamus has four functional members, though the median pair are longer and stouter than the lateral. Compared with those of /lotheriwm they are relatively shorter and much heayier. In Sus there are also four metacarpals, but the laterals are much reduced, while the median pair, which carry most of the weight, are yery short and thick, and the distal carina surrounds the entire trochlea, dorsal as well as palmar. The mode of articulation between the carpals and metacarpals is quite different from that found in either Hlotherium or Hippopotamus, the head of me. iii being much broadened and articulating extensively with the trapezoid, so that me. ii is cut off from any contact with the magnum. This is what Kowaleysky has called the “adaptive method” of digital reduction, and it is in decided contrast to the inadaptive method exemplified in L/otheriwm. The phalanges, which are quite short, as compared with the length of the meta- earpals, are developed only in the median pair of digits. The proximal phalanx of digit iii is relatively elongate, straight, broad and depressed ; its proximal end is both wide and thick, and carries a concaye facet for the metacarpal trochlea, which is deeply notched on the palmar border for the carina. Toward the distal end the phalanx narrows but little, though diminishing much in the dorso-palmar diameter; the distal trochlea is low, wide, depressed and only slightly notched in the median line. The second phalanx is short, broad and thick, and of quite asymmetrical shape ; its proximal trochlea is obscurely divided into two facets, of which that on the radial side is the larger and extends more in the palmar direction, while the median dorsal beak is not prominently developed. The distal trochlea is much thicker than that of the first phalanx, is reflected much farther upon the dorsal face, and is more distinctly notched in the median line. The course of this surface is oblique, so that it faces somewhat to the ulnar side. The ungual phalanx is curiously small and nodular in shape, and is short, but quite broad and thick; the proximal trochlea is imperfectly divided into two slightly coneaye facets. The palmar surface is nearly plane, except for its rugosities, while the dorsal margin descends abruptly to the blunt distal end. THE OSTEOLOGY OF ELOTHERIUM. 509 In Anthracotherium (Kowalevsky, ’73, Taf. XI, Figs. 53, 54) the phalanges are of the same general type as in Hlotherium, but are proportionately much shorter and stouter. In Hippopotamus they are short, broad and yery heavy, while the unguals are reduced and of nodular form. In Sus the three phalanges of a digit are together con- siderably longer than the metacarpal, which is far from being the case in Elotherium ; they are also of quite a different shape from those of the latter. The proximal phalanx is much thicker in proportion to its length, and its proximal trochlea is deeply grooved across 1ts whole face for the metacarpal carina. The ungual phalanx is longer, broader and more depressed and pointed. Measurements. (CETDES INGIGIING cos00000 nod on 0sg nesses anandnanason0sdHSaD;BoseEouDaqdeBd00qbq0200e-0RKonddD oAaoDDaDAGGNSAUEDELOOOLODE 0.072 CHIROTIS, TWAlaltHNscscoonsca9asc0800 s¢scqoDo000 000000000 He0oaandad BOOTH. GdoDNCHOdOODUSGDOADaddd BddobOHODGODASHHSOGuRGOGND 077 Scaphoid, height... Scaphoid, breadth ...... Se NNONGL, TAMONEES io. conccaes stoooe0don002 09000000 nbossscoOsEaboRoCOBUDCbbROsScoOONSHODEREONNAE snonscaodonbo90n05000 047 ILPIAATE, MASE condo ccnccocodeDscdo ooo nss00900G00025Go0sDSaNsHOIEccONe DasOUdoODadODENBODdADEHODIECDNCAdaCDNoODUOHSHOS 047 ILADHTEIE, | HREAGT AN, copsann cocoon cocoon DvD NAEDSOONAHDDOHOVESsOATDIGONEBODOISoCBUSHaADEOSoLORGAdHNDBSOEODOCONH oDoCeD ¢ 036 ILATMAFATR,, UW oNKE) SHYESS, 5 cooonn0: Gaotob qadaoouDDoodH Bove ccHenoabaHadooo sco0dadbodow Tad: qoBoos0d00NNoBuD GoNOeBDEAse DodBOe A 050 IP nee], NSH MHo0090 oooddspoosecq00000009 nonoDsadoo5uSaDOSaDDoCOSOHEUASSoAAEEHOBOGEDNDDLOHODONIEdDOOVCHDEBOCAGEOS Ll 033 IP agen ball. ]RREEYOKD «|, 2c06c008000000000900000n Hoo }osoHosesboSodDRE;odcoDDREq00000 sonedaq0HGEOBONHHNdos soD0DADGER00000 027 IP WELL, THOVORINESS cpocdo00 000090090 bscacDpDECHODoGODOGGOHoBaEAEAToOBeNEONHODodBGREDSGOOvOGODND ooGODDSAbSHEHOnGeS ~C 039 isiformeplen ythwadsajasccrreesersesceneceercecsecee cress eee ertereece 042 ARE IEZADHG), TGHEA Nh; ovoconvooedensdscacoode0D60050 0 coapabosHDebooNadocopoUdAaoUDODdoONEACSDEESOOENOSOONOBODHOBEDDOBED. ¢ 025 MRE OEYAONGL, TORREAXETLELA coosoneo coppos onde anos 0c co ponSanboBaC4bEHADECKoA00900200000 ceovedoodcos oonn0qD0boD0onD6e009008000 012 AE DEAONGL, (HMOESTESSoo09ocoososun0000099090 00D00G0S909N5EA0HORKC. SaddonscoBoDA|.addd0R ceoo}oDODOONnED}HOReBeRdRoOOE .019 Macnumphel chin (excleolheat)ircnned-mncadsaceelmacosceratseeeetsseeacce ees teeecetcer tree erect tone 025 Nilenearayaned, LSWERKEAN occocdocpaceboo00d sconcnoonboposoooonScdbAceasca0cA Soo noCoDDCbODbDDRHH0D0BI60000 sobvadoccoadqs00000 035 Magnum, thickness. 048 Unciform, IN@IE| MFhoo o.oo ssonbasascnSpo9ODGUEDoDboGaRSqa0NG000 .036 Whmerbiforetins [oneal @oco000e516000c0006n00000450Ga0se0000 9096000008000000 09000000 bonoDoNGRADODAEAaGDoBOsobAROSESEEHDERODE .037 (Whaverbiirarn, TLV AaVeSS) oconocne0000000000500900000000005000 950006 DosDDaG OO DOUUGODOsooSOHOSBdaBOOnOOBdbodDSSUeoSeDBdEEN A 050 Metacarpaleiimleng thy Guipoaedianeline))spccccots>> 200000 000000000s9n9e0 doodNDsDASIDSZNGDDNs9ADbDOTORKCSoB0conq55Ka spo9ncONeONNdEDOAGEO500H0q550000d 0.083 Astragalus, width proximal trochiea....... -045 Weratewllere, IGT 01h, o902¢0000000000000009600080020000006000950000500N6854 aaR9HNNGe NSbocODbEAC sesadouGGeG0000N00050000 024 INFAVAIEMI ETE, WIELD Dodo cocdococoeasccacaqsn990D0EE00C: 480600000 SanSdaTaNSeeHOaSod josAToHDoSuAcSeABaoROgHSAdSOdenq000000 029 INFANBKCTUEN Dn CUKe) 610 2\SShanae deat ase dood eadsaeArianccadc ao Saanadtocdadedo-. Gb SE qneEaascsdosacodo oases amocBecasecsocee .044 IBMT UMAGA, XS FOOT socessaagoGsass cagdeooooEDBoAHO0cooDoRHadecoode sododsqdaoSAsegNosOUBADEGDNDGAsREaRan00G50 .037 Entocuneiform, width... -014 INIESOCTINGTIORTIN, INET EA RcaccocoscossecondeoDnonaso5ceasqabanDHSaGdoCONs DAncaDNSEDOD SEO soNSEOSaoKondaOGoENAEHOAOGOOE .016 JRO OUINE WIGAN, EKA F-sooao03o0050030000/d9N900097000009000000000500e000006000000000000000 >eoeconacqonadecoo500G6~ -022 PM CLOCTIN ETFO WIG bh x oc ce veateseenich swe suscecceeecale sccchnasebe ceatisues nadie sues seine soem eb casemate es aewemeee a 025 (CaOONGL, INES oi ecoscccoponnosccouesoscopandandEopodoSBecanEdeaoDHSENACOD Fe eee aioe er ee eee Ee .045 Cuboid, width... napooonasnoobancoodeo0 . .038 C@uboid thickness. 5 -cceeepeatnernst ssc cde ne'sd cia nu ancti see bale nea et ales Chioe mension etieneinciney octet eee oie cuiscwacecse 047 Metatarsallpint elem otleencectreesta-pems steerer ceattel a ssacrcerterecttstecte steer et serreerethe -eeeeeeseesse tees 181 Metatarsal iit) width proximal Cnd-..---...--.-0.c-++.cneeeaeceneseccnee \uescassccercassssesecsccessierseonees -029 Metatarsa laine wa dibhydistallWem derccmeccesrcasssedersehocasceenenacectoteceiecectceeitccsaseettadsccescsesiccecs .032 Metatarsal iii, thickness proximal end.... -O41 INIGIDTERESEM Tity, GIVEN dorccooncnece sscocodonncessonbenn vos093290300n00000095052000NETes0DEs vagEnDHceBNI00000 00000000 181 Metatarsal iv, width proximal end.....-..........+-sssscccceccsnesecsessccancsesccccceasssssanssccensescanees 033 NICHI ESAL Tiny, walsh Ghistell GrN(Cl..9000G009006000000000000000600000 600005900 050959n0N000 s9D0900H99B0600059005C00008 .031 Metatarsallivs thickness proximal Cndl-s-rcccr ere seecorersinnnstereasecsheasrcaeeesnassceressceseeeerc senses .046 Proximal phalanx, length 060 Proximal phalanx, width proximal Cnde...--..1....sccersaasereseliserecesesensee esreaccneceeesecaeecceses .032 Broximalysphalanxs width distally end scceccescesscecars-snerencecnseresseeereseeeiieceataseesscsesscseeseossec ss 027 Sieeoinel RIES, MEINE D ceescocd9d506000000000000800000900000005030 xzsoasnoscoODEnn||RDOvEOSORceASSdE2qG000Is FONEO .042 Second phalanx, width proximal end -030 Second phalanx, width’ distal! end)... .-..-..-2...eceensormsseeceeosenenseasmsersoserssosrcrsarsecsessecemcses| « 024 [URSA TAME, ETE Necascoscas oandocoss6s990oGaDAD20q00990G0 cOccDSeHJanSaConoNnqEdONSos0D0s86Ns0R05600080000008 -032 Uneualyphalanx. widbhsyproxamallenls.esss.-ssene en: -ceraasceaesscssteceesceeeeceeteasiescesteceseeeeceesel 022 A. P. S.—VOL. XIx. 2 0. ey) i) S THE OSTEOLOGY OF ELOTHERIUM. X. Restoration or Erornertum (Plate XVII). The skeleton of this genus has a remarkable and even grotesque appearance. As in so many of the White River genera, the skull is disproportionately large, and the immense, dependant projections from the jugals, together with the knob-like protuberances on the mandible, produce a highly characteristic effect. The long, straight face, the prominent and completely enclosed orbits, the short cranium, the high sagittal crest, and the enormously expanded zygomatic arches give a certain suggestion of likeness to the skull of Hippopotamus. The neck is-short, nearly straight and very massive, with prominently developed processes for muscular attachment. The trunk is short, but heavy ; the anterior thoracic spines are yery high and heavy, while those of the posterior region are short and quite slender. In consequence of the sudden shortening of the thoracic spines, a conspicuous hump is formed at the shoulders. The thorax is of moderate capacity and the loins are short. The tail appears to be of no great length, though the individual vertebrae are greatly elongated. The limbs are long and rather slender, and the fore and hind legs are of nearly equal height ; the humerus and femur are almost the same in length, as are also the radius and tibia, while the pes is somewhat longer than the manus. The scapula is very large, especially in the vertical dimension, which considerably exceeds the length of the humerus, and has a short but promment acromion; the pelvis, on the other hand, is rather small, the ilium having a long and slender peduncle, and only a moderate anterior expansion. The elongate limbs and slender, didactyl feet are in curious contrast to the huge head and short, massive trunk, and form a combination which would hardly haye been expected. Prof. Marsh has published, with a very brief explanatory text, a restoration of Elotherium (94, Pl. 1X) which differs in several details from the skeleton here figured. It is difficult to tell from the data furnished exactly how much of this restoration is con- jectural, or to determine how far the discrepancies to be mentioned are the result of the association of parts of many different individuals in a single figure, and how far they are due to actual specific characters. On comparing the two figures, one is struck by the following differences: (1) In Marsh’s restoration the skull is somewhat smaller in pro- portion to the length of the limbs. (2) The neck is more slender and the spines of the cervical yertebree, notably those of the sixth and seventh, are much less developed. (3) The trunk is decidedly longer and twenty thoraco-lumbar vertebre are figured. No reason is assigned for this departure from the well-nigh universal formula of the artio- dactyls, which is nineteen, and we are therefore ignorant of the evidence by which it is supported. (4) The spines of the thoracic vertebree are much more slender and decrease ‘more gradually in length posteriorly, so that there is no such decided hump at the THE OSTEOLOGY OF ELOTHERIUM. 321 withers. These spines are figured as having curious expansions at the tips, which are either absent or much less distinctly shown in the skeleton described in the present paper. (5) The lumbar region is longer and has neural spines which are lower and incline more strongly forward. (6) The conjectural restoration of the presternum is entirely different from the specimen herewith figured. (7) The scapula is relatively shorter and broader, and has a less prominent acromion. (8) The ilium has a shorter neck, expanding more gradually into the anterior plate and with the acetabular border of an entirely different shape. The ischium is much more slender, is more everted and depressed at the posterior end, and has a much less massive and prominent tuberosity. Materials are yet lacking to detcrmine how wide is the range of variation in the skeleton of the different species of Elotherium. So far as I have been able to observe, there are no important differences between the species, save those of size and proportions, the larger forms haying more massive as well as longer bones. In particular, the great John Day species have exceedingly heavy limb and foot bones. XI. Tue Retationsures oF ELorHEerium. There has been a very general agreement, among those who have made a study of this genus, regarding the systematic position of Hlotheriwm. The acute, compressed pre- molars have, however, led some observers to see affinities with the Carnivora and de Blain- ville went so far as to include the genus in his carniyorous family Subursi. Almost every other writer has referred these animals to the suillines. Leidy says of it: ‘‘ Hlothe- rium is a remarkable extinet genus of suilline pachyderms. .... Its allies among extinct genera are Cheropotamus, Palwocherus, Anthracotherium, and among recent animals the Hog, Peccary and Hippopotamus” (769, p. 174). Kowalevsky expresses the same idea in a more definite and specific way: “Schon bei dem ersten Anblick der Bezahnung bleibt kein Zweifel tiber die Familie zu der diese Form gehdrt, niimlich den Suiden ; sie bildet aber darin wegen des auffallenden Baues der didactylen Extremitiiten eine sehr eigenthtimliche Gattung. Pl6tzlich konnte eine derartige Form sich nicht bilden, das Entelodon hatte gewiss Vorahnen, deren Knochenbau einen allmiiligen Uebergang von der tetradactylen zu der didactylen Form vermittelten, bis heute aber sind uns solche noch ginzlich unbekannt” (76, p. 450). Zittel refers the genus to the Achenodontine, a subfamily of the Suid (94, p. 335). Marsh erects a separate family for the genus, and says of it: “‘The Hlotheride were evidently true suillines, but formed a collateral branch that became extinct in the Miocene. ‘They doubtless branched off in early Eocene time from the main line which still survives in the existing swine of the old and new worlds” (’94, p. 408). Schlosser has expressed a somewhat different opinion 322 THE OSTEOLOGY OF ELOTHERIUM. and has referred the genus to the bunodont division of the family Anthracothervide, which family he derives*from an Eocene stock common to the Anthracotheriide, the Anoplothe rude, the Hippopotamide and the Suide (87, p. 80). The complete account of the dental and skeletal structure of Hlotherium is now before us and yet it is hardly less difficult than before to determine its phylogenetic relationships and systematic position. The genus is so far specialized that it implies a long ancestry, not a member of which is, as yet, certainly known, although there are certain Eocene genera which throw some light upon the problem. In the absence of this ancestral series, we are without any sure criterion by which to distinguish parallelisms from characters of actual affinity, since only by tracing, step by step, all the gradations of a differentiating phylum, can we safely determine the true position of its members. However, some facts seem to bear a clear and definite significance. In the first place, it is plain that Marsh is right in forming a separate family for this genus, as it belongs to a line which diverged very early from the main stem, whatever that was. In the second place, the relationship of this family to the Swidw must be a very remote one. When we compare the skeleton of Hlotheriwm with that of the swine and peccaries, point by point, the only notable resemblance between the two groups is found to consist in the bunodont character of the molar teeth, and this resemblance, standing by itself, cannot be regarded as at all decisive. The selenodont molar has been independently acquired by several distinct lines, and so far as the artiodactyls are concerned, the bunodont pattern is almost certainly the primitive one. That two widely separated families should each have retained a common primitive character is too frequent a phenomenon to excite surprise. In all other structures, skull, vertebral column, limbs and_ feet, no particularly close cor- respondences between the Hlotheriide and the Suidw can be detected, though that a common early Eocene progenitor should have given rise to both families is altogether likely. Between Llotherium and Hippopotamus, on the other hand, are many points of resemblance. The likeness in the dentition is here quite as great or even greater than between either of these genera and the Suide. In the skull there is much to suggest relationship, though combined with many striking differences, which may perhaps be referable to different habits of life, such as the enormous massiveness of the premaxillary and symphyseal region in the modern genus, the peculiar development of the canines and incisors and the elevated tubular orbits. In the skeleton the two genera are widely separated ; Hlotherium is a long-limbed, long-footed, didactyl creature, with small thorax and slender ribs, evidently of terrestrial habits. Hippopotamus, on the contrary, 1s a short-limbed, short-footed, tetradactyl and isodactyl form, with immense thorax and broad, almost slab-like ribs, which is chiefly aquatic in its habits. Whether the resem- THE OSTEOLOGY OF ELOTHERIUM. 323 blances in skull and dentition indicate any relationship between the two families can be determined only when their history has been worked out. In any event, it is not prob- able that the relationship can prove to be closer than that both lines were derived from a common stock which separated from the other Artiodactyla at a very early date. As has already been observed, no direct ancestors of Hlotheriwm haye yet been recovered, but there are certain Eocene forms which seem to be related to these unknown ancestors in such a way as to suggest the character of the latter. The Achawnodon (Elotherium) uintense of Osborn (95, p. 102) is such a form and differs from the A. robustum of the Bridger in the “ great elongation of the face and the shortening of the cranium, both of which characters relate it to Elotherium” (/. ¢., p. 103). This species is more specialized in several respects than the White River Elotheres, and like its fore- runners of the Bridger, A. robustum and A. insolens, it has but three premolars in each jaw, and hence is not at all likely to be ancestral to the later genus. In the Wasatch Achenodon is represented by A. (Parahyus) vagum Marsh, which likewise has but three premolars, and, so far as it is known, differs from the Bridger species only in its smaller size. There is some reason to think, as Osborn has pointed out, that even A. wintense had four functional digits. While it is very unlikely that Achwnodon can haye been the direct ancestor of Hlothe- rium, there are, nevertheless, so many suggestive resemblances between the two genera, and the types of their dentition are so nearly identical, that we can feel little doubt as to their real phylogenetic relationship. In this case, Achenodon will represent a somewhat modi- fied side-branch of the stem which culminated in Elotherium. uccaDpecNdesebadGnD000 .008 V. THE Manvs. Of the carpus the only element preserved is a single scapho-lunar of J). vetus, inter- esting as showing that the coalescence of these elements had already taken place. This bone differs in a marked way from that of both recent canines and felines, but resembles the seapho-lunar of the White River sabre-tooth, Hoplophoneus. It is broad transversely and thick in the dorso-palmar diameter, but very low proximo-distally, even more so than in Canis; the tubercle at the postero-internal angle of the bone is well marked, but smaller than in the felines or modern dogs. The radial facet is simply convex in both directions, not having the postero-internal saddle-shaped extension which occurs in the recent dogs. This radial facet is reflected far oyer upon the dorsal and internal surfaces of the bone, converting the inner side into a thin edge, formed by the junction of the radial and trapezial facets. On the distal end of the scapho-lunar are three plainly distinguished facets, for the unciform, magnum and trapezoid respectively. The very deeply excavated unciform surface reduces the ulnar side of the scapho-lunar to an edge, not yery much thicker than the radial border, and hence there is no well-defined facet for the pyramidal, such as occurs in Canis. The shape and proportions of the unciform and magnum surfaces are very much as in the latter genus, but that for the trapezoid is not demarcated from that for the trapezium, though there can be little doubt that the latter element articulated with the scaphoid, as it certainly does both in Cynodictis and in Canis. The general 346 NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. shape of the seapho-lunar, recalling that which we find among the mustelines, strongly suggests that Daphanus had a plantigrade or, at least, al semiplantigrade gait. The metacarpus (Pl. XX, Fig. 17) consists of five members, which bear little resem- blance to those of the recent Canide. Schlosser (88, p. 24) has pointed out the essential characteristics of the metacarpus among the modern forms, and it will be well to quote his description, in order to make clear how widely Daphenus departs from the arrangement which has been attained by the later representatives of the family. “Die Metapodien haben sich auffallend gestreckt und sind zugleich kantig geworden. Sie zeigen nahezu quadratischen Querschnitt, in Folge ihres gegenseitigen ... Die distalen Gelenk- fiichen haben das Aussehen yon sehr kurzen Walzen und sind beiderseits scharf Druckes ; sie liegen einander niimlich ungemein dicht an. abgestutzt. Es lisst sich eine freilich sehr entfernte Aehnlichkeit mit dem Fusse von Hufthieren, namentlich yom Schweine—nicht verkennen. .... Die Anordnung der Carpalien ist scheinbar primitiver als bei den tibrigen Raubthieren, wenigstens als dieselben unter einander und mit den Metacarpalien nur reihenweise artikuliren, statt wechselseitig in einander zu greifen. Auch hat nur das Scapholunare eine etwas betriichtlichere Grisse erreicht, Magnum sowie Trapezoid und Trapezium bleiben sehr kurz und enden sowohl oben als auch unten simmtlich in einer Ebene. Demzufolge liegen auch die proximalen Facetten der Metacarpalien so ziemlich in einer einzigen Ebene.” This description of the structure of the manus in the recent Canid@ does not at all apply to Daphenus. In this genus the metacarpals are remarkably short and quite slender; they are not very closely approximated, but diverge somewhat toward the distal end, and hence they have not acquired the quadrate shape which Schlosser mentions as so characteristic of the modern dogs. The general appearance and character of the meta- earpals, and their mode of articulation with each other and with the carpals are very much as in the wolverine (G'wlo). The first metacarpal, even of the large D. felinus, is actually not much longer than that of the coyote (@. latrans), but is much longer in proportion to the other metacarpals, as well as much stouter and in every way better developed. The proximal end is thickened both transversely and antero-posteriorly, and bears a large facet for the trape- zium, which must have been a relatively large bone; this facet is convex in the dorso- palmar direction and is very slightly concave transversely, while in Canis it is deeply concave in this direction. In OD. vetus the articular surface for the trapezium is more oblique and inclined toward the radial side than in D. felinus. There is no other well- defined facet for any carpal but the trapezium, nor for me. ii. The shaft is short, slender, of oval or subcireular section, and arched toward the dorsal side. NOTES ON THE CANID# OF THE WHITE RIVER OLIGOCENE. BAT The distal end is large and has a well-developed trochlea, which is much more strongly convex than in Canis and of a different shape, the modern genus haying here a trochlea which is more like that of a phalanx than of a typical metacarpal. In Daphwnus, but not in Canis, there is a well-defined palmar carina, and the lateral processes for ligamen- tous attachment are more prominent than in the recent type. The second metacarpal is much longer and stouter than the first, though very short with reference to the size of the animal and to the length of the other segments of the fore limb. The proximal end is not much expanded transversely, but has a great dorso- palmar extension, the head projecting much farther behind the plane of the shaft than in Canis. The facet for the trapezoid is less concave transversely than in the modern genus and is of more uniform width, narrowing less toward the palmar side ; the ulnar border rises more above the head of me. iii and has a more extensive contact with the magnum. Though larger than in the recent Canide, this contact with the magnum is much smaller than in existing felines, and is of about the same proportions as in the early sabre-tooth, Hoplophoneus. The combined facets for the magnum and for me. iii form a broad, curved band upon the ulnar side of the head, which is made slightly concave to receive the adjoining metacarpal. No distinctly marked facet for the trapezium is visible upon the radial side. The shaft is short, weak, of transversely oval section, and is arched toward the dorsal side. The distal end is expanded, and made broad by the large, rugose pro- cesses for the attachment of the lateral metacarpo-phalangeal ligaments, processes which are much better developed than in Canis. The distal trochlea is of a quite different shape from that seen in the modern genus, being narrower, higher and of more nearly spherical outline, and is demarcated from the shaft by a deep depression, such as does not occur in the existing members of the Canide. The palmar carina is prominent and thins to a narrow edge. The third metacarpal is incomplete in the only manus found in the collection (D. felinus, No. 11425, Pl. XX, Fig. 17) as it lacks the distal end. The portion pre- served is, however, as long as the whole of me. ii and the complete bone was evidently considerably longer. The shape of the proximal end is much as in Canis, except for the relatively greater dorso-palmar diameter. The magnum facet is narrow, but deep, some- what concave transversely and strongly convex antero-posteriorly, but less so than in existing dogs. The facet on the radial side for me. ii is larger, more oblique and more prominent, and is more extensively overlapped by me. ii than in the latter, and the surface for me. iv, while not so deeply concave, is larger. When the third and fourth metacarpals are placed together in their natural positions, it is seen that the former rises higher proximally than the latter and has a contact with the radial side of the unciform, which, though narrow, is larger thanin Canis. The shaft is somewhat more slender than 348 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. that of me. ii and is of a more quadrate section, the dorsal and lateral surfaces forming distinct angles. The fourth metacarpal has a narrow, but deep head, which projects prominently behind the plane of the shaft; the facet for the unciform is slightly concave in the transyerse and strongly convex in the dorso-palmar direction. Compared with the cor- responding bone of Canis, the following differences in the shape of the facets for the adjoining metacarpals may be observed. The surface for me. ili is, as in the recent animals, divided into dorsal and palmar portions, but they are not completely separated ; the dorsal moiety is much larger, but not nearly so prominent, and the palmar portion is much smaller. The facet for me. y is of about the same shape in both genera. ‘The shaft is slender and nearly straight, but slightly arched toward the dorsal side; though relatively short, it considerably exceeds me. ii in length. The prominence of the lateral ligamentous processes gives great proportionate breadth to the distal end. The trochlea is like that of me. ii, except for its greater size and presents the same differences from the modern type. The fifth metacarpal has been lost from the specimen. The phalanges are yery remarkable, but can be most conveniently described in con- nection with the pes, with which the most complete specimens are associated. Measurements. No. 11424. No, 11425. Scapho-lunar, breadth 0.015 24 «depth (Gorso-palmar)......ccccceccecesececeeseseerennscssneeeecceeseccesccccscccnseesecasccesessenssseseces 011 Metacarpal i, length......cccccccceceecceceeecececnecearteeeeeceeecnseeccesescaseesenessssseassseacesessteaeeneseeeercrsreccercs .023 .026 i breadth of proximal end........... Fecer CBRE cee aceccp sbonasenancspadsnaconqgceooe00g .007 .009 oo GE GINS HAIL G16 | po6conpc00snocob canon ooASEN gosDODODOSDSODOOsHCODSCODHODOsbODAbdSSGONoOSoCoNSSq9000056" .006 os €€ Gistal trochlea........sscsssessseecessceesencccesccssuceccaaeseceecuscsessesenssenserssccsscsens .0045 Metacarpal ii, length.........sccccccnceceeccnneeeeeecceneccesccescneanscecssseeccececesersrssesecesseesannnenss poac00t000000005 0395 es “ breadth of proximal end .009 UG ef 6G install @iaVél-coosoonneeanndcqn 000005000 pnodenodOD DO sDODODONdHEUOSDUSOUOSUBNBAOGSAOSDDASDOQcOSEN000 -012 a af ie GG TRGYEL HIKE jronconcccenscn nonoDononoDDqGOsODSNDSECENBDOS0R00CC Suneeaevesecseceseccesseseces .009 Metacarpal iii, breadth of proximal end........--eeceteeesssesesseceeeeeeeeeeneeeseeceranecasenaaeereeeeeeseeenecererces -0105 Metacarpal iv, length -050 se Dreadth of proximal end.........cccssseeecccccseseececsrsneeececeeececeseserencnsecsescassaseeessecenans .0095 ss ae GB GIS EAT (1116! coococooopacdbeug2o0dD50aHeccKO6 oc¢nod danoacansobapAAabocoacconopdaneocq—ICo0adaGn 030 .012 o ns a BO OCHeatenrateseeantecteeeesnetsester-teerane Spqnodqoaadqassonnq50bcanSSNOnDDSEeNeS .010 NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. 349 VI. Tue Hinp Line. The pelvis is represented by several specimens belonging to D. vetus, D. hartshornianus and D. felinus, all of them incomplete, but so supplementing one another, that the shape of the os innominatum may be determined, with the exception of the anterior border of the ilium, which is unfortunately missing from all the individuals. So far as it is preserved, the pelvis is rather feline than canine in character, both in its general outlines and in its details of structure. The neck or peduncle of the ilium is wider and shorter than in Canis, narrower than in Fe/is; the anterior plate expands to its full width somewhat more abruptly than in the latter, but enough of the broken fossils remains to show that the iliac plate has the narrow form which is found in the cats and does not expand so much at the free end as in the modern dogs. The gluteal surface is not simply concave, as it is in the two recent genera mentioned, but is divided into two unequal fossee by a prominent longitudinal ridge, such as occurs, though not so prominently developed, in certain viverrines. This feature is repeated in another White River dog, Cynodictis, and is almost duplicated in the contemporary sabre-tooth, Dinictis, another of the many correspondences between Daphenus and the early Machairodonts. The sacral surface is placed much less in advance of the acetabulum than in Canis, and oceupies about the same relative position as in the cats. The ischial border of the ilium is, for most of its length, nearly straight and parallel to the acetabular border, but descends more abruptly than in either the recent dogs or cats, and follows a course more like that seen in Viverra. As in Canis, the acetabular border is more distinctly defined than in the true felines, and ends near the acetabulum in a long, roughened prominence, the anterior inferior spine. The pubic border is very short, and hence the iliac surface is not well defined. The acetabulum is of moderate size and has somewhat more elevated borders than in the cats. The ischium, which in the existing Canide is much shorter than the ilium, is very elongate, and is proportionately even longer than in the felines. The anterior portion of this element is straight, rather slender, and of obscurely trihedral section; behind the acetabulum the dorsal border is arched upward into a convexity, the spine of the ischium, terminated abruptly behind by the ischiadic notch, which is as conspicuous as in the cats, while in Canis it is very faintly marked. The posterior part of the ischium is expanded into a broad and massive plate, which is very rugose upon the external surface. This posterior portion is not so strongly everted and depressed as in the modern dogs, and there is no such stout and prominent tuberosity, which, again, constitutes a resemblance to the cats. The pubis is L-shaped and its anterior, descending limb is unusually long, broad and thin, much more so than in the felines or modern dogs. The obturator foramen is 350 NOTES ON THE CANIDE OF THE WHITE RIVER OLIGOCENE. very large, forming an oval, with its long axis directed antero-posteriorly, in shape and size agreeing much more closely with the condition found in the cats than with that of the recent dogs. The femur (Pl. XX, Fig. 18) is stout, and long in proportion to the length of the fore-limb bones, but not very long as compared with the size of the animal. While not differing in any very marked fashion from the thigh-bone of Canis, it yet has some resemblances to that of the felines. The small, hemispherical head is set upon a longer neck than in recent dogs and has a smaller, deeper and more circular pit for the round ligament, than in the latter. As in Canis, the head projects more obliquely upward and less directly inward than in Felis. The great trochanter is large and has a very rugose surface, but it has no such antero-posterior extension, does not rise so high and is not so pointed as in the existing forms of Canide. In consequence of this shape of the great trochanter, the digital fossa is smaller and much shallower than in the cats or recent dogs. From the great trochanter a sharp and prominent ridge, the linea aspera externa, descends along the external border of the shaft. Whether a third trochanter was present cannot yet be definitely determined, because in the only two femora preserved in the collection, the outer edge of the shaft is broken away at the point where the third trochanter would be, if present. In all probability, however, Daphenus did possess this trochanter, at least, in rudimentary form, as may be inferred from the analogy of the sabre-tooth Dzinictis, and still more from the little contemporary dog, Cynodictis, which in many respects approximates the structure of the modern Canide more closely than does Daphenus. The lesser or second trochanter is larger, more prominent, and of more decidedly conical shape than in the recent species of either Canis or Felis. The shaft of the femur is long, slender and nearly straight, though slightly arched toward the dorsal or anterior side; it differs from that of the modern dogs in its lesser curyature, and in broadening and thickening more gradually toward the distal end, and from that of the true cats in being more slender and of more nearly cylindrical shape. The rotular trochlea is rather narrower transversely than in the true cats, or eyen than in Dinictis, but is characterized by the same shallowness, and resembles that of the latter genus in its shortness vertically and lack of prominence. Trans- | versely, the groove is but slightly concave, and it has much less prominent borders than in the existing species of Canis ; these borders are slightly asymmetrical, the external one rising a little higher and being a trifle more prominent than the internal. A decided difference from both Canis and Felis consists in the fact that the trochlea hardly projects at all in front of the plane of the shaft, the anterior face of the latter gradually swelling to the level of the groove. In both of the recent genera mentioned, and especially in the canines, the trochlea projects prominently in adyance of the shaft. Jt —_ NOTES ON THE CANID#® OF THE WHITE RIVER OLIGOCENE. De The femoral condyles are feline rather than canine in shape; they are small and of nearly equal size, though the outer one is slightly the larger of the two, and project much less strongly behind the plane of the shaft than in Canis. They are also less widely separated and less expanded transversely than in the latter genus. As in so many features of the limb bones, the whole distal end of the femur is more like that of Dimetis than it is like the corresponding part of the modern dogs or cats. In Dinictis, however, the rotular groove is shorter proximo-distally and broader, and the condyles are even less prominent. The patella is very different from that of the recent Canide, in which group this bone is small, narrow and thick, but has more resemblance to that of Dinictis. It is quite broad, but very thin in the antero-posterior dimension; the anterior face is more roughened than in the Machairodont genus and the proximal end is more pointed, not so abruptly truncated. The facet for the rotular trochlea of the femur is, in correspondence with the shallowness of that groove, but slightly convex transversely and slightly concave proximo-distally. The tdia (Pl. XX, Figs. 19, 20) is relatively short and slender, and bears consider- able resemblance to that of Dinictis, more than to that of Canis. The proximal facets for the femoral condyles are small and but little concave ; the outer facet is somewhat larger than the inner, and projects farther beyond the line of the shaft, both posteriorly and laterally. On the distal side of the overhanging shelf thus formed is a facet for the head of the fibula, which is much larger than in the recent dogs and more rounded in shape than in Dinictis. The spine of the tibia is very low and is more distinctly bifid than in the Machairodont genus, though much less so than in Canis. As in the former, the cnemial crest is not very strongly developed ; it is far less prominent than in the existing Canide and does not descend so far upon the shaft as in them. The tibial shaft is slender and nearly straight, not displaying the lateral and antero- posterior curvatures seen in Canis ; proximally the shaft is of trihedral section, becoming approximately cylindrical below and transversely oval at the distal end. The latter is shaped much as in Dinictis and is conspicuously different from that of Canis; the astragalar facets are less deeply incised, and the intercondylar ridge is less elevated than in the latter, but the facets are deeper and the ridge higher than in the Machairodont, in correlation with the deeper grooving of the astragalus. The large transverse sulcus, which in the recent dogs invades these astragalar facets, is not shown in Daphenus. The internal malleolus is very large and resembles that of Dinictis, save that its posterior border is more inclined and the process is thus distally somewhat narrower. The sulcus for the posterior tibial tendon is very distinctly marked, more so than in Canis. The ANS 1, SE —AVOlby SADE, MHS 302 NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. distal fibular facet is quite large, beimg much as in Dimctis and consequently much larger than in ‘the recent Canide. The fibula (Pl. XX, Figs. 19, 20), which is greatly reduced in the modern dogs, is in Daphenus much stouter and has heayier ends, both proximal and distal. In Canis these ends have the appearance of being reduced and simplified from the condition seen in the White River genus. In the latter the proximal end of the fibula is relatively very large, especially in the fore-and-aft dimension, in which it considerably exceeds that of Dinictis, though the excess is principally due to a large tuberosity which projects from the hinder border, and which is present, though much less prominent, in the Machairo- dont. The facet for the head of the tibia is longer antero-posteriorly and narrower transversely than in the latter, forming a long, narrow, irregular oval. The shaft of the fibula is slender, though very much thicker both actually and proportionately than in Canis, and has about the same proportions as in Dinictis ; it is laterally compressed, the ‘principal diameter being the antero-posterior one, and of oval section, though its size and shape vary from point to point in an irregular fashion. The distal end of the fibula resembles that of Dinictis, though it is somewhat smaller, in proportion to the length of the bone. ‘The enlargement is both antero-posterior and transverse and gives rise to a very stout outer malleolus, at the postero-external angle of which is a deep sulcus for the peroneal tendons. The distal tibial facet is rather larger than that of Dinictis, while the surface for the astragalus is somewhat smaller, the two together making a high narrow band. Measurements. No. 11421. | No. 11424. | No. 11423. Femur, length (fr. head) .........:scecccccecnseecececccces sesseeesecseneeenreoes adi toe acy ea 0.195 «« ‘Dreadth of proximal end........ ....0cceceeececcceeecececeececcecerencnerneeeeenereneeseeeseenee: - 044 es GO" _GbETHIL r0\6 ld scosocnbsoomnpnossodcodadedodonsos ods OD0ocd osa0DodDede: NoaDobaosOI0oDNGoD00 .038 “fs €€ POtUAT ZTOOVE.--------eeeeceeeeereneeeeeeeeccnessnenceseneccsetcserecessraeeseneeeees .014 Bis DAN Terie thiva set emma Sk es ai Boncas i tcc antalat cae ang ae ean ee ema 149° ¢ “‘preadth of proximal end .........-.-cscccsneeeccecensercnscecensercnesererscsensserensstecensesee .031 .036 cs OF AIP EYL Bia ly cacona nodoeacosnonadcosqecmedodaogss0N060579-0 nos oDsoOSaONBODaROnUBeNOSNOSS .021 .021 -025 Fibula, ant.-post. diameter prox. CN .....eeeceeeeeseeeeeeneeeesccereeeeseseescssaeceseetaareessecees 019 ef as ee Alii CO Sggecoeeesonencengoodbdbascodabononcocdscuencosasddcesocca00q000 .0145 .017 VII. Tue Pes (Pl. XX, Figs. 21, 21a, 22). The pes, which displays structures of the highest interest, is much better represented in the collection than the manus and may be more adequately described. As a pre- NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. d00 liminary, it will be useful to cite Schlosser’s account of the salient characteristics of the hind foot among the recent Canide. “Die Anordnung der Tarsalien und Metatarsalien weicht natiirlich weniger ab yon jener der tibrigen Carnivoren als jene der Carpalien und Metacarpalien, doch finden wir auch hier immerhin einige nicht unwesentliche Modificationen. Es hat sich das Navi- culare ziemlich betrichtlich yverschmiilert, so dass es nicht mehr die Aussenseite der unteren Astragalus-Partie umhtillen kann. Das Metatarsale II, das sonst nur von zwei Punkten mit dem Mt. ITI in Beriihrung kommt, legt sich hier seiner ganzen Breite nach an das Oberende desselben. In Folge der Verkiirzung des Tarsus ist auch der aufstei- gende Fortsatz des Mt. V sehr kurz geworden. Die Phalangen haben gleich den Meta- podien nahezu quadratischen Querschnitt, die Krallen sind sehr spitz, aber wenig gebogen, haben jedoch ziemlich bedeutende Liinge. Die Hunde sind die ausgesprochensten Zehen- gadnger unter allen Carnivoren ” (’88, p. 22). In Daphenus the astragalus is decidedly different both from the astragalus of Dinictis and from that of Canis, but approximates more the latter. The trochlea is low and but moderately grooved, decidedly more than in Dinictis, but less than in the modern dogs, and the articular surface does not descend so far upon the neck as in the latter. The trochlea is asymmetrical, the outer condyle considerably exceeding the inner in size. The neck of the astragalus is much longer than in Hoplophoneus, Dinictis, or even than in Camis, and is directed more strongly toward the tibial side of the foot; the head is depressed, but yery convex. The external calcaneal facet is hardly so large or so oblique in position as in Dinictis, but it is more like the facet seen in that genus than like the facet of Canis. The sustentacular facet is shorter and wider than in the latter, and the sulcus separating it from the external facet is very much shallower. In Dinictis the sustentacular facet has a posterior concave prolongation, such as is not found in Daphenus, nor does the latter possess the distal accessory facet for the caleaneum which is so distinctly shown in Canis. The navicular facet is depressed, but very convex, and there is a small facet for the cuboid. The caleaneum is more like that of Dinictis than that of the recent dogs ; though the tuber calcis is longer, thinner and more compressed than in either of those groups, and its dorso-plantar diameter is more uniform, increasing less toward the distal end; its free end is less thickened and more deeply grooved by the suleus for the Achilles tendon. Along the outer edge of the dorsal border is a quite deep and conspicuous groove, which occurs also in Dinictis, but not in Canis. ‘The external astragalar facet is very like that of the Machairodont, being more angulated and more oblique in position than in the modern dogs, presenting inward as much as dorsally. The sustentaculum also resembles that of Dinictis in being less oblique,much more prominent and in having its facet much 54 NOTES ON THE CANID# OF THE WHITE RIVER OLIGOCENE. (Js) more widely separated from the external astragalar facet than in Canis. In the latter genus occurs a third astragalar facet, which is distal to the sustentaculum, and which is found in neither Dinictis nor Daphenus. The distal end of the calcaneum is occupied by the large cuboidal facet, which is more regularly oval in outline and much more deeply concave than in the existing forms of Canide. In these forms we find a facet for the nayicular, which adjoins and forms a right angle with the accessory astragalar surface already mentioned, but is not present in either of the White River genera. On the external side of the caleaneum, near the distal end, is a prominent projection for liga- mentous attachment. This process is not present in Canis, but it recurs in Dinictis, less markedly in Hoplophoneus, and is found in many of the recent viverrines, mustelines and raccoons. The cuboid is not peculiar in any noteworthy way; it is longer proximo-distally than in Dinictis and is proportionately narrower and thinner (7. ¢., in the dorso-plantar diameter). The long, thick and rugose ridge which on the fibular side of the bone over- hangs the suleus for the peroneal tendons is more prominent, especially on the plantar face, than in the Machairodont, but lacks the great, rugose plantar protuberance, which occurs in the recent Canide. The facet for the caleaneum is more convex than in Dinictis, very much more so than in Canis, in which this surface is almost plane. On the tibial face of the cuboid are three facets, a narrow proximal one for the navicular, and a median and minute distal facet for the ectocuneiform. The facet for the head of the fourth metatarsal is very much more coneaye than in the modern dogs, while that for mt. v is smaller than in the recent forms, and lateral rather than distal in position. The navicular, as compared with that of Canis, is short proximo-distally, but broad transversely, not having undergone the reduction in width which Schlosser mentions as characteristic of the recent members of the family. The astragalar facet is not more concave than in the latter, and there is no such stout tubercle on the plantar side of the bone as occurs in them. Two very small facets articulate with the cuboid, one near the dorsal and the other near the plantar border of the fibular side. The distal facets for the three cuneiforms have nearly the same shape and proportionate size as in Canis, but they are more in the same transverse line, the surface for the entocuneiform being less dis- placed toward the plantar side. The entocuneiform is of similar shape, but relatively better developed than in Canis, as would naturally be expected from the presence of a complete hallux in Daphenus. The bone is long proximo-distally, thick antero-posteriorly, and narrow, though broader than in Canis, and its proximal and distal facets, for the navicular and first metatarsal respectively, are relatively larger and more coneaye. The only other facet is an obscurely marked one on the tibial side for the mesocuneiform. NOTES ON THE CANID& OF THE WHITE RIVER OLIGOCENE. = BOO The mesocuneiform is a very small, wedge-shaped bone, broadest dorsally and thin- ning to an edge on the plantar side. The nayicular facet is concave and yery different from the curious oblique surface which we find in Dinictis, As is well-nigh universal among the Carnivora, the proximo-distal diameter of this bone is much less than that of either of the two adjoining cuneiforms, an arrangement which allows the head of the fourth metatarsal to rise above the level of the first and third. The ectocuneiform is, as usual, much the largest of the three, though it is not so large proportionately as in Dinictis. The shape of this element is very much as we find it in Canis, but with certain minor differences. Thus, the proximal end is less extended in the dorso-plantar diameter, and the navicular facet is more concave; the plantar tubercle has a more constricted neck and enlarged, rugose head; the facets on the tibial side for the mesocuneiform and second metatarsal, and on the fibular side the inferior facet for the cuboid are more distinctly developed, while the distal facet for mt. iii is more concave and has a shorter plantar prolongation. As a whole, the character of the tarsus is rather more machairodont, or viverrine, than canine. A conspicuous difference from the tarsus of the modern Canidw, is to be seen in the fact, that the articulations which in the latter are nearly plane (e. g., the cubo-calcaneal) in Daphenus retain their more primitive concayo-conyexity. The metatarsus consists of five members, which are longer and relatively more slender than the metacarpals, though an exact comparison between the two cannot yet be made, because the collection contains no specimens in which both metacarpals and meta- tarsals are represented by anything more than fragments. The first metatarsal is considerably longer and stouter than the corresponding meta- carpal. In this case we can determine the true proportions, for of the species to which the finely preserved hind foot (Pl. XX, Fig. 21) belongs, D. hartshornianus, we also possess a pollex, though associated with a different specimen. The almost exactly similar skulls of the two individuals show that the animals were of approximately equal size. The head of mt. iis enlarged in both the transverse and dorso-plantar diameters, and bears a roughened tubercle upon the plantar side. The proximal facet, for the entocuneiform, is large, and strongly convex antero-posteriorly, nearly plane transversely; no other facets are visible on the proximal end. The shaft is slender and arched toward the dorsal side; in section it is transversely oval, expanding somewhat at. the distal end, where the breadth is increased by the prominent tubercles for the lateral ligaments. The distal trochlea is small, but well developed, and of irregularly spheroidal shape, with plantar carina. The first metatarsal of Dinictis is like that of Daphenus, and certain viverrines, such as Cynogale, also have a hallux of much the same proportions, but in all the recent Canidae, with the exception of certain domesticated breeds, mt. 1 is reduced to a nodule. 306 . NOTES ON THE CANIDHZ OF THE WHITE RIVER OLIGOCENE. The second metatarsal is much longer and stouter than the first, but it is much shorter and weaker than mt. ii in Canis, and rather resembles that of the viverrine genus Cynogale, though it does not have the peculiar shape of the proximal end which charac- terizes that genus. In Dinictis mt. ii is somewhat heayier than in Daphenus, but is other- wise similar. In the latter the proximal end of mt. ii rises considerably above the level of mt. i and iii, owing to the shortness, proximo-distally, of the mesocuneiform, and is firmly wedged in between the ento- and ectocuneiforms, an arrangement common to all fami- lies of the fissipedes and already general among the creodonts. On the fibular side is 2 wedge-shaped projection which is received into a corresponding depression on mt. iii, thus making a very firm and close connection between the two bones. Above this pro- jection are two facets for the tibial side of the ectocuneiform, one near the dorsal border and the other on the plantar projection. The shaft is straighter than in Canis, but is slightly arched dorsally, the distal end not curving toward the tibial side, as it does in . the modern genus. In section the shaft is transversely oval, while in the recent dogs it has become trihedral for most of its length, owing to its close approximation to the shaft of mt. iii. The distal trochlea resembles that of Dinictis and differs from that of Canis in its more spheroidal and less cylindrical shape, and in its demarcation’ from the shaft by a deep depression ; the lateral ligamentous processes are likewise more symmetri- cally developed. The third metatarsal is much longer and stouter than the second, the difference between the two being greater than in Dinictis or the viverrines, or even than in Canis. The proximal end bears a facet for the ectocuneiform, of the usual shape, but the plantar prolongation of this facet is shorter and broader than in the last-named genus, and it resembles that of Dzinictis in being oblique to the long axis of the bone, inclining decidedly toward the tibial side of the foot. The tibial side of this facet is deeply incised to receive the wedge-shaped prominence of mt. ii, an incision which does not appear in the recent dogs, but occurs, though somewhat less conspicuously, in Dinictis. On the fibular side are two facets for mt. iv; one near the dorsal border, which is a deep spherical pit, and the other a small, plane surface placed upon the plantar prolongation of the head. The shaft, when viewed from the front, appears quite straight, but when looked at from the side is seen to have a slight curvature toward the dorsal side. The distal end displays the same differences from Canis as do the other metatarsals. The fourth metatarsal forms a symmetrical pair with the third, very much as it does in the recent dogs and cats, though in Daphenus they are relatively shorter and weaker. In Canis these two metatarsals are closely pressed together for most of their length, and their shafts have thus acquired a more or less trihedral section, with the approximate surfaces flattened, while the distal ends curve away from each other, somewhat as in NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. 307 Poebrotherium. In Daphenus it is only the proximal portions of the two shafts which are thus closely pressed together ; for the greater part of their length they are not in contact, and thus preserve the primitive oyal section. As their divergence is due to the relative positions of the tarsal bones, there is no necessity for the lateral curyature of the distal ends. The two metatarsals are very closely interlocked and in much the same fashion as in Canis. On the head of mt. iv are two facets for mt. ili, of which the dorsal one is a stout hemispherical prominence, which is received into the pit on the head of mt. ii, already described. The plantar facet is actually upon the plantar rather than on the tibial face of the bone ; the prolongation from the head of mt. iii extends around and embraces this facet, and by means of the double articulation a very firm interlocking of the two bones is effected. On the fibular side of mt. iv is a large and deep depression which receives the projection from mt. y. The facet for the head of the latter is large, slightly concave, and continues without interruption from the dorsal to the plantar border, while in Canis there are two distinct and quite widely separated facets. The shaft resembles that of mt. iii, but is somewhat more slender. In both of these meta- tarsals the distal carina is placed symmetrically with reference to the trochlea, but is less compressed and prominent than in Canis. The fifth metatarsal is not completely preserved in any of the specimens, the only representative of it being the proximal end, belonging to a large individual of D. vetus (No. 11423). As the specimen is incomplete, nothing can be determined respecting its length, but probably this was equivalent to that of mt. ii, the two forming a symmetrical pair, much as in Dinictis, though mt. y, so far as it is preserved, seems to be somewhat the stouter of the two. On the fibular side of the head is a very prominent projection, ending in a roughened thickening, and directed obliquely outward and upward, the “ascending process ” (aufsteigender Fortsatz) of which Schlosser speaks in the passage already quoted. In the recent dogs this process is very much reduced, while in Dinictis it is of quite a different shape. In the Machairodont the process is a long and promi- nent ridge, extending along the whole dorso-plantar thickness of the head, and projects much more proximally than externally, while in Daphewnus it is a blunt hook which projects more outward than upward. ‘The Machairodont Hoplophoneus has the process developed in very much the same way as in Daphenus. The facet for the cuboid differs from that of Canis in being quite concave transversely and in presenting as much toward the tibial side as it does proximally, while in the modern genus the facet is small, plane, subcircular in outline and altogether proximal in position. On the tibial side is a rounded protuberance which fits into the pit on the head of mt. iv; this protuberance is more prominent than in Canis and decidedly more so than in Dinictis. What little of the shaft is preserved is transversely oval in section, with a 308 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. sharp ridge running down the fibular side, and is thus quite different from the trihedral section, with flattened tibial side, which is found in Canis, and is much more like the corresponding metatarsal of Dinictis. The parallel arrangement of the metatarsals which we observe in the modern Canide is in Daphenus replaced by a radiating arrangement, the bones diverging toward the distal end. This distal divergence is, however, less decided in the pes than in the manus. The phalanges display a very curious and surprising combination of characters. They are long, both actually and proportionately ; compared with the tibia as a standard, they have about the same length as in the recent species of Canis, but they are decidedly longer than in that genus when compared with the length of the metatarsals. A proximal phalanx of one of the median digits is long and depressed, but quite strongly arched upward or dorsally. The metatarsal facet has quite a different shape from that seen in Canis, the transverse diameter being relatively greater and the dorso- plantar less. The facet is also somewhat more oblique to the long axis of the phalanx, presenting rather more dorsally and less entirely proximally ; the notch for the meta- tarsal carina is less deeply incised. Similar differences are observable in the body of the bone; its breadth being proportionately greater and its thickness less. The distal trochlea, which in Canis describes a semicircle from the dorsal to the plantar surface, is in Daphenus much more restricted, projecting less prominently from the plantar side and not reflected so far upon the-dorsal face. On the other hand, this trochlea is more deeply cleft in the median line than in the modern genus and the tubercles for the attachment of the phalangeal ligaments are larger. Tn all the differences from the modern Canidew which have been mentioned, we may observe resemblances to the corresponding phalanx of Dinictis, in which the bone is somewhat shorter and broader than that of Daphenus, and has rather more prominent ligamentous tubercles, but is otherwise very like it. The proximal phalanges of the lateral digits differ from those of the median pair only in being shorter, more slender and less symmetrical, and in having a lateral curva- ture which becomes yery pronounced in the hallux. The second phalanx is of about the same length, with reference to the first, as in Canis, but is broader, more depressed, and more asymmetrical than in that genus. The proximal facet, for the first phalanx, is more distinctly divided into two depressions by a more prominent median ridge, and the beak-like process of the median dorsal border is much more pronounced. The distal trochlea is reflected farther upon the dorsal side and projects more”from that side, but extends less upon the plantar face ; it is thus more con- vex in the dorso-plantar direction, but much less concave transversely than in Canis. 259 NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE.. oo The asymmetry of this phalanx is quite marked: its tibial side is straight, while the fibular border is quite coneaye, and the dorsal surface is hollowed, or cut away, near the distal end, allowing a retraction of the claws, to a limited extent, as may be readily seen when the second and third phalanges are put together. This asymmetry of the second phalanx is much less conspicuous than in Dinictis, not to mention the modern felines, but it is, nevertheless, unmistakable and is certainly one of the most surprising features in the whole structure of Daphenus. That an animal with the skull and dentition of a primitive dog should prove to pos- sess even imperfectly retractile claws is not what our previous knowledge of the early carnivores would have led us to expect. So unlooked for was this character, that at first I was strongly inclined to believe that the association of the hind foot shown in Pl. XX, Fig. 21, with the skull of D. hartshornianus was an accidental one, and that the pes must belong to some genus of felines or Machairodonts as yet unknown. Fortunately, how- ever, the collection contains a number of other individuals with more or less well-pre- served hind feet, and the agreement among them all is complete. Curiously enough, the characteristic second phalanges are preserved only in connection with the specimen figured, but other specimens haye parts of the tarsus, metatarsus, proximal and ungual phalanges, and a comparison of them shows that the reference of this particular hind foot is not open to question. The fact that the pes and the skull were found enclosed in the same block of matrix corroborates this inference, though, of course, such a fact is not of itself entirely conclusive. The ungual phalanx is hardly less peculiar than the second, being short, very much compressed laterally, and bluntly pointed ; it is very little decurved and has a plainly marked groove on the plantar face near the distal end. The narrowness, compression and straightness of this claw are in very decided contrast to the heavy and strongly decurved ungual phalanges of the modern Canidae, though among the latter there is con- siderable variation in these respects. The articular surface for the second phalanx is much more strongly concave than in Canis, permitting a greater freedom of motion in this joint, as was necessary in order to provide for the retraction of the claw. The sub- ungual process is not so large as in the modern genus and does not project so promi- nently upon the plantar face of the bone, but it is produced much farther proximally, extending beneath the distal end of the second phalanx, when the two are in their nat- ural position. The long hood which envelopes the base of the claw is of about the same size and shape as in Canis, though the space between this hood and the body of the ungual phalanx is narrower. The ungual phalanx of Dinictis is shorter, more compressed, but deeper in’ the dorso-plantar diameter than in Daphenus, and has a decidedly larger subungual process, in correlation with the more complete retractility of the claws. The A. P. S.—VOL. XIX. 27. 360 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. few specimens of these phalanges which I have seen are without the bony hood around the base of the claw, having much the appearance of the unguals in the viverrine genus Cynogale. It is possible that the apparent absence of the hood may be due to the break- ing away of that delicate structure, but this does not seem very likely. Measurements. No. 10546. | No, 11421. No. 11424. No. 11423. No. 11425. is C@alcanenmemlen otiltscanae eaeeare) leben series eee tentel see seiaeeeeleee rena eter | 0.045 0.044 0.051 0.055 of dorso-plantar Ciameter.....-..-..cceeeeeeeeeeseeeeee eee eee 016 | -015 -020 020 “ Tenet Grit aber ee eee ie. eee tee coche ie. | 028 036 040 gs extreme distal bread th........-.0..sssseceescecereeeenneneee Ole | 017 -022 022 Astragalus, length ........... os gdaQoasdounocamnaqassoanoTH0OO usanSCo nEReeDOoG -027 -031 031 m PLOXimal Dread th. .....-......2.0-c-veceeeees eacceseerecneeere .018 .021 -022 = width of head..... nace 014 -016 -019 CITTSORGL, THEI] 118, .oxoscn9c0060 426500000 AnasaqnocAobsoanARoEtaasboDNSsDsedns0CO 015 .016 G0 SGVAKE I a cosace osedosasnnosqnoo¢bono pescSayaacan 59ND Den GaboSssaxED|TOOOIe | O11 -012 Navictlar, Width..........-:.cccesccsscneeecenccnereennecuseesenewnccsrecenees | O17 O19 Ectocuneiform, width ..... 010 O10 Metnbarsal to longthe.dvss.cte.ctee st teeta there echroes ccna] Weneost s Dreadth prox. CNd..........cccscccsecneceneeensereerecscnees -009 010 | i oe dist, °° 007 Metatarsal ii, length. .-...-2...0...0.c0ssccnssesecaecceecessccnercoenecrenre .044 a breadth prox. end ........c.sccecceeeceeeeceererereeecseneee .006 007 | “ ia) Gddsbaoek hed arate ec are meee | .009 | Metatarsal iii, length | -054 = Dreadth prox. CNd......-...ssccessceseceeeeeceseneceecerens -009 O11 " eFIn Hi cEMEIND naa us atte TE eae 0105 | Metatarsal iv, length ....--...scceesssccneecccsceceneesceccectensssenseeseane | -056 ss breadth prox. end... 006 ie G6 GbE, © joononsmocoocadanoccbcesaceHbcanacaqccNeC .010 Metatarsal vy, breadth prox. Cn) .......-..ceeeeeereeeereeccneeneeesseneers -O11 The species of Daphenus hitherto recognized are three in number, two of them, D. vetus Leidy and D. hartshornianus Cope, from the White River stage, and the third, D. cuspigerus Cope, from the John Day. Two additional species are described in the sequel, one of which, however, can be referred only provisionally to the genus, until more com- plete material has been obtained, though the species in question is evidently very closely allied to Daphenus, if not actually referable to it. Dapuenvs vetus Leidy. Dephanus vetus Leidy, Proc. Acad. Nat. Sci. Phila, 1853, p. 393. Amphicyon vetus Leidy, ibid., 1854, p. 157; 1857, p. 90.. Hxtinet Mamm. Fauna of Dakota and Nebraska, pp. 32, 369. Cope, Tertiary- Vertebrata, p. 896. This species has a skull about equal to that of the coyote (Canis latrans) in size, NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. 361 but the vertebrae are much larger and the tail is longer and stouter. The tubercular molars of both jaws are relatively larger than in the other species. The inferior sectorial has a low anterior blade, and the internal cusp of its talon is reduced in size. The hori- zontal ramus of the mandible is long and slender and has a nearly straight inferior bor- der. White River. DAPHENUS HARTSHORNIANUS Cope. Daphenus vetus Leidy, Amphicyon vetus Leidy, in part, loc. cit. Canis hartshor- nianus Cope, Synopsis New Vert. from Colorado, 1873, p. 9. Ann. Rept. U.S. Geolog. Surv. Terrs., 1873, p. 505. Amphicyon hartshornianus Cope, Tertiary Vertebrata, p. 896. This species is somewhat smaller, and the tubercular molars of both jaws are propor- tionately smaller than in the preceding species; the anterior triangle of the lower secto- rial is high and acute, and its talon is basin-shaped, with the internal cusp as large as the external. The horizontal ramus of the mandible is straight and slender. Both this species and the preceding one have been found in the middle division (Oreodon beds) of the White River formation, but not as yet, to my knowledge, in the lower (Titanothe- riam beds) or the uppermost division (Protoceras beds). DaAPHENUS CUSPIGERUS Cope. Canis cuspigerus Cope, Proc. Amer. Phil. Soc., 1878, p. 70. Amphicyon entoptychi Cope, wid., 1879, p. 872. Amphicyon cuspigerus Cope, Bull. U. S. Geolog. Surv. Terrs., Vol. vi, p. 178; Tertiary Vertebrata, p. 898. D. cuspigerus is much the smallest known species of the genus. The sagittal crest is very short and inconspicuous ; the cranium is fuller and more rounded, the postorbital constriction is shallower and more anterior in position than in the White River species, and the mandibular ramus is nearly straight and very slender. The inferior sectorial is very robust and has a low anterior triangle and basin-shaped heel. John Day stage. DAPHENUS FELINUS, sp. nov. The inferior dental series of this species slightly exceeds in length that of D. vetus and the sectorial is larger. The lower tubercular molars are inserted in the border of the ascending ramus of the mandible, and, judging from the alveoli, were reduced in size. The horizontal ramus is not much longer, but much heayier than in D. vetus, and has a more sinuous ventral border, which rises more beneath the masseteric fossa. The limb 362 NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. bones and vertebrae are somewhat larger and heavier than those of D. vetus, and the neu- ral spines of the lumbar vertebree are very high and incline strongly forward. In size D. felinus is the largest and most massive species of the genus. The type specimen consists of a fragmentary skeleton (No. 11425) with which are associated both mandibu- lar rami, and which was found by Mr. Gidley in the Oreodon beds of Hat Creek Basin, _Neb., in 1896. ? DapHznus Doneet, sp. noy. As already intimated, the reference of this species to Daphenus cannot yet be defin- itely made, but the material so far obtained, consisting of lower jaws, affords no sufficient ground for separating it from that genus. The inferior dental series is relatively short ; the premolars are much smaller, especially in the antero-posterior dimension, than those of the later species from the Oreodon beds, but, at the same time, they are proportion- ately thick and heavy. The lower sectorial has a low, massive anterior triangle and a basin-shaped talon, with the inner cusp much smaller than the outer. The horizontal ramus of the mandible is short, but relatively much stouter than in any of the other species, and has a more sinuous ventral border, which rises steeply toward the angle. This species is dedicated to my friend, Mr. Cleveland H. Dodge, of New York, whose liberality has made possible much of the work undertaken by the Princeton Museum and to whose kindness I am under the greatest obligations. The type specimen (No. 11422) was found by Mr. Gidley in the Titanotherium beds of the Hat Creek Basin. Before proceeding to an examination of the next genus of White River Canide, Cynodictis, it will be necessary to introduce a brief description of a species which has been found in the Uinta stage of the upper Eocene (or lower Oligocene) and which ap- parently represents the forerunner of Daphenus, though more perfect specimens will be required before its position in the canine phylum can be definitely determined, MIACIS Cope. This form differs from Daphenus in the construction of the upper tubercular molars. M1 has an exceedingly broad external cingulum, forming at the antero-exter- nal angle a very large projection ; the internal unpaired cusp found in Daphenus and in all subsequent genera of the Canid@ is absent in both m+ and m2. The upper secto- - rial is of yery primitive and undeveloped character in the shortness of the posterior cut- ting ridge and the great transverse breadth of the crown, NOTES ON THE CANIDE® OF THE WHITE RIVER OLIGOCENE. 363 Mracis vrinrensis Osborn. Bull. Am. Mus. Nat. Hist. N. Y., Vol. vu, p. 77. Size rather less than that of D. hartshornianus; upper sectorial relatively small and tubercular molars large ; premolars short and thick. Measurements. MM. Men cths pe scOMMeniMClusivetc-sseccons cecccsetanascneeressuccacssceessescecasccsssceescscccostcvecuitecservoseans 37 P * length P+ length P= widthes:-<---ce Sa0dD 5 ROESDOE COSC ANS ISHSOS OS DUCEE SAEED” -o/aucH UNE BoC Gac0 Bocas Bode Su SuSoCeaae eee aceaee aaEeReEeere 11 M + length... Me Dev dthv eens faeces ae cea behest RELL. Se oatbee ee gd i tad Feil ceded lal ty aU EAS ay Tea fl Nogebeaecoaaho I eaco- bo eas ageo NSO RO Repo TeSUer ao Jon ec ERE EE ne Ere SRS EERE INCE TIGh it soccccs bo aencacbSdanc sad See qeande Yondos ed aacosaconcc EISEN GELS a cicee ecSCeHCuG Ie ne SUR OEE ene eee Hed) . Fie. A.—First upper molar of the left side : 1, of ? Miacis wintensis. 2, of Daphxnus hartshornianus. 3, of Canis latrans. x, cusp usually regarded as the protocone. If Ihacis be rightly regarded as having a place in the canine phylum, then the structure of its upper tubercular molars is of great interest and will require a revis- ion of the current views concerning the homologies of the cusps in the upper molars of the dogs. In Cunis, according to the usual interpretation, m 1 is composed of two external cusps, the para- and metacones, and at the apex of the triangle of which the para- and metacones form the base, an unpaired internal cusp, the protocone, with the proto- and metaconules on the anterior and posterior sides of the triangle respectively. Internal and somewhat posterior to the protocone is a large crescentic cusp, which is commonly regarded as an enlargement of the cingulum, although in unworn teeth a faint cingulum may be traced all around this crescentic cusp and is continuous with the prominent cin- eulum which bounds the anterior wall of the crown. If this interpretation of the cusps be correct, and further, if Jfacis is ancestral to the Canide, them min the Uinta genus is without a protocone and has only the para- and metacones, minute conules and the large inner crescentic cusp. Itseems much more rational to conclude that the lat- ter is really the protocone and that the cusp which has been so named in Canis is an additional element subsequently developed. In Daphenus this inner crescentic cusp and 364 NOTES ON THE CANIDH OF THE WHITE RIVER OLIGOCENE. the conules are relatively smaller than in the modern representatives of the family, which goes to confirm the conclusion that the name protocone should be given to the innermost cusp and that in Canis the middle part of the crown has undergone a special increase in complexity. CYNODICTIS Gervais. Amphicyon Leidy, Marsh, in part. Canis Cope, in part. G'alecynus Cope, non Owen. It is with much hesitation that I employ the name of this Kuropean genus for North American species, for there are certain constant differences which Schlosser (’88,) appears to consider as being of generic value. An actual comparison, however, of the American forms with specimens of Cynodictis lacustris, Gervais’ type species, and from the typical locality, Débruges, has failed to reveal any important differences between the two, and, therefore, for the present at least, I retain the name of the European genus for the American species, which are very closely allied, if not positively referable to it. The structure of these small carnivores, especially of the John Day species, is much better known than that of Daphenus, though our knowledge of the White River species has hitherto remained yery incomplete, and even of the better known John Day forms — only Cope’s brief descriptions have as yet been published. Despite the fact that Cyno- dictis is one of the commoner White River fossils, well-preserved specimens are com- paratively rare and of these the greater part consist only of skulls. The bones of the skeleton are so small and so fragile that it is exceedingly difficult to obtain more than fragments of them. By dint of great care and attention paid to these small formis, Messrs. Hatcher and Gidley have succeeded in gathering some very fine specimens for the Princeton Museum, and others I owe to the kindness of Mr. John Eyerman. Together, these various individuals represent nearly all parts of the skeleton and enable us to reconstruct the animal and to compare it with the better preserved and more abundant species of the succeeding John Day formation. I. The Dentition. The dental formula of Cynodictis is: I 3, C4, P 4, M 2, differing from that of Daphenus only in the absence of the third upper molar. A. Upper Jaw.—The incisors are very ‘small, simple and antero-posteriorly com- pressed, giving them chisel-shaped crowns; they increase in size from the first to the third, but the latter does not greatly exceed the others; not nearly so much, for exam- ple, as in Canis or Daphenus, and hardly more than in the viverrines. A very short ciastema separates the lateral incisor from the canine. The canine has a stout, gibbous fang, which produces a marked convexity upon the side of the maxillary ; its crown is quite elongate and somewhat recurved and much com- NOTES ON THE CANIDEH OF THE WHITE RIVER OLIGOCENE. 365 pressed laterally. The tooth is relatively smaller than in the recent dogs and thinner transversely, and has therefore quite different proportions from those seen in Daphenus. The premolars increase in size posteriorly ; in the unworn condition they have high, compressed, thin and very acute crowns, but in old individuals, without showmg much appearance of wear, these teeth have low crowns, elongated in the fore-and-aft direction. The first premolar is very small and simple; it is inserted by a single fang and follows immediately behind the canine, without a diastema, which is a difference from Daphenus. The second premolar is much larger than p+; it is implanted by two fangs and has a perfectly simple crown, without posterior basal tubercle, though the cingulum is thick- ened at that point. The third premolar is still larger, especially in the vertical height of the crown, and is distinguished by the presence of a posterior tubercle in addition to the thickening of the cingulum already found in p 2. The fourth premolar is a very effectively constructed, though small, sectorial blade, being much more compressed and trenchant than in Daphenus. The anterior cusp of the shearing blade (protocone) is relatively higher and thinner and has a sharper point and edge than in the latter genus, and the posterior cutting ridge (tritocone) is better developed and more efficient. On the other hand, the internal cusp (deuterocone) is very much smaller (hardly larger proportionately than in Canis) and occupies a more posterior position. In the Euro- pean species of Cynodictis the deuterocone is not so much reduced and is placed as far forward as in Daphenus. The first molar is large, particularly in the transverse dimension, and is of subquad- rate outline. The outer cusps are high and quite acutely pointed, and the central cusp (usually called the protocone) is lower and of crescentic shape, and the internal cusp is a broad, crescentic shelf, which occupies about the same position as in Canis. The ecnules are very small, but of nearly equal size, a difference from the modern genus, in which the metaconule is large, while the protoconule is rudimentary or absent, and eyen in Daphenus the posterior conule is much the larger of the two. The cingulum is very prominently developed upon the outer side of the tooth and forms a large projection at the antero-external angle, as in Daphenus, though not in Canis, a reminiscence of creo- dont ancestry. In the John Day species, C. geismarianus and C. lemur and still more in C. lati- dens, the first upper molar has a much more distinctly quadrate crown, due to the enlarge- ment of the metaconule, which has become as large as the central cusp, and to the more symmetrical development of the internal cusp (? protocone). In the typical European species, C. lacustris, on the contrary, the crown of this tooth retains a more trigonodont character. The second molar is very small, being relatively much more reduced than in Daphe- 366 NOTES ON THE CANID# OF THE WHITE RIVER OLIGOCENE. nus. It is composed of the same elements as m4, but has a different shape, owing to the greater proportionate length, antero-posteriorly, of the inner portion of the crown. In appearance this tooth is a miniature copy of that of Canis. B. Lower Jaw.—The incisors are very small and closely crowded together, so that the fang of i 5 is pushed back out of line with the other two. The canine, which is even more compressed laterally than the upper one, is long and recurved ; it is separated from p ; by a very short diastema. The first premolar is a very small, simple cone, inserted by a single fang. The sec- ond is much larger and is supported by two roots; it has an anterior basal cusp, which is formed by the cingulum and is subject to considerable variation, being much larger in some individuals than in others. The third premolar has a high, compressed and sharp- pointed crown and bears three accessory cusps, anterior and posterior basal cusps formed by the cingulum, and a third developed upon the posterior edge of the protoconid, very much as in Canis. The fourth premolar is slightly larger than p 5 and has more dis- tinetly developed accessory cusps, but on both p 3 and p q these cusps are subject to much variation and in some specimens they are feebly marked or eyen absent. The European C. intermedius has very similar premolars to those of C. gregarius, and in both species the anterior basal cusps (which are not present in Daphenus) give a somewhat viverrine character to the dentition. The first molar has a quite elevated anterior triangle, with a high, pointed proto- conid and a well-developed paraconid, both of which are more compressed and trenchant than in Daphenus. The metaconid is smaller than in the latter and is placed lower down and more posteriorly, so that it is visible from the outer side, much as in the mod- ern dogs. The heel is basin-shaped and is composed of a large, crescentic external cusp and a smaller internal cusp. In the European species may be observed certain differ- ences in the structure of the lower sectorial from the White River form, though these differences are not great. In the Old World species the anterior triangle is higher and the protoconid less compressed, while the metaconid is larger and occupies a more ele- vated and anterior position; in other words, the anterior triangle resembles that of Daphenus. Another difference from the American forms consists in the presence of a second internal cusp in the heel of the sectorial, which may be observed in most of the individuals figured by Schlosser and Filhol. Howeyer, in a specimen of C. /acustris from Débruges, which the Princeton Museum owes to the courtesy of Prof. Gaudry, this sec- ond cusp is not visible. In perfectly unworn teeth of Daphenus hartshornianus a feeble indication of this second cusp may be seen. The second molar is tubercular and of a narrow and elongate oval shape ; in consti- tution it entirely resembles that of Canis; the paraconid has disappeared, while in NOTES ON THE CANIDH® OF THE WHITE RIVER OLIGOCENE. 367 Daphenus it is still distinctly visible, though very small. The proto- and metaconids are of equal size and placed on nearly the same transverse line; these cusps are higher, more sharply pointed and more slender than in the recent Canidw. The talon, which is somewhat lower than the anterior half of the tooth, retains a distinctly basin-like form. In the European species we find a more primitive character of m 5 in the retention of the paraconid. The third molar is yery small ; it has an oval, roughened crown and is car- ried upon a single fang. As Cope has pointed out, this tooth is usually missing in the fossils, and occasionally a specimen is found which has not even an alveolus for it. The dentition of Cynodictis gregarius is, on the whole, a little more modernized and advanced than that of the European representatives of the genus. This adyance is shown in the reduction of the inner cusp of the upper sectorial ; in the somewhat more quad- rate outline of m 1; in the less elevated shearing blade and more posterior position of the metaconid on the lower sectorial, and, finally, in the more complete reduction of the paraconid of m 3. In the John Day species, especially in C. geismarianus and C. latidens, the departure from the European type is even more marked. Measurements. No. 10193. | No. 10513. | No. 10939. | No. 11012. | No. 11382. | No. 11432. Upper dentition, length I1 to M 2... 0.044 0.044 | 0.0485 0.0435 Upper canine, ant.-post. diameter............sseeeeeeeeees 005 005 005 -0045 is «« transverse Mae roasts Se ste fesesevstee dente: 0035 .003 003 Dee premolar series, lena this--c-esensebeese)le-ecerereseo== 025 | 923 025 “¢ molar series, length 010, | ~— 010 O1L LOOM OLO -010 aeeSPe lel erie tilivesscstyss-snsee cacatsseoveseeavepauesnersoscee: .0035 003 | .003 .003 C8 IPB SFO capsdiaansoodsoscodcnsadsoncobsnsocHnsnoesnonnoce -005 .0045 2.004 | .0045 003 SpTep svt) use Bae ales NES Toe Dery, 2 0055 005 | 0055 CIP AL GB Aeaeesacancctcopercosesccbanonue sedabsdooonecee -010 | -009 -009 .0095 | 0085 -009 COL SETBIL STORE LN ec pO Eocene 006 | 2.004 .0055 | .005 Sri Mp (og lerieet tio sstecet xeon sect cet tce,tace Se coco: 0065 | 007 .006 006 = —-.006 .006 “« M1, breadth..... ae ccd| 00) | .008 008 OE WB Ueya1N2550c 3:0ce odcccaauaneeouensocs! OsnaqoedaKeecos .0035 | .003 -004 004 -003 -003 CC WEB Toren chD cocossosconcosocecosoabs sonopososacesabecadHe -006 -006 0055 | .004 -005 Lower premolar series, length.....--- 02.00.00. +:11eeeeeeeeees | | 1021S eae OL9 ‘“« molar series, length... 017 ~+| ~ .016 017 -015 4 SP evlenathvans tn. tac Sana ee enn | 003 | 003 | TBO NRCP ae oBS sec cooper RSE ee aE S005 gm en 005i ia 004: ~ | oC SI2Gy ete ei taie seteeisetetelentetseiseteetciseecicecissiemettcstetest -0055 -005 | .006 -005 OC aad colt: OC Sectodins SaQsos0Oc Uo oTOSHaSdCScosEC CaS BASOSONeCS -0065 -007 -006 | 0065 | .006 PaREAMIV aT pactne: Kates Say eee Ni 010 | .0095 0095 | .010 | .009 BO NED OG nance seocascacdoosbongaconoagsagsDasbAGECeGa5 005 -005 5005-00 | 0045 | « a S oe = te . Sf i fe ‘ b KS 1 a / y ; : | : | | re : il q ; 1 5 1 y 7 1 : i te a = _ { io. ; : i he i ; . St / Ri i : % 1 : : ? : ‘ 4 i = j ; : ; &. { i te - , 1 / i Le fs Wi ; ; s i ¥. a 6 t= : ; 4 ' \: ; y i 4 ib t * - ‘ . . in ii 1 : we * id 4 : : ; = f ; - 7 ~ / . ‘ ‘ 7 fi , a 2 | : : : j 6 } ; , il iD y EB { i . : : ; , i F re a Ps : iy fa : ’ i - , he Na . / ; i ie 7 4 \ " " a a 1 y 4 ; ’ iW : 1 a Trans. Am. Phil. Soe., N. S. XIX HPL UE fe} ® age an) oe) &9, Ss lbc-cpri TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE IV. Hital moment} of momentu ads 1 _ Fig.E. eh DIAGRAM FOR THE CURVES OF A SYSTEM OF EQUAL STARS, UNDER THE INFLUENCE OF TIDAL FRICTION. Lower Curve illustrates increase of Eccentricity as the Stars separate. =, TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE V. H, 1408 H. 20°02 Drawings of double nebule according to Sir John Herschel TRANS. AM. PHILOS. SOC., N.S XIX. PLATE VI. GLOSSOPHAGA SORICINA. — sei a 2 ras rig TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE VII. GLOSSOPHAGA TRUEI. - a ss wail S ae E ciese TRANS. AM. PHILOS. SOC., N.S. XIX. PLATE VIII. 21 MONOPHYLLUS REDMANI. TRANS. AM. PHILOS. SOC., N.S. XIX. PLATE IX. BRACHYPHYLLA CAVERNARUM. TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE X. BRACHYPHYLLA CAVERNARUM. TRANS. AM. PHILOS. SOG., N. S. XIX. PLATE Xl. 47 LEPTONYCTERIS NIVALIS. 7 = ze = See ae od _— ed. a) oe wee Per Sh. ee TRANS. AM. PHILOS. SOC., N.S. XIX PLATE XIil. WERE a a \\\ \ Shoe ASSES 53 54 CHGERNYCTERIS MEXICANA PLATE XIll. TRANS. AM. PHILOS. SOC., N. S. XIX. LONCHOGLOSSA CAUDIFERA. TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE XIV. Lilia MWY B WI Qy on R iS fi RS i Hl Hg SSA Ned ‘j . Wr | \ Nal moire eTT NN ‘ \ iia if NX y) AY BZ w\ a aim itt oa fy iodle il byt i iT Lent yn i vis ' 1! Hla \ | (are TUTE TY y\3 m Vue ay ty t 70 ANURA WIEDII. TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE XV. Wed Hy ii \ \ 79 76 VY PHYEEONYCPERIS SEZBCORNIT TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE XVI. EX ECTOPHYLLA ALBA-CEPHALOTES PERONI. a r patric OncDhil Soa. WS aa Viol ttt Lie XVI. ve a inth.v Werner & Winter, Frankfort°M ad oad } hat Pb 4“ a NW a = 5 5 fi Ho a 5 5 S & wv ae —— , Ved tide Sy Z: Lol. Soe F / Ff Y lod, ALE, el y LATE GF, XIX li u fh. Werner & Winter, Frankfort°M . S Wolk we 2 on. Gil Soe Tia tibe uM rtOM h. Werner &Winter, Frankfur ia TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE Xx\l. RHOADS—-NORTH AMERICAN BEAVERS. TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE XxXIil- RHOADS-NORTH AMERICAN BEAVERS. oa a et Pod - TRANS, AM. PHILOS. SOC., N. S. XIX. PLATE XxXiill. RHOADS—NORTH AMERICAN BEAVERS. = at SS ta ee : Raat oy TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE XXIV RHOADS—-NORTH AMERICAN OTTERS. TRANS. AM, PHILOS. SOC., N. S. XIX. PLATE XXvV. RHOADS—-NORTH AMERICAN OTTERS. NOTICE, ee Ree eee Preceding Volumes of the New Series can be obtained from the Librarian at the Hall of the Society. Price, five dollars each. A Volume consists of three Parts; but separate Parts will not be disposed of. A few complete sets can be obtained of the Transactions, New Series, Vols. JI—XIX. Price, ninety-five dollars. Address, THE LIBRARIAN. Vie nage ‘6 wi ss) u VE at ae a a ve ied (t AAA Saag os ™ ay Xa