te ewe eee = Ce a a ne ne en AO atm aR NA AP eee ar te en ee anne rename TRARSSerTONS AMERICAN PHILOSOPHICAL SOCIETY BED (AT PAE Ee, PE A, FOR PROMOTING USEFUL KNOWLEDGE. VOL. XIX.—NEW SERIES. PUBLISHED BY THE SOCIETY. > o ob = ees Philadelphia: MACCALLA & COMPANY INC., PRINTERS. 1S98. CONT EME S7O ER VO. XOX. PAIR ARTICLE TI. A New Method of Determining the General Perturbations of the Minor Planets. By William MeKnight Ritter, M.A. . ARTICLE II. An Essay on the Development of the Mouth Parts of Certain Insects. By John B. Smith, Se.D. (With 3 plates) ASE EL. ARTICLE III. Some Experiments with the Saliva of the Gila Monster (Heloderma suspectum). By John Van Denburgh, Ph.D. . ARTICLE IV. 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 Glossophagine. By Harrison Allen, M.D, (With 10 plates) ARTICLE VI. The Skull and Teeth of Eetophylla alba. By Harrison Allen, M.D. (With 1 plate) PAO Lrr. ARTICLE VII. The Osteology of Elotherium. By W. B. Scott. (With 2 plates) ARTICLE VIII. Notes on the Canid 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) Si] 199 267 325 TRANSACTIONS OF THE AMERICAN PHILOSOPHICAL SOCIETY. ARTICEEHST. 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 coirdinates, 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 zweckmdssigen Methode zur Berechnung der absoluten A. P. S—VOL. XIX. A 6 A NEW METHOD OF DETERMINING Storungen 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 Lagranen. Hansen himself seems to have felt the force of the objections to his method, since in a posthumous memoir published in 1875, entitled Ueber die 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 HansEn’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 HANSEN’s original method made by that author himself, by Hriu 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 ;,. 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 PLANRTS. fi 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 coefticients of the various arguments. These coefticients have been tabulated by RuNKLE in a work published by the Smrrnsonran Institution entitled Mew 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 involved In the analysis we use the equivalent functions of the eccentric anomaly for those of the true anomaly, and when making the numerical computations we cause the eccentric anomaly of the disturbed body to disappear. This is accomplished by dividing the circumference into a certain number of equal parts relative to the mean anomaly and employing for the eccentric anomaly its numerical values corresponding to the various values of the mean anomaly. Having the expressions for the odd powers of the reciprocal of the distance in series in which the angles are the mean anomaly of the disturbed body and the eccentric anomaly of the disturbing body, we derive, in Chapter II, expressions for the J or Besselian functions needed in transforming the series found into others in which both the angles will be mean anomalies. In Chapter 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 meananomaly. 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 I” by integrating the expression in which 4, and ZB are factors easily determined. 8 A NEW METHOD OF DETERMINING From the value of JV we derive that of JV by simple mechanical processes, and then the perturbations of the mean anomaly and of the radius-vector are found from nM. 02 = nf W . dt W bh dt, i in © y being a particular form for g. The perturbation of the latitude is given by integrating the equation (’ being a factor found in the same manner that A and B were. It will be noticed that in finding the value of n. 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 circum- 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 Hansen’s derivation of the function i’, 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 THE GENERAL PERTURBATIONS OF TITLE 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 formule, 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 inyestiga- 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 )' 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- UNS Te SNOB OI OEE 10 A NEW METHOD OF DETERMINING dQ aie apie ane nents, we would at once have a i by differentiation. And since dé Bio. AC . : 5 dw only two multiplications would be needed in finding the value of » qo Whose expres- ‘ -C¢ 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 Althea, the planct 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. in 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. mal 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 : : : P\ 2 : will be necessary to find a convenient expression for () 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 LA? = 7 + r? — 2rr’ H. If we introduce the semi-major axes a, a’, which are constants, and their relation a’ - a = 7, we obtain =() + @) #—2() (ee (1) I 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 II, 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 (7” + I’) —(f + I); and the angles f + H, f+ UW, being in different planes, we have H = cos (f + II) cos (f’ + Il’) + cos Zsin (f + MT) sin (f” + WW), (2) I being the mutual inclination of the two planes. To find the values of TH, 1’, Z, 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 ¥ 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, Q, 2’, the longitudes of the ascending nodes on the fundamental plane adopted, which is generally that of the ecliptic, we have i) SS = ec NY Se a (3) The angles ®, ¥, 2 — &’, are the sides of a spherical triangle, lying opposite the angles 7’, 180 — 2, “’, being the inclination of disturbed and disturbing body on the fundamental plane. The angles J, &, J, are found from the equations sin $ Zsin § (Y + ®) = sin $(Q — Q’) sin $ (@ 4 7) sin 4 Jcos$ () + ©) = co LPG = ena ot shy) ; ; : (4) cos 3 Isin § (J) — ®) = sin 3 (Q — Q) cos 3 (@ + 2) cos § Tcos$ (Y — ®) = cos § (Q — Q) cos § (¢ — Z) In using these equations when Q is less than Q’ we must take 4 (360° + 9 — 9) instead of $ (Q — 9’). We ee a check on the values of J, ®, 4, by using the equations given in HAn- SEN’s posthumous memoir, p. 276. Thus we have cos p. sin g = sin 2’. cos (Q — Q’) \ COS Pp. COS Y = cos 2 cos p. sin 7 = cos 7’. sin (Q — Q’) cos p. cosT = cos (8 — 8’) sin p =sin? sin (Q — Q’) sin Jsin ® sin p sin J cos ® sin Jsin (} — r) = sin p . cos (¢ — q) sin 7 cos ay — >) = sin (7 — q) | cos p. sin (2 — g) I| cos [ = cos p. cos (t — q) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 13 : J To develop the expression for (=) we put eos, 7. sin Ul = rein. 1 sin Il’ =k, sin 44, ) cos Hl’ = k cos K, cos Zcos Il’ = k, cos K,,} and hence II = cos f.cos f’.k cos (11 — K) + cos f.sin f’.k, sin (11 — A) — sin f.cos f’.k sin (11 — A) + sin f.sin f’.k, cos (Il — A). Introducing the eccentric anomaly ¢, we have ~ a ; Aya G ' cos f = — (cos e—e), ain) == . COS p . sine, e being the eccentricity, and ¢ the angle of eccentricity ; and find r HE cos «. cos &.k cos (IlI— AK) — cos &’. ek cos (II — K) — cos ¢.¢k cos (11— KH) + e&k cos (II — K) + cos €.sin ¢.cos ¢’.k, sin (Il — A,) — sin &’.e.cos ¢’.k, sin (Il — A) — sin «.cos «&.cos ¢.k sin (11— A) + sin ¢.é.cos o.k sin (11— KX) + sin «. sin ¢’.cos ?.cos ?'.k, cos (II — A). 9 : - r : 3 . Jj\2 Substituting the value of —.-,. // in the expression for (-) we have fy —1-+4 a—2e.cos e+ e cos *« — Zaeck cos (11— K) + 2a¢k cos (II — K) cos ¢«—2ad cos p.k sin (Il — A’) sin ¢ — [2a’e — 2aek cos (II—A) + 2ak cos (Il — K) cos ¢ — 2a cos ¢.& sin (11 — X) sin ¢€] . cos ¢’ — [ — 2ae cos 9’. k, sin (11 — AG) + 2a cos > cos p’. k, cos (11 — 44) sin ¢ + 2a cos 7’. k, sin (11 — /,) cos €] . sin & + a? é”.cos *’. Putting 7, 2, y, for the coefficients of cos ¢’, sin «’, cos *e, respectively, and y, for the term not affected by cos «’ or sin «, we haye the abbreviated form aN , . / See!) 17 (-) = Yo — y - C08 & — By. sin & + 72. COS €. (7) 14 A NEW METHOD OF DETERMINING In this expression for (5); Yey V1 and 3, are functions of the eccentric anomaly of the disturbed body; y, is a constant and of the order of the square of the eccen- tricity of the disturbing body. In the method here followed the circumference in case of the disturbed body will be divided into a certain number of equal parts with respect to the mean anomaly, g. 9, 360° 3 360° 1,360° a. Ki Go 6 6 Hand * ? nm? n ? . . 360° The various values of g will then be 0°, - For each numerical value of g, the corresponding value of ¢ is found from g = e—esine. Before substituting the numerical values of cos ¢, sine, for the m divisions of the cir- cumference, the expressions for 7, 7, 2o, will be put in a form most convenient for computation. Let p.sin P = 20? Z — 2ak cos (11 —K ) (8) p.cos P = 2a cos 9’ k, sin (I — Ay), and Bo=f.sin# ) (9) y =f. cos#’s ) we find Go =fsin # = 2a. cos >. cos 9’. k, cos (Il — A). sine + pcos P. cos e — ep. cos P i =f COS (2072 —psin P). cos e—2x.cos p.ksin (Il — K).sine + ep.sin P. And from these equations we find, since jf .sin(#— P)=f.sin Fcos P —f cos F.sin P J .cos (#— P) =fcos F.cos P+ fsin F’. sin P, j.sin (#— P) = [2a.cos >. cos ¢’. k, cos (Ti — K). cos P 2a.cos >.k sin(II—K). sin P]. si See ae + 2a.cos 9.k sin(TI— K). sin P]. sin e + [» 2a : sin P |. cose ep f. cos (#’— P) = [2a.cos >. cos 9’. k, cos (11 — Kj). sin P ~ . 14 — 2a.cos @.k sin (I—K). cos P]. sine + 2a?.* . cos P. cose e THE GENERAL PERTURBATIONS OF TILE MINOR PLANETS. ] cr [f we now put vsin V= 2a.cos¢.ksin (Il— KX) ) vcos V = 2a.cos.cos >’. k, cos (11 — A,) of ° wsin W= p—20’?.~.sin P 3 \ (10) weos W=v.cos(V— P) | w,sin W,= v.sin(V— P) w,cos W,= 2a’. _ cos P, | we get J.sin(#’— P) = w.sin(e + W)—ep f.cos(#— P) = w,.cos (e+ I,). (11) Further, if we put R=1-+ a— 2a’. €”, (12) we have ¥o = R— 2e.cose + e?. cose + ey; or, y — R—2e.cose+ ¢.cos*% + .fcos F. (13) We find the value of y, from The constants, /, Ay k,, Ay, p, P, w, W, w., W,, 2, are found, once for all, from the equations given above. For every value of ¢ we haye the corresponding value of J and F from equations (11); hence, also the values of sin #, feos /, which are the values of 3, and y;. 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 : ; . (4\2 : y. being constant, we thus have given the values of ( ) for as many points along a the cireumference as there are divisions. We can put in the form ( 4 a A NEW METHOD OF DETERMINING A\? , = )} ve ( ) =Yo— 71 COS € — By.sine’ + y..cos.é a = [C—q.cos (e’ — Q)] [1—q . cos (¢ — Q,)], (14) in which the factor 1 — g, . cos (e& — Q,) 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.qsinQ.sin Q, Y=@.cos 0+ q,.C cos Q, ¥2=9-G.cos(Q+ Q,) . Cbo=gq.sinQ+q.C sin Q, 0 =sin(Q+ Q,). The last of these equations is satisfied by putting 2. = —@.: The remaining equations then take the form iC —¢ .g\ sien vi — (9 + 9,-C).cos@ | Y2= 9-H | bo=(q—a.C).smQ | The expressions g.snQ=2,+¢& ] q-cosQ=y,—7 | ( | J (15) (16) satisfy the relations expressed by the second and fourth of equations (15‘, where C= Yo oir ‘ Ta sie ta + = W ele . We have now to find expressions for the small quantities &, 7, ¢ found in these equations. 17 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Equations (16) give q-%. Cam’Q = (65-4 &) .&: The equation ; ¥ = C—q.qunsin’Q then becomes (Y+OGC=(BotEE (a) From (16) we have, also, ¢-n-C=(B+4E+(nm—n)%, from which, since y,= q.q@, and C= y+ ¢, we obtain e+ %)-¥2 = (Poe) + (1 — 2) x. (b) (c) Equations (16) give again (mn —gMe=Co+ €)2- When ¢ is known, is found from (a) ; and the difference between (a) and (4) (d) (yo +S) G2—$) =Qi—7)-% gives 7 when ¢ is known. The equations (a) and (c) give Be + 4:(7o +6) 5 = (Go + 28)? B+ 2§=Nn--5 / and hence — ‘tr ae Be +4 (oF S)o=y'- A. P. S.— VOL. XIX. C. 18 A NEW METHOD OF DETERMINING Deduce the values of 3, + &, y: 7 from (a) and (d), substitute them in (¢), we find rw 1! te The last equation then takes the form 6). 6: (¢) This equation furnishes the value of ¢; and with ¢ known, we find £, 7, from equations already giyen. ‘The three equations giving the values of the quantities sought are te ee Wee he: Ot (Yo— 2) C+ 4(yr + Co —4 0-2) 6 —4- Co -Y¥2 =O | Pins — Gas OS =O} CfA P—yent (+9) (%~—Ss) =03 Finding the values of ¢, £, 7, from these equations, and arranging with respect to y., preserving only the first power, we have nh / ¢— o- : Yo ( (g) o—_— > Dae 2? a P Po f ee Ye fiesta op Substituting these values in equations (16), they become q.sin@Q = Bo+ ais : Bae 2 re tr Po qzcos QO = 7 — (pe ape h 17 i S30 ras = nO EN ( ) gq, Csn Q= 3%, ine applet qn Coos Q= 5.92 40 noting that C= y, + ¢. ¥ If more accurate values of ¢, £, 7, are needed than those given by equations (q), we proceed as follows : Substitute the value of § given by (g) in the second term of the first of equa- tions (f°), we find, up to terms including y.’, — = Fa" = 4. fo: Fo 2 Yo: By" 2 Saar i =o oe ° -¥2 At ~ VYoe (18) io S na Ge 1 Parla y Gas == Ei: z THE GENERAL PERTURBATIONS OF The last two of (/) give also SX | x | Introducing the values of f/, /, given by (11), putting G=Y2t 4.4 A. cos" F L="i— 4. ye SF we have Po te hore Ci a. Sim so that C=y+7.sin 7. Moreoyer, since ¥.—6 = 7 .cos’F, we find from the expressions for £, %, given above, if Aa | —_ 4. > AS) NS —s Substituting these in the expressions for qsin Q, gcosQ, they become gsin Qe=F. sin L’ gq cos V=Ff.7'. cos LF. THE MINOR PLANETS. 19 (19) (20) 20 A NEW METHOD OF DETERMINING The value of q, is found from gu a (23) The quantities g, g;, @ can be expressed in another manner, The equations (22) give gO —*,.ta F ig Q= 7-49 =i... ain ig COs has from which we derive = , Q=F+ a a .sin 2+ 4 (= = a .sin 47+ ete. apes Sar 4, log. g = log. f + § log. (&?. sin *£’ + 7° cos °F’). Since y° and y” agree up to terms of the third order, the equations for & and 7’ give fi CA Sea aaa or, = = Se lait Ts (2% — 12) cos 2F é + 7 je + ZF? = Fe Di Further E° sin?’ + 7x? cos*#=1+4 2 7 (z.sin °F’ — 7’ cos *F’) — (4) and Log. (£2 sin 24+ 7? cos ?F’) = - (zy sin’? F—y' cos °F) —F (vy sin°’ Fh —y’ cos°*F’)?—$ (4) Substituting the values of y, 7’, C, given before, we find (G} . 97 , ey yw 2 Yo" a 7, 2 =a (x sin *P— yz! cos *F) = ee 4 (G4 be ) cos 27° i. 47 ie 2f° To Te ie (ae = cos 4 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 21 The equation y, = q.q, gives log. ¥, = log. ¢ + log. q Putting log. ¢ = log. f+ ¥, we have for q; log. gq; = log. aay: 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+ex log. gq = log. f + y (24) log. g, = log. = y in which — To Y2 Ta e 7 3fo Ya" fs <7 Pe == s (48 Pig :) sin 27+ 3 (“as — 5) sin 4/ (25) = BE (Tate Ta. Oe (eres __ ae cog AW Y = oT do (193 ay: cos 24 2a (“Eh ya) cos 4F And for C we have from the first of (15) C=yo+y72-sin *Q. (26) By means of the last three equations we are enabled to find the values of Q; % di C, with the greatest accuracy. The equations (17), where not sufficiently approximate, will, nevertheless, furnish a good check on the values of these quantities. eee : A\? Sas : All the quantities in the expression for (“) are thus known; and substituting their : : 4\2 values corresponding to the various values of g, we have the values of (“) for the different points of the circumference. 22 A NEW METHOD OF DETERMINING Using the values of OQ, 9, q:, @Q, just found, Hint, in his New Theory of Jupiter : : 4 : and Saturn, has given another expression for ) which we shall employ. = a To transform’ ()= (C—q.cos (¢ — Q)) (L—q. cos (¢ + Q)) a into the required form we put 1 sin 9 Sle, Gua Og Al a=t93%, b=ty3n (27) sec 3 7.sec2 7 i = Al VG: Then ey = (0) | 1—sin x. cos (e — Q) | |1—sin 1 -cos (é + Q) | _@ [ sec” hy (1—sin x . cos (e’ — Q)) | [ sec” ky, (1 — sin x. cos (e¢ + Q)) | sec 57 se 271 _ ELA tg? bx — 2g bx cos (2 —Q)| [1+ tg” $4, — 2tg Sy, cos (e! + () | - sec’ 3y sec’ 37, Substituting the values of a, b, V, we get (1) al E + @ Sg cos Ce — Ql: [1 + b — 2b cos (¢ + Ol? (28) We compute the values of a, 6, 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 (¢ — Q)i\m = E 6b” + B® . cos 0 + 6 . cos 29+ 6°). cos 39 + ete. | [1 + b —2b cos (& + ay =|5 B® + B® cos(24-@)-- B.cos 2 + Q) + ete. | nt ' . where s = “, # = & — Q, we are enabled to make use of coefficients already known. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 23 : 1 For 2.cos 6, writer +, and then we haye ae [1 + a — 2a cos 0)” = & + a@—a (@ + =) | = [1 — ax |” [1 "|" Expanding we haye = $s § S-+1 55 s stl s+t+2 94 13. 4 — = - ax -- mB -_— —— . 0” - aa [1 aa | Do Ge og EEE Te oy 3 aie 8 er Peseteho gat O's 8) 3 ae FS ee eS xe (—-—z— -@a? + ete. ial Bf s stl @ s sl 's-/ 2 a s stl s--2 s-+ 3° a’ [1—*| = a a i ia 8 s ¢---2 “s -- 3 -s'=- "+ ete 1 ad b . . 5 * 9? . t _— pan — - &% et to = ~~ eis Pao - nh an me es i<} 2. Sa | a to n =|5 aS) ed tw ~ 2 oO aN a to|=- vp bo tk < to a P @ => ° s s\2 s+ 1 3 s s-1\2 se-2 —; -_—— .@ é -_—— - a eet (t) gp + GS) eeee aae9\2 3-4-3. gt ete: 1 a G peu ) ; ae vee | (w+ =) s s+l 5 $\2 s+l1és Zi hea s stl\?2s+2 s+3 ¢ +[+- 3 @+(t). Sak ae at. 5 Ne ee ey s -+- See\ces oF Si 4 : 9 ] - (= — De a =) See tae a + ete. | (2+ =) s stl s+2 23 s\2 s+1s+2 s+3 5; + /F: Spates +(}). a eee $ sai + ete. yk | 1 bed = 2) two %. % a2 ~t (fe) < ic) LSJ ——s 3 eee, = ae sae bul fe : But wtf =2cos#, w+ a = 2. cos 2), a+ = 2.cos 38, ete., 24 A NEW METHOD OF DETERMINING =! and hence 4 6 —1 + (\.a+ (; He : =) at + G e: : bY = 2sa[ 1 h ; e 5 Ena t= ‘ (; ; = 2 : 2 a! +5 a 12)" $13 af + ete, | 2 1 Mere Se i ; Bis GY Tet eg ee ee ti 3 us 2 9 s § LS pee Ais ooieae a b = 2.5.5 a 1+. s AU = 5 ol Sy So aie a ghee \p ate S'S a+ ete. | ry 2 3 ) 4 ; \ Tas Me (29) ten, Sucks | 3 s 3 9 §, s-h1 s-- 3" si 2) al (2t b =2.5--5 a [1 +3 7 Qaatea So es and generally Osab. oS Sse! (Seu) a 8 8 t fon SuSsat Soe Siete ] b = 2.5.7 : .@{ i+ re ty a a a + ete. | : n 2 : : ae Since s = ”, we find from these expressions the values of the b" coefficients for different values of 7. Runk te has tabulated the values of 6° in a paper published by the SmrrHsonran Instrrution. Thus the value of [1 + a — 2acos (e—Q@Q)] ? is obtained with great facility. The value of [1 + 6°— 2bcos (e+ (%)] ° is found in the same way. We now let 9 = 1. 1. B°.c0824@ ) (30) =}. BY sin2.69} And hence have Oo = t NN. B” eV=1.N. B”. cos2Q 30 = 4.N. B.sn2Q o=1,N. B®. cos4Q s?—1.N. B°.sn4Q ete.= ete. bo ou THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Multiplying the series [5 6° + 6”. cos 0 + b®. cos 26 + b”). cos 36 + ete.] by [3 B® + B® cos (e+ Q) + B. cos 2(e + Q) + ete.], noting that 6 = ()— ¢, and arranging the terms with respect to cos 7, sin 70, we find (“) = $B. 6 + 6, co + 42, J + [6™. c + (6 + 6) cD 4+ (6 + B®) c™] cos 4 +[ + (6 — 6) s + (6% — 6) s®] sin 6 + [6.6 + 6 +.B) + (6 + B) c®] eos29 \ (31) +[ + (6 — bv) s® + (6 — b) s@] sin 26 + (6. co + (6% + 6) & + (6M + 6) c®] cos 30 +[ + (6% — 6) s + (6%: — 6®) s] sin 36 Le etc. ete. Now let k,cos K, = AOR ¢) a (6%) =|. Berd) cD ae (Oo) =t AG )) ¢®) ) (32) k, sin K, at (64 — pf) gs) ame (Bl?) =e) 5) ) and we find | (“) = k, [cos K;. cos 7# + sin K;. sin. 76] = k, cos (0 — K,) = k;. cos (¢ — te’ — K;). (33) Subtracting and adding the angle zg, this becomes (‘) = k,cos [¢(Q—g)—K as (ig —t’) | = k;,cos [¢(Q—g)—K;| cos .i(y—e’) —k;.sin| 1(Q—g) Salen sg) 64 If we put ae = * «C08 [2(Q.— 9.) —K,,,] | (35) (s) 2 A,,.=—k,,.sin [¢(Q.— 9.) — KJ, ] A. P. S.— VOL. XIX. D. 26 A NEW METHOD OF DETERMINING n being the number of divisions, we find (s) (“) = Ae C087 (9,—é',) —A;,,. Sint (9, —€&,) (36) If now, for the purpose of multiplying the series together, we put (c) (c) Ai; i y« COSUY = aye Casamvg | ay (s) (c) (s) ¥ ( ) A,, = >8,,.cosvg + > 8;,,.sinvg j : se we have G = [= C,, cosrg + C., sinrg] cos 7 (g—e’) __[S G,, cosvg-+ S,, sinrg] sin 7 (g—e’) (38) Performing the operations indicated we get * hs (ce) (e) > cos (tg —te’).C,, cosvg= =4C,,cos[( aes )g—te JH=25 C, /08 [(i—v) g—%" ] (s) => cos (tg —t’).C,, sinveg= > See sin[(¢+7) g—%’ ]|— = “30 , Sin [(¢@—v) g—#e" ] . . (c) (c) CY . . —22 sin (7g —te’) S,, cosvg =—S> $8, sin[(¢+v) g—vde’ ]— > 1S. sin [(¢—v) g—te’] (8) —* sin (4g — te’) S;, sin vg = SS18, * cos | ( t+v) g—te’ |— S50, cos [(¢—vr) g—%e'] Summing the terms we find (c) Eyes O. ag) ) cos [Gg — ie | F4SS(C,48,,) sin | (=F 7) g—te' | (39) (ec) _(e From the formula of mechanical quadrature just given, we have C;, o, S;,o. when (c) (c) v= 0; but we know that they are $. C,,, 5 S;,, as shown by their derivation. Thus (c) (c) (ec) Sle gey Pa A,=3$C,,+ C,, cosg+ C,..cos 2g + ete. 48) eee oe on. = >C,, cos vg + >C,, sin 7g + C,, sing + C,.. sin 2g + ete. (s) (c) A; = 38. + Scones iS. » cos 2g + ete. | (ce) (s) 2 ® . () 2S;,, cos vg + SS;,, sin vg. + §,, sing + S;,. sin 2g + ete. j Hence where » = 0, each series is reduced to its first term. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 27 In the application of the very general formule care must be taken to note the signification of the yarious terms employed. In case of (c) 2 Ay = = hi Con [a Q: 6) Sie] n A,,=—k,,.sin [i (Q.—g.) — Ki, : Sunt : Eanes ns 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 ¢ shows the term in the series 1h + B cos (e& — Q) +b”. cos 2(e’ — Q) + LD”. cos 3(e — Q) + 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 (c) (s) A,,., A;., when we give to » values from » = 0, to the highest value of » needed in the approximation. 2 f =. : In © .k,., 7 Q. — 9.) — Ki, for each value of 7, there are 7 values of each quantity. (ce) (c) (8) (¢) (8) The next step is to express the » values of Ay , tA,, Apia As tA, , 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 1’, of which the special values are Vio) | So. - Teens we haye Y= he, + ¢, cos g + ©, cos 2g + ete. ) (40) +s, sing +s, sin 2g + etc. ) The values of ¢,, s,, in this series are found from the 7 special values of Y. 28 A NEW METHOD OF DETERMINING From A, ,or A, =}$q+¢, cos g + © cos 2g + ete. + s, sing + s sin 2g + ete., (c) (s) (c and similarly, for every other value of x in A; ,, A; ,, we have a check on the values of ¢,, S,, In each series. Thus if in case of sixteen divisions of the circumference we take g = 22.° 5 and find the value of the series, the sum of the terms must equal the ) (s) value of Ae 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@ (c) (s) extends to 7 = 9, we have ten separate checks for the values of A; ,, A;,,, respectively. In the equation Y=3eq+ ¢.cos g + ¢.cos 2g + c;. cos 3g + 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 9 = 2H +G+a+6-+ ete: Y, = 3c, + ¢. cos 30° + ¢,. cos 60° + ce cos 90° + ete. + s5, sin 30°+ s, sin 60° +s, sin 90° -+ ete. Y, = 4q+ ¢.cos 60° + c,. cos 120° + e, cos 180° + ete. +s, sin 60° + s,. sin 120° + s, sin 180° + ete. Y,, = 3¢q + ¢.330° + ¢,.cos 300° + ce; cos 270° + ete. + s,.330° -+s,.sin 300°-+s, 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 (0. G: ) == Piste wes () =Y—¥F, OS NOG ae as (6) See a @8)=hty%, (par x 60=%t Wey, Then 3(¢ + 2c.) = (0.6)+ (2.8) + (4.10) 3(¢— 2e) = (1.7)+ (3.9)+ (6.11) 3(@+ e)= (0.6)—| (2.8) + (4.10) | sin 30° 3(@— «)=[(1.7)+ (5.11) ] sin 30°— (3.9) 3(s.+ s,) =[(1.7)— (6.11) ] cos 30° 3(s— s,) =| (2.8)— (4.10) | cos 30° Bla + o)= (%)+[(3)—G) | sin 30° 3(¢q,— ¢;)= [(a— (Fp) | cos 30° 6.q = ()—@)+G) s3)=[(4)+ + (4) | sin 30° + (3) 6 = A= ae (i) cos 30° (4) 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 )” as given in the series for r r am Oy oy lp D ie oar 6! a by lls When we divide the circumference into sixteen parts, each division is 22.°5. We find the values of Yj, Yi, ¥2,.... ¥is, as in the case of twelve divisions. To find the values of c, and s,, in the case of sixteen divisions, we put O28 j= y, tae) =r G9.) — Y,-peeee a) = Vi— 2, (2.10)= Fit Yo ()= Ye— Yo (ied5)\ = Y, +e G.)= Y— Vs 30 A NEW METHOD OF DETERMINING (0.4) = (0.8) +(4.12) (0.2)=(0.4) + (2.6) (1.5)=(.9) + (6.28) «0. 3)=] G4 o-EG. 7) (2.6) = (2.10) + (6.14) (S.2%) = (8. DEE (7 Ab): Then A(cy + 2.¢3) = (0.2) 4(ce, — 2.6.) = (1.3) 4(eg +c) = (0.8)—(4.12) 4(q,—G) = $[ (1.9) — (5.18) | —|(3. 11)—(7 .15) |} t cos 45° A(s. +s) = §[(1.9)—(6-13)|+[(8.11)—(7.15) |} cos 45° 4(s,—s;) = (2.10)—(6.14) 8.c,= (0.4) — (2.6) 8.5; =(ie5) — (357%) 4(c, + ¢,) = (2) + [ G2) — G8) ] cos 45° 4(a—«) =[()—G5) | cos 22.°5 + [ (G3) — G§) | eos 67.°5 4(e;++ 6) = (%)— [( 5) =| £;) | cos 45° A(c,—e) =[()—(G5) | sin 22.°5—[| (4) — G4) | sin 67.°5 4(,+s,) = (4) + (45 )| sin 22.°5 + [( (35) + (85) | sin 6% 5 4(s,—s;) =| Gs) + Gs) | cos 45° + (G4) 4(3-+8,) = [ (4) + (5) | cos 22.°5 —| (4) fe + (5) | cos 67 .°5 A(s;—s;) =|(2)+ (8) | cos 45° — (,4). When the circumference is divided into twenty-four parts, each part is 15°. Let (0.12)=%+ ¥' (6) =(0.12)+ Grisy SG) = (0.412) — (618) (1.18) = ¥,-+ Ya: G4. 13) + Gay 1G 1s) = es) (2.14) = ¥, 4-7 ae oS: eee 0) (=. aa 20) (11.23) 39,4. vg! Ga ity | Ge) PS fe THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Then 6(¢ + 2 6(¢,—2.¢ 6(c, + eo) 6(¢, — ey) 6(c, + «, ) 6(c,— ¢, ) 6)s, + $1) 6(s, — 8) 6(s, + s5_ 6(s,;— ss ) LZ: 12. Further, let Then C2) = (0.6) + (2.8) + (4.10) 2) = (1.7) + (8.9) 4+ (5.11) = (8) + |G) — Gir) | sin 30° = | (4) — G*%) | cos 30° = (0.6) —[ (2.8) + (4.10) | sin 30° = | (1.7) + (5.11) | sin 30°— (3.9) = | (4) + G4) | sin 80° + (4) = (2) + Gs) | cos 30° =(G ty — (37 ) | cos 80° =|@ y= Gs) | cos 30° = (t)—-(@) saa s = (4) — 4) + G2) qos) =Y— Vp (+15) = Y¥, — Vi, (=2z) = ea 6(e + en) = GY) +[ (2) — GY) | cos 80° + [ (445) — (5) | cos 60° 6(¢ — en) = [ (4s) — (44) | cos 15° + | (44 Se Ge -) | cos 45° + | (4%) — Gy) | cos 75° 6(c, + ey )= Gs ar. Ga) ar ig x0) 6(¢—« ) = {G) -GD)-[G)-@ +) |— Sele = ()—- [ (2n) — || 49) | cos 80° + [G is) — aa cos 60° fr) — (dy) | sin 75° ) | sin 15° + [@ ) + (5% )| sin 45° + [ + (45 ) | sin ¢O° 6(e;-+¢,) = 6(es—c, ) =| Gs) —G4) 6(s: + sn) = [ (as) + GH 6(s, —s8,)= (2p) + (49) qy r) | cos 45° | sin 15° —[( =) — (3%) | sin 45° + | ( i sin 30° + [ Gt t) + ( $y) | sin 60° + (,8; G(s; + %» ) = § (fs) + GD) + Gs) + GY —[G) + Gs) |} eos 45° 6(s;— s, ) = (4 6(s, + s,) =| (ts) + GS) 6(s,—s;) =| (2e) + G! >) 8) + (49) | cos 15° —[() ae (Ex )| cos 45° | sin 380° —[ G4 5) + (3) | sin 60° + (58; 31 + Lr fr) + (qz) | cos 75° A NEW METHOD OF DETERMINING When the circumference is divided into thirty-two parts, each part is 11°. 25 Let ( 0.16)= Y+ Ne (1.17) = ¥,+ Yy ( 2.18)= %4+ Vs 5.31). = i re Then 8 (¢)+ BY) — 8 (eo— 2. Cys) —— (O. 2) = 8 (e, + ey) 8 (¢,— ey) 8 (¢4 + Giz) 8 (e,— ep) 8 (¢5 + Cio) 8 (¢5— e,0) 16.¢, 8 (8: + Sus) 8 (s2— Su) = Gn) = Ge ) | cos 45° + G4 -) 8(s,+ 8.) = [® + (3) | cos 45° 8 (s = Sp) = (2) 8(s5 + Si) = te) + (5) | cos 22 ©5—| (34) + (+5) | cos 67 25 8(s5s—S) = = [@s) — Gtr x) | cos 45° — (+4). (0.8 )=(0.16) + ( 8.24) (0:4)=(0.8 )+ (4.12) (1.9 )=(1.17) + (9.25) (1.5) =(1.9 ) + (5.138) (2.10) = (2.18) + (10. 26) (2.6) = (2.10) + (6.14) Go) = (32 y-ray @.15)=(7.8)-@5.3 0.4) een (.3)=075 ) Gane (2) = (0.16) —( 8.24) (2) = (0.8 )— (4.12) (§) = 1.17) —( 9.25) @=(.9 )—GAs) (§) = (2.10) — (6.14) (Ji) = (7. 23) — (15.31) (2) = (3.11) — (7.15) (0.2)+(1.3 (1.3) =(%) + | @)—-( 3°) | cos 45° =|4)— (45) | cos 22. °5 + [| | cos 67 .°5 = (4) =(@—- (2) | cos 45° =@—|G)— Gren cos 45° = |) —Gs) | sin 22.° 5 — [( 3) — (5) | sin 67.°5 — (0.4) — (2.6) = [(4) + (es) ] sin22.°5 + [68) + Gs ) | sin 67.25 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Further, let And besides, let =|) — Gi B =(G)—-(@)| A =((%s) —G4)] 204)! A’ =|) — 4) | BY =(Ga)— (4) ] A Gi) | Gt) — )| cos 11°.25 + [ Gs) =e (em sin 22°.5 is) = Y,— Vig tr) = ¥,— Ny, (Gs) = ¥:— Ya L ) | cos 78°.75 sin 11° 25 —| (i ay) — (3%: =) | sin 78°.75 cos 22°.5 + [Gd — (4 ) | cos 67°.5 ~[() — Gp] sinors cos 33°.75 + | (4) — (44) | cos 56°.25 sin 33°.75 — Kez oz) — (24 )| sin 56°.25 (RR ) | cos 45° o_o ) | eos 45° C =(Gr+@) D =(Gh)+GD| : eee D = | (#5) + (Ft) Cc” =[(Gs) + GP] D” =[(es) + GD) C” = (Gh) + GD] |G) Gt) | P. S.— VOL. XIX. E. | sin 119.25 + [ (sy) + (435) | sin 78°.75 cos 11°.25 — [ ¢ g 23 ) + (5) | eos 78°.75 ] sin 22°.5 + [ (a) + (48) ] sin 67°.5 cos 22°.5 = [ Gs) + (42% )| cos 67°.5 | sin 33°.79 + [GD + ( +4) | sin 56°.25 | cos 33°.75 — | (5) + (44) | cos 56°.25 cos 45° + (.8;) cos 45° — (,8;). 34 A NEW METHOD OF DETERMINING Then 8(¢,+¢5) =A” + A’ 8(¢— 45) =A AY 8 (¢3 + ¢is) = B’” + B’ 8 (¢;— Gs) = [A— A” + B+ B’] cos 45° Cie ac Be ak 8 (¢;-—¢n) = [| A— A” —(B+ B’)| cos 45° Sh (Gy aten chy) ws ve 8(¢,—« ) = B—B"’ 8 (s, + 55) = C+ C” 8(s,— $5) = C’’ + C’ 8 (8; + 8s) =| D+D” —(C—C”) | cos 45° 8\(6;—=s) = 2. -- De 8(s, + su) = [D+ D” + C—C’’] cos 45° 8 (s;—syu) = DB’ — D’” 8(s; +s) = D—D" 8(s,—s )=—C”"+ C. The expressions for the determination of the values of ¢, and s,, just given, are found in HAansEn’s Auseinandersetzung, Band I, Seite 159-164. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 30 CHAPTER II. Derivation of the Kxpressions for Bressew’s Functions for the Transformation of Trigonometric Series. GN ve : : : 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. HANSEN, in his work entitled Hntwickelung des Products einer Potenz des Radius Vectors et 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 T= =u y = Ca we have yy’ = (cose + sm €) (cos e+ /—1 sin «’); also yy’ = (eoste + /—I1 sinze’) (cos? e+ »/—1 sin?’ &’) = cos (te— 7 e’) + /—1 sin(¢e’ — 7 &’). Denoting the cosine and sine coefficients of the angles (te—Ve') by (24,7, €) and (7, 7’, s) respectively, the series F=S3 (i,7,c) cos (te—? & )—EEV—1 (2,7,8) sin (te — 7’) (1) ean be put in the form F= 135 $(4,7,c)—V—1 (4,1,8)t y'y”. (2) 36 A NEW METHOD OF DETERMINING In a similar manner we get IP = where sh z- SE $((4,h',c)) —V—1 (Rh, 8)) y*-2-*, Zc", We have now to find the relation between y and z. Let Then from g = the mean anomaly, and « = the eccentric anomaly. = e—esing, introducing ./ —1, we get Since we find Now from we obtain gV/—1 = 6 / —1—esine Ne 2/— 1 .sine = — 9, g fat = /—1—5 (y—y""). (3) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Bye and ; (y—y') = log. (c 2 yy") : (4) Thus G/—1= lox 2= log: (y ee la ss) and hence z=y.€ 7 Uy) (5) From 2G (yy), we haye f=) aay) (6) and yimd.cr IY), (7) Let 5 be denoted by 2; then —Ky—y) =e Wy .y, , (8) and PC y*) = ey g-Uey, (9) But hs i320. g ne ait eee oly .cy'=(1—h.y +75 -¥ pas: ¥ tizsa ¥ FN (1 + ha.yt+ e ytt am yt a yt ete.) 38 A NEW METHOD OF DETERMINING and chy gy = (1+ aa.y + ye 4 ety + rasa y' + ete.) i a . 2 is a. + Fy?! aa a ae .y-' + ete.) Performing the operations indicated, we have iy aes on 9 hips Rene h§28 cha@—y) = (1— ee + ea — pay + porgep + ete.) 3 78 ane ae a (4 — a5 + ta — pee = ete) (yy) (+ te —pas + pea Fete) (yt ty) he}? He) : ( 1.2.3 eos = etc. 1 ) (+ ht} : ae ete} (y ++ 9) . 1.2.3.4 c ? a JHe7Ue hz? hij be aa 1.2. (1 lm--1 a 1.2.m--1.m-+2 + ete. i; d A wg 588 oxy yJ=1—7V +5 un ah a2 292° 12,22.3? als 12,27.32.42 == ete. (4 + Ba Fee) (P47) cs eee (+aaaa Fete) (Ara) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 39 As we may write h in place of 7, we haye, thus, also given the value of ¢”2(Y—Y")» Now put +e (—m) chsy-y?) = =, Im on) +o (m) (10) cea(y—y7) == ra J. y Then, from the preceding developments, we see that (—™m) m (m) Ji = (—1) Jn 5) (m) m (m) Oia = (—1) Ain > } (11) _ (—m) (m) —ha = ha + Again +o _(—m) (0) (—1) : "(—2) i (Cs) 2, Jn Y “= JS_yptden -Y + J_»-Y +tdu-y + ete. (12) ~ (1) (2) 5 (3) 4 + Jim-y tJdmy +Jnm-y + ete. +o (m) (0) (1) (2) ‘ (3) . In -Y® =Intdn -y tdn -Y +In -y + ete. . (13) (—1) (—2) > (—3) ‘ =F Jy -. 1s B= Sia ° OfRa AYR . as +- ete. 2) (—m) } € = Comparing the values of >_) J_,, .y~” and c~" a(y—y™) we have a, @) ; naz eis Wi sabe Im =JIy =ha— 22 a= [2223 — 12.92.34 +ete, for y”, (1) (1) E h373 hei hie : : i Sg ha—- = ODE Me EEE + ete, for y’, 42) Wee nis her ae Jn =In = [9 pes + pasa t ete, fory”, (2) (2) h2}2 his Asie Sao Jn =In = a7 pss t pest ete tory, 40 A NEW METHOD OF DETERMINING m) : foo 2m) € Si Comparing the values of ¥_) J, .y” and clay)» we get the same expressions for y” and y~”. (1) (2) We see from the values of J, , Ji, ete., found above, that the general term is (m) my 19 nl n aes Amjm hm t 2m +2 hm +4 Jm+4 i Vom | ema a1 a 17;23...m.m--1.m--2 ay ie Amjm We Aint =a al ee lm--1 a 1.2.m+-1.m--2 ae ete.) (14) Further, we have gh i h s(y—y) : y' and, by putting m= h—z, this becomes a= J, ae (15) Let a) | Multiplying the second of these equations by 2". dg, we obtain i aK +o (i) y 2" .dg= %_ Py .dg. Integrating between the limits + 2 and —z, we have i aes I Ch, (17) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. From z= 0e’'"'=ecosg+ /—1 sing, we have dz = (— sing + /—1.cosq) dq; also z2/—1l=/— 1 cosg —sin g. Therefore dz=z»/—1.dg, and (17) becomes (i) 1 ays" Pp, = ——! eye eae = Ges 274 {Jc e-7V¥—1 In like manner we find ‘(h) 1 etry—1 On aa sie 27 —|] ea7y-l Integrating by parts we have (h) Ca ¥=1 ] } he — h— a= J aif De ale Ir —!] z eat V1 (i) th) Comparing this value of (, with that of P,, we obtain h (—i) (i i.Q—=h.P,=h.P,, or (i) v (h) - (h-i a Z P= Q); —7. Ji. A. P. S—VOL. XIX. F. 41 (18) (19) A NEW METHOD OF DETERMINING 42 Thus we have, between the mean and the eccentric anomaly, the relations ot +. Bae i | (20) (=e { _— h sha 0f In the application of these relations, since (=) —h! —i) y v— ps Psy a 4 the expression for /’ is changed from into The other value of /” is F=3333((@h',c)) —V—1 (GR, 8)) y'. 2. A comparison of these two values gives : i rot Fa (ww) (4, h',¢)) => P_y (4,7,¢) = 2. Suey (2, 2, ¢) (21) In transforming from the series indicated by (7, 7’, ¢) into that of ((z, h’,c)), it is evident that h’ is constant in each individual case, and 7 is the variable. Thus we find, beginning with 7 = h’, hi eee diene (2%, h’—1, ec) + ete. ey are) (42, ¢)) = 5 - Tuy (4, h',.¢) + + hos 1 pee) ae ih (7, +1, ¢) + ete. h' . WN THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 43 To transform from ((7, h’, ¢)) into (7, 7’, ¢) we have —N') (h’'—7) ( \ 75c)— OS (GC) es dy (2, h’, c)). Here, 2 is the constant, and /’ the variable; and for the different values of h’, begin- ning with h’ = 7, we find (0) ((#—1) —#)) (4,0, ¢) = Sy ((4, 0 €)) + Se ayy ((z, ’—1, ¢)) + ete. ((#+-1)—7")) + Stay (4% v + 1, ¢)) + etc. The expression 5 jm im (1 her? ae hii hie + ete.) 72m \ Lmtl ' 12mplm-2 ~~ 1.2.8.m1m+2.m+13 (m) enables us to find the value of -/,, for all values of m. A simpler method can be obtained in the following manner: Op) - Putting o's -Y™ in the form _(0) hs(y—y) a ae ee -2 c's =F te ¥—-Grey sp at! 4 -y* + ete. we haye, for the differential coefficient relative to y, (1) (—1) €fy__y— E (2) - (2) - hs(l+y~) yy) — Jie +2. Jie -ytete + Jie i 2S y* + ete. If we multiply the second member of the first equation by h5(1-+ y~), we have an expression equal to the second member of the second expression, and by comparing the two we find (m+1) (m-1) (m) hid . a =m. J (22) BU hs (a ie’ > 44 A NEW METHOD OF DETERMINING Let (m) h x — p ~ =) mame, ( 93 ) Sl hy then I | SSE lo 'S | XN hy Pi>P2 (24) Cie3— 9) CLC:, — sere (m) From the values here given, since wa is put equal to p,,, we have, by increas- hs ing m by unity, (m-+1) h> STA = = Pn Pn at hs . m . Putting 52 = Im equation (22) takes the form Pm + Pm =e if aT » Pm: From this we find 1 Pn = a 2 1 a 1 Tm =" l THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 45 We also have Dm = Vy, == Pn +19 ( 25 ) a form more convenient in the applications. (m) The general expression for J), is hs Tne = Je Dyfi Der.-» Dray (26) where = —i+ 3 itp See (27) if we put T= ha. From the expression (-—7’) a he E Zz (h'—7') fae (Cehe6) > Pie (iene) — => > June (25.4, 6) 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 kh = 0 we have 46 A NEW METHOD OF DETERMINING The equation el —t z=y.6 WW —-Y¥) gives dz _ dy Oxy ow ¢ FS ee ae ) dy. (28) Hence (2) ~y—l a GFF 5) a. 1 +€ 271 —)| J aay When p is a whole number bry =1 (e _ y?. dy = 0, . Cat except when p = 1, when this integral is 274/—1. Hence it follows that When 7 = 0, we have Using the expression (—v . (1) . *, y ) fe * (i?) . * (Gh',.¢)) = 3 Beat, >¢) = P_) Ce he, Gea) =e peal 4, J= 1, C), we have ((0, 0, ¢)) = (0,0, ¢) — 22’ (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 (1, 0, ¢)) = ,¢c) —W (1,1, ¢) — 4’ (1, —1, c) (GO) ash (eige\— 27 — 1s) ((2,0, c)) = (2,0, c) — wv (2,1, ec) — 4’ (2, —1, ¢) ((2,0, s)) = (2,0, s) — 1’ (2,1, s) — 4’ (2, —1,s) ete. = ete. In what precedes we have put and obtain g = the mean anomaly, é = the eccentric anomaly, ¢ = the Naperian base, gi Came. an pou y=, z* — yf hs(y—y 2) (= 4 40. of = 2.24. y ); hely—y) - : : siete where ¢72U—9) is expressed in a series, the general term of which is m 5m ee Wk hen (1 lm--1 Thus hee oy aR Ae (1 — We have also put and since hii her — ete. 1.2.m+1.m-+2 1.2.3.m-+-1.m--2.m-+3 = Nik hes a as : ete. ) es lm-+1 1.2.m-+-1.m-+-2 1.2.3.m-+-1.m--2.m-+-3 st (—m) on (YY *) , doy .coshg, (29) h=oa (h—*) Sa a J, . sin hg. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 49 We notice that (1) (—1) P, = Py = -!¢, (0) (26 el For all other values of 7 [Pp == (0), 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 \ a i If we wish to determine any of them independently we have from (m) jmjm R222 hi. J! HS. js 7 RS ea [ me ES MA Sees (bee ; : | wes 1:2... 1.m-+1 oP 1.2.m-+-1.m-+-2 1.2.3.m—--1.m+2.m--3 ce ete. |, (0) es : : \ fee" ht et ten \ oy PO [ a eae sd iat dees a hs 1 fia teemerGhe S610 G4 =i ete (1) e€ @ 52 a hx i? é& hae ieee e° ee a [ == =). —— : et 3] hy 1 1 >a re ae lag gg = Cte () libs) ? é& pees ‘ ee i SS ete. | (30) hs 1.2 1 3g See aly Sen ies) fe?) ees ae 23 [1- La aan F ete. | ® ro (n.5)' h? é& Sree Je = 1934 [1— Bae +ete. | In these expressions we have written for 4 its value 3¢. (m) Since h# has all values from h = -+-o to — we find any value of J, by at- tributing proper values to A. From equations (29) we find the values of the functions cos ¢, sin ve, in terms of cos hg, sin hg, and the J functions just given; always noting that when h = 0, we have only for 7 = + 1, — }¢ as the value of the function. We can employ equation (22) when only a few functions are needed, or as a check. A. P. S.— VOL. XIX. G. 50 A NEW METHOD OF DETERMINING It may be of value to have 7' in terms of 2" and the J functions. From the sec- ond of equations (20) we have — Jj, .2'—4J,.2"*— 1d, .27 — ete. ( — dd, .2 —t4dy.2 —Ady 2 ete. BS (1) (@) (Oy Ve —2J,.2 +2dy.2 + 33.2% + ete. (8) (4) (5) (1) (0) (1) —— —3S, 127° + 2d. 27° + 8d. 2 + ete. (3) ‘ (4) " (5) —#d, .2 —#J,.2? —2J,.2° — ete. Then from y' + y~ = 2 cos te yi —y* = 2 /—1.sin se we find the values of cos ¢, sin ¢, cos 2e, sin 2e, ete. In case of the sine, as for example when 7 = 1, we have YY = 2y/ Sse; but in 2— 2 += Jn een, we have the same factor, 2 4/—1, in the second member of the equation. From r= a(l1—e cose) we find (=) = 1— 2e cose + & cos *e (; )* = 1 + 2¢ cos e + 8? cos *e + 4é' cos *e + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 51 r 22 For (<) we have r\2 ‘ 9 (") =1-+ $e’ — 2e cos e + he cos 2e But d i 9 . x dz == 9 . ( ) = 2 sin e (1—e cose )7-= é sin ¢, dg oe and i= Sine = ape sng + Se 15 sin 2g + 4 Ge + Jat) sin 3g + ete. Multiplying by 2e. dg we have for the integral of ty (3) (0) (2) rae (A +4, a 2 Qe (1) (3) ; 2Q¢ (2) (4) cos J — - [ J + Jy | cos 2g — “y [ Zo + J, eos 3g — ete. where c= 1+ 3e’. By means of (22) this becomes 9 r\2 , (1) (2) (3) (“) = 1+ 3e— 4, cos g —4J, cos 2g — 4, cos 38g — ete. r\—2 In case of () , we have d€ .cos *« = $e?(1 + cos 2c), 4¢ cos “« = & (3 cos « + cos 3e), Se. cos *« = 2e' (3 + 4c0s Ze + cos 4e), Ge’. cos *« = 58€° (10 cos ¢ + 5 cos de + cos 5e), Te’ cos °e = x56 (10 + 15 cos 2e + 6 cos 4e + etc.) and hence ye at ae 3e at 15¢4 ae ar ae ete. + [2e + 3e’ + $8e + ete.] cos ¢ + [Be + 28e! + 192° + ete.] cos 2e + [e+ 2% + etc.] cos de + [Re + 48e'+ ete.] cos 4e a2 A NEW METHOD OF DETERMINING Attributing to 7 proper values in equation (29) we find the expressions for cos ¢, cos 2s, cos 3e, ete. We then multiply these expressions by their appropriate factors and thus have the value of io Tht a ») (—2) The following are the values of R; and FR, to terms of the seventh order of ¢. ( Rk, =1+3¢e Be Tl 3 oy) 1 if Se Re FP ee t3 = — yor et 2608 é (2) R, = — ge (2) E Rk, = — Aor’. RR, See 14+ @+ 3¢ 4+ 15¢5 4+ ete 0 — ae = ra 8 ° R, = 2+ je + Soe + poTse! = 32 (—2) ; R, = e— He + 3940 (2) Ry, = iese'— ssi (—2) (—2) R, = 1223¢ —2) Ite = re se See HAnsen’s Fundamenta nova, pp. 172, 173. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 53 3 a x : (2) 2) 3 We add also the differential coefficients of FR, , ; , relative to e. = 52 = —24 je — she + sigs Fete = —e+ #¢— le + ete. = 32 45! 5 u = — 46 + He — pene + ete. = — 2¢ + 4¢ F ete. — 254 3755 = — Hie + tiie Fete. A NEW METHOD OF DETERMINING The value of s found by integrating d (") = 2e.sin e.dg, is (1) (2) (3) , cos 2g —4eJ;, cos 38g — ete. se —4J, cos g — tJ, (2) In terms of the ?; functions, , 0) (2) 2 ay (2) (2) cos g — R, cos 2g — R, cos 3g — ete Again, since we have 2 (—2) 1 if a . € _=-/, Hs) = S= , =. T : 141 —e*® dg Let +0 ee ft — 9-2. Claimvzges then df +0 « - i 1+ ,,7C; cos, and hence 2) TAGs V1—e The coefficients represented by C; designate the coefficients of the equation of the centre. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 55 Using the values of the C, coefficients given by Le Verrier in the Annales de ? Observatoire Impérial de Paris, Tome Premier, p. 203, we have f—f7 = [4 (4) —2(5)° + §G) Toe (Cg)! + SPagt ()? | sin g +[5G)— GY +H GY +HG) tete. | sin 2g + [38 ()°— 42 G) + 9G) — EG) + ete. | cindy + [122 (¢ yi — 92 (F)% + 4123 (£)8— ete. | sin dy [2887 GG) — 2882 GS)’ + et)? ss a [1pea (<)' — 15825 (£)§ + ete. sin 6g + [#5852 (5) — neat ($)' | sin 7g ee Jae ae [2988828 ($ sin 9g Converting the coefficients into seconds of arc, and writing the logarithms of the numbers, we have for the equation of the centre, f-9g= + | 5.9164851 (5) —5.6154551 (5) + 5.5362739 (5) + 5.787506(5)’ + 6.25067 (5)° |sin g + | 6.0133951 ($)— 6.1797266 (5)‘ + 6.067753 (5) + 5.59571 (5)* | sin 2g + | 6.2522772 (5)’— 6.6468636 (5)° + 6.690089 (5)’ — 6.22336 (;)']| sin 3g + | 6 5491114 ($)'— 7.093540 (5)°+ 7.27643 (5)*| sin dy + | 6.8775105 ($) — 7.533150 (5) + 7.82927 (5)"| sin 5y + | 7.225760 (5)’—7.96973 (5)'| sin 6g + [7.587638 (5)'—8.40484 (5)"| sin 7g + [7.95944 (5)°| sin 8y + [8.33880 (5)"| sin 9g 56 A NEW METHOD OF DETERMINING CHAPTER III. Development of the Perturbing Function and the Disturbing Forces. 5 c . 7 a By means of the formule given in the preceding chapter, the functions u.(*), 9f Aa\3 . . . . u .a?(“) , ete., can be put in the desired form. The next step is to determine the com- plete expression for the perturbing function, and also the expressions for the disturb- ing forces. If k° 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, z, 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 o= m’ iE __ wal al we Em A 7 Now AY = (a —2) + (y yl + (2-2), =r+r?—2%r.H; hence ps a ar aa= [Sea 1m If a © is regarded as expressed in seconds of arc, and if we put 5 206064.8, EE" og a Se (#) =". (2) ©. : 1m ~ we have or ca | THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Finding the expression for (/7) first by the method of HansEn, we let h= _ .k.cos (1—K), LES — cos p.cos ¢’.k,. cos (I1— Aj) = “+ cos @.k. sin (11—K he = cos 9’. k,. sin (Il — A)), and have, if we make use of the eccentric anomaly, (17) = h. cos e(%)- . cos f’ —eh(”)" .cos f’ —l.sin «. (eh cos f’ IN Tee os 7 V2 sj if - 3 a'\2 sin rad + I’. cos e(*) aes eee, (3) a th’. sin « ( ;) 25 = cos ¢’ r’'} °cos ¢’ cos ¢ Putting “\ cos f’ =y',.cos g’ + y's. cos 2g’ + y';.cos 3g + ete. ‘ y UY asa a ! 7 [ ye Lee = §',.sing’ + 82. sin 2g’ + 4,’ .sin 3g’ + ete. r cos ¢ . . we find (77) = 4 (hy’; —h'8',) cos (— g’ —e) + (ly. — l9,) sin (—g'— 8) —ehy', cos (— gf et eld’; sin (— gf ) + dhy', +88) cos( gy —e) + Lidy’, + UN) sin( og’ —) + 2(hy’, —W'8'2) cos (— 2g'—«) + 2(Ly’.— 19'2) sin (—2g' — €) re —A4.ehy', cos (— 2g )+ 4.el'S’, sin (—2g’ ) 4 2(hy's + h's’,) cos | 2Qo'— e) + Aly’, + U8’.)sin( 29’ —e) a ete. == ete., where (0) (2) (0) (2) Oy == She -+ The > 1 — Jy —— eke (1) _ (3) 1 (3) <= | J+ Jan is a= | Jay — Jn | ete. ete. A. P. S.— VOL. XIX. H. A NEW METHOD OF DETERMINING Cr os) When the numerical value of (/7) has been found from this equation we trans- For this form it into another in which both the angles involyed are mean anomalies. purpose we compute the values of the -/ functions depending on the eccentricity, e, 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) (m—1) m In (m (0) (1) we are enabled to find -J,, if -J,,, -7,, are known. It must be noted that the argument of BEsSEL’s table is 2.5, or 2.hA, or he. (1) Thus if it is sought to find the value of -/,,, we enter the table with 2.24 or 2e as the argument. When we need the functions for 2 from h =—1toh=4, we must find the i (3) 1 2 1 (1) (0) e 2) = q if if i 1 : aS al values of ad, bs aT es apie eich eae and —1J .. OF 29 < r4 =F 5) to (1) (0) (3) The values of 3.7, . and-J. we take from the table. To find Je we have (2) For J. we have ) And for J, we have THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 59 The expression for (7) can be put in a form in which both the angles are mean anomalies. Thus, resuming the expression for (/7), CE) = hi cos € Ge cos f’— ch (yr cos f’— l.sin « (Gy - COs f* ? ‘ ; B a\2 sin Ff’ ,fa\2 sin f’ amy a\2 sin f’ +1.cose(“). —a(“).* +h .sin e. (“) pea g cos ¢ cos ¢ in which h = ~,.k.cos (1—K) == r -0S V y= bs . GOS .cos @'.k,. cos L v COs a* co ? ccs ? y ky - COS (11 K) gilt " a —? . fa v sin V l * COS >. k.sin (11— FX ) Lu. s a ‘ a I’ eee eat Tee ees K. en cose = (21 COS 9. ,. sin (II — 4}) ye. : ; a\? v\2 sin f’ r we find the expressions for (“) cos f’, (5) sin 7 as follows. We put as before f cos ¢’? (5): cos f/ = y’,cosg’ + y’2 cos 2q' + y’, COS ag + ete. a’\” sin /” bes ; pO yst PO.) ( ,) ““ = §,sin g’ + 0. sin 29’ + 0’; sin 39’ + ete. Uh cos ¢ t : : 5 2 Ca : - If we differentiate |, cos f relative to g’ we have d r” .COs us eos f' dr’ rh & > af’ sin /” Ca Ue) = COST, CR rca fo ene dg’ a dg a : dg cos ¢ < di’ ae’ sin f” df’ a” oat — 7 =—— == —,,-- COS Ne since ig ia 7 _ D; and hence @ (% cos J”) a? - = — —,-COS dq” ee J 60 A NEW METHOD OF DETERMINING Similarly, in the case of ” SID we have y> , i>) a’ cos¢g a@ e eee = a?) asin dg” \a’ cos ¢ r? ~ cos g” r ies cr r > ; r’ sin 5s But + cos f’ = cose’—e', and - 2 == Sie’. a + a’ cos ¢ Hence d?(-, eos 7’) qa”? 5 ad’. cose = = = —= co f = — a dg” ie ; dg” jo (r’ sin f’ ap, 5 , 2 : , a, =a) yc eesinw d*. sin < dy” = 7? ° cos ¢ = dg” Now (0) (1) _(3) cos & = —A’ + [- Fe aay cos g’ + ae — J, | cos 2g’ 4+ ete. zr Sine = 7 se Tks ‘| sing’ + 3 ier + a, sin 2g’ + ete. / From the values of cose’ and sine’ we have (1) = cos f’ = Bee ie cos g’ + 2 [oie »_| cos 2g’ + 3 [Y= Fw COs aie 7 (2) 12. os a? 7 fs SE =) eed, | sae Rae in ae sin 2g’ + 3 pe Til sin 39’ + ete. r COS © We now assume 1 (i—1) _(t+1) (i—1) (i+1) Yi= i a — Ji, i = Ji, + Sy | (v1) (i’+1) 1 (7’—1) (i/+-1) , / YS Sin: Tiny i y v7 Tu! =n Tin: |. Comparing these expressions for y’», 6’), with those found in the expression for Cie eeSinayae Ul Bae given above, we see that the relation between them is 7” a cos ol THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 61 The expressions for cos ¢, sine, are the same as those of cos ¢’, sin «’, if we omit the accents. Hence if we perform the operations indicated in the expression for (JZ), we have = 30? [hyy’’'yv EN), 8] cos (41g —77') — 20° [By Ely 8)] sin(+ ¢g@—'q') (2) Z and 2 having all positive values. Attributing to 7 and 7 particular values, we find, noting that 4, = 0, and 4, = 0’, (A) = 8 fh. yy + 88,01 Jeos( g— g’')— 3 [yi tls] sn( g— g’) +4 [he nyi—h's51, ]cos(—g— g') — $ [Byi1—l’n91] sin (— 9g — 9’) + dh.yo.y' COS ( — g)— fly sin ( — 7) +2 [he ny’, t+ h’.d,0] cos( g—2g’') — 2[bdy'2.4 U'y18'2] sin( gy —29’) + 2 [h. yy’2— Rh’. 8,02] cos (— g — 29’) — 2 [1.by’.— U'y7,8’2] sin (— g — 29’) + 2h.yoy’s Cos ( — 2q') — 21.78’ sin ( — 29’) + 2 [h.yy’s + h’.6,0';] cos( g—dg') — $ [0.8.y’s+U.718',] sin( g—3q’) + ete. — ete. + [h.yoy's + h’.885] cos( 2g— g') — 3 [bby +U 728%] sin( 2g— 9’) + 3 [he yo7'1 + W881] cos (— 2g — 9’) — 3 [L.8:y'1—U-28's] sin (— 29 — 7’) + 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 (/7) from the preceding equation, it can be checked by means of the tables in BrssEw’s | erke. . L . . . . . The expression for « (“) is known; and with the expression for (//) just given, we obtain the value of 4.02 = (") —(/). 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 7 perpendicular to this plane is found by differentiating © relative to z and afterwards putting z = 0. Thus from 2=,7,|5-5 =ts ee = | je a we Teer 4 ip A? = (a-w')? + (y-y')? + (z2')’, Orn a we find apo ps [ Lie AS ae = | thy aeioe A? dv r? * du | CKO ey [ ] (" — ue we H adr Ww 1--m a 4 r?_|? nv’ 9 a d2=," [—3-dA—-2 a 1A dH dA dA va ASS arr A— = r—r' #7, — dv dv? dr 2 dz 4 Hence a2 nv 1 1 ant Et dv i1tmLlé ws | rr Hf dQ-__ m’ 1 u e Fe eerie asco Speer aa mn’ 1 1 ‘eee D , = em [= a4 sin 7.7’ sin (f’ + Il’) where Tf = sin (f + Il) cos(/’ + I’) —cos Lcos (f+ I) sin (f’ + WW’) /=—r’.sin Zsin (f’ + I’). rn | 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. 12 d If we differentiate the expressions for Ts - c 2 . ~, we find » 2 dQ mn Sais ” Ges __ dr? 1 Gat ater q 7A emule 20 mn u 1 a, g mW a ag 1tm S =) rr “ttm ~ 2 2 = mn 3 2 op ae it , , ae an (7? — rr’ H) sin Ir’ sin (f’ + I’) 2 _ m 83 a2 =~ \-Em ~ A 1 he fereinia A pe m ; sin *Jr”? sin *(f’ + Il’) — aa dQ nv’ 1 1 : pe , a = om = — =) sin Z.r sin (f+ Il) #2 ne 3 2 , A . ‘ dQ _ =— = 9 Ss (Cte hy! « a I - 4 dr.dZ’ limé 4 (1 m H) a ae (7 au ) a5 dZ GEQP i m 3 rei eee : E =f n m ( se ) eee dzaz’ =~ Teme 0 I .rr' sin (f + I) sin (f’ + I’) + Tem (a 78) ©08 vg To eliminate /7 from some of these expressions we find from = ye 2 . Hf, that vie = m [ J=2 {a es H| dr |--m 24° 24 “ithe From the value of A° we have, further, r—rrH __ ry 1 J aoe ' 22° 64 A NEW METHOD OF DETERMINING and hence CQ z nm rr 1 | 5 mulls , 7 rT — — 3 = sin /.7" sin il drdZ 2 m [ a a S ( iii SF ) r = Re ae = = | sin 7.7 sin (f+ I) + a the latter of which, by means of the expression for becomes r — =3 aes [ as — =| sin Zr sin (f + HW) — = sin [ sin (f+ I) The expression for A’ also gives aE Na Mai eon, Wen et me | HRT 24 44? by means of which we find tk ta eee ae — ele ee If we put, for brevity, (H== oe sin J ) sin (f= It’) a (Tysaaepin (2): (\eeim fn) a IN 3 (ye Xcos 7G) a ae THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 65 the expressions which haye been given for the forces, together with the perturbing function, are a= u(“) — (IT) fee eG) el eG e *( tos dZ ie ae (a = —ya?(“)". = Bi dh sin (f’ + 1I’) + (2) a s ar(“*) = Fua'(“ i [2 aa) (=): = +4 (2 (fT ) dr re 4 (a “ow ‘i uae = a” 4, Le ‘ ‘) a 3 p74 “\" [ Hk 1 =| Sin’ lara ¥, ; — Sua — -— sin I] at c a” aca a’ Cf at ) a ar” oar d = Z. ae 3ua2(“) a I ae sin ( f' 41 ) aa( ) = wa(“)". a d 2 sin (f + Il)—(Z)’ (aed 8 (ues zi sine tee ‘\araz!) = 2 = 3-3 i aS ea aoe 4 a” TA a a Bim Cf ar ) 1 Pose gua*(“)’. = £ sin (f +1) —( I y aa (oe) = — Bya'(“)’. sa sin (f 22 HW)" sin( f+) + uc? 2(< i susie y Gy a The form given to these expressions is the one best adapted to numerical compu- tations; and the equations are readily derived from the preceding in which the magni- tudes occur in linear form. Thus from r dQ = m’ [ ar 5 = 1 oe x H | dr 1--m 24° 24 7 i A. P. S.—VOL. XIX. I. 66 A NEW METHOD OF DETERMINING we have where, as before, — eu Si =a ee = a 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 — a poe aQ=u(")—H and the forces J 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 (Gs wae(G) » wai(G) we need expressions for the magnitudes r\? 1 7s SES 7a ory ; SHIP os ie (G). aa, —— Gein (f' +10), === = sin (f+ 0), (i), (2) 7 Gee) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 67 When these are known we multiply the function wa?(“) by r\2 i , Sinviena ea ; a sin/r_. , earl: at gsin(f + I), asin (f +) r'\2 cos! . a’ ? a b the function wa (=) by oes 1 r* 2 O Sine ee , . r l + 4 [= = =| 2 ein | Gx oe (f “Ai IT’) E — ap sin? J ip 5 ae , = 2 sin] r .. 2 ry”? 1 7 Soa) oa SI (GF Se IM). — gem (7 + TI) [“.— | ao bo] Ge sin? 7 4 = = sin (f + M1) “sin (f+ I’). 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 c Ci sinh (aie sin (f+ 1’). Putting, for brevity, b = — “cos 9’sin J cos I’ = Seneiesin el’, and noting that a\2 sin /’ ) Ome lia) 7 arb gts ( :) —~ =|J, +d, | sing’ + 2| Ja + J» | sin 2g’ + ete. ”/ cos¢g ({) cos = he — Ti] cos g’ + of tay — Bin | cos 2q’ + ete. 68 A NEW METHOD OF DETERMINING we have (D)\i=6 ace Is sin(— g')+ OU Es Teo Coss) (1) (3) (1) 3 + 2b [eee + Joy, | sin (— 29’) + 20’ Ee 4 cos (— 29’) (3) + 88 [ Say + Jey | sin (— 3g’) + 88 [Jy — Jey | c08 (= 89’) + ete. + ete. The value of (/)’ is found from (7) = Saint (ie sin (f + 1). From = = Gas & ») we find = (1—€' cos &’)™. ES | Expanding, a’ Sees ] Qa, 7 BI8 “ , ( :) = oe + (de + e+ ete.) cos g + ($e-+ fe" + etc.) cos 2g’ + 38e" cos3q’ + 28e" cos 4g’ + ete. ; which, for brevity, we write, ey =" Porate 2 P1 COS g’ + 2 2 COS 2g’ + 2 Pp; COS 3g’ + ete. 7 But . = 5 (0) (2) (1) (3) 7 Ee Be + J; | sing +4 Re + J, | sin 2g + ete. (0) (2) : (1) 7 2 « 7 5 .cos f = —fe+ | J, —A, cosg +4 | Jn (3) Ses | cos 2g + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 69 Putting L =~ .cos¢sin J cos I, ies = _sin Z sin TI, (0) (2) 0) (2) = I,.—I, eae ERY 8 (1) (3) (0) (2) y= “gs — do | d. — 4 Ex + Jr | ete. etc., we haye (Co — $he. -{- L. 9.0 sin g =p Ly. po. COS J +1.9,.d.sn( g— g') 4-h.pi-y,c0s( g— g’) —l.9,.d,.sin(—g— yg’) +h.9:y%, cos(—g— g’) — 2hep, cos ( — 7) (4) +l.p..d, sn( g—29') +h.~m.y,cos( g—2g’) —l.p,.6, sin(—g—29’) + 1..y, cos (—g— 29’) — 2he.p. cos ( — 29’) + ete. + ete. For (7 )” we have the expression 3 (Ey = * cos I (“) Putting i Se 5 cos J, and using the p; coefficients as for (Z)’, we have fy t ah + l;.p, cos (—g’) + 4, . p, cos (— 29’) +- ete. (5) To obtain an expression for the factor [G)- = =] it is only necessary to have that for ae 70 A NEW METHOD OF DETERMINING In terms of the eccentric anomaly we have, at once, Gy — 1 — 2ecose a. € COs e = 1-4 $e — 2ecose + te’ cos 2e. Substituting the values of cos, and cos 2, we haye r\2 2 (3) ( ) — 1+ 3é aN Gos. g ans cos 2g — 4-/;, cos 3g — ete. sin J To find an expression for the factor == a sin( f+ II’), for brevity, we let sin I sin I = . cos d’ cos IT’, C= .sin I’, : eT Sie ieee > and from the known expressions for —,“—“,, — cos f’, we get a cosg’’ a ee sin i 7! - 7 ‘ (0) (1) (8) 3 F =a gt a CE-E Th) Ee + ae vi e,sing’ + 3 Es + Jig, c, sin 29’ + ete. a ¢ (0) (2) (1) — $e’ + [. —dJy, | 6, COS g’ + x wis c, cos 2g’ + ete. |= In the same way, if in I Sinisa He = -. Cos P cos H, — .sin I, a we find Elne/aeee 5 = (0) (2) 5 (1) (3) ‘ ——., sin( f +I)= J, +4, | essing + ae +», | ¢,sin 2g + ete. (6) (0) ) Q) (3) — gee, +| J, =e c.cosg +4| Jn +n ]o.cos2g + ete. By means of the expressions for the factors r\? yee) eae Cae a €), =. 5ain(f +I), = —.-.sin(f+M), THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. s just given, we can form those for 3 - 1 al AEG ow ONSLNe eerie , ; i Ge eae ae (7 +11) [= x | 3 sin? J 7 2 ie sin’ ( f’ + II’) a a Sar SINW OLA veh Ae (le a7? 2 a ma a (Sf ar i) FE @ =| 2 an ey eI) a - 1 ~] bo A NEW METHOD OF DETERMINING CHAPTER IY. Derivation of the Equations for Determining the Perturbations of the Mean Anomaly, the Radius Vector, and the Latitude, together with Equations for Finding the Values of the Arlitrary Constants of Integration. HLANSEN’S expressions for the general perturbations are Ng = Mt + Jo + mf |W + ah . oz + 2° |dt "Td W, LW, = —= i Saud oO Jat 4 C 2 dt + dt? oe GB — fy be in OS VS ) co} Ay where d W, als ye ae 9) =i ne p gee — dQ a = = hy} 2° cos (f—w)—1+42 Wa coe Se icon — a) — \ (5) + 2hy = sin (f—o)7? (= JE 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 x 2 ee rie + h#(1+m) a= 0 a +#(1+m)4=0 - + kh (1+ m) 2 = 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 17,2(2—=x in! ely’ —y y’ of 2 —z 2! mI? ( 7 iG =i) m val a a r); ale \ ae =) Introducing these into the equations given above we have in the case of dis- turbed motion Pax 2 : Fa, T2 c—T a dé ale I (1 ae 1) 7 mk ( 4 oar =) ay 2 ( Y — pp qpefy—y _ (1) a k? (1 +m) a = Mk ( 7 v) Mz 2 Z 7790f 2—z Zz — + #? (1+ m) — = m'k*( = 3) dt? ; ip AS 7? The second members of equations (1) show the difference between the action of the body m’ on m and on the Sun. The action of any member of bodies m’, m’’, m’”’ etc., ean 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 oS” Ca =) 1+-m \4 vit Differentiating relative to x d2_. om ( 1 d4 a! ) dz ~ 1--m “a dz y/* ] But since we have A. P. S.— VOL. XIX. J. 74. A NEW METHOD OF DETERMINING and hence dQ ; = £ m) — =m ~———). a sD) dx : A® a) In the same way we derive the partial differential coefficients with respect to y and 2. The equations (1) then become d= 4+e21+m){=ePA+m)” dt? : ee dx +2 (1+ m)4 =P (+m) ae dQ cs + kh? (1 + m) = =k (1+ m) oe Let X, Y, Z, be the disturbing forces represented by the second members of equations (2), ZF, 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 7 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° + 7. We then have Sein 7, Ss cosa. In case of A, we have =X | Yoke if 7 and for S, From these we find THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 75 If we wish to use polar coirdinates we have 12 , 5 = Reos f — S sin f aL % dQ : 4 a sin f + S cos f. ; From Li= 71 COS, fs Yen fe we find dx = dr cos f — rdf sin f dy = dr sin f + rdf cos f Px = dr cos f — rdf sin f — 2dr df sin f —rdf* cos f @y = dr sin f + rd’f cos f + 2dr df cos f—rdf* sin f From the expressions for dx and dy we find dy cos f.—dasin f= r df de cos f. + dysin f = dr, and hence aQ __ 1 €d&& sin f+ d2 cos f dae r af dr Q 1 dQ shapes do FF - COS f+4 ~ sin f; dy r from which we see that Oo R=F(1+m) ~ S=—F (1+ m) : rT If we multiply the expression for de by cos /, that of geet by sin f, and add, we obtain @xcos f + @y sin f = dr —rdf-. re ‘6 A NEW METHOD OF DETERMINING In a similar manner we find dy cos f — @asin f = rdf + 2dr df: Operating on equations (2) in the same way, we have cos f + = - sin f + © “ r™) — Xvcos f + Yosn fe oy cos f — % sin f = Y.cosf—X snf=S Comparing the two sets of equations, we have raf odr df __ 72 1 dQ dt a made. wae Ss (1 a> m2) r df (3) dQ ar hit ke ad in m) af agp oe = ee) 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 i 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, a the angle between the axis of x and the perihelion, v, the angle between the axis of « 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 for¢=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 y in place THE GENERAL PERTURBATIONS OF THB MINOR PLANETS. 77 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 + gy = ¢—esine rcos f = acose— ae rsin f = acospsing v=ft+x a'n? = k’ (1 + m) Now let nz 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 nz be designated by a superposed dash, and let the true disturbed value of r be given by the relation r= r (1+ 1). We have then Ne = E—G sine r COS f = Ay COS E — Ae r sin f = d) COs py Sin é v=fon ayn =k’ (1+ Mm). We will now first give BkuNNow’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 : di da ae = a“ = mea =| (Ya— Xy) dt+ ¢, C being the constant of integration. Introducing cosy = =» and sin f = = 78 A NEW METHOD OF DETERMINING into equations (2), neglecting the mass m, we find Px , k.cos f _ ax io cosf _y Gt mo) 4 Py 4. sin f __ Y (*) dt? Ga I We haye also TAM ESOL ES Or Ber S df ane ‘Ss aeat sin f ore Gp 2s dr : df . af = Si set cos f «a, 5 and hence _ dy Chip Gy Gf Ce) dk a or o af td We 7 sr = ((% — Xy) dt + C; and 2 jf eran, - Tie, =f Sr.di-— C: In the undisturbed motion we have po being the semi-parameter. Hence PF — { Sr.dt +k py = ka/p. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. From these relations we derive a ee j Sr. dt Po ky Poe and also V Po 1 “V/ Po a Vio Sr. dt VP kV Pod \/p If we eliminate -- from equations (4), noting that rn l d vA 1 a = kp, < eal =;: we have da Asnf sin f : ae or ct ge! flr oa neglecting the constants of integration. Since r = r (1+ 1), we have also a=a(14+y), y=y(1+>). The equations (7) then become pee es) eee ( (x ae dt VP p —~ w@ dy __keosf _ ((p4 cost Sir) dt etc”) gemma SY topes From the equations & = My) COS E—AEy Y = Ay COS Pp SIN ky 79 (5) (6) (8) 80 A NEW METHOD OF DETERMINING we have dx = —q sin «de dy = dy COS $y. COS é de. Then since dg " de, df = cos o.“ dg, x = hy = 0 r eC 9) 2 U7 VA Po using the values of sin «, cos «, in terms of sin /, cos Ff, we find Cee sin f dy > cos fe dz 1 /p, az V Po And these give ke sin f —_ __ dt Pr VP. dz VP keosf 2 dy WP 2 Be VP dz 1p 7/B — dy V Po sia key =a { dep / P The equations (8) then become a® + @)atr)% ve] = ((x—2. &r) di dt dz VP p ye a. dy [a ae ee vibe VB =f yee sents Sr) dt, dt \ /p ke, 5 5 6 : the constant — —~ being included in the integral. VP We will now transform equations (9), and for this purpose we multiply the first by 2, the second by = , and noting that 5, ee dz dz a kA/P , THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Sl we have a = Tf (XK — A. Sra Lf (yp Lt) op a (10) Now multiply the first of (9) by y, the second by «, putting for '/" its value VP given by (6), noting that we haye (1+ 1) de = ; De Sr dh ix ane Sr)dt dt kVp’ P key/ por Pp (aba) Bet f (YS Syas ky/ poe Pp . Zz = = We ean write ‘ 7, in the form ( dz dz \o. dz > dz —~=—2(1+ 7)——(1+ 7)?.— 4+. dt 2( a ) dt ( ais ) dt aie dt We have If df dz df “ 1 LN) Ee: — 5 — = #— COS @. aay: a? ae dz dt” dt r ?, df ay yeep? Sint = Ny. —, -COS Poy WN = Ay NM. ¢ r Making use of these relations we find ee and for |, given above we have dz dz ) td VP S914 1»). ee ites ir Vr T 4) V Po A. P. 8.—VOL. XIX. K. io) bo A NEW METHOD OF DETERMINING The equation (11) is thus changed into 1 gs z ] ¥ Ny 2 : si Pe oY a= 1——_f (14+ 24) sra—___J (x —™2£ Sr)at dt ky Po VP hy Po P (12) 2G) (7 aosiy (COS fen iG: ? VP (¥ + — Sr)dt es Rare =J p A ) =f (1+) V Po The equations (10) and (12) can be put in briefer form. 1 I Let Nes XE Sr Ve ae Ce ee, P p Then dy cos f--e, ( sinf (+ = |) gat | Yas, dt ~Po e jth (13) dz 1 . 1 ; y a =e G2) See haar key Pow VP V Po V Poe The values of «, y, found in these equations we get from eu 1 s hae daz, 9 = te o(e—7) +3 qe (2—) + ete. (14) ly, ly, \9 : y¥=yta (e—t) + 3. (¢—t) + ete. ml 5 5 ax di From the expressions for is : es , we have also cosf+e _ _! (ae a Ay ea ) ne pes cap, \dt ae ee ee (15) sin / = 1 dx, 1x, : aie > kyr G Tag t)) uae 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. 85 as constant if we designate all magnitudes in these factors dependent on ¢ by a Greek Jetter. . We thus obtain CC es Uae AA tee one (sy ca RE Ge = yp d (b+ OVE) Srat— | f( Rev YBa 2(z— t) i = (eh) > de 2p : ae = J (x2 = 2.5 ) dt +. i (16) = f(x v.48) at 9 f(x. ya These equations include terms of the second order with respect to the mass. If we put ai 1 VP\ o. Be eee Lee W=—7 J+ Vo). Srat—_ = \ (.v— ¥,.£) dt we get Noe = Mt + gy +r fall W a oon Y"] o f (17) “rdw, @w . ae om) L dt aly dz” sz | dt In equations (17) g, is the mean anomaly for {= 0; NV is the constant of inte- gration in the value of ». From the value of I!” given above, we haye r l ) 2 = = A Li a N (1 +20) Sr — (X, eel di hy Po VP ky Po Now since > > as . = TE Sys KO) CO ° ——" 0) : - x i di f rT df = = dQ > l dQ VMSsin f. = + cos f. a ay 7 adi dQ i dr AQ sa! 84 A NEW METHOD OF DETERMINING neglecting the common factor k (1 + m), we have oy 4 aE A on fh inf) sche (itn 7+ Mit con) e+ Gta[ A ne OE a And as v =psinoa, & =p cosa, this becomes = = a [(- 1—2 ad 5 — 2psinw. cos f. a + : .¢ sino sin Pe FF + 2p cos o. sin f. a + 20. : 7 8 w cos f + 2p. mel Sey) Fy oe ee — a = p COS a [1a yin atom 8 +2" cos (f—o as + 2e,.° eos o =| P df P d/ But 2e, 9 COS. - — 2p. z == g (€ p COS @ —p) =—p-2 a also h k THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 85 Hence since * (1 + m) is included in X, Y, R, S, we have CMe es p ee 2p .h? / <7 ae dQ > a7 [2 : cos CH @) i} 4b Re (eos ( f —o) 1) | oF (18) dQ +> 2hop . sin ( j= 0), If we write h,°. a) cos *@ in place of # in equation (18), we have the same ex- 4 1 pression for : a € 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 as that given by HANSEN. a. _ ltecosf r \ cOos*¢g also 7 cos’ ¢, a an L-Pexcos yas hence ra __ 1-e cos f cos *¢, rd, cos*g ~“1+e,cosf Using: ™ —y in place of 7, and developing, we get D 0 iG « D>? 5D ra __ r--reos f.e cos (y—z,)-+rsin fe sin (7—7) Aly , COS *¢, Let us put esin (y — 7%) = 7 C08 "op, (19) e cos (y — 7%) = E cos"d + 3 since eé = sing, we have cos *p = cos *p, (1 — 26 & — cos *@) &° — cos “py 7"). 86 A NEW METHOD OF DETERMINING With this value of cos °p,and 7 = a, cos") — 7 Cos f, we find ra a, cos *¢,—e,.7 cos f +-r cos f (Ecos *¢,+¢e,)-+-r sin f .7 cos *¢, ee d, COS “Gy a cos*g,-+-7r cos f. &cos*¢g,+7 sin f.7 Cos *g, , a, COS *¢, (1—2e,E—Cos *¢, E°— cos *gy7 y and hence z Ps 1-25 cos -7.— sine r a Wa et. Gar rd, 1—2e,€—cos’¢,&—cos’¢,y” . From lr ChE eh Gk ak Wadia. mdz andbe and df __ ky/p(+m) ae. Te r ? we have df x a A@ =. ,2 COs Gi Tn like manner we find Ce 3 Co ——a Po: We have therefore dz = n.a@i7"Feos ¢g dt Nyy - 1". COS Pp THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 87 s n ; — 0 ) If we put | =1-+ 4, substitute the values of “—, and cos “9, we get C) rT. (1+6.".cos f+." sin F )? dz 0 0 2, dt — =U b) = = ” (1—2e,5—cos *¢,5°—cos "47" )3 Further, in the case of », we have r 1+v7=- z Then since Pers mi 6} n Un = Ay Ny, = (1 + 3b), ° and SOS? = (1— 26, € — cos "hy &? — cos" 7”), COS “$9 . we have 1—2e,6—cos *9,.€’—C0s 7¢,.777 (1 Fa eee re + —eos f.€4-—sin f.7)* (1-2 C+ cos fe, sin f)§ (+0) If we let in * - jf . a—1 | cospae jo. BID. 7, a, My B= 1— 26 — cos "py & — cos “gy 7’, h _ (+6)? hy = RB e we find a= 5) a) — —— a B — A(1-++b)4° A NEW MBTHOD OF DETERMINING 838 From the latter we have (=) = 1—2(1+ BS + G+ bt. =. Hence (gq) B= 2049404 Oe ee If we put = h h h r : h Lg WE 2 ae 3 —1+2,.& a, 8 -f 27, Ws, sin f, we have ai We hee We have yet to express |, in terms of the elements. From — &. cos") — 7° . COS “Gy = y bo —_ Soy B & and from we have h =| ie COS “cos ¢’” (21) THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 89 or Pa COS g hy COS AN If we put a, v7 h, 0 fo x COs ¢, we have an ~ cose 5 5 : 5 = (than 3 These values of h and h, being substituted in the expressions for I, = is found ( expressed in terms of the elements and of 7, in a very simple form. To find the rela- : dz : tion between a and 7, we use the equation ¢ Be a = as Fic hy and as this is also equal to 7. , hia dt we find Gholi 1 ie i (CS For the purpose of keeping the formule 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 ¢, r, « f, 2 v, 2 Y, we write, T, ~, 2, @, 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. A. P. 8.— VOL. XIX. L. 90 A NEW METHOD OF DETERMINING If in equation (21) we introduce ¢ instead of ¢ we shall have = 1 ee (23) where 7 | ) = h . eee ee + 2 ee oso 2 Soe h, h hy dy hy My We have also dt I . ee (24) dz n(1+-A)? 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. Codrdinates 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 be a function of ideal coérdinates, 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 r. The function dependent on ¢, 7, and the elements we designate by A. Then dL dA (hp Se (he? or da oe where the superposed dash shows that after differentiation 7 is to be changed into ¢. Let us write the equation (24) in the form THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 91 Differentiating relative to t, we have vy B 1c? Clear ae Gea dz 9 i Net dz The differentiation of (23) also relative to t gives G6 dw dé h, 28 df d?* d¢ dr i, (Caza te Eliminating x by means of (24), we have de dw 28 dp US tm chs 1+8 dr ° eee cs - ip Substituting in the expression for oa we have at dp ral dw ia bole dr Since » is an ideal codrdinate, we get from this JV 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,F — cos ’¢,.6* — cos *¢.77 LS = 14 pyeeees ey. sin FY a, a o Since 2 is also an ideal codrdinate, we have from (23) —— hy y \ ¢ net = mot +g + mf 1 +3: (=) dt (26) go 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 t, and we have Ne = ME+ G+ Mm { W, dt ! | aes \ (27) dW, = ee | v= N af ( dr ) e J where qs Se Brg Oe Seine oe eae an 7 egies Ty ea i and p and @ are functions of t, being found from MtT+G = y7—AQSINY 9 COS @ = My COSY — My & pSIN@® = A COs Sin x. : e thiae p Also in the last two terms of W,, “ is put equal to unity. ] h q 0 When terms of the order of the square and higher powers of the disturbing force are considered, ¢ cannot be changed into rt. In this case let My t= mT + J + néz. Likewise let My F = MT + Go + NOG where nog is a function of ¢ and ¢. According to Taylor’s theorem we have W=Wo+ ae Ngee = 62 + ete. the value of W, being given by (28). THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 93 We then have dv 1 2W, x; oe = Me 4 SM ae 4g SM 8 + ete. dg dz d= ir Retaining only terms of the second order, the equations (25) and (26), replacing 8% by dz, give MZ = Mb Jo + mo f [Mo + om be + v* | dt | (29) y= 3 f[% 4 OH] a dr The equation (26) has been put in simpler form by Hii. For this purpose from (21) and (22) we haye (= + WH). Hence Developing the second member and adding IV, we have My = Mt + go+ my f Ut aie (30) The next step is to express ome and “* in terms of the disturbing force. From (19) we find oo = bess ale = cos*¢, os (x 7M) cos2¢, 94. A NEW METHOD OF DETERMINING Using these values of & and z, and ep coso = a cos’ —, in equation (28), we find 20 WwW, = —— = Recon (y—m—o@) + —“— Ly Mest een Ny COS* = cos *g, h Since CISGie was hy/1+-m “cos Gap we have from the expression of h already given, k?(1--m) h= rr. ey a : dt By means of f=f—0—(4—m— 0), P—1=eccosf, we may transform the expressions = =~ | mcs 2, a == ieee e sin f, into r. F ” —h = cos (f —o) .he cos (y —™ —o) + sin(f—o) .hesin (y —m—o) dr See (f —). he cos (y —™ —) — cos (f —o) .he sin (y —™m— @) 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—™m—e) = (c= —h) cos (f— o) + = sin (f—o). Substituting this value of h.e.cos (y— 2, —.) in the expression for W,, noting that eel = h, hyd. cos*g, -2(1-+-m)” we have ZeRes dv 2h,. SF , ar Woe SSN pee ee is Vo qe amt Olt apts Geen) irr Ole, oe SOS ai dos\( fa) = Wine Nig. COS? go h Differentiating relative to the time ¢ alone, t remaining constant, and haying care that all the terms of the expressions be homogeneous, we have dW, 2hyp d*v 2h,p : ar = . CO — @) r—_ + ——" _ . gin (f—o).- FTG = Paes ies) ee (1m) =O) 2p ; . dh h, ah 0 —«)—I1 =e hyd, COS® ¢, [¢ . (F 0) ] at Hi atee and dh__ +m) d_ hr’ d*v di (er ‘tde a kB(1tm) d? : ic- = dt Substituting (1 + m) 5 ( =) for La s , 2 a? (1 + m) (=) for ~~ , 96 A NEW METHOD OF DETERMINING we have dW, q 2h? dQ ai (i 12 cosa) ian aa a [cos (f —o) — 1)’ (= : _) (30) + ho” sin (f— o) 2 (=) diya of a2 zi a eae ie : ; aW,. eet . This expression for), is the one used by HANSEN in his Auseinandersetzung. It is given in a much simpler form in his posthumous memoir, and as the latter is the form in which we will employ it, we will now give the process employed by HANSEN to effect the transformation. Substituting first the value of h, omitting the dash placed over certain quantities, noting that in the posthumous memoir ¢ takes the place of o, and remembering that we are here concerned only with terms of the first order with respect to the mass, we have aWe an eS | 2 & cos (fo) —1+ 75 [eos (f —0)—1] | (7) 2 ain) r( — ) i= - From the relation pe = a(1l — &) —ep cos ow we have (Cie = —_ ep COS w a(1—e?) a(1—e*) * 6 An inspection of the value of —- shows that its expression consists of three parts, one independent of +, the fis two multiplied by p cos w, and p sin o, re- spectively. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 97 Put aW de aY dt sd dt ae (2 cos @ + je) Bee sin ie a = dt a and we have V 1—é 1—e? 1—e? af 7 dS _- = a ae cos f e cos f 1 dQ ae sin f n.dt 3 ale r is si tie 3|( \+ “5 ( dQ of dr ) i) a@ jf acosf (cos f+-e dQ a sin f dQ = A pegl[SE + ett Y(t) ene (ety), V1—e r 1—e df 7 dr / \ a 2 4 | [ a sin f J sin f ie )\— acos f i ) ndt vie r 1—e? df r : dr / J But df a 5 ae cos f e cos f l a A ee ae = ie tt ry/1—e + (1—e’) ue Uae ENE Gs) 1 4 Sa hi i 1 . de” \r a <2) SE dr = ap = cos f; hence cin on (a2) ndt ~~ da. dg/? iB eel al je 1 dQ ar ar Ma ig) ee * (=) |; 2 @) ndt . fl—e? de Again from Ca) = ar) Magali A. P. §.— VOL. XIX. M. ) (e ) dr dg 98 A NEW METHOD OF DETERMINING we have (= = @) ie . (= resin f = rs dg ay le? dr a(1—e’) ealys A 12 3 U2 Eliminating (‘ =) from the expression for 2 if Vi , we have Tn the same way we find ey ly aed tla eae = rie sin dQ ee ace But if we employ the relation 1 r recos f a(1—e*) a(1—e’) : acos f 5 , 5 in the term, /1—e, of the preceding expression, the whole term becomes (= + e re dQ a (1—e?)3 ame ure ea Bt er Using the equation 0=—recosf—r+a(1—e’), multiplying by e (— FANE a (l—e*) aa)? <> © THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. adding to the preceding, it becomes E r cos f 2e d2 — = Vom ( = ). 1 (=F ap 1 Al dr Further, we have cos f e sin *f resin?/ = a ain fp a |=" cosf VI—e sf + aa 1/ 1—e? a(1 e)i- Reducing this expression in the same manner as employed before, it becomes > sin f | 2rcos/-+3ae 7 =[- sin f + aaa e") 2 ay/l ( ' A c at Multiply this by dg, the last expression for a becomes dt 2 2rcosf+3a a dQ r cos J }2ae dQ = — = (0 7 =-ar iF ) ndl 1—e? J a7/1—e* y ( a a 1/1—e? ar}? the integral to be so taken that it vanishes at the same time with g. dS dY av Substituting these values of —,—,—, in ndt’ ndt’ nat dW da aY¢p ‘ ar nae SS SS Se SS 3e sin « ndt ndt ndt ( an Sa = ) zis nde a, od this expression can be made to take the simple form dW dQ d2 : = Ad eee (—), (31) ndt dg dr in which e)—r 2osinw i p a (1 { \ (2 cosa +3 e) ~ a 1—e? a a I e ay A= ( r sin f 2psinw (" 15 Sem — (-! cos +30) —— == a >/ 1—e a7/1|—e* 100 A NEW METHOD OF DETERMINING Since a7 rsin f ae dqmnay ae n= 2" cos f ——— eos a’ e.de a Ss we have 1 d.p r° — a? (1—e?) dp i dears A=—3 [ = ; : = [ a d er 1—e? a’. de se ae a’ e dy J a’ .de a J 2 1 dep: det d.p Ghote fz = eae E —4e|—| = — Be |< 2 (1—e*) |@e.dy La’. de a .de w@e.dg These expressions for A and B can be much simplified. Thus from a = Fh e e* =1+ $¢&—(2e— fe’) cosg— (3 e— fe’) cos 2g — Fe cos 3g — cos 4g—ete., 2 2 a” 2 . . . p and a similar expression for —, we get a” ape avn eS (2— =) sin y, a@e.dy 4 d.p° — — $e = —(2—__2 &) cos a.de ( 4 ) i) d.r EN Js BN on 0 8 alae 2 Bat nS (2—") sin g + (e—,,) sin2g + Ze*sin 3g + 36 sin 4g + ete., ae.ag 0 Df Choire eee 2 BN de Cas Rola J [= ae | dg =—(2—#e)sing— ( —;) sin 24g — | sin 3g — ; sin 49 — ete., a. r* — a? (1 — e?) . e zs e e& = = $¢— 2—") —(5—;) 2g — — cos 3g —— cos 4 i 5@ (2 7) cosg—(,—,;, ) c0s2g9 —; cos 3y — Js Ps S32 2 3 322 2 28 aa —4e= —e—(2— 2) cos — (e— 3) cos 2g — $e cos By — 3 cos 4q. a. ae . THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 101 From which we obtain A=—3+(4+4 2) cos(y—g) B=—(2+é)sin(y—g) | +(¢+ 1) cos (y—2g) —(e+") sin (y—2g) —(5e =f —) cos y —(e is ) Ae e 3e2 > (32) + = eos (y — 39) —~ sin (y—3g) see cos (y —4q) — = sin (y —4g) cat a +5, sin (y + 2g) These are the expressions of A and 6 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 + 7, with very little labor. dW Let us resume the expression for =o Ti that is, Ws dQ _(d2 a Aa (a) + Bar i) A and B having the values given before. Since “ can be put in the form ay ha f , = 28 eosikg, @ we have a 7 ist d.— 2 r sin f a k ae r : Py d R® > in eg, 2—cos f = — =— cos kg, ay/1—e e.dg e a de de 102 A NEW METHOD OF DETERMINING 7) d fa iF € ( 2 (i) . 1R© { | & ) seh ag = 22° 5in tg +2 de g—3e@q. But since P= 1 + §e—(2e— Je + ye) cos g — (he — he! + alse") cos 2g — (1¢ — Ye’) cos 3g — ete. we have dR© eS gD) Gp = I(k : ; : athera IR® Hence the integral just given is simply es sin kg. Cr e e A and B ean then be written 1 2/1 —e?) — 72 2 OD (k) A= — 3) ieee: se) eS en iy | =; ; Oe ay/l—e kde u e wad 2 psinw 7 dk) = — — | (2% e080 + 3e ) k Re sin kg —— = ai za cos kg — 2e) | 1—e’ a 2 a/1—e\ de Putting == BR cosxy, we have likewise a, AR(*) ae, 9 Pees o = — — = = eoszy, 22 sing = —— es = 20 a = Ae Dore ie ay c= A gin xy. Introducing these values of 2 f COS ©, and 20 sin @ into the expressions for A and BL, after integration relative to y we can write JV in the form W = a cos (xy'+ BE) , THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 103 where dRi(«) Ri«) i a) = U — J} de e bt=wg +79’, Uand V being two functions depending alone on #. Putting x = + 1, and —1, we have / a (1) - a) — dk” U Bee Re J de e ay Re R® >, A oe ee Er de e ? and hence av Lg qa — a) DS ~ dk® ? a aes RY = dew e Thus we find dR) dAR®) de R&) de_ R&) in or putting dR) = de Re) n° = gd R™ + x IR ~ de dR) : “de R(x ON = FaR® —* am? ~~ de we have a) = yw aM 4 8 ao. (33) 104 A NEW METHOD OF DETERMINING The values of 7 and 6 are readily found from ae 3 3 5 a 1 4 le 4G ,=14$é@—(Qe—jfé4+ Ge)cosy—Ge—te + 7; &) cos 2y CEN Mee ier —(té—, &) cos3 y — ete. = > BR cos x y. We have RO=1+2¢ R® =—(2e—4 E+ 4 &) RO=—(L@—1ié+76&) R® = —(46—fe ) AC = ete. — ==ONG te Ci ha Ue =—(e— he + 3¢) a ee ee ue =—(g¢— $2) ete. = ete. For 7 we have Oe (e—Z e+ ge) yy GS aaa 4=sesan4) @e—tep ee)’ Qe — 31, ¢) Hee ees or n® =he—lé—ahye. (34) For 6 we get at once 629 = —7é— he THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 105 In a similar way we have a(S) Sue 5 (4) 5 1) = 2 ?—1h A (if) eee 8 > 2 — 3g ©. (55) In case of the third codrdinate 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 haye d R® aR g@® + go (0) — = —_—s ! (1) (—1) we — de C= de 5d Re == yn”? (a - Oo xy de where dR de 7° —= Sel R® ~~ de For x") we then haye 79 = —(Ge+ 3 e +ete.). (36) Perturbation of the Third Codrdinate. Let ) 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 b = sin 7 sin (v— he dQ 9) oe) " r pais! 2, —,=ltht ek hy { (1+ 2) (—,) ae. And since the quantities under the sign of integration do not have any constant terms we can write ath hy : 5 2 sae "= 1 +h) + ek, + periodic terms to U h, Re i= 1+, + periodic terms THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 115 . h : , E Since (— 1) is a quantity of the order of the disturbing force we have baie (ha) (hay ce from which we get Now putting = (eee etc. = H, + periodic terms, substituting this expression and those for the preceding expression for gives, preserving only constant terms, kz; =—4 (k + ek) + 3 A. Introducing this value of /:, into the expression for JV it becomes N= —1( 4h 4: eh +38H) +4(8V, + 24,—327). Preserving only the terms of the first order we have N=—1(4kh + ek, + 57%,). . .* . . . h To find the value of A, the constant of integration in case of 4 , 7 We have = 1+ A + periodic terms, hy ill) A NEW METHOD OF DETERMINING also No — 1 + ky + periodic terms. h. | From these we get bo 4h 1] kh me Hence [= ee H, =4(h ea) 3 45 or, neglecting the term of the second order, K= 4(ky a ck, ). log n log a THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. ALTH@A 119. CHAPTER Y. g = 332° 48° 53”.2 ea WBA 211: ) oe 20m bi. 51,5 1894.0 r= 5 44 46 | @= 4 36 249 n = 855’.76428 = 2.9323542 = 0.4117683 The mass of Jupiter is jjyssy- in the Berliner Astronomisches Jahrbuch for 1896. are for 1890. To reduce from 1890 to 1894 we employ the formule of Warson in his Theoretical Astronomy, pp. 100-102. The epoch is 1894 Aug. 23.0. Numerical Example Giving the Principal Formule Needed in the Computation Together with Directions for their Application. JUPITER. g =63° 5 486 nm = 12 36 594 ) Q= 99. 227 59:9 1894.0 ie gees 36.9 | p= 2 AB bie n’ = 299” .12834 log v’ = 24758576 log a’ = 0.7162574 v=21+7c0s(2—8) dd : Fe V=A24+ (U—Fb) Zi —7 sin (Q—?) cot .2 d nm =xa-+ (t—t)- 1 l i 1. Fas 7 sin (Q—4) tau $7 The elements of Jupiter are those given by Huu in his New Theory of Jupiter and Saturn, in which the epoch is 1850.0. Applying the annual motion of 57.9032 in 2’, of 36’.36617 in Q’, to Hrw’s value of 7’, and of 9’, we have the values given The elements of Althzea are those given The ecliptic and mean equinox 118 A NEW METHOD OF DETERMINING where 6 = 351° 36’ 10” + 39.79 (¢ — 1750) — 5.21 (t’ — 2) 7 = 0.468 (t' —t) dl fy) na 50.246. These expressions for 7’, Q’ and a’, 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 most recent data, gives the following expressions for 6, 7, and — when referred to 1850.0: § = 358° 34 55” 4- 32’.655 (¢ — 1850) — 8.79 (t — 2), 7 = 0.46654 (¢ — 1850), —- = [ 50".23622 + 0.000220 (¢ — 1850) | (t — #). Let «= > ; we have then uw = 0.34955 2u = 0.69910 3u = 1.04865 4u = 1.39820 5u = 1.74775 6u = 2.09730 Cte ete: Hence 1 —3u = — .04865 , 2—6u = — .09730. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 9 This shows that the arguments (g¢— 3g’), and (2g — 6g’), have coefticients 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 — 3q’) and (2g — 3g’), since in the developments the coefficients of these affect those of (g —3q’). From sin 3 Z.sin 3 (¥ + 0) = sin} (Q —Q’)sin} (¢— 7’) sin 3 I. cos 4 (¥ + ©) = cos (Q— QM’) sin § (¢—7’) cos $ 7. sin 3 (® —®) = sind (Q—Q’) cosd (7 +7’) cos 3 J.cos $ (% —®) = cos $(Q — Q’) cos $ (¢ + 7’) where, if 9’ > &, we take 4 (860° + 2 — Q’), instead of 1 (Q— &’), we find eA PS T1615 300 o> tt 50) 339 = 6) Ad rsa An independent determination of these quantities is found from the equations cos p sing = sin? cos (Q — 9’) cos p COs g = Cos 7 cos psinr = cos?’ sin (Q — 9’) cos p cos rT = cos (Q — 2’) sin p = sin?’ sin (Q — 9’) sin Jsin b = sin p sin Jcos ® = cos psin ({— q) sin Jsin (‘/ — r) = sin p cos (¢— q) sin Zcos (/ —r) = sin (?—q) cos I = cos p cos (t— ). 120 A NEW METHOD OF DETERMINING =z —AQ —® WV’ = a’ — Q’ — 0 we have Tr = 156° 11’ 55.7 , Tl’ = 156° 58’ 22’.8. Then from k sm K = cos /sin Il’ CO kG = cos II’ fis, Sion AG = sin IT’ k, cos AK, = cos J cos II’ t . 9 é€ aa psin P = 2a’ — 2ak cos II — & ) é p cos P = 2a cos 9’ k, sin (11 — 4) vsin V = 2a cos¢ ksin (I1— 4) v cos V = 2a cos ¢ cos 9’ k, cos (1I— 44) / . ir 9 € . w sin W = p— 2a’ — sin P é weos W= v cos (V — P) w,sin W, = vsin( V— P) w,cos Wi= 2a?* cos P, (7 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 12] we find Ke = 157° bo 93646 log k = 9.999614 i — 1b6. 45 7.4 log k, = 9.997849 d Zia! em (Al 0) log p = 9.932748 ye =359 6 2.4 log v = 0.601463 W=266 4 39.5 log w = 0.605196 We=266;) 15. 38:0 log w, = 0.601352 Then from R= 1 + e262 5 Sa €", we have log R = 0.702855, logy. = 7.976024. The values of the quantities from II 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= E—esin L, where ¢€ is regarded as expressed in seconds of are. Substituting the sixteen values of ¢ in the equations fsin (/— P) =w sin (#L— W )—ep fcos (fF —P) = w,cos(#H+ W,), we obtain the corresponding values of / and /. ASP. S.—VOl. XX. P. 122 A NEW METHOD OF DETERMINING Then in a similar manner from = 7 O= ha C=y + y2sin*Q logq= log f + y 2 Ce bere) 2 praree Yo V2 V2 o.. ) FAI ay Seon ie a0 iy as ( ae Tal sin2 #'-+ s ( af af ) sin 4 / where s = 206264”.8, log” = 9.63778, we find the values of Q, C, log q, x, and y. Thus we have found all the quantities entering into the expression (*) = (C—q cos (L’ —()) c= Ge ge Q)). Instead of this, we use the transformed expression (°)' = N" (1+ @ —2a 008 (H'—Q))? (1+ ¥ —2eos (E’ + QY*, and have, for finding the values of N, a, b°, the equations ( ox sin 7 B == S17 sl a=tyhy b=t93% Ny Sea V C THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 123 To find the value of S) we put 7 (1 + a?— 2a cos (E’ — Q)) 2 = Te + b,. ‘cos(H’—@Q)+ Hee cos 2(#’—Q)+ ete. | (0) (1) ; 2) 2B, + B, cos(#'+ Q)+ B, cos2(H’+ Q) (1+ 8 —2b cos (H’ + Q) 2 = + ete. | For finding the values of the coéfficients in these expressions we use RUNKLE’S Tubles 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 (dl) (2 (3) (0) bi by bx as (0) @® (1) @ (2) eo 6, , then those of —, —,—, ete, ete.; —.b, ,{.b, , .b, , etc., ete., where G* is found y a’a'a 6 5) & 5) & 5) . . De a from the sixteen values of 9°= —. 1—a?’ Since 0 in (1 — 26 cos (” + ()) is very small it will suffice to put : By = bao n > > (1) { 1 ) B= See — os 0. » Dy Then from ‘ (7) n (i) . Cc, = £ Weel cos27 0 os yy (?) =n @) é Se = 5 Neem 27 0, us my . . {a we haye, in case of 1 (5), (0) (1) 2 , a i » WE cy, = 7; bcos2 Q, Sz, = ag bsin2Q; 124 A NEW METHOD OF DETERMINING : » fa? and, for ua? ( ) ahs (0) 3 (1) 3 (1) 3 N, te = 7,N 3bco2Q, 345, = 7, 3bsin2Q. ©) t s b t\i— We divide by 8 to save division after quadrature. (i) (i) (7) With these values of ¢,, s,, and the values of the coefficients 6,, we find the D> y wy y r values of k,, A, from ma () (0) (i—=1) (i+1), (1) i COS, = O07 Ge = (0, +b, Ve Dp yD yD > a z z For 7 = 0, we find & from (0) (0) (1) (1) v= Grae Wn Cn o . 2 ae . L Then in case of u (“) from k * | th im sk,cos [t(Q—g)—K,] — A,, = im'sk,sin [¢(Q—g)—A,], where m’ is the mass of the disturbing body and s = 206264.’8 ; and from A, = $m sa’k, cos [t(Q— 9) —Ki] Die m sa°k,sin [¢(Q— g)— Ay], a 3 (c) (s) in case of wa? ( “) , 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. (c) (s) Again, since A;,., A;, are given in the forms (c) (s) A, = =C,,cosvg + >C,,snrvg (c) _(s) . A, = 2S8;, 087g -- >S;,, 8» 9, we have the following equations to find the values of the coeflicients (;,,, Si, ve (8) (0.4) (1.5) (2.6) (3.7) (08 )=¥,4 ¥ Ciy= Mo a9)=¥4+¥% (2.10) — DS =F Vio (ee tee Co ye = (08 ) + (4.12) = (1.9 ) + (5.13) = (2.10) + (6.14) = (3.11) + (7.15) (0.2) = (0.4) + (26) (1.3) = (1.5) + (8.7) 4 (cq + 2¢,) = (0.2) 4 (ce, — 2e3) = (1.3) G) = (0.8) — (4.12) cs) = §| (1.9) — (5.18) | — [ (8.11) — (7.15) | eos 45° s.) = ${ (1.9) — (5.18) ] + [ (8.11) — (7.15) ]{ cos 45° s;) = (2.10) — (6.14) 8e, = (0.4) — (2.6) 8s, = (1.5) — (3.7) Ou 126 A NEW METHOD OF DETERMINING eM (hse )) = (69) se [ co) oa 4. (ee) = [( 4) — Gs) | cos 22°.5 + | (3) — G's) | cos 67°.5 4 (¢, + 65) = (8) —| (2s) — (a's) | co 45° 4 (¢;— 5) = | (4) — Gis) | sin 22°.5 — [ (3) — Gy) | sin 67°.5 4(s +s) = [ ( 1)+ (7s) | sin 22°.5 + [ -+ (5) | sin 67°.5 4 (s,— 8) = | (2s) + (tx) | cos 45° + (44s) J(Gh ae Ss) = || (Ce ar Gd cos 22°.5 — LGD “+ (+5) | cos 67°.5 4 (s:—83) = [ (Py) + Gir) | cos 45° — (ay) The values of c¢,, s, must satisfy the equation (c) (s) A, or A;, = 4+ ¢,cosg + ¢ cos 2g + ete. + s,sing + s,sin 2g + ete. (i) 7 answering to? in b,, and x being any one of the numbers, from 0 to 15 inclusive, > (e) into which the circumference is divided. We use ¢,, s, as abbreviated forms of C,,,, {s) . é : ae ae A (c) (c) () C,,, ete. Having found the values of ¢,, s, from the 16 different values of A), Aj, A,, (c) (8) (ec) (8) a : A,, Aj.-...0\» Ay Ay both forse ( =) and wa* ( as we have the values of these func- tions given by the equation ay (¢) (s) Re ; eels) (c) Re E ; = 155 (C,,+ 8,,) cos | (tv) 9g 1H’ | F322 (C,,+ S,,) sin| (FF v)g—1# ac WE Ig The yalues of the most important quantities from the eccentric anomaly £ to ¢,, : a o [a3 : : 2 s,, needed in the expansion of (G) and ja? (“) , are given in the following tables, first for u (“) , and then for ua? ae when not common to both. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 127 Values of Quantities in the Development of « ¢) and ua(“). : g | E | H+W | F+W, | F—P F o / ” iS / ut C / uy 12) / Ad Cc / ad ( 0) 0 0 0.0 266 4 39.5 266 15 38.0 266 21 17.2 309 24 44.9 (G2) 24 24 4.2 290 28 43.7 290 39 42.2 290) PSs has 23 11 34:8 (2 48 26 37.2 314 31 16.7 314 42 15.2 313 40 58.4 46 44 25.4 ( 3) 71 52 24.9 Sa vO) et s38h 7B 229 336 53 39.3 69 57 6.3 ( 4) 94 35 14.0 } 0 39 53.5 0 50 52.0 309 41 1:3 92 44 28.3 ( 5) 116 36 51.7 22 41 31.2 92) 52 29.7 2159 7:8 hora esas ( 6) 138 4 29.4 44 9 8.9 4420 1.4 ] 43 47 3.8 136 50 30.8 ( 7) 159 8 19.6 65 12 59-1 65 23 57.6 | 65 8 48.4 158 12 15.4 ( 8) 180 0 0.0 ly eS REY 86 15 38.0 86 13 41.4 UTD) ies S24: ( 9) 200 51 40.4 106 56 19.9 107 7 18.4 107 15 14.8 200 18 41.8 (10) 221 55 30.6 128 0 10.1 128 ll 8.6 128 28 47.5 221 32 14.5 (11) 243 23 3.3 149 27 47.8 149 38 46.3 MSO ash PALS 243 11 54.6 (12) 265 24 46.0 W129" 2535 171 40 24.0 172 23 51.4 265 27 18.4 (13) 288 1 30.1 194 12 14.6 194 23 V3.1 195 17 19.4 288 20 46.4 (14) 311 33 22.8 Dis Smee Die49! 108 218 43 0.9 311 46 27.9 (15) } 335 35 55.8 24] 40 35.3 241 51 33.8 242 28 57.5 335 32 24.5 s 1613 47 17.9 yy | | 1433 47 18.6 g Log. f. | y x Q Thor. g. |) Loe.’ ©. fo] ad O / ” ( 0) 0.612427 | —.001251 — 12.2 | 359 24 32.0 0.611176 0.706582 (1) 0.612078 —.000860 +4315 | 23 18 46.3 0.611218 0.706349 ( 2) 0.609315 —.000081 598.0 46 54 23.4 0.609234 0.705934 ( 5) 0.605242 | -+.000981 +390.0 10 3 36.3 0.606233 0.704403 ( 4) 0.601312 +-.001292 — 58.6 92 43 29.7 0.602604 0.703241 ( 5) 0.598569 | +.000846 | —476.9. | 114 54 37.9 0.599415 0.702241 ( 6) 0.597310 +.000091 —626.7 136 40) 4.1 0.597401 0.701493 ( 7) 0.597194 | —.000956 —435.1 158 5) 0:3 0.596238 0.701011 ( 8) 0.597621 | —.001322 — 165.7 179 16 52.7 0.596299 0.700788 (ee) 0.598109 ——.000997T | 408.7 | 200 25 30.5 0.597112 0.700494 (10) 0.598532 —.000152 618.1 ) 221 42 32.6 0.598380 0.700021 (11) 0.599177 | +.000777 | +4966 | 243 2011.2 | 0.599954 0.699872 (12) 0.600584 +.001978 | + 96.7 | 265 9855.1 | 0.601862 0.700504 (13) 0.603163 +.001032 | —363.1 | 288 14 43.3 | 0.604195 0.702020 (14) 0.606734 +.000148 | -600.1 | 311 36 27.8 0.606882 0.704038 (15) 0.610302 —.000825 | . —452.4 | 335 24 52.1 0.609477 | 0.705810 s “EEE DYE | MEE Th vs | heya? 4.823838 | 5.622201 x! 4.893834 — 2 aii) | 1433 47 17.9 4.823842 | 5.622200 Values of Quantities in the Development of u(“) and ua*(“) : A NEW METHOD OF DETERMINING g Xx uM Log. 0. Loe.ta. A | a. Log. N. =a Oo ; / ad at a y) ai | = lie = ( 0) b3i23 4503 T 57.83 7.063818 9.701484 0.502902 | 9.695669 @ w) 53 26 41.3 T 57.78 7.063792 9.701945 0.503437 9.695880 ( 2) 53 14 15.6 7 59.97 7.065778 9.699988 0.501173 9.695892 ( 8) 52 54 33.7 8 3.30 7.068781 9.696876 0.497594 9.695837 4 52 28 55.6 Siso 7.072405 9.692804 0.492951 9.695616 t 5) 52 6 31.2 8 10.95 7.075601 9.689226 0.488907 9.695421 ( 6) Si Gs} Ai 8 13.23 7.077613 9.687169 0.486597 9.695400 (C0) 51 46 50.0 8 14.55 T.0TST74 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 T.0TTI13 9.688321 0.487889 9.696120 (10) 52 18 36.9 8 12.12 7.076635 9.691160 0.491089 9.696905 1 52 36 21.2 8 10.34 7.075061 9.693986 0.494294 9.697532 tis) 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) 53nd) 1225 8 2.58 7.068133 9.698559 0.499527 9.696354 (15) 53 13 54.4 TS osT0 7.065534 9.699932 0.501109 9.695743 = te 3.956815 77569096 a (a 3.956845 17.569088 = ; (0) ; (1) ; (1) (0) (1) (2) g Log. + Cy Log. sey Log. $81 Log. 1 Log. De Log. 6, One = 2 2 2 2 = Dita 2 ( 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 ( 2) 8.792802 4.98551n 6.16173 0.331867 9.746235 9.326571 ( 3) 8.79275 6.05070n 5.97267 0.831369 9.742375 9.319511 ( 4) 8.192526 6.16734n 5.14693n | 0.380730 9.737346 9.510298 (a5) 8.792331 5.982190 6.05562n 0.330182 9.732946 9.302224 ( 6) 8.792310 4.93934 6.17378n 0.329872 9.730425 9.297590 ( 7) 8.792340 6.03383 6.016142 0.329707 9.729076 9.295111 ( 8) 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 0.830477 9.735322 9.306586 @ap 8.794442 5.948120 6.07618 0.330914 9.738805 9.312970 (12) } 8.794541 6.164660 5.36611 0.331246 9.741407 9.317738 (13) 8.794051 6.07296n 5.942027 0.331460 9.743073 9.320808 (14) | 8.793264 5.23742n 6.16200n 0.331637 9.744461 9.323327 (15) 8.792653 | 5.97789 6.041547 0.831858 9.746165 9.826445 z 2.647715 7.912926 74.508222 uf 2.647721 77.912945 74.508268 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. Values of Quantities in the Development of u(“) and ua?(“ ) : 129 | Coinale (4) (5) (6) CHG) pene CS) a |G) g | Log.b, | Log.b, | Log.b, | Log. by | Log. 6, | Log. b, | Log. 8; ( 0) 8.954999 8.60017 8.2570 7.9215 7.5915 7.2654 6.9426 6-1) 8.956515 8.60214 8.2594 7.9244 T5947 7.2691 6.9468 i) 8.950082 8.59373 8.2490 7.9120 7.5804 7.2528 6.9286 ( 3) 8.939865 8.58036 8.2326 7.8926 7.5578 7.2271 6.8997 ( 4) 8.926521 8.56292 8.2110 7.8668 7.5280 7.1932 6.8617 @5) 8.914818 8.54760 8.1921 7.8444 7.5020 7.1636 6.8285 ( 6) 8.908100 8.53882 8.1812 7.8314 7.4870 7.1466 6.8094 ( 7) 8.904506 8.53411 8.1754 7.8244 7.4789 7.1373 6.7991 ( 8) 8.906000 8.53606 8.1778 7.8273 7.4822 7.1411 6.8033 (9) 8.911861 8.54373 8.1872 7.8386 7.4953 7.1561 6.8201 (10) 8.921142 8.55588 8.2024 7.8565 7.5160 7.1796 6.8464 (11) 8.930392 8.56797 8.2172 7.8742 7.5367 7.2031 6.8728 (12) 8.937298 8.57701 8.2285 7.8875 7.5520 7.2205 6.8923 (13) 8.941742 8.58283 8.2355 7.8960 7.5618 7.2317 6.9048 (14) 8.945388 8.58760 8.2415 7.9030 7.5700 7.2410 6.9152 (15) 8.949898 8.59349 8.2488 7.9117 7.5800 7.2524 . 6.9280 ay 71.449530 68.55219 65.7484 63.0060 60.3071 57.6402 54.9995 yy 71.449597 68.55223 65.7482 63.0063 60.3072 57.6404 54.9998 3 7 (1) . ane Secon (1) | (2) (3) g |Log.t+N | Log. $ Ca Log. 4 83 Log. 4 bs _ Log. b; | Log. Be Log. Oy 8.183917 5.49374 3.738372 0.280319 0.417421 0.200612 9.961097 8.184550 5.25307 5.29293 0.281000 0.418474 0.202090 9.963016 8.184586 4.24928n 5.42550 0.278120 0.414013 0.195824 9.954877 8.184421 5.31430n 5.23627 0.273612 0.406981 0.185917 9.941987 8.183758 5.43028n 4.409877 0.267827 0.397890 0.173060 9.925223 8.183173 5.24454n 5.3179%n 0.262860 0.390004 0.161858 9.910585 8.183110 4.20163 5.43607n 0.260054 0.385513 0.155458 9.902210 8.183200 5.29621 5.27852n 0.258559 0.383116 0.152039 9.897732 8.183797 5.43847 3.83805n 0.259184 0.384116 0.153464 9.899598 8.185270 5.31804 5.25490 0.261621 0.388024 0.159038 9.906900 8.187625 4.49962 5.43747 0.265530 394254 0.167901 9.918485 8.189506 5.216817 5.34487 0.269488 0.400515 | 0.176758 9.930076 8.189803 5.43364n 4.63509 0.272484 0.405223 0.183435 9.938754 8.188333 5.340477 5.20953n 0.274429 0.408267 | 0.187732 9.944350 8.185972 4.50257n 5.42714n 0.276036 0.410773 0.191265 9.948948 8.184139 5.24121 5.30466n 0.278037 0.413885 | 0.195644 9.954643 65.482568 2.159554 3.209203 | 1.421019 | 79.449192 65.482592 2.159606 3.209266 1.421076 79.449289 A. P. S.— VOL. XIX. Q. 150 I AR RR OO at — tals A NEW METHOD OF DETERMINING of a\3 Values of Quantities in the Development of « (“) and ua*(“) : 9.70884 9.71121 9.70116 9.68524 9.66450 9.64638 9.63600 9.63043 9.63276 9.64181 9.65617 9.67052 9.68125 9.68816 9.69382 9.70087 ea ue 37462 (5) (6) (7 (8) (9) | Log. b, Log. 6; Log. bs Log. bs Log. bs Di 2 2 2. 2 9.4484 9.1822 8.9118 8.6383 8.3621 9.4512 9.1854 8.9155 | 8.6423 8.3665 9.4393 | 9.1716 8.8998 | 8.6247 8.3471 9.4203 9.1496 8.8747 | 8.5965 8.3158 9.3955 9.1207 8.8418 | 8.5595 8.2747 9.3739 9.0956 8.8131 8.5273 8.2389 9.3614 | 9.0813 8.7968 8.5089 8.2184 9.3549 9.0735 8.7880 8.4991 8.2077 9.3576 9.0766 8.7914 8.5030 8.2119 9.3684 9.0893 8.8058 8.5191 8.2298 9.3856 9.1093 8.8287 8.5449 8.2585 9.4028 9.1292 8.8515 8.5705 8.2868 9.4156 9.1440 8.8684 8.5893 8.3078 9.4937 9.1537 8.8791 8.6015 8.3213 9.4305 9.1614 8.8882 8.6118 8.3329 9.4389 9.1711 8.8992 8.6240 8.3464 75.2339 73.0471 70.8269 68.5804 66.3134 75.9341 73.0474 70.8269 | 68.5803 66.3132 g | Log. k, | Log. k, | Log. k,; Log. &, | Log. k, | Log. &; | Log. k, | Log. &, ( 0) 8.824187 8.54492 8.12562 7.750420 esQo DON (e0D23 6.7168 6.4105 ( 1) 8.824302 8.5443 8.12588 7.751220 7.39678 | 7.0540 6.7190 6.4054 ( 2) 8.823605 8.53875 8.11916 7.742693 7.38634 | 7.0416 6.7046 6.3714 ( 3) 8.822665 8.53172 8.10982 | 7.730361 TeSTO9Ie |) 720232 6.6832 6.3298 ( 4) 8.821701 2E 8.09963 7.716100 7.35261 7.0007 6.6565 6.2932 ( 5) 8.821143 8.09246 7.705215 7.33807 6.9826 6.6849 6.2764 ( 6) 8.821183 8.09009 7.700585 7.33130 6.9737 6.6239 6.2809 (Gi) 8.821397 8.08981 7.699023 7.32855 6.9698 6.6187 6.2913 ( 8) 8.821810 8.09164 7.701551 7.33151 6.9732 6.6226 6.3027 ( 9) 8.822444 8.09567 7.707159 7.33895 6.9824 6.6337 6.3093 (10) 8.823323 8.10077 7.715298 7.35002 6.9965 6.6506 6.3129 (11) 8.824009 8.10550 7.723069 7.36070 7.0100 6.6669 6.3147 (12) 8.824233 8.10915 7.728940 7.386874 7.0202 6.6793 6.3196 (13) 8.824055 8.112338 7.733450 7.387462 7.0274 6.6879 6.3342 (14) 8.825809 8.11622 7.738311 7.88053 7.0345 6.6960 6.3608 (15) 8.823826 8.12113 7.744423 7.88795 7.0433 6.7062 6.3901 Dy 70.583851 | 68.25726 | 64.85258 | 61.793910 | 58.89655 | 56.0927 | 53.3503 | 50.6520 a! 70.583841 68.25722 64.85260 | 61.793920 | 58.89653 56.0926 53.3505 | 50.6512 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 131 Values of Quantities in the Development of u(“) and ua?(“)’. Log. k, Loe. ky KK K; DADA OT i WY | ! / ! ! ! / / 6.0606 5.7378 || —0.6 —04 — 03 03 —03 03: =0S. =—.0:3 6:0636 5.7413 || +203 + 12.9 i 4. SEV iiail S065 9 a 10 eI =e F833 6.0454 5.7212 |} +27.9 117.8 15.6 Nee 114.6 +140 -+13.5 -+12:5 6.0178 5.6904 = 118.4 TE) By ENOL) SOON) =e Oly 1 94 -1 919 -+ 8.8 5.9830 5.6515 — 2.8 1.8 — 1.6 — 1.5 13) — 1.5 —l]5 —1.5 5.9541 5.6191 OT 145 aT 12 es.) ee tlt — 11) 5.9391 5.6019 OK) 19'0) = 687 losis 14.9 =114°5 13.7 POSIG! 55984 — | —20.7 13:9 106010!) 08 if) est) esol DISD) i — OT 0.5 ae et 04. — "0:4 — 0.3 =—023 =~ 0.3 b.9p12) 5.61051 «|| —+-19:3 119-3 SEN) Eye SE ys SE yt E90 + 8.2 Didier 5.6405 || --29.1 118.6 See Soy ey See eal) Pi41 13.3 5.9959 5.6656 || 123.4 114.9 P32 o ees) 107 See See eles 6.0124 5.6849 |} + 4.5 + 2.8 ae pis Son +94 93 - 93 = 21 6.0251 5.6968 —17.0 10.8 — 9.5 — 8.9 = $ehts) — 8.7 8.6 8.4 6.03841 5.7083 —28.1 17.8 —15.7 — 147 —14.3 —13.9 —13:6 —I3:0 6.0468 5.7224 |) —21.0 Mise —11.8 =X) 11026 — 0.2 — 98 — 9.0 etre 45.3459 | — 5 ee ee oe Se =a 45.3441 || 0 1 Qs = il Log. k lLog.k, Log.k Log. k Log. k, Log.k; Log. k, Log. k, fo) ( 0) 8.465272 8.60289 8.38621 8.14674 7.89481 7.6341 7.3679 7.0975 ( 1) 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 T.05TT ( 4) | 8.450550 8.58006 8.35509 8.10719 7.84645 T.5TT4 7.3026 7.0237 ( 5) 8.445362 8.57214 8.34391 8.09259 7.82837 7.5559 7.2776 6.9950 ( 6) 8.443224 8.56872 8.33868 8.08543 1.81922 7.5446 7.2645 6.9800 ( 7) | 8.442508 8.56750 8.33651 8.08224 7.81495 7.5399 7.2581 6.9726 ( 8) | 8.444020 8.56954 8.33902 8.08521 7.81840 7.5433 7.2623 6.9771 ( 9) 8.444679 8.57452 8.34564 8.09354 7.82847 7.5501 7.2760 6.9925 (10) | 8.453274 8.58206 8.35573 8.10632 7.84401 T.5734 7.2971 7.0165 (11) 8.458368 8.58906 8.36522 8.11851 7.85895 7.5912 7.3176 7.0400 (12 8.461465 8.59345 8.37153 8.12680 7.86927 7.6036 7.3320 7.0564 (13) 8.461922 8.59532 8.37468 8.15126 T.87506 7.6105 7.3405 7.0660 (14) 8.461886 8.59651 S.8TT04 8.13471 T.ST9IOT 7.6163 7.3472 7.0739 (15) 8.462852 8.59905 8.38088 8.13992 7.88616 7.6242 7.3564 7.0845 z 68.69172 66.90360 64.93175 —«62.85706 = 60.7165 58.5297 56.3095 a! 58.5300 56.3096 68.69184 66.90364 64.93185 62.85719 60.7166 132 A NEW METHOD OF DETERMINING 3 Values of Quantities in the Development of u(4) and ua°(“) : g | Log.% Log. || HK KK EK WQ=9)-% 2(Q@-9)-K 3(Q-9)=K | tA ran : " ° = : Oo / ur (0) | 68940 65478 || —01 —o1 —o1 ||. 359 25.1 358 49.5 358 13 55.0 (1)a) 6:8980y — Gasmeueei a 9 hae Be ale omaely 1 24.6 9 14 57.0 (2) | 68092 6.5817 ae, 6.0) — S60 1 265 3 31.0 5 OF 344 (-3)| 6.7795. 64988. || 489 139 13:9 2 15.3 4 55.5 7 30 33.9 CAS 624d Ses eta —- 016: ~ =20l6val| sad, aes 5 28.8 8 12 24 (5) |] 6.7093 6.4209 || —4.7 ly) al) 2 47.3 5 13:8 i 26 316 G61 G:600r. = Gabe || eRe 269 = ago || weray wag 3 39.1 5 16 51.0 i) |eessr. “6iggoza ieee” = 43 es | oie 1 23.2 1 56 39.0 (8) | 6.6887 63976 || 09 —0.9 09 |) 359 17.6 358 843 Shee 5n Gee (9) | 6.7058 6.4165 H40 =14.0. ~ 240) || 357. 969 Sha aaa, . gbaesoamerais (10) | 6.7327 «6.4463 ee. bed 6.1 I) aS56) S| OS5R. 6:5. = 349 SINS (11) | 6.7589 6.4752 || +5.0 +450 +50 || 355 968 351 255 347 17 294 (2) | 6.7773 6.4958 || +1.0 +1.0 +1.0 || 355 244 350 55.0 346 94 128 (is); | “eises 6080. |) ean 23.5 25 I bee cies 351 40.2 347 23 40.2 (14) | 67976 65187 | —60 —60 —60 || 357 46 353 30.7 350 5 38 (15) |. 68098. GSR eee 24 — as 358 15.9 356 3.1 353 56 22.5 Y | 54.0630 51.7961 || 0 0 0 || 1793 47.8 1781 22 3.6 y | 54.0698 51.7957 | + 8 (4.3 + 8 || 1488 47.3 1421 21 591 g 4(Q—g)—K,5( Q—g)—4, 6 Q—9)— K,, T( Q—g) — 4; 8( Q—g) — 4, 9( Q - g) — Ky 7 fo} / Oo / fo) / oO / °o / ( 0) Sia tsi 357 3.0 356 27.5 35D) 5221 Bi) UAai 354 | 4g C1) 3 Se) 3 53.2 4 49.5 5 31.8 6 21.1 7 10.5 ( 2 1 JRA ies: ale alg! 133. 733 Tse S223. 16 Site ( 3) 10 4.4 12 38.3 Ilse pA) 17 46.0 20 19.8 PR ash ( 4) IO), Gye) 13 39.0 16 22.5 19 6.0 DPA GES 94 33-0 ( 5) 9 50.6 19 1520) 14 39.4 ili(e Bee) 19 98.4 91 52.8 ( 6) 6 55.9 8 35.6 ig) ilsy8? TW 754.9 133 By bS HY sae ( 1 Oe Bx) Oye Lys) 3 40.1 4 14.7 4 49.3 by 2B!) ( 8) 357 8.0 356 24.9 Boy Ui 354 58.6 Byyf 16es) 353 «329.4 ( 9) Byanl Byles} 349° QT 34% 23:6 345 19.5 By hey tay! Ba eS: (10) 346 34.9 Bits Wie SAO Oe 336 43.7 Seo Ost) 330 9.6 (a5) 343 @©68.5 338 58.9 334 49.3 Eis | 326 30.1 322 20.5 (12 Buhl GBYY) Bei, ADT Seq Gls! 328 20.0 323 48.9 319 17.9 (13) 343 et BBis) 532.33 304 36.9 Say ls 326 6.1 321 50.7 (14) 346 40.5 342 16.6 839 52.7 336 28.8 333 4.9 329 41.1 (15) 351 50.4 349 44.9 a4i 394 345 33.8 343 28.2 341 97 Di 1144 65 pall 13884 6.0 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 133 4 a In the expansion of (a): (e) (c) (s) (c) (s) (ce) (3) “(c) (s) oa ea 0S 1 1 A, ae A, A, A, 4 a 4 Mt ur ” A ur A iad ( 0) 13.13109 6.9027 —.0701 |+-2.6281 —.0539 +1.10745 —.03418 -++.4889 —.0201 (by 13.13458 6.8933 +-.0571 2.6294 +.0647 1.10917 +-.04356 A901 +-.0262 ( 2) 3.11352 6.8033 --.1712 2.5849 --.1588 1.08348 4-.10356 AT51 +.0615 ( 3 13.08513 6.6912 -|-.2633 2.5254 -|:2176 , 1.04890 +-.15827 45538 4-.0809 ( 4) 13.05615 6.5922 -+.3192 2.4646 -+-.2364 1.01333 +-.14604 4353 -.0840 (5) 13.03939 6.5457 +-.3187 2.4959 2150 } 0.99004 +-.12935 4224 -+-.0733 ( 6) 13.04058 6.5584 +.2479 2.4172 +.1543 | 0.98367 +-.09095 4190 + .0509 (Co) 13.04700 6.5880 -+.1067 2.4198 + .0585 0.98375 +-.03339 4190 -+-.0184 ( 8) 13.05942 6.6190 —.0816 | 2.4317 —.0606 0.98937 —.03712 | 4218 —.0211 ( 9) 13.07850 6.6377 —.2779 2.4464 —1863 | 0.99667 —.11189 | 4249 —.0633 (10) 13.10500 6.6498 —.4389 2.4645 —.2979 1.00593 —.18002 4287 —.1023 (11) 13.12573 6.6578 — .5301 2.4816 —.3742 1.01487 —.22886 4322 —.1310 (12) 13.13248 6.6727 —.5359 2.4991 —.3995 1.02497 —.24789 4373 —.1431 (13) 15.12612 6.7090 —.4658 2.5224 —.3693 1.03984 —.23254 4463 —.1354 (14) 13.11967 6.7727 —.3458 1225559) — 2907 1.06142 —.18555 | -4600 —.1090 (15) 13.12018 6.8478 —.2074 2.5954 —.1791 1.08668 —.11537 | A760 —.0683 sy LOL75791 | 5315708) =.7340 | 422010460 —.5531 8.26962 —.344910| 3.5661 —.1992 Pe 104.75663 53.5705 —.7354 | -+20.0463 —.5531 8.26992 —.34409 | +3.5662 —.1992 (c) (s) () Gs) (e) Om oe (3) @ ee A, 5 6 A, A, 7 A, A, 9 “Ay a a ie au na ” a) i ur jiu Ta uM at ate A ( 0) +.2217 —.0114 | --.1023 —.0063 + -0505 —.0036 +-.0226 —.0019 --.0107 —.0010 lh) 2293 +.0151 1027 -+-.0085 0498 --.0048 0226 +-.0025 0108 +-.0014 @2) -2138 -+-.0350 0978 +-.0194 0451 +-.0105 0211 4.0057 .0099 +-.0030 ( 3) 2028 +.0454 0916 -+.0249 -O401 +-.0128 0192 +.0071 | .0089 +-.0038 ( 4) 1916 +-.0465 0856 -+-.0252 -0365 -+-.0126 0176 +-.0070 | .0080 4-.00387 ( 5) 1848 +-.0401 .0821 +-.0215 0356 +-.0109 | 0167 +.0059 | .0076 -+-.0030 ( 6) 1832 +.0277 -O815 --.0147 03868 +.0078 | .0166 +-.0040 076 4-.0021 CD 1833 +.0099 0816 +-.0052 0384 -+-.0028 0168 +-.0014 OTT +-.0007 ( 8) 1847 —.0116 0823 —.0062 0394 —.0035 0169 —.0017 0078 —.0009 ( 9) 1860 —.0346 0826 —.0185 0388 —.0102 0168 —.0051 .00TT —.0026 (10) 1870 —.0561 0827 —.0301 0372 —.0160 0166 —.0083 .00TS —.0048 (11) 1880 —.0722 0827 —.0389 03854 —.0199 0163 —.0108 0072 —.0056 (12) 1904 —.0793 0837 —.0429 0350 —.0216 0163 —.0120 | 0072 —.0062 (13) 1956 —.0T56 -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 By +1.5765 —1105 | +.1077 —.0598 | +.3219 —.0318 | +.1467 —.0168 | -+.0674 —.0087 yw 1.5768 —.1106 +.7078 —.0598 +.8218 —.0318 -+.1467 —.0168 | +.0674 —.0086 134 A NEW METHOD OF DETERMINING 5 pi fa Ne In the expansion of ar Gi . (c) (c) (s) (c) (s) (c) () =| (c) (s) g | 0 1 A, to A, | 3 3 | aA A, = ey wi, yy | ”" 7) ” ” | ” ” ( 0) 23.3520 | +32.0569 —0.3301 | +19.4613 —0.4009 | +11.2092 —0.3464 | + 6.269 —0.258 (as) 23.4045 32.1423 +0.4199 | 19.5273 +0.5272 | 11.2569 0.4603 | 6.300 --0.347 (2) | 93.9107 31.7192 +1.0033 | 19.1618 +1.2486 | 10.9731 ++1.0737 | 6.096 0.802 ( 3) 99.9239 | 31.1043 --1.3503 18.6375 1.6470 | 10.5748 +-1.4097 5.813 +-1.041 ( 4) 29.5137 30.3821 +1.4503 | 18.0367 +-1.7240 | 10.1342 -+1.4580 | 5.516 -+-1.063 (5) 22.3056 29.8387 1.9952 | 17.5937 +1.5122 9.8190 +1.2644 | 51310 =--0:912: ( 6) 22.1960 29.6180 -+0.9110 | 17.4156 +1.0505 9.6988 0.8734 | 5.239 +-0.626 ( 7) 22.1595 995473 +-0:3342 | 17.3564 -+-0.3782 | 9.6618 +0.3118 5.219 +.0.222 ( 8) 22.2368 29.6861 —0.3713 | 1(.4552 04367 | 9.7264 —0.3654 5.959) —=0:204 ( 9) 29,4949 30.0100 —1.1187 17.6808 —1.3068 9.8617 —1.0915 583k —Oli86 (10) 22.7157 30.5036 —1.8033 | 18.0224 —2.1155 | 10.0630 —1.7762 5.436 —1.285 (11) 22.9837 30.9679 —2.3042 18-3471 —2257150 | 110!9558' ——2:-9962) | 5.536 —1.667 (12) 23.1482 31.2707 —2.4810 18.5835 —2.9616 LO4121 — 25144: 5.627 —1.839 (13) Pay RA) 31.4193 —2.3026 18.7500 —2.7837 10.5580 —2.3763 5.139 —1.148 (14) | 23.1706 31.5386 —l1.8212 18.9291 —2.2155 10.7412 —1.9027 5.895 —1.409 (15) 23.2229, 31.7564 —1.1097 OSES —— es Gl6 | 10.9764 —1.1843 6.091 — .882 oc gl 182.6038 946.7758 —3.4493 147.0656 —4.1071 82.9580 —3.5000 +45.337 —2.564 pa 182.5968 246.7862 —3.4356 | 147.0719 —4.1125 82.9644 —3.4985 +45.339 —2.563 (c) (s) (c) (s) (c) (s) (c) (s) (c) (s) g 5 A, 6 A, 7 7 8 8 9 9 eA 4 ty liad AA yr 4} ta A (Ke iad ( 0) -+3.440 —O.177 | +1.863 —115 +1.000 —.072 +.532 —.044 282, —.027 (@m) 3.458 —+.0.240 1.874 -+.157 1.005 —--.098 -085 —--.060 2983 2036 ( 2) 3.318 10.550 1 7ete™ O56 944 1.221 AQT 134 260. +.076 (3 3.130 0.706 1.660 -+.453 868 +.279 450 -+-.167 231 —-.098 ( 4) 293. 0. ila 1.540 -+.453 197 +.276 | .409 +.164 .208 +-.095 ( 5) 2.812 0.606 1.467 —-.381 156 JOBoee SbSiz: 159 196 +.078 ( 6) re SE DIE; 1.448 -++.260 148 +.157 300 —+.092 195 +.053 Co 2.766 0.146 1.446 +.091 750 +.055 386 1.032° 197 +.019 ( 8) 9.789 —0.175 1.459 —.110 157 —.053 389 —.039 199 —.023 ( 9) 2.824 —0.522 1474 == 399) | .160 —.199 Sts) STILT 197 —.06T (10) 2.870 —0.855 1.491 —.540 | H1D9 ab sis) <= liso) S93 .——ah (11) DAM tay TES) 1.505 —.705 sory See sos) = 2! 181 —144 (12 2.963 —1.235 E5298 0So Gin —— Ass 382 —.280 188 —.162 (13) Sts —— eng 1.582 —.753 803 —.457 404 — 9712 201 —.158 (14) 3.164 —0.957 1.670 —.615 86. —= sis 446 —.227 isles (15) | 3.312 —0.604 1.775 —391 949 — 943 495 —.147 5 9r— 0a y 24.253 —1.798 12.780 —1.094 +6.639 —.648 | 3.493 —392 - | 1.752 —939 Dy, 94.959 1.799, 12.783 —1.095 6.641 —.660 +3.415 —.395 ile Gl OP} (c) ae The Quantities $C, , (s) (c) : Ci. > 4S;, () THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 155 (8) : , 2S; , arranged for Quadrature in the Expansion of E=0 a = Na) ¢=4) 95 1—=16 (c) " : " us | ” - ” ” sl 40 |-+-3[209.51454]} +-53.571 +20.046 | +8.26978 13.566 | —+-1.576 +107 ws (c) i | | Sie — 135 —DosnNe emegaerde|) | 199) || = tNOn| | 060 l | a EO) | : SS | OF +-.25653 548 +.382 +.99949 +129 +.071 +.038 (s) ; | Si: +1.706 1.273 +.78997 +456 +. .253 +138 y=1} (8) i == I02i —.122 —.046 —01129 | +.002 +..005 +.006 (3) il +.022 +.017 +.00807 -+.003 001 000 f _ (ce) : > | 42 +-.00463 = 251 +.096| +.05847| -+.038| +.094] -+.013 (s) : i2 sili) —.008 +..01835 E01 +.007 +..004 v= (8) | 12 -+.12279 1 198 +-.080 +.04667 -..026 015 -+.001 (c) |S.» 1.065 +048 +.03063 +.018 +.010 +-.006 l 7 : en) i ; | Cis 4.03070 1.020 +.007 | +.00662 4.005 +002 4.001 | (s) | Sis —.003 1.002 | +.00216 +.002 1.001 1.001 i (8) | OF +..05945 +041 +-.023 --.01319 --.006 -+-.0038 +.002 (ce) . Sis 000 —.001 —.00217 —.002 —.001 —.001 l = Ss stare (ie 1(C) ; 4 | Ge, 00037 +.001 1.00030 (s) se ba 000 +.00052 i, (s) 4 00055 000 +-,00076 (c) [Si 7 —.001 —.00103 156 (c) The Quantities $C,, ,4C,, ,4 A NEW METHOD OF DETERMINING (s) (c) (8) S,, ,348,, , arranged for Quadrature, in the Expansion of m(() i=0 t=1 (=2 (=38 (=4)7=5/t=6 (=TI=8li=9 if (c) tl Wt ‘7 = iad B | uy il is ” ur | Co -+-3[ 364.6002] +246.7810) +-147.068) +-82.9613|+-45.338|4 24.256|+-12.781/+ 6.640|+-3.419)/ 11.751 v0; | | | (ce) | CF —3.4388] —4.110| —3.4992| —9.562| —1.729| —1.095) —.654) —.399| —.998 l | { (c) | Cia +4.3500| 44.6277) 3.873] +2.8862| 11.956] 11.953] +.771] +461) +.970] 4.154 | | | | (s) | Si: +17.8438] +9.373| +-7.9505] +5.816] /+3.910| +-2.488|+1.514| +.898] +.521 y=1} (s) | |G —1.8014) —1.1511) —.801 —.3643) —.106, +.017) +.062| +.078| 4.058) +.049 | (c) Sin = 1015 +.104, +.0731) +.043 +.024) +.011| —.008) +.003) +.001 { (ce) | | Cir —.2566] +4.0899} 4.994] +4.3888| 1.384) +1.397|/ +.959] +.193] +134) +086 IO) | Ls +-1010} +.296] +.3297/ +.302) +.239) -+.173) +.116} +.078] +-.047 a 5 | | Cie +1.1803) 41.1209) +.883] +.6281) 1.418] +.266| 1.162) +.093] +.058) +.031 (c) | |S: +.3367/ +.400} +.3459) 1.955) +.170) +.106) +.065) +.034) —.018 f | | | Gs +1118) +.1140/ 4.099] 4.0809] +.066| +.049] -+.035|. +.024] +.013/ +.012 | | | (s) | | | | | Sis —.0170 .000| +.0059} +.019} +.015] +.015) +.015| +.013] +.008 v=3} \ | | Cos | +5132) 4.6602) 4.317] +.2097] +.130| +.076| -++.043| +.020| 4.019] +.002 | (e) | | | Sis | —.0138} —.030} —.0344] —.932) —.027/ —.020| —.005) —.010] —.005 f _(e) . ce. - Y | Cis +.0177) +.0085] +.003} +.0028/ +.009| 1.002 000] +.001; .000] +.001 | 6) Sis +.0117) — +.005) +.0061] +.005! +.006} -+.004] +.004} +.001] +.001 yv—4 a | he +0182} +.0172} +016, +.0134) +.010) 4.006} +.005] +.003) —.009| +.001 (c) | Sis —.0109} —.022} —.0182] —.016) —.012/ —.008] +.002] —.oo3} —.oo1 | THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 137 c) eA (s) . ie The quantities C,,, C,,, etc., of the preceding tables have been divided by 2 to save division after quadrature. To check the values of these coeflicients we will take the point corresponding to g = 22°.5, using the equation (c) (s) : in One on | COS %, cos 2 ete. 47,01 A 54+ C, g+C. 2g + et + S, sing + S, sin 29 +- etc. noting that the tables give one-half of the values of these quantities. Thus we have all — i= 5 = ¥ (c) " " | (ec) " " Co == 153.571 120.046 || 2319 = — 0.735 — 0.553 (c) (s) 141 = + 1.013 HOG vie este | 1.306 a ae! (s) (c) Ge — .094 == 20382 Sit = + -040 - .031 (ce) (s) 12 =+ .363 1.135 fie = 240 — 004 ~{*) (c) Co =-+ 181 114 19 = + .092 - .070 (c) (s) hs + 015 005 S,3 = -— .005 + 004 (s) (c) Gs =+ 077 043 So = 0 — 001 (¢) (s) Y 1,4 u i| Si4 = 0 (s) |} (c) 1,4 = 0 eA u a 7 =a 4 4; wan 5 / = Ta a 3 95.126 121.018 > = 0.458 L 0.521 +> == + 6.891 2.627 |} 25 = + 0.057 - 0.065 (c) | (s) iA = 2. 6.898 | 2.629 A | 0.057 + 0.065 In this way we check the values of these quantities for all values of 7, in case of both « (), and ua?(“). Applying to the coefficients of the two preceding tables the formula n _{c) (s) a z (8) (c) L P ean (5) = 322(C,, S;,) cos LC Fv)g— iB’ | + 423(C,, + S;,,) sin Lc Fv)g—ik’ | noting that } has been applied, we have the values of u(“), ua (“) that follow : AS b. S—VOL. xix. BK: 38 A NEW METHOD OF DETERMINING g LE’ cos sin cos Ld A 4‘ tv) +4/209.51455] +1 364.6002] 1—0 0.25653 —0.25027 +-4.5500 (i) | +0.00463 +0.12279 — (0.2566 a0 | -L0.03070 -L0.05945 0.1113 4—0 | 0.00037 0.00055 +0.0177 2—1 | +0.023 —0(.041 0.1310 1—1 | 0.427 — 0.193 —0.0112 0—1 | —1.158 0.101 S=seeul(ail 1—1 “53.511 -+0.735 + 246.7810 2—] +2.954 —0.144 +12.4716 3 -L0.087 0.063 0.1909 Bee )| +0.016 40.041 +.0.0970 1—2 --0.099 G—2 +0.098 —0.129 — 0.001 ee | —0.891 0.029 —5.500 =F | +-20.046 =.0.553 +147.068 2 +1.656 — 0.063 13.246 12 +0.093 +0.032 +0.590 vt +0.00446 —0.01101 +.0750 1—3 0.04011 —0.07730 +0591 2—3 —0.56048 + 0.00322 —5.0643 3—3 +8.26978 + 0.34414 + 82.9613 4—3 1.01947 —0.01936 10.8367 o9—3 +0.07682 +0.01603 +0.7185 —-s 0.00879 0.01536 0.0868 4 0.003 —0.004 +0.053 Dt! 0.020 —0.044 0.082 3—4 —0.326 —0.005 —3.859 4—4 13.566 +0.199 45.338 5 0.585 —0.001 IEC 6—4 -1.0.055 -L0.008 -L0.687 7—4 +-0.078 2—5 0.005 +0.045 0.0383 35 0.016 —0.025 0.088 4—5 — 0182 —0.007 2.657 H— 5 +1.576 +0.110 494,956 65 -+.0.325 0.004 +5.163 7—5 +-0.0381 +-0.004 +0.567 4—6 0.009 —(.008 +0.079 5 — 6 —0.100 —0.006 —1.717 6 —6 0.707 0.060 +12.781 6 +-0.176 0.005 +3.260 Saete —0.005 0.426 0.018 sin ad —1].8014 +1.1803 +0.5132 0.0182 —0.6464 —1.4577 11.0496 +3.4388 —1.2526 0.7842 +.0.6740 —0.287 —1.283 +0.697 +4.110 —0.905 +0.483 —0.1753 —0.9741 0.2912 13.4992 —0.4375 +.0.2822 40.2441 —0.098 —0.674 +-0.062 +2.562 —0.149 +0.163 0.162 —0.049 —0.095 —0.041 +1.722 —0.006 +.0.436 —6.269 —0.073 11.095 0.050 10.057 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 139 ib oli N@ = We have next to transform the expressions for u(“) and ua? (2 ust given i b ys | t A D5 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 p; = —. Putting m= 4, we find the values of 7, as in the case of 7;. Then we get p, from i —— 4 Ps T.—0s Pt (0) We proceed in this way until we finally have the values of p,. Then we find J,,. or (0) (J, — 1) from (0) 9 it (he Ji) e se: tho [2 -f 1 = 36 4- ele. where /=/’'<, (m) and J We from > (m) (0) Jy) c= Ji é + Pi + Pa- Ps + Ps Ds The details of the computation are as follows : 140 A NEW METHOD OF DETERMINING Computation of the J functions. — ze e Ze 2e 2 de té 4e log. 1 8.38251 8.68354 8.85963 8.98457 9.08148 9.16066 9.22761 9.28560 log. 7; 9.31646 9.01543 1.839384 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, 9.21955 1.91852 1.74243. 1.61749 1.52058 1.44140 1.37445 1.31646 log. 7, — log. p;|| 4.53601 3.93395 3.58177 3.33189 3.13807 2.97971 2.84581 2.72983 Zech —1 = He i) — 31 = 45 = 62 ell 9.91954 1.91847 1.74231 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. 7; | 2.09461 1.79358 1.61749 1.49255 1.39564 1.31646 1.24951 1.19152 Diff. | 4.31415 3.71205 3.35980 = 3.10984 2.91591 2.75741 2.62334 2.50737 Zech | —2 =f) 1) oe ey Saf(§ is) = | 2.09459 1.79349 1.61730 1.49221 1.39512 1.31570 1.24848 1.19017 log. Ps 7.90541 8.20651- 8.38270 8.50779 8.60488 8.68430 8.75152 8.80983 log. 7, | 1.91852 1.61749 1.44140 1.31646 1.21955 1.14037 1.07342 1.01543 Diff. 4.01311 3.41098 3.05870 2.80867 2.61467 2.45607 2.32190 2.20560 Zech | —4 == i = 38 SSG Sa SS aS 1.91848 1.61732 1.44102 1.31579 1.21850 1.13885 1.07136 1.01974 log. pp» | 8.08152 8.38268 8.55898 8.68421 8.78150 8.86115 8.92864 8.98726 log. 7; | 1.61749 1.31646 1.14087 1.01543 0.91852 0.83934 0.77239 0.71440 Dif. || 8.53597 2.93878 2.58139 2.33192 218702 1.97819 1.84375 1.72714 Zech = 15 Sl) ie 003) ie eee 1.61736 1.51595 1.138923 1.01341 0.91537 0.83480 0.76621 0.70633 log. 7, | 8.38264 8.68405 8.86077 8.98659 9.08463 9.16520 9.23379 929367 log. if: 3.53004 4.73716 5.43852 5.93828 6.32592 6.64964 6.91044 7.14240 4 log. 9.92798 4.13910 4.83646 5.33622 5.72386 6.04058 6.30838 6.54034 _ log. P 6.76502n 7.367082 7.71926n 17.96914n 8.16296n 8.32132n 8.45522n 8.57120n Diff. | 3.837 3 14684 2.03086 3.83704 3.23498 2.88280 2.63292 9.43910° 2.98084 2. Zech =i — 95 = 51 — 0 = 1 SO BI 0) log.(—P +7) | 6.76495n 7.36693n 7.71869n 7.96813n 8.16139n 8.31905n 8.45214n 8.567187 | 3.93505 2.63307 2.98131 2.03187 1.83861 1.68095 1.54786 1.43282 Zech | <50¢ *=Srom ~— 9971: ao —2ieob = 2 eo6) = aor aes log. J | 9.99974 9.99899 9.99773 9.99599 9.99375 9.99104 9.98787 9.98495 log. pi | 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 9.07838 9.15624 9.22166 9.27792 log. po || 8.08152 8.38268 8.55898 8.68421 8.78150 8.86115 8.92864 8.98726 log. ef 6.46390 7.06572 7.41748 7.66679 7.85988 8.01739 8.15030 8.26518 log. ps | 7.90541 8.20651 838270 8.50779 8.60488 8.68430 8.75152 8.80983 log. J | 4.36931 5.27223 5.80018 6.17458 6.46476 6.70169 6.90182 7.07501 log. py | 7.78046 8.08153 8.25769 838271 8.47973 855905 8.62617 8.68415 log. J | 2.14977 3.35376 4.05787 4.55729 4.94449 5.26074 5.52799 5.715916 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 141 Noting that log. (J —1) = log. (— P-- se i ef and 1=h’/1’, we form the following tables : h’ Lo ° it 6.7649n 2 T.0658n 3 T.2415n 4 7.3661n 5 7.4624n 6 7.5409n 7 7.6070n 8 7.66410 1 >” 1 Log. 7 Jy Log. Pa Suit 1 (3) Log. i Shes Log te — 1 fy —-F 1 2 8.38238 6.4639 4.3693 8.38201 6.1647 4.9712 8.38138 6.9404 5.3231 8.38052 7.0647 5.5725 8.37941 7.1610 5.7658 8.37809 7.2392 9.9235 8.37656 7.3052 6.0567 8.37483 7.3621 6.1719 ” (h’—i’) Value of in dew bss feed fas Se 1/4.9712n 6.4639n 6.76495n 2 | 3.35387n 4.6703n 8.68341n 3 6.9410 4.9714n For h’ —0, 6 7 8 we have 9 8.38201 7.36693n 8.85913n 7.36675 5.67T02n 6.9404 8.68241 T.71869n 8.98344n 7.6393 6.1012n 5.5725 4.2455 7.3657 6.0668 4.7835 8.85764 7.6381 6.4006 7.96813n 8.98147 7.8413 9.07949n 8.161l4n 9.0TT06 7.8432 9.15756n 8.5190n 6.4176n 8.0061 6.6689n 8.1423 6.8TTin 9,22320n 1 Yi ed 2.1498 3.0527 3.9807 3.9551 4.2456 4.4826 4.6828 4.8562 | Mentl (Ak es 5.1598 6.6588 5.4583 8.0042 6.8709 9.15471 8.1402 8.4521n 9.21993 9.27965n 8.5672n 9.32905n In computing the values of the J functions, the lines headed Zech show that addition or subtraction tables have been used. For convenience, (J? —1) is em- ployed instead of J‘, its values being found in the line headed log. (— P+ ). 142 A NEW METHOD OF DETERMINING From the expression 4 (h'—i') (@h)) = 2 Iw (% 2), h’ being the multiple of g’, and being constant, and 2’ being variable, we have (h'—1) 5 (h’—2) (2) = dew 8 (tg —E') + Iuy 88 (ig — 2B) + ete. 1 (W'+1) yee + Fun sn (0g +B" \—", Syn es (ig + 2H’) ete. Now for h’ = + 1, we have, if we write the angle in place of the coefficient, (—1) ((¢g — g)) =4Iy 8 (ig — BE’) + 2Jy & (ig —2E’) + ete. Ldy % (ig + B')—2 Jy % (ig + 2B’) and for h’ = —1, we have —2) (—3) (ig +9')) = —td-n & (ig — EB) —3 I, & (ty — 2") — ete. O) ctae =x sin (29 tL penne vas (7g + 2H") + ete. Since (—m) ‘ (m) (m) (m) (—m) (m) Jy = al” Jy b) J_y a (ail Si, ? J_y —vVwy 5 the last two expressions give (0) (1) , (ig —g’)) = Jv & (tg — EB’) — 2d, & (1g — 2B’) + ete. (2) 3) — Jy 8 (ig +E!) — 2d, & (ig + E’) —ete,, (2) ‘ (3) (9 + 9’) = dy as (1g — BE’) — 2d sn (09 — 28) — ete. (0) (1) + dy sin (29 + HE’) — 2d, Sm (tg + 2H) + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 148 And for the particular case of ¢ = 1, we have (1) y (g—g))= Ty, 93( Ge tn 9 ae ek 2h ces ( ¢ — 32’) + ete. WV sin (2) (4) Ty 2( g + E')— ely 88 (g + 2H) — Bil, 2% (g + BE") —ete. 3) (g +9')) =—Iv &(g— El’) — QI | (g —2B") — 3d, 8 (g —3L’) — ete. (0) Q) (2) + Sy Sin (9+ EB’) —2dy sit (9 + 2H") + 38d y se (g + 3.4’) + ete. (0) (0) Instead of -/,, we use (-/,, — 1), as has been noted. If we put h’ = + 2, we have (1) (tg — 2y')) = $A SK (ig E'\) +3 ay cn ig — 2H’) + ane oe (7g — 3E’) + ete. (4) — 4 Iuy 2° (ig +E!) —¥ Inv & (ig + 2H") — ete. Fi (h'—i') In the table giving the values of | 7 wx , we have, under /’ = 2, which applies to the equation just given, (1) (3) fore = 1, log. $4. = 8.38201 log. (— 3d) = 4.9712n; (4) for v2, log ( ¥hy—1) = 736693n log. (= BA, ) = 3.8597: for 7 = 3; log. (— $ Sn ) = 8.85913n ete. = ete. ete., ete. = etc: We find the values of — } a oe — 3 be in the table under h’ = —2. We see that these are the forms of the function e Jue alten i= —2, and #= Land 7 = 2. In the expansion of the coefficient of (¢g—h’g’) indicated above by ((7g —h’g’)), we have coeflicients of angles of the form (¢g +72’). These can readily be put into the form (— 7g —7 #’), but the form employed is convenient in the transformation. 144 . : a a\’. : Arranging the functions 1 ( 52 a” ( 2) in this form, we haye — Go Oo Go tO = ee He He en on O71 -T -T +1 A NEW METHOD OF Ef’ cos — jl 0.0637n =—% 8.9912 =a) 7.6493 +1 9.6304 —] 1.72893 aD 9.94997 —3 8.6032 — 4 T4771 1 8.3617 — 1 0.3530 a 1.30203 eS 9.7486n can 8.3010 — 5 6.6990 —1 8.9395 —2 0.2191 == 3 0.91750 =! 9.5132n —5 8.2041 | 8.2041 ere) 8.9685 — 0.0082 ae 0.5522 = i5) 9.2601n = 7.9542 a 8.8855 =f 9.7672 ay 0.1976 == (2 9.0000n 129 7.9440 =i 8.7404 eis SEO TS) — 6 9.8494 — a — 6 —7 —8 sin 9.0043 9.1106n 8.04187 9.2856 9.8663 8.4624 8.8882n 7.6021n 8.6128 9.1584n 9.7427 7.5079 $.6435n 7.6532 8.7993 8.7993n 9.5368 7.6990n 8.3979n 8.6128 8.5051 8.2869n 9.2989 7.8451n 7.9093n 8.2049 7.0000n 9.0414 T.T782n 1864 90381 7.6021 8.7782 T-1 0 DETERMINING 0.5074n 7.0000n 8.8751 DH aT -T TO bok Oe HS co OS Ss He UO 9. = o 9.1173 1.0959 2.1675 0.7045n 8.9138 9.2808 1.6565 0.4244n 8.8976 9.8564 0.8905 1.3848 0.2347n 8.9385 9.8370 6.7129 1.1066 0.0224n 0.5132 0.8222 9.7973n 0.0210 0.1082n 9.2437n 0.1637 0.5364 9.8432 9.98860 8.9912n 9.8105 0.09T8n 0.6138 9.4642 9. 828Tn 9.8944 9.9566n 0.5440 8.7924 8.9TTIn 9.8287 9.6839 9.6410 0.4085 8.6128n 9.4298n 9.4506 0.1732n 0.2560 8.8633n 9.3876 9.2122 7.7782n 0.0394 8.8451n 8.6990 9.8156 8.7924n THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 145 We will now give examples to illustrate the application of the tables for trans- . . . . L forming from eccentric to mean anomaly, in case of the function « (5). For the angle 3g — 39’. (h'—i’) a Of “ (5) yon g £ cos sin (3) Log. Product. Product. ” ” 3—1 8.9395 8.7993 6.9404 5.8799 5.1397 + .00008 + .00005 3— 2 0.2191 8.7993n 8.68241 8.9015 7.6817 -+- .07970 — .00808 3— 3 0.91750 7.5368 T.T1869n 8.63862n 5.2555n — .04327 — .00180 3— 4 9.5132n 7.6990n 8$.98344n 8.4966 6.6824 -+- .03139 + .00048 3—5 8.2041 8.3979n 7.6393 5.8434 6.03872n + .00007 — .00011 48.26978 +0.34414 +8.38775 0.33973 For the angle g — og. (V’ = 0) = 1—1 1.72893 9.8663 8.38251n O.11144n 8.2488n —1.29259 — .01773 1+1 9.6304 9.2856 8.3825I1n 8.01297 7.6681n — .01030 — .00466 0.25653 —0.25027 —1.04636 —0.27266 For the angle g + q’. (h’ — 1) 1—1 1.7289 9.8663 6,4639n 8.19287 6.3302n — .016 .000 40.427 +0.193 10.411 +0.193 A. P. S—VOL. XIX. 8. 146 A NEW METHOD OF DETERMINING For the angle og — og. 0—1 0.0637Tn bas ays 8.3825n 8.4462 For the angles represented by (7g — gq’), there may be cases when there are sensi- ble terms arising from g + #H’, g + 2H’, ete.; 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 (2g + g’), there may be terms arising from the product of the numbers in the column h’ = 1 and the coefficients of the angles g + ’, ete. This will be made clear by an inspection of the two expressions (0) ) el (¢g —')) = Jy & (ig — EB’) — 23, SS (4g — 2B’) + ete. (2) (3) ; —- JS, SF (tg + BE’) — 2dy & (ag — 2H’) — ete., (2 3) =, , oS = Ud ( cos *: , (eg + 9')) = — Sy & (tg — BE’) — 2S, & (tg — 2H’) — ete. (0) (1) +Jy So (ig + EB’) — 2, 8 (ig + 2H") + ete.; where ((¢g — g’)), ((4g + g’)) represent not the angles but their coefficients. In retaining the form (¢g + 7’) instead of the form (— cg — 7’ #') we can per- form the operations indicated without any change of sign in case of the sine terms. Making the transformations as indicated above, we obtain the following expres- . p > a 2 3 sions for the functions u(“), and ua*(“) : wnNr Oe oe 6 bo bo bo bo to See eo oo ww OO OO OD | | cnn en on ee | | aa — 6 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. cos --104.78521 +- 1.04636 0.05031 GU. 02860 0.411 1.162 53.583 1.286 0.014 0.070 0.399 20.093 1.056 0.027 0.00815 0.04342 0.40733 8.338 0.675 0.028 0.027 0.275 3.628 0.397 0.021 0.020 0.167 1.623 0.224 0.012 0.092 0.731 sin uf 0.27266 0.12527 0.05793 —0.193 0.107 0.734 —0.171 0.066 —().127 0.053 -+-0.551 —().086 +-0.033 —0.01707 —0.07447 0.03392 -+-0.340 —0,036 40,010 — 0.043 -|-0.023 +0.197 —0.013 -+-0.008 —0.023 +-0.012 +0.109 0.004 —0.008 +0.007 +0.059 +182. 3777 — 1.6046 — (0.5606 + 0.1067 0.1274 0.0830 | = steht +246,9027 + 5.3656 0.3758 — 0.085 + 0.456 -+-147.392 + 7.214 I 0.086 0.0718 0.0041 + 0.079 | 0.050 + 2.174 | 46.016 |. 4.828 0.156 + 0,080 + 1.762 + 24.329 + 3.306 + 0.077 4.535 -+- 13,312 ofa = war( sin —1,9194 +1,1949 +0,4943 —(0.6468 —1,4558 -+1,1107 -+-3.4023 1.4496 LO.S8304 —1,242 {-0.848 4,049 —1.137 0.537 —0,2352 0.9231 0.5514 43.432 0.659 0.449 zl —0.637 OH 92 2.512 —().323 0.188 —).074 0.241 +1.565 —0.148 — 0.250 0.150 1.085 147 148 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 — #’), ( 0O— 2H’), (0 —3#’), ete., ete.; also (0 — g’), (0 — 29’), 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 haye for the logarithms of the sums before transformation, and for the sums after transformation the following : g cos sin Gq GJ cos sin 0—1 1.85407 1.62090n 0—1 - 70.548 — 40.188 0 —2 1.25778 1.51473n 0—2 + 19.809 — 32.318 0—3 9.7024n 1.26993n 0—3 + 0.906 — 19.352 0—4 0.7101 0.9147n 0—4 — 4.540 — 9.263 0—5 0.6632n 0.3899n 0— 5 — 4.707 — 3.318 0— 6 0.438Tn 9.0934 0 — 6 — 3.059 — 0.330 O— 7 0.1222n 9.8069 = 4 — 0.623 + 0.739 0—8 9.5965n 9.8865 0—8s — 0.071 ++ 0.615 For the angle (0 —1), (0 — 2), 0— 3. " 1 “Lait See ae 1 1 — 0.041 + 0.024 + 1.722 — 1.007 + .062 — .037 —= WER Fe IRE = (VO 4k sine Se HOLS He! 000 = — 0016 + .03T + 1.346 + 0038 + .097 + 71.462 — 41.774 = 9 — wig aE yl te + 70.548 — 40.188 + 18.104 — 32.714 ———9 020) — = Oa + 70.573 — 40.196 = 119'809 ——32'318 — .504 — 18.618 See + 19811 —32.319 40.906 + 19.359 a + 0.902 —19.355 The numbers in the last line of each case are the sums of the divisions after con- version when zg 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 u(“) and ua?(“) be retained. In the expressions for these functions given a large number of terms and some groups of terms have been omitted as they produce no terms in the final results of sufficient magnitude to be retained. In transforming a series it will be convenient to have the values of the -/ functions on a separate slip of paper, so that by folding the slip vertically we can form the pro- ducts at once without writing the separate factors. . . of @ 2 . The numerical expressions for u(“ ) and uoa?(“) being known, we need next to haye those designated by (JZ) and (7), which represent the action of the disturbing body on the Sun. To find (27) we use two methods to serve as checks. We have first (HZ) = S[hyyy + h'8,8/] cos (g — 9’) = — 3m’ + Usd] sin (g — 9) LUhyyy — hd!) cos (— g — f’) — [by — Udy] sin (— 9 — ) 3 hyoyy Coss (—— J) My dy sin (— gf) 2[hyiy — W',5] cos (— gy — 2g) — 2[ ly’ — Vy:0,'] sin (— 4g — 27) Zhyvys’ cos (— 29’) — 21y,d,' sin (— 29’) af it + 2[hyyys + 25,81] cos (gy — 29) — 2Byy2 + Uyr82] sin (gy — 29’) at i + 8[hysys’ + 88,82] cos (g — 39) — [ys + Uysds] sim (g — 89’) + etc. where 3 Q) (3) Y2 = > [Jo — Jy J i + [Jo + Sx ] (2) i ; (2) (4) n= 1 [Ja a Js, ] ds = ; [Sa = Js ], and similar expressions for 7;’, 5;’, 72’, 4:’, ete.; noting that 7. = — de. 150 A NEW METHOD OF DETERMINING The other expression for ( ZZ ) is (7) = 3[hy, — h'dy] cos (— H— gg’) + Bly’ — Us] sin (— FH — ’) + shy + hd] cos (#— g) — 3[by + Us/] sin (2 — 9’) — ehy,’ cos (— g’) + el’dy sin (— 9’) + 2[hy’ — Wvd,'] cos (— BH — 29’) + 2[ly’ — U5,] sin (— # — 29’) + 2[hy’ + h’b,'] cos (EH — 29’) -- 2[hy’ + V6,'] sin (# — 29’) — dehy, cos (— 29’) + 4el’d, sin (— 29’) + ete. + ete. In both expressions for (#7) we have h =" keos(I1— XK) eae poh TaN COSays f= =, C08 @ cos >’ k, cos (II — Ay) = gu — oe : - v sin V 1 = “cos¢ k sin (1I— K) = == a a Ul = “ cos@’ ksin (1— K) = 2 2 cos P Lu — where as before , w= oe 206264."8 and a=”. ™m a Tn 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 We have in case of Althea 4€ (0) Log. (J—1) — 7.207407 (0) Log. J — 9.99930 (1) Log. J 8.60344 _ (2) Log. J 6.90632 (3) Log. J 5.0329 (4) Log. Ai 3.0347 T.80894n 9.99719 8.90341 T.50TT 5.9356 4,2¢ 384 THE MINOR PLANETS. 151 Be 2e $.16025n 8.40890n 9.99368 9.98872 9OTTT4 9 20016 T.S587T 8.1068 6.4630 6.8365 4.9418 5.4408 (h—i) Z . . From these values we may form a table of : J), as was done for the disturbing U body. ‘The values of these quantities can be checked by means of the tables found in ENGELMANN’S edition of Brssn’s Werke, Band I, pp. 103-109. Finding the numerical value of (/7) first by the second expression, we get Eg 1— 1 ] | 0 l | 2 —l 2 0 2 1—3 —l1—3 0—3 cos 48.154 0.188 3B.884 4.044 0.018 O.38TS00 0.00141 0.03048 sin 0.651 —(),102 0.044 0.062 —O0.010 0.004 LO.00510 0.00081 —().000386 To transform we change from (h#—7q’) into (7’g’ —h#). 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 9g bo COs " 5.826 0.560 0.04566 + 0.149 +48.076 0.186 + 0.011 A NEW METHOD OF DETERMINING sin Vr — 0,066 —0.006 —0.00057 —0.108 0.650 0.062 -+-0.00502 -+-0.026 +-0.002 0.000 (fH) —— (Nay — (0.04575 + 0.180 +48.079 sin Vy —0.066 —(0.006 —0.108 0.650 | +0.062 0.00510 0.030 +0.002 0.000 (I) sin cos 4} VW +4.799 +9.043 0.463 +0197 -0.038 10.016 To find the numerical value of (7) where (1) = b= — : : 9 (d2 needed in case of the function @ (|). we have bs’, sin(— g')+ J'y',cos(— g') + 460’, sin (— 29’) + 4b’y’, cos (— 29’) + 9 bd’; sin (—3q9’) + 9 by’; cos (— 39’) SP KO, abe: ” cos 9’ sin J cos I’, ba = sine a a Having the values of u (5), lua” Gy. (/T), and (7), we next find those of THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 153 from a =u(%)—G ae = ee (5) [B55] — be (G)— aa = te (Gy a pa 0) + CL) where 2 (1) (2) (3) As 209 _—1+ 3—4J, cos g —4J2 cos 2g — 4, cos 3g — ete. SDF: (0) (2) , ; (1) (3) ; a (f’ +’) =—[4 +4] asing’ — s[Jw + Jw] ce sin 2g’ — ete. , a (0) (2 : _(2) (1) (3) + $e, — [Jy —dJy ] & cos y' — [Ay — Joy ] & cos 2g — ete. ¢, and c, being given by the equations sin] ; : ¢, = —— cos d’ cos II a sin]. ; = sin II’. a We find 4[ 71" | = [95769400] — 2[8.38238] cos y’ — 2[6.46366 Jeos 2y'— ete. + 2['7.99450] cosy + 2[6.29667] cos 2g +- ete. —sintr’ sin( f’ + I’) = [7.18046] + 2[8.39074] sing’ + 2 [6.77809] sin 2’ a , — 2[8.01941] cos g’ — 2 [6.40668] cos 2y’ A. P. 8.— 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 Ax in case of the sine, (2, those of the angles ux in case of the cosine, y. those of the angles 7y in case of the sine, and 4, those of the angles py in case of the cosine. The following cases then occur : a0, Sin (Aw + py) + 4,6, sin (Ax— py) tl a, sin A@ . 6, Cos py = Buy, Sin (ua — vy) l= Buy, sin (ue + vy) — |= 3, Cos ux. y, sin vy = Pu My 8,8, Cos (ua + py) + 33,6, cos (ux — py) t|— GB, cosux.d, cos py = = — bay, cos (Aw + vy) + Zany, cos (Ax — vy). a, SIN Aw. y, Sin 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. c ; 5 7e dQ 9 dQ Performing the operations indicated, we have the values of aQ, ar = w that ar az follow : 0—0 1—0 2—0 3 — 0 1—1 0—1 1—1 2—1 3— 1 0—2 1—2 2—2 3 — 2 4— 2 0—3 1— 3 2—3 3 — 3 4—3 5-3 2—- 4 3 — 4 4—4 5 — 4 6— 4 4—6 5 — 6 6 — 6 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. aQ cos sin vw Ud +104.78521 ona — 1.04636 —.27266 — .05031 12527 -+- .02860 +.05793 + 231 —.090 + 4.662 Shaye) = 5.504 +084 — 641 —,201 aS Kilts +-.066 + .632 —.121 — 4.206 —.009 + 19.907 =| .549 + 1.056 —.086 + 027 -+.033 f -05390 —.01764 — .83396 —.07957 + .39221 --.03380 + 8.338 -.340 == 675 = 036 = 028 +.016 -— .027 —.043 = ae +°.023 -+ 3.628 +.197 + 397 (1153 + .021 -+.008 + .020 —.023 Sealy +.012 + 1.623 +109 a .224 —.004 092 +.007 4+ 0.012 —.008 -t- Se ATI 1.059 cos +-16.5202 2.4398 .B040 .02T4 054 880 4-15.430 +} .883 013 -034 281 8.605 L 1.061 ated HIS on © Orb -+T oouc 155 cos — 16940 3998 1494 — .355 + .481 + .190 — TT + .288 414 .200 -1.270 441 + .180 — .0602 — .3306 + .1339 +-1.087 — .269 + 157 — .210 -+- .908 +- .882 — .137 -+- .063 — .078 + .044 + .543 + .064 —(0.095 + .026 + .386 i -|-0.2828 —2,6311 .059 O17 46.0177 4. 939 — 01 2 rt§ —§6.095 ZEX0BT 42.011 + 194 Ay bg — 166 —3.658 — .134 +-1.099 + 123 ==. 146 —2.078 — 06 + .586 + .083 —— Woo ——1'.150 —— 27 + .311 156 A NEW METHOD OF DETERMINING : : ; ‘ eee) Having a we differentiate relative to g, and obtain a g We then form the three products, 4. at we find A, Bb, C, from Bar (i G:a(©). To this end dg 2 ] dz A =—3+2[2+ e] cos(y— g) B= —2[1— FJsin(y — g) + 2[§ + ¥] cos (y 2g) —2 [$+ $] sin (y — 29) —2 [Bs + 48] cos y — 205 + she] siny + 2$ cos (y —- 39) — 22€ sin (y —3q) + 2© cos (y —4q) — 2¢sin (y—4g) +- ete. — ete. C= 2[i—te] sm(y— g) + 2[¢— 6] sin (y — 29) + 2[—#e+ é] sin y 3 sin (y — 3g) See: sin (y — 49) + ete. The numerical values of A, B, C in case of Althzea are A=—8 + 2 [0.302429] cos (y — 9g) B = — 2 [0.001399] sin (y — 4) + 2 [8.604489] cos (y — 29) — 2 [8.604489] sin (y — 29) — 2 [9.304508] cos y — 2 [8.606234] sin y + 2 [7.2076] cos (y — 39) — 2 [7.3836] sin (y — 39) C= + 2 [9.697567] sin (y — g) + 2 [8.380066] sin (y — 29) — 2 [8.77953] sin y + 2 [7.08265] sin (y — 39) For the three products we then have > i LO or mw Dee oO Ow 09 09 bo DO ee © Om ROO ONDER OS | eed feed fed ped ed pet pet et et i bo bo bo bo LO bb OO OO be ST TIES Oo oo co Go Go te Oo Oo Co OO THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. -+ .462 — .266 —10.992 + .462 + 3.680 + 1.119 — .342 —11.3801 + 2.360 — .033 + 2932 + 6.837 —80.684 — .848 -+ 1.633 116.433 + .422 —79.078 — By — .408 5985 — %.6517 — .0661 —50,140 + (828 — .380 +. 3.482 + .263 49.676 — 6,395 uM 0.5371 .565 0439 -299 97 457 000 026 sin cos uw Vw VAsay —().6804 4021 2 Bye} — 32.9502 + 0549 —— 0153 Se lt ay — 1.1310 + .6821 — .1263 —" 3 0a0 - 439 Se he se aR AGL —18.335 187 — .477 349 1. 929 a stai") — .449 — 476 + .306 2b wie 18.336 ales — Hee) + (559 — 264 — 276 — .232 -+- 7.300 a BE — 45.419 | 1.264 JE say + 406 — o410 eT! (SILL f/ — 935 + .048 + 168 +-45,412 —1].264 — 213 + 384: aE eeoS — .163 | 4644 — 3261 | 1.1042 + 1641 + 0541 = Oat —27.2994 1.0854 —— DUS SE BOS — 2.8964 — 2201 — 1.1112 — 1645 well Ag +27.299 —1.083 + 3.899 SERCH IY — 0049 11.2995 + .083 —— Ni —1.881 I epy! — 298 =—— ~S15 -+-1.906 +- .067 — .1i8 157 cos wf —3.0038 ak 1 a= 2411 A802 0228 2.9712 i 2404 157 .8160 0180 174 136 .558 818 .005 .206 534 158 A NEW METHOD OF DETERMINING 1O ) = ) AP ee) B.ar S) GE v(e ) dq dr dz yg) 1G) sin cos | sin cos sin COS Ww Wy ai zi WA Ww 1 1—4 — .165 = i CCR — .038 a a 3) ies eed — 2.229 AS H@e Wy) Se ere + .939 —=.889 1.029 —j Bas + .011 se (oily aaiete ects | .008 + .014 1 38—4 —29.0382 +1.564 —15.481 tS )iliy + .022 — .083 —l 3—4 + .058 — {jl == 089 sk lis) — .024 + 051 1 4—4 — 1.063 — 287 —— ilgay0):! —— {i)s}s — .140 — 300 =| 44 + 1.268 — 024 a 022 == 71938 + .390 —1.033 == (5 4 —98\751 1.597 +15.479 95 + .033 = 199 == 64 — 4,543 =—— 108 -++- 1.506 + .098 tL) 9=5 == IO) S360 |} + .002 088 ho R= == IN! + .132 | — .063 + .063 — .206 SE 5y(() —l 3—5 ae Ole + 014 —— =001 — .003 + .001 + .008 1 4—5 —16.185 +1.082 | — 8.661 + .544 + .034 + .038 —l 4—65 - 015 — 148 {I == 076 | — .035 + .004 1 5—5d5 — 1.061 — .158 — 1.412 —— {05i5 = 080) — .168 —l 5—5 + 294 —= 01% + .062 — .063 SE PAG == .063 —l 6—5 —16.038 -+1.100 + 8.661 — .b44 1 S86 = ,llPil — .063 Vee Seen = 1,088 + .086 + 2.052 + .038 1 ©—6 — 8.707 + 7103 == Ll + 3887 —!] a0 — 8.818 + .711 + 4.516 = wey | Next from OQ 12 ES A. a(S ) + B.ar i‘. ) dg dr we find the value of ce Then we find W and —. from nat cosz We ee ndt THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 159 We first form a table giving the integrating factors. From log. n’ = 2.4758576, log. n = 2.9323542, we have ” — 0,34954524. nt eee |e N 2 - n'y | 1 || Sector ° ca) | Se TN 1 = oe. (2-1-2 — Ss Es ix od (=e, eae: Log. (7+ Z ") Los. )| ev)i+ee Log. (i+: 7) Log. (a —2 — 1] —2.34954 0.37098n 9.62902n 3 —3| +1.95136 0.29034 9.70966 —] — 1] —1.34954 13018” 9.86989n ||4—3] +2.95136 0.47002 9.52998 V1) —= ByiG ay 9.54350n 0.45650n ||/5—3] +3.95136 0.5968 9.4032 1— 1} + .65045 9.813217 0.186783 4 S988 9.60008n 0.39992n 2 — 1) +1.65045 0.21760 9.78240 9—4| + .601819 9.77946 0.22054 3 — 1] +9.65045 0.4233 9.5767 3—4| +1.601819 0.20461 9.79539 4—1] 13.65045 0.5624 9.4376 4— 4] +9.601819 0.41528 9.58472 —1] — 2} —1.69909 0.23021n 9.76979n 5 — 4] +3.601819 0.5565 | 9.4435 0 — 2} — .69909 9.844670 0.1554n 6— 4; +4.601819 0.6630 9.3370 1— 2) + .30091 9.478423 0.521577 2—5| + .252274 9.40187 0.59813 2— 2) +1.30091 0.11425 9.88575 3—5| 1.959974 0.09770 9.90230 3 — 2] 19.30091 0.36190 9.63810 4—5| 9.959974 0.35263 9.64737 4— 2] +3.30091 0.5186 9.4814 5 — 5| +3.959974 0.5122 9.4878 5 — 2] +4.30091 0.6336 9.3664 6—5| +4.259974 | 0.6286 9.3714 0— 3! —1.04864 | 0.02062n 9.97938n 3—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 (4g + vg’) = ( 47 ) g= (7 4 7 .) nt. The differential relative to the time is (i 44 E) ndt. g n n The preceding table is applied by subtracting the logarithms of the column headed v . 2 1 log. (7 +7 : \ or by adding the logarithms of the column headed log. (=): dw ndt? grations the angle y is constant; after the integrations it changes into g. . . u . . . We will now give the values of W, and ea? remarking that in the inte- OSt 160 A NEW METHOD OF DETERMINING au W — ndt cosa | Yo dG | sin cos cos sin cos sin a =aes| ” Sour Vi.) =) ane " ” 00 + 8.93876 —1.2175 | — 1.21%5né + 3.93876nt| — 3.0038nt — 1.3464 né ibe a0 See Oe oss) | -— BESO0le ate O03 =) 198 ee eee =191—0 | s3) 3016972) Se 0988 | — 382:6979' 4 20988 381i ae ee 1 2—0 | — 2073 +4 4647 | + .1036 + .2393 at (0024) wal 2014 0 SS EPSteO et eOSh) | )—/ § 24 740m eee (0405 =~ 6497. - “a0 486 Si Bea Se Omer n0850) |). == | 0450 ueueies euoes = 028 + 0801 he = fe Rod een TiGy 2" | 3383 ae iy == 4038 ein 1==1.— 1 -- | 18% <= .446 + .115 — .330 | —0.62 — 1.60 : Dyno 4 — 29.397 =. .840 — 83.900 = he | +1.013 + 1.84 =i ai Sa sy E258} ee its; Sas 2] ae — 1.64 ly Bees + 4609 —1.374 ony = An) SEIeyen ain lee Ble = 1 670 — .489 = 030 ae a= 1e370 = oi | 1 yea |) = BG LR aia 1022 ae 5 Gln ee O40 SL 08 al Pia SEB SE GI — 4963 a OB.) te LTO == Bil eS 1 | Oo eA + 007 + .096 | —j] 22.7 =) 430 gees e540 =~ 0h = Qe + .670 tet = 297 2 ea05 ce Saeost e029 | emt a9 = sos eee | | i Op e445 eer + 20.207 = Be =iaG — 4,33 ie heer —126.276 +3.459 +419.660 11.503 yy) SOS ie M19 116 + .408 + 2.380 + 1.356 + 46 oe) %) 1 Ee <= Sai eo + 1.410 SS 150 + 36 = 7S STO 9) |) eee gare? Saar = 00s — .365 =—=95 eal 1 oie DSS eT mA NE +° .210 = Lil aor SS eye — 33.666 -+ .990 + 14.632 + .480 + .02 ee 1 SS =) LOT ee ete + 005 + .038 Sh Or = C50 an + 9.469 == (185) =U! sb 8h) HH] HN) SI ee 050 = PI a= 3 == 1.0629 —=saei4-| 1.0186" == 4591 == ANS = oe 1 ws ==) 1545) Eeessbs: | — 81.8180) © ==eiesD —14.56 Sieh Sie SS — . 70120) yaesss. | = 9459) 9 aaiean ZL. D5 aS jee er — 77.4394 12.9904 | + 81.400 + 3.139 NL eis ieee 1) oo ese | —— S194 ST + .06 ie ee — 3.2764 — 7121 | + 1.679 365 els = 8 il eee) 42 93706) ears | — 1.216 ee ee = 9 it eS) 2) 448 ea — .050 SSS ies .00 00 =i 4——3- | — 905i eee) 1.413 + .338 = ill = on 3) | a ee 62K = Je = Ae TE ee ce eer | RA ais == 096 ees 1 94 — 1.96547 +1.126 + 3.965 Seaesail a ee) SS aaa fo ra | — 44.518 ~+9.479 | 27.790 + 1.548 =e eee —1 3—4 == 03) a + 019 == iy Sle eS Ae: 1 4-4 | —- on6Ts eos: By E986 = 48 ZL (54S Ae —l1 4—4 | + 1.002 — .9638 = NS = eon el 0 ee lt b—4 | = 920m eros + 006 + .016 =~ 54) || 1807 eeeecee + 3.686 + .190 =e 009) aes =) beet SSS a + 660 O02 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 161 ae Ww wl nat cost Bay eg. Gg! | sin cos cos sin COs sin ” ee ” in j ” : ” 0 STU, 5 SEN95: + 1.374 + - 156 Sey ee st.) JS eal Sey = 009 See 4. = 2 00l a col 5 — 24.846 +1.626 + 11.030 + 799 = Ogee eet 08 te A 5 == 080) 072 Serie 032 + 016 00 io Sos) == 104 E760 rico | 2 08a — 08 ee + 356 — .080 = ud Lp a eeyis *SSeeah: 6 — 5 = ei = Sia A021 =e .038 —1 6—5 — F3tT —-+ 556 + 1.135 + «130 int — 5 + 1.413 + .036 ee) 0G) + 007 56 HE 964) Se 24. = Si ae UT | 6 — 13.223 +1.090 4. 4.555 ass meee | 6; 4 008- | 0ST 06 | i 6 —6 = 946. — 002 nee) 00 =e — 6 = 9.098. —= 040 =) 58 =a Oil 6 — 3302 + .$24 ee i + .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- 3a CG is the following: Gig cos 7 al |! A cos sin 1—o | + 81302 + 8181 ie eo) — Bre +14 2—0 | 4. 1509 — .8i5t poo || —. al —.06 SU —— 20858 = — GSE Q—4 | — 22% +.43 ha 151 + .20 S|, =f 5a = 18 Ve 95:39 =e | — 16% Sn ely 2 2:88 + i Bee. |) 165 +-.05 morte | .04 — ag fee |-— 08 05 Ne Stee 2a 934 + OOGNEa es 5. | 214 +.16 2—2 —91.80 —2.53 | Mant ar agg ==1N8 Bete] 13 + .34 bey || — 1.49 —.50 SP a a 12 ao |) — 95 +.02 1—3 | —20.6020 —4.9099 2 a +.05 — 3 | = Bais — 210 m6 | — 48 —04 3—3 | —38.46 Sie — 6 | — 3.35 —27 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 + 7g + 7g’), and of (Oy + 7 + 7q) in the case » u of cost The expressions for this purpose are 1 = $e — 36 — seq 7 = 26 — jet ny) = 1¢° n” = — (e + 3% + etc.) For Althzea we find log. 7 = 8.60309 log. 2 = 7.38368 log. 7 = 9.08196n We multiply the coefficients of (+ y + 7g + 7g’) by 7, and x", respectively, to find those of (+ 2y + #?¢+q'), (43y +74 +4 79’). In case of (Oy + zg + 7g’) in the expression for = we add the coefficients of (+ y + tg + vq’) to those of (— y + zg + 7g) and multiply the sum by 7. dw dy* With these two we give at once also their integrals, which are néz and v respec- We will give a few examples to show the formation of J’, and — $ tively. W “Ad id Ww 2 dy (0 — 0) cos sin sin cos ” " uw u =i = =s9097 +-0088 +16.3486 + 0494 =a) ss = HTN) SEY + .0190 -.0017 == : —32.7162 | +.0511 " | ” — 32.1162 nl +.0511n¢ W ad W 2 dy (1 — 0) “ u" | “u ” ae 0d 042 | Se RN e021 He) eeeigde. Sc Sie = - ee Oe Sid 004 | —1.314 ++ .004 MeO OL TMG rs PS ORTG ne! ers COMA tees S80 ” u” ” ” 7 ” 5 nu” 41.351 — 1.2175nt -+ .856 +3.2376nt —1.077 —.6087nét + .025 —1.6188nt " ” ” " uw ” " ” +459 —1.21T5né —2.01T —3.2376nt —0.54 +.6087nt —0.58 —1.6188n¢ (—1—1) ur ” | wu ” | 2a 383 20m) hel MSE ion —.035 —l1 o-1 — .045 —1.516 ++ .022 —.158 —2 1—1 — .041 030 +.041 —.030 ete sds == 2513 + 200 z = she — =| 2 —0.216 —1.246 ee sone == 823 u" ” “ ur ae allt? = op +.19 +.61 (f—*) ” ” | “ur ” —2 3—1 — 022 — .004 | + 022 —.004 Senieeeo =I = | 4.963, +1 038 | 9.131 ++.019 ee 251890 390 | = = Tee 83-900) —. 973 —41.950 +486 —113.574 —1.329 —39.798 +.501 ” ” uv u” —174.61 42.04 461.19 10.77 : . 2 1W In the integration we apply the proper factor to each term of I, ota yand obtain the values of ndz, v, except in case of the terms (7g + 09’). Let us take the term (gy — og’) or (1 — 0), and let « the integrating factor to be applied. Let c, a, d, b, represent the cos, sin, nt cos, nt sin terms respectively. A NEW METHOD OF DETERMINING 164 Thus we have Cc d a b “ ul ” ul 11.351 —1.21 Tint 1.856 +3.2376nt; and hence uc wb ud —ua wd —ub aye “WT " " " +1.351 +3.2376 1.21 75nt —.856 —J.2175 —3.2376nt or, since u is unity, ur ‘tf Vt Vt +4,59 —1.2175nl =i — 3.2376. In case of the term (2 — 0), u is 3. u In the case of 2 cos a In the way indicated we derive the values of ndz, and ». we have the values at once without another integration as was necessary for ndz and ». In the value of JV given above the arbitrary constants of integration have not been applied. We give these constants in the form ky +k, cos y + kesiny + 7° k, cos 2y + 7k sin 2y + ete. 5 TW Then in case of —1‘ 7 we have 2 aie 3k, sin y — $k, cos y + 7° ky sin 2y — 7° k, cos 2y + ete. : r : : dW W7 r Having |W from the integration of aa? we form JV from the value of W and converting y into 4. We thus have from the equation el + Wie) at +(1’.351 + k,) cos g¢ + (0’.856 + k,) sin g ~— 1”.2175nt cos g + 3”.2376nt sin g + (—'.284 + 7 k,) cos 2g + (0.589 + 7° k,) sin 2g + ”.1298né¢ sin 2g + ete. —" Q488nt cos 29 + ete. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 165 In the second integration the constants of ndz and y are designated by C’ and V respectively, and the complete forms are C+knt+ ksin g —k cosg + 37%, sin 2g — 3n kb, cos 2g + ete. NG — 1k, cos g — $k, sin g — 3) k, cos 2g — 37” k, sin 2g — ete. In case of the latitude the constants of integration have the form l, + 1, sing + 1, cos g. We thus find nz = C+[l + & —382”.7162]nt + [4.59 + k,] sin g + [—2’.07 —k,] cos g — 1”.2175nt sin g — 3” .2376nt cos g + [—07.11 + dy &] sin 2g + [—0’.31 — 37 k] cos 27 — 0”.0244nt sin 2g — 0'.0649nt cos 2g + ete. + ete. y = +0".0511nt + NV + [—0".54 — $h,] cos g + [— 0.58 — fh] sin g + 0” .6087nt cos 4 — 1’.6188nt sin g + [07.05 — 4,%k,] cos 2g + [— 7.24 — gn b] sin 29 + 0'.0244nt cos 2y — 0”.0649nt sin 29 + ete. + ete. — = 1, + 0.3616 + 0" .3623nt cost + [1.52 + 1] sin g + [—0”.68 + 1,] cos 9 —1’.3464nt sing — 3’.0038nt cos 9 + 0.32 sin 29 — 0.16 cos 2g — 0’.0539nt sin 2g — 0’.1204nt cos 29 + ete. + ete. g 9 | 00 0 0 oil (i=—2 (= t=) 2—2 3—3 4—4 5 — 5 6 —6 1— 2 2—4 1— 3 9—1 2— 3 3 — 2 3 — 4 4 —3 4—5 5— 4 —ijh = il (=p) A NEW METHOD OF DETERMINING The complete expressions for 75z, v, —“— in tabular form are the following : cost noz sin cos +k, nt _39,7162nt — 459 ky = 2.07 — k, — L.21'T5nt _- 3. 2376nt — 0.11 + 44%, — “31 — 47%, — 0.0244nt —- “0649nE BO 3109 53:00 ae ey) + 0.93 eG —174.61 + 2.04 +263.97 — 7.21 + 95.15 == O81 + 6.71 — 0.35 + 1.64 017 He 49 = %05 1185.18 + 2.10 — 1.10 — .j1 +-410.16 —87.44 — 5.25 ne 8h — 37.24 = 18.03 + 6.17 + .04 B eagiae:1) = 286 af 2 re 04 SIT =. 68 a) Ba + 01 = 16 — .92 Cos sm ” 511 nt “ 0.54 — th, 0.608Tnt sin 58 — Lilt, 1.6188nt 24 — dyke? -0649nE cos 7 4 is $4 — 1.3464nt a ‘32 — "0539nt — 4.83 + 1.30 — 387 + 2.69 — 1.15 — 1.60 Se 8 1, oe pee -- .3623nt 68 + 1, 3.0038nt 16 i 204nt 2.03 61 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+ ksing— k,cosg + $y%k, sin 2g — $y” ky cos 29g + ete. 4 (ndz)o = Yo k& + kecosg+ ksing+ 7k, cos2g + 7k, sin 2g + ete. + nat (n0z)o = 0 N — $k, cosg — $k, sin g — $n k, cos 2g — $y” k, sin 2g — ete. + (rp = 0 ° D 5 ‘ d 4 + $k, sin g — 3h, cosg + 7k, sin 2g — 7 k, cos 2g + ete. + cai (ao = 0 + 4 sing + 1, cosg + 7 1, sin 2g 4+ 7 cos 2g + ete. + ( Je = l, cosg — l,sin g + 7 1, cos 2g — 7 l, sin 2g + ete. + d ( e ) = 0 ndt \cos i 0 To find &, and k, we have k, [cos g — e + 7 cos 2g + cos 3g + ete.] + hk [sin g + 7 sin 2g + etc. | = 37, + 6 (eo 4 4 (ndz)y = 0 K, [sin g + 27,” sin 2g + 3,” sin 3g + ete.] — [eos g + 27 cos 2g + etc. ] d ndt + 2—(v),= 0 where N=—%y—jh—34, %=— 32.7162, ky being found from hy = eh, + 3%, — 3 (ndz)y — 6 (vo We have also ts —_—_— él,. The symbols (ndz),, (v)o, etc., represent the values of ndz, v, ete., at the Epoch. 168 A NEW METHOD OF DETERMINING To find the values of the angles (7g + 7g) 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 coeflicients of the perturbations by their appropriate sines and cosines. : Dy : : : In forming mea (ndz), ete., we can make use of the integrating factors, multiply- na S ing by the numbers in the column (7-4-7’“ ). Having their differential coefficients we 5 - DS proceed as in the case of (ndz), etc. We thus find (ndz), = + 401”.7, (v7) = + 180”.6, (ee ) = 99 0G COS? d ¢ ane ” d + es an d U . 3 ndt (ndz)q = 391 .G; wai (0 NF a a 70 -D, rie. () = a 41 5. And from these we have k, = + 412.8, k. = — 82'’.9, iy == — 262215 ik =0'20) = — 45”.2, f= -+ 0.4, N= + 287.3, (Oa 332° 44’ 12 6. The new mean motion is found from (1— 32”.7162 — 26”.21) nt, which gives n = 855".5196. With this value of n we find the only change is in the coefficients of the argument (1 — 3), having + 405’.29 instead of 410.16, and — 86”.30 instead of — 87.44. The constant 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 P OSt Ne docs AA Gis + 855.5196 ¢ + 417.4 sing + 80.8 cosg — I1’.2175sng — 3.2376 cosg + 16.4 sin 2g + 37.0 cos 24 — 0°.0244ntsin2g — 0.0649 nt cos 24 + ete. + etc. i E28 + 0”’.0511 nt — 206”.9 cos ¢ + 40”.9sing + 0.6087 nécosg — 1”.6188 ntsing — ~ 8'.2cos2g + 1.3 sin 2g + 0.0244 nt cos2g— _ 0.0649 nt sin 29 + ete. + ete. aa eee Eee 4+ 07.3623 nt cos? — 44’ 2sing — 0.7 cosg — 17.3464 ntsing — 3.0038 nt cos ¢ — 1’.5sin2q — 0’.2cos 29 — 07.0539 nf sin 2g— 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 haye an approximate value of the distance; its logarithm is 0.14878. A. P. 8.—-VOL. XIX. V. 170 A NEW METHOD OF DETERMINING Reducing the above date to Berlin Mean Time, and applying the aberration time, we have, for the observed date, 1894, Sept. 19, 72800, G=1339° 19 3820, Gk= ba° 241. Forming the arguments of the perturbations with these, we find noz = 4-4’ 43.2, 7) = Sb Bb. 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 haye from » = + 3’.6, the correction, + .000008, to be applied to the loga- rithm of the radius vector. u , In case of _= — 2”’.8 we have cost (a —— 8 SX aicos2 === Converting into radius as unity, we have ¢z’ = — .000035. The coirdinate 2’ is per- pendicular to the plane of the orbit. As we will use codrdinates referred to the equator we haye, to find the changes in a, y, z, due to a variation of 2’, which we have designated by 42’, the following expressions: 6% = (sin Zz sin &) 62’ dy = (— sin 7 cos Q cos ¢ — cos 7 sin €) 62’ oz = (— sin? cos Q sine + cos 7cos €) 02’ where ¢ is the obliquity of the ecliptic. For 1894 we find da = (— .0404) dz, , dy = (—.8128) de’, de = (4.9491) de’ And for the date we have éx = + .N00001 dy = + .000011 dz = — .000033 7 THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 171 With 4 = 5° 44’ 4” 6, OL = 2OS%b1" bv -55 = 2a wel AOS; we compute the auxiliary constants for the equator from the formule 5 = tg tv cots A = — tg Q cos 2 (a By = = fo) J 06 ? Wf Ho ane Q? - ee COS)? cos (£, + «) cote B= —— tg Q cos E, COS € eots C = cos zt sin (£, + &) > — 5 aries aan ’ ty 2Qcos £, sin ¢ . cos 92 : sin §2 cos < : yan (Sint) sinte ing in — & , sin C= —2— sin A sin B sin ( The values of sin a, sin 0, sin ¢ are always positive, and the angle E, is always less than 180°. As a check we have sin b sin ec sin (C — B) tg = sin @ cos A We find A = 293° 45’ 29.3, B= P22 bg 46" 9, C=A07 45" 5520 log sin a = 9.999645, log sin b = 9.977735, log sin e = 9.498012 Applying néz = + 4 43’.2 to the value of g, we have nz = 339° 24’ 21.5 By means of g or nz = E — e sin E we find BE = dal? a9'2a".4 Then from J/rsin bv = Jfa(1+e) sins E /1, COS 3 /a(l— e) cos $ E 172 A NEW METHOD OF DETERMINING we find where v is the true anomaly. Calling wu the argument of the latitude we have (ee ea Os EER Se ech Hence AP 77°38) Lae B+ u= 346° 52' 28'.7, C++ u = 354° 38' 36'.8: And from «= rsinasin (A+ w) y = resin 6 sin (B+ wu) z2—=rsinesin (C+ w), where log r = log 7, + 6 log r = log 7, + .000008, we have x = + 2.331894, y = — .515438, z= — .070208. The equatorial cobrdinates of the Sun for the date of the observation are X = — 1.002563 Y= + .045198 Z= + .019611. Applying the corrections da, dy, 62, we have w+ da-+- X= + 1.829332, y+ dy4+ Y= — 470224, 2+ 04+ 72=— .050630. THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 173 Then from Sey Se eth ye Se Z ° A SRA (ae ee ig SS ia : _ COS 4, w+ da 2 y +82 + J x Ou > Z oz + Z ) a a sin 0 we have, giving also the observed place for the purpose of comparison, Gee cele le 4. Noss = 2 6) 2A log A = 0.149514 oi, = all) sai Zeal = — BY Bl where the subscript ¢ designates the computed, and the subscript o the observed place. Both observed and computed places are already referred to the mean equinox of 1894.0. If the observed position were the apparent place we should have to reduce the computed also to apparent place by means of the formule Aa f +g sin (@ +a) tyd Aan g cos (G + a), the quantities f, 7, and G being taken from the ephemeris for the year and date. If the observed position has not been corrected for parallax we refer it to the cen- tre of the Earth by means of the formule zpcos¢’ sin (a 0 Aa = = Se ( a a cos 0 tg ¢! tyy — g¢ : cos (a — @) ea LE sin ¢’ ; sin (r =) J sin 7 where «a is the right ascension, } the declination, A the distance of the planet from the Earth, ~’ the geocentric latitude of the place of observation, # the siderial time of 174 A NEW METHOD OF DETERMINING THE GENERAL PERTURBATIONS, ETC. observation, p the radius of the Earth, and ~ the equatorial horizontal parallax of the Sun. For the difference between computed and observed place we have C— 0 = — 2 37.7 in right ascension, and C'— 0 = — 57.7 in declination. By the method just given we have found the positions of the planet for several dates and have compared with the observed places. The comparison shows outstand- ing differences too large to be accounted for by the effects of the perturbations yet to be determined, which are the perturbations of the second order, with respect to the mass, produced by Jupiter, and the perturbations produced by the other planets that have a sensible influence. We have therefore corrected the elements that have been used in the computations thus far made, by means of differential equations formed for this purpose, employing as the absolute terms in these equations the differences be- tween computation and observation for the several dates. A solution of the equations has given corrections to the elements that produce quite large effects on the computed place. Thus recomputing the position of the planet for the date given above with the corrected elements we find a, = 340° 33’ 44.5 , 6. = — 2° 2 156. And since a = 840° 33’ 49”.1 , 6, = — 2° 2 25" 4 we have, for the difference between computed and observed place, C— O = — 4’.6 in right ascension, and C— O = + 9”.8 in declination. ARTICLE II. AN ESSAY ON THE DEVELOPMENT OF THE MOUTH PARTS OF CERTAIN INSECTS. BY JOHN B. SMITH, Sc.D. Read before the American Philosophical Society, February 21, 1896. Since the publication of my paper on the mouth parts of the Diptera, printed in the Transactions of the American Entomological Society for 1894, I have continued gathering material, have examined the oral parts of a very large number of species of all orders, and am more than ever convinced that in all essentials the conclusions already published by me are correct —revolutionary as they seem at first sight. That my ideas have not found unquestioned acceptance is not surprising; but no one has, to my knowledge, published anything that disproves the points made by me. It has been suggested, however, because I have not made continual reference to the works of previous authors, that I was ignorant of the literature, and several papers have been cited as contradicting my conclusions. As a matter of fact I believe I am fully aware of all that has been written on the subject, and have, in each case where my attention has been called to a paper, studied it carefully, and found nearly always that the facts given bear me out, though the con- clusions are adverse ; simply because no author has seriously questioned the univer- sally accepted homology of the mouth parts in the various orders. My own studies have been made on a basis so radically different from any heretofore accepted, that my results must stand on them alone, and my conclusions, if valid, must stand on the facts as they appear to me. I have used principally the dissecting needles in my work; but have not neglected the section cutter. This latter instrument has been rather too much used at the expense of the needles, and its results, though undoubtedly accurate as a record of facts, are easily misinterpreted if the basic homology which is assumed 176 AN ESSAY ON THE DEVELOPMENT to exist is inaccurate. For the reasons just given no references to previous writers will be made, except incidentally, and as I have in some respects modified my views as to the homology of certain of the parts, I will go into the entire subject in such detail as is necessary to prove my point ; but without reprinting my first paper, which should be herewith consulted. I do not expect denial at this day, when I claim that no explanation of the homol- ‘ogies of the mouth parts of insects can be considered satisfactory which will not stand the test of criticism by the theory of evolution. If we assume the origin of all insects from one original type, we must, necessarily, assume that all the mouth structures are derivatives of one type, and we must so study them as to be able to explain, step by step, just what specializations have occurred. We may not be able to complete en- tirely each link in the chain of evidence, but we can, at any rate, reach a result con- sistent with all the facts known to us. Any explanation which satisfies all the require- ments of a regular and natural development is to be preferred to one which demands an unexplained specialization of any part, not in line with its function in other series. It is therefore necessary to study carefully the make-up of every separate mouth organ, and of every sclerite in each, to become thoroughly familiar with its uses and to ascertain the lines in which it varies or develops. It may be premised that the mouth parts of the Hemzptera in their present con- dition are not included in the range of these studies. I have examined numerous specimens and have devoted especial attention to Cicada and Thrips—the latter classed as hemipterous for present purposes only—and | believed at one time that I had made out the remnants of a mandibular sclerite, and so published it. Mr. C. L. Marlatt questioned my conclusions and asserted that the mandibles are represented by one pair of bristles. While I believe that I was wrong in my identification of the man- dibular sclerite, | am yet convinced that I am correct in claiming that beak and setze are all maxillary structures. I have concluded, however, after a careful review of all my preparations and of what has been written, that the Hemzptera in the mouth strue- ture are not descended from any well-developed mandibulate type, and that no trace of true mandibular structure occurs in any present form. In other words, the Hemiptera equal all the other orders combined in rank, for all others are mandibulate or derivatives from a mandibulate type. The archetypal Thy- sanuran with undeveloped mouth organs varied in two directions—toward the haustellate type now perfected in our present Hemiptera, and to the mandibulate type: and there has never since been any tendency toward a combination. The haustellate type proved ill adapted for variation and there is, in consequence, a remarkable same- ness throughout. This kind of structure must be studied on an entirely new basis to OF THE MOUTH PARTS OF CERTAIN INSECTS. 177 get at the steps by which the present “beak” was developed, and my material is not sufficient for that purpose. The mandibulate type, on the contrary, proved well adapted for variation, and its differences and modifications are here traced. For convenience, Kolbe’s figures of the mouth parts of a grasshopper are repro- duced on Pl. III, Fig. 22, and may be referred to in connection with the following explanation. In a well-developed mandibulate mouth we have, forming an upper lip, the lab- rum, often notched in front or toothed; but never a paired organ, never with appen- dages, and never mechanical in function. It is articulated at base to the clypeus and serves to shield or protect the mouth in front; as a matter of fact, not a functional mouth structure at all. It is marked /dr in all figures. More or less intimately associated with it on the inner side is the epipharynx, which is compared in function with the palate of vertebrates, and is furnished with sensory hairs, pegs or pittings. It may be so closely united with the labrum as to form, prac- tically, a part of it, or may be entirely free. If free from the labrum, the epipharynx is more closely united with the other mouth parts, and in such cases its supports go to the mentum or labial structures. Not infrequently it has attachments to both. In form it may be a mere pointed process, or it may be a more or less divided, plate-like organ; but its functions are gustatory or sensory in all cases—it never becomes a functional mechanical structure, and I have never found it without a more or less de- veloped labrum to shield it. It is lettered ep? in all figures. Just below these covering and gustatory organs is a pair of mechanical structures —the mandibles—set, one on each side of the head, and attached to the inferior margin of the epicranium or an extension from it. These mandibles are never jointed, rarely bear appendages, and never such as are functional, rarely have a movable tooth, and are usually solid and highly chitinized. They are actually made up of a number of sclerites, laterally united, but distinguishable in certain types like Copris, Pl. I, Fig. 8. I haye elsewhere named and homologized these sclerites; but as the matter is not in dispute, and of no importance here, a simple reference to the figure in which they are named is all that is necessary. The position of this pair of mouth structures is inva- riable. They are completely disassociated from the maxillary or labial structures and remain attached to the head when all the other parts are removed in a body. They attach by socket joints to the epicranium and their tendons and muscles attach to its inner surface. They never change in function, never become united with or attached to the other mouth organs and never become internal structures. When not needed for chewing or biting the tendency is to obsolescence: never toward a change into a thrusting or piercing organ, so far as my observations extend, A. P. S.—VOL. XIX. W. 178 AN ESSAY ON THE DEVELOPMENT Below the mandibles are found a pair of maxill:e, made up in all cases of a number of sclerites, and nearly always supplied with palpi or jointed tactile organs. The more particular consideration of these organs and their parts may be somewhat deferred. Forming the lower lip and closing the mouth inferiorly is the labium, also made up of a number of sclerites and usually furnished with palpi. It is never entirely paired in existing insects, but is assumed to be made up of two more or less united structures, similar in essential character to the maxilla, as has been well stated by Prof. J. H. Comstock. This labium is an exceedingly important structure-and forms the oral termination of the digestive tract or the mouth of the cesophagus. Attached to the inner surface of the labium is the hypopharynx, a variably devel- oped structure, which is supposed to be the remnant of another originally paired organ, the endo-labium. I have never seen the genera in which it is said to be well devel- oped, hence have no well-founded opinion to offer. I find it uniformly a single organ, often highly developed and gustatory in function, sometimes a merely passive structure more or less closely attached to the ligula, usually very near the opening into the digestive tract. Briefly recapitulated, the insect mouth, when most fully developed, consists of two pairs of lateral jaws moving in a horizontal plane between an upper and a lower lip, which are furnished with gustatory structures forming the roof and the floor of the mouth respectively. This mouth is adapted for biting and chewing and varies to types adapted to lapping, to sucking only, and to piercing and sucking. The problem before me is to ascertain by what modifications these different changes in type have become established. If we examine the head of a well-developed mandibulate insect from the under side—Copris carolina, Pl. I, Fig. 7, may serve as type—we find, centrally, the gula or throat, bounded laterally by the gen:e or cheeks, extending to the posterior margin of the head and bearing anteriorly the labium. The labium when carefully dissected out is found to consist of a broad basal plate, the submentum, more or less firmly articu- lated to the gula and never, in existing insects, a paired organ. It bears anteriorly another plate, the mentum, also a united organ, though sometimes traces of a division are apparent. It is usually smaller than the submentum, sometimes membranous, often entirely separated and frequently so united with the latter part that the two are not separable. Though the submentum is the most persistent and dominant structure it has been customary to use the term mentum to apply to the united sclerites, and it will become convenient for me to so use the term hereafter when no confusion or mis- understanding can be occasioned. The structure is lettered m in all the figures. OF THE MOUTH PARTS OF CERTAIN INSECTS. 179 Attached and articulated to the mentum anteriorly are the central ligula, a pair of paraglossa bounding it, and a pair of palpigers, one at each outer edge, bearing the labial palpi. The ligula or glossa, marked gl in all the figures, is a paired organ only in the more generalized orders, and is usually present as a single, central structure, which may be either chitinous and rigid or membranous and flexible. It is the most persistent of all the labial structures, is never attached except to the mentum, and always has asso- ciated with it the hypopharynx where that is present. We always find at its base the opening into the alimentary canal, or cesophagus, as this part of it is termed, and this must eyer be the test of labial structures—that they are attached to the mentum and have at their base the opening into the alimentary canal. The association is never broken, and the base of the ligula, whatever its form or however it is modified, always marks this point. On the other hand, by tracing the alimentary canal to its external opening, we can always recognize the ligula by its position, however little it may re- semble normal types. The paragloss:e are sometimes intimately united with the ligula, sometimes com- pletely separated from it: they may be of the same or a different texture; but they always arise from the mentum on each side of and close to the central structure. Their tendency is to obsolescence, but they may become united and form a bed for the ligula which remains the inner organ. ‘Their range of variation is not great; they are never jointed, and never become mechanical structures. The palpi are tactile in function under all circumstances, though they may lose this function in great part and may, by coalescence, form a sheathing to the ligula. They are never, under any circumstances, attached anywhere except to the mentum, directly or indirectly, and their location must be constantly the same. They cannot, without losing their essential character, become disassociated from the mentum, nor can they ever form an envelope or covering for it, or for the submentum, with- out a change entirely at variance with any reasonable theory of development. ‘To accomplish this they would first lose their character as labial appendages. In brief, the labium is the external beginning of the alimentary canal, and none of the parts ever lose this association. Whatever their modification, no labial structures can ever be joined to the sides of the head outside of mandibular or maxillary structures. As an illustration of the most generalized form of labium at present known to me, the roach ( Periplaneta orientalis, P1. II, Fig. 16) may be selected. Here we find the mentum with a well-defined impression resembling a suture, and bearing a broad paired structure, from which arise the slender, two-jointed ligula, the broad, fleshy 180 AN ESSAY ON THE DEVELOPMENT paraglossxe, and the three-jointed labial palpi. This generalized structure fixes the relation of the parts, and from it we may pass to more specialized types. In Harpalus caliginosus (P\. II, Fig. 7) we have a case where the ligula forms a single, central organ, laterally bounded and on one side completely enveloped by the softer paraglossxe. The location of the palpi remains essentially the same. We have here two cases showing the change of a two-jointed membranous paired organ into a single, rigid, chitinous structure, and the identity of the parts is not questioned, nor I believe, questionable. If we carry our dissections one step further and from the fresh specimen remove not only the highly chitinized parts, but also the softer attached structures, leaving maxillze and mandibles undisturbed, we find in all cases the cesophagus in the cavity below the mentum and submentum, and these sclerites afford attachments for neces- sary muscles. They also form, by means of chitinous extensions and processes, a chamber or cayity protecting the cesophagus and supplying muscular attachments when a sucking or pumping structure is needed. Thus the mentum and submentum, whether separated or united, are always inferior coverings to the cesophagus. To sup- port this structure, processes sometimes extend almost or quite to the upper or anterior surface of the head, and in many cases, where the epipharynx is separated from the labium, it is connected by means of long processes with the mentum. ‘This is true in many Coleoptera, quite usual in the Hymenoptera, and occasionally found also in the Diptera. In PI. I, Fig. 6, is a lateral view of the labium of Copris carolina when completely dissected out, and the clubbed processes, loosely attached to the inferior prolongation of the submentum, normally support the epipharynx. In PI. I, Fig. 9, and Pl. II, Fig. 18, we note similar processes in Andrena vicina with part of the epi- pharynx still attached, and in Polistes metricus, where the structures are complete. Precisely the same structures occur in Stmulium (Pl. I, Fig. 1°), as will be more fully noted hereafter. It may be stated that I have adopted the term ‘“ fulerum,” used by Macloskie and others, to designate the structure formed by the mentum and submen- tum and containing the beginning of the alimentary canal. In Polistes metricus (P\. I, Fig. 18’) I show the labium completely dissected out, with all its attachments, viewed laterally. It will be noted that here the mentum and submentum are united, highly chitinized, and form a scoop-shaped structure, bearing at one end the labial structures and enclosing normally the beginning of the cesophagus. Attached by long chitmous rods to the posterior angles is the epiphar- ynx, so that hypopharynx and epipharynx are borne on the same base, are closely op- posed to each other and may be manipulated by muscles arising close together. The origin of the palpi is shown from the mentum. On PI. II, Fig. 18%, are shown ligula OF THE MOUTH PARTS OF CERTAIN INSECTS. 181 and paraglosse of this same Polistes. The structures are here membranous, some- what bladder-like, and well adapted for lapping by means of flattened, bent processes, set in series on the entire inner surface. The paraglossze are completely separated and the mouth opening is shown at the base of the figure, as well as the chitinous ring marking the beginning of the cesophagus. In Andrena vicina (Pl. I, Fig. 9) we find a similar yet quite different structure, 7. é, the same parts, used for much the same purpose, yet considerably modified in de- tail. The mentum is here much longer, more shallow, but similarly bears the epiphar- ynx on chitinous rods. The ligula is more inflated and the paraglossie are much reduced, but the palpi originate as before, and we have simply an illustration of the variation in form found in this united mentum and submentum. It is important to note here that in Polistes, Andrena, and indeed the Hymenoptera generally, the labial structures are free from all lateral attachments to the head and may sometimes be pro- jected forward quite a distance. The attachment to the head, indeed, is muscular and membranous entirely, and there is no direct articulation to any point by chitinous or rigid processes. ‘There is nothing therefore to prevent the growth of the head sclerites around the mentum, which would thus become an internal structure—as has actually happened in the Diptera. Another feature upon which Dr, Packard rightly places great stress is that a salivary duct opens into the hypopharynx at the base of the ligula, which he thereby identifies. As this ligula is always attached to the mentum, it follows that this struc- ture may be identified in the same way, while no structures not originating from the same point can be labial in character. Before studying further the specializations of the labial structures, it may be well to say that they sometimes tend to become useless or obsolete, or so much reduced that they are difficult of recognition ; and, curiously enough, in such cases the palpi seem to be the persistent organs, Thus in some species of Scoliide among the Hymenop- tera the mentum bears only little, feebly developed palpi. A striking case is in the Panorpide, where on Pl. III, Fig. 4’, the mouth structures of Bittacus strigosus are shown. Here ligula and paraglosse have disappeared entirely; but the palpi are dis- tinct and the curiously developed hypopharynx marks the beginning of the opening into the cesophagus. A modification of this type is to be found in the Lepidoptera, where practically in all cases the palpi alone, attached to a plate of variable size and shape, represent the labial structures. It seems a long jump from the reduced type in Panorpide to the fully developed labium of the Apide ; yet, except for the fact that all the parts are much elongated, 182 AN ESSAY ON THE DEVELOPMENT there is no difference from Andrena or Polistes, which have been already studied. I have found no species which shows all the parts more fully developed than Xenoglossa pruinosa (Pl. Il, Fig. 15). Here all the parts are equally developed and all are func- tional; hence it makes a good starting point. The mentum is not shown in the figure except at the point to which the other parts are attached, and surmounting it cen- trally, we find the ligula; here a united, though extremely flexible organ. Lying cen- trally upon it, so as to close a groove, is the hypopharynx, in this case not easily separ- able from the ligula. Arising close to the central organ on each side are the para- glosse; almost as long as the glossa itself, flexible, unjointed, flattened and a little incuryed at the margins so as to form, when closely applied to it, a partial shield for the ligula. Outside of all, situated at the outer margins of the mentum, are the palpi. These are four-jointed; but the basal joints are enormously elongated in proportion to the terminal two, and they are also flattened out, broadened and infolded, so that when at rest they cover and almost conceal the other labial parts, though not extending for- ward as far as they. In this insect the structures just described are almost entirely covered by the maxillz, and a transverse section (Pl. I, Fig. 15*) is interesting and instructive. It represents the structure at about the middle of the combined maxillz and labium and illustrates the relative position of the parts. The tendency in the bees is toward a loss of the paraglosse, which shorten grad- ually until they disappear altogether, as represented in a species of Bombus figured in Pl. I, Fig. 15. Every intergrade is represented in any good series of bee mouth parts, and in their rudimentary condition, without function, they appear in Bombus sp., represented on PI. III, Fig. 6. The palpi retain their unique development, and in the figure just cited are seen to be as long as the ligula itself, the basal two joints en- folding it almost completely, while the terminal joints are much reduced in size and set near the tip of the second joint, on the outer side. In other species these terminal joints are proportionately yet more reduced and are sometimes difficult to find. The essential point to be noted is that at their best development the paraglossz are not jointed and that they tend to complete obsolescence in the most highly specialized types. The palpi in Bombus require a little further examination: Reference to the figure last cited will show a short segment between the mentum and the first long joint, and this is membranous in texture. The mouth parts in Bombus are folded when at rest and the hinge is at the mentum; hence the necessity for some such pro- vision to enable the palpi to bend safely. Now let us assume that the ligula of this Bombus became rigid and chitinized, and that the edges of the palpi enfolding it became united to form a complete cylinder ; and then let us examine Hristalis tenax (Pl. ILI, Fig. 5) in the light of this assump- OF THE MOUTH PARTS OF CERTAIN INSECTS. 183 tion. First let me say that I have already shown that a change from flexible to rigid ligula is not uncommon, and the suggested union of the palpi is a much less violent requirement than that imposed by the current explanation of the Dipterous mouth. Referring for a moment to Pl. I, Fig. 3, we see the entire mouth structure of Hristalis tenax. Above is the mentum and submentum, very like the structure already de- scribed for Polistes and entirely homologous with it, and at its tip we find arising in a group the structures further enlarged at Pl. IIT, Fig. 5. Centrally we find the now rigid ligula, deeply grooved in the middle, the channel closed by a flattened, also rigid and chitinized hypopharynx. Loosely enveloping this central ligula is a more mem- branous cylinder, evidently made up of two lateral halves, two-jointed, and the ter- minal joints separated or paired except at the base. As in Bombus the mouth of Hris- talis is hinged, and the joint is also at the base of the ligula. The latter organ is so articulated as to allow of the flexion; but in the palpi we find again the provision already noted in Bombus—a flexible, membranous, pseudo-segment. Now if we sec- tion the Bombus and Hristalis at the middle, we find the cuts alike, except that in Fristalis the palpi are completely united over the hypopharynx and closely approxi- mated at the opposite side. If we section near the tip, the cuts in both cases are identical. That this united structure in Hristalis is the united labial palpi seems to me beyond doubt. In the first place, the point of origin is normal, next to the ligula and at the tip of the mentum; and, secondly, it is a jointed organ and therefore can- not be paraglossa. It is in all points the structure of Bombus, with the terminal joints lost and the two halves united for the greatest part of the distance. That the parts named mentum and submentum are really such, is proved by the fact that the hypo- pharynx, which is not in dispute, originates from and that the cesophagus originates within it. In Bombus fervidus the ligula is unusually developed and much longer than the labial palpi, while the paraglosse are wanting. In PI. III, Fig. 12, is a camera lucida sketch of the labial parts of a carefully mounted specimen. The structures here are exactly as normally held when at rest, and only the mentum is a little crushed by the cover glass on the shallow cell. Now chitinize this whole structure thoroughly, and then compare with the drawing of Chrysops vittatus (Pl. III, Fig. 13) made in the same way. The magnifications are different, of course, the Bombus being drawn at short range with a four-inch lens while the Chrysops was drawn at long range under a one-inch objective. The object was to get the two of approximately the same size for conyenience of comparison. In the Tabanids the mouth parts are rigid and not flexed, and no sort of joint or hinge is required ; hence the structures are all rigidly united at the base to the mentum. In Bombus fervidus the palpi are reinforced by a heavier 184 AN ESSAY ON THE DEVELOPMENT chitinous rod a little to one side of the middle, and just this sort of structure we find everywhere in the Tabanids, lying outside of the ligula at base, articulated to the outer edge of the mentum. This, in fact, first led me to suspect the true nature of the structure. If now we section Bombus and Tabanus near base, the cuts will be alike, save that the palpi in the latter are united at one margin. If the cuts are made toward the tip, the sections are alike—ligula and hypopharynx alone appearing in both cases. We have then, in Chrysops also, a complete labium, save that the paraglossve are absent and the palpi are united on one edge. In the Simuliide are many interesting species with generalized mouth structures, and of these I have studied the “ Buffalo gnat,” from material kindly furnished by Dr. Riley, an undetermined Simuliwm sent me in numbers by Prof. Aldrich, and an undetermined little midge collected by me at Anglesea, N. J. The species are prac- tically identical in the labial structures, and here again the mentum and submentum strongly recall Polistes and other Hymenoptera. The hypopharynx is well developed and the ligula are nearly divided; but I haye no satisfactory sections of this insect and the relations of the parts are not clear to me. At Pl. I, Fig. 1’, the labium of the “ Buffalo gnat” is shown. In the species sent by Prof. Aldrich I succeeded in getting a dissection illustrating the connection of the epipharynx with the mentum, and this is illustrated at Pl. 1, Fig. 1°. This is really an exceedingly interesting speci- men and it clears up the relation of the frontal prolongation of the mouth. That the structure so labeled is really the epipharynx there is little room for doubt, and the location of the little, chitinous, toothed processes, and their character, leaves no doubt in my mind that they are mandibular rudiments—exactly as I claimed in my firet paper. That they can be dermal appendages, as has been claimed, does not seem rea- sonable to me. They are too highly chitinized in comparison with their surroundings, and why should they so completely resemble miniature mandibles? I do not know of any case of dermal appendages of a similar character, and it is at least passing strange that such should be developed exactly where, normally, mandibular rudiments might, be reasonably expected. The tendency in the piercing Diptera is constantly in the direction of simplicity of labial structures, and so we gradually note the loss of all trace of accessory labial structures, leaving the ligula and hypopharynx as sole representatives. In the As- tlide there are no other attachments to the mentum, as shown in PI. III, Fig. L’. These apparently single structures are sometimes interesting in section, as appears in Stomoxys calcitrans, Pl. I, Fig. 11. Here the cut shows two crescent-shaped struc- tures connected at one edge by the thinnest kind of a chitinous shell, and closed oppo- site by a hypopharynx, which is almost tubular in structure. OF THE MOUTH PARTS OF CERTAIN INSECTS. 185 Very interesting is the modification found in the Hmpidz, illustrating the extreme in the loss of parts; for here the hypopharynx is also wanting, though the salivary duct remains, opening into the grooved ligula, as shown in PI. III, Fig. 2". In this case the hypopharynx is replaced by an extension and peculiar modification of the labrum. ‘This sclerite is elongated so as to extend to the tip of the labium, and is very much dilated, somewhat bulb-like at its base. In PI. III, Fig. 2’, labrum and ligula of Rhamphomyia longicauda are seen from the side, while in Pl. IT, Fig. 13, are shown the same structures in Hmpis spectabilis. The edges of the labrum are turned under sufficiently to leave a central channel just large enough to receive the ligula, with which it then forms a closed tube through which the food is taken. In most of the Muscid flies we find a structure approximating Pr/stalis with the labial palpi removed; and the parts may be longer, or shorter, or differently developed, while adding nothing to what has been already shown; they are, essentially, reduced piercing structures, no longer functional. We have, however, in certain other species, where the mouth structures are short, very poorly developed labial structures. So in Hermetia mucens (P1. II, Fig. 14) the broad and large mentum bears only a short, scoop-like ligula. The specimen from which the figure was made was somewhat distorted in mounting and the ligula is turned just half round. Similar structures oceur in the Bibionidw, and Huparyphus bellus (Pl. I, Fig. 12) is not essentially different. Heretofore the hypopharynx has been referred to mainly in species in which it was feebly developed and played but a passive part as a covering structure. It is sometimes a highly specialized sensory structure, though it varies greatly, even when functional. A very curious type is found in Bittacus (PI. III, Fig. 4’), where it takes the form of a simple cylindrical process, set with spines, almost like an odd joint of some slen- der palpus. In Copris carolina, Pl. I, Fig. 4, showing the epipharynx, may be accepted as a fair representation of the hypopharynx as well, save that the latter is on amuch reduced scale. The opening of the salivary gland is in a dense mass of spe- cialized spinous processes. In the Libellula, among the dragon flies, we have an inflated, somewhat tongue- like organ (Pl. I, Fig. 10”), in which the salivary duct is plainly traceable to its open- ing among a mass of crossed, specialized spines. The surface is richly supplied with sensory pittings and tactile hairs. It is a great modification from a structure of this kind to the simple, ribbon-like form of Bombus, or the flat, slender, chitinous form in Tabanus ; but the intermediate stages are all present. To recapitulate concerning the labial structures. The mentum and submentum A. P. §.—VOL. XIX. &. 186 AN ESSAY ON THE DEVELOPMENT cover the wsophagus. They may be united so as to form a single organ, and their tendency is to become internal head structures. The ligula has at its base the opening into the alimentary canal; it is rarely paired, may be rigid or flexible, and has closely associated with it the hypopharynx, recognizable by the salivary duct which it shel- ters. The paraglossie arise on each side of the ligula or glossa, and may be chitinous or membranous. They are neyer jointed, never developed for any specific mechanical purpose, and their tendency is to become obsolete. The labial palpi are essentially tactile and never become mechanical save as they may form a covering or sheath for the ligula. From the most generalized type found in the Blatt¢de the modification is first from a divided to a single ligula; next to a disappearance or obsolescence of the para- elossie ; later the labial palpi also disappear, and finally the hypopharynx is also dis- pensed with. There is no break, and nowhere is there any violent change of structure or function. We are now ready to take up the maxillee, which, though composed of a larger number of sclerites, are usually more easily understood in the ordinary type of man- dibulate insect. The organ is usually paired and never so completely united as the labial structures. The two parts are always external to the labium, which it is their tendency to enfold, and they never have any direct connection with the alimentary canal. Though the maxillary structures tend to form a covering or sheath for the labium and its appendages, there is never any intimate connection between them. No part of the maxilla ever unites with any part of the labium or with any of its appen- dages. The maxille are essentially mechanical structures, and their range of variation is sufficiently great to meet the most diverse possible demands made upon them. A dis- tinct and fundamental characteristic is the fact that each set of sclerites has its own peculiar possibilities and limitations, and once these are understood the most highly specialized type becomes simply explicable. On PI. III, Fig. 17, is a copy of Prof. Comstock’s figures of Hydrophilus, show- ing the maxilla from both surfaces, and these may conveniently serve as a text to explain the sclerites composing it. At the base is the cardo or hinge, giving attach- ment to muscles and tendons articulating it to the head. It is to be noted that there is no firm or chitinous articulation to any head sclerite, and except by muscles or ten- dons no direct attachment. This we found the case also in the labium in the more specialized forms, and in the Hymenoptera, for instance, labium and maxillze together are easily dissected out without cutting any but muscular tissue, and without breaking any chitinous connections or joints. This is in marked contrast with the mandibles which, when functional, are always firmly articulated by chitinous joints to the external OF THE MOUTH PARTS OF CERTAIN INSECTS. 187 head sclerites. Supported upon the cardo is the stipes or foot-stalk, deriving its mus- cular attachments largely from the cardo; but to some extent from the head itself, and this feature is a variable one. Surmounting the stipes is a palpifer or palpus-bearer, to which is attached a palpus, varying in the number of its joints. This derives all its muscles from the stipes in the typically developed maxillee. On the inner side of the stipes is attached the subgalea, deriving its muscles from the head in large part; and this bears a two-jointed galea or hood. It is a matter of some importance to note that this galea is never more than two-jointed under any circumstances, and that the tendency is to maintain that number; though in many instances it is reduced to one only. It is the most persistent as well as the most variable of the maxillary struc- tures, and is present when any of them exist at all. Inside of the subgalea, and attached to it as arule, is the lacinia or blade, which may or may not bear a digitus or finger. In the figures just cited we find what may be termed a normal or proportionate development of all the parts, in which no one sclerite is unduly developed or special- ized. Before attempting to study specializations it is important to note that, when carefully examined, the sclerites are seen to be arranged in three parallel series. That is to say three separable parts have grown together laterally, and this union bears with it the possibility of future disunion or separation for special purposes. We have as the inner series lacinia and digitus; as the middle, subgalea and galea; and as the outer the cardo, stipes and palpifer with the attached palpus. Now if we examine some of the Neuroptera, e. g., Stalis (Pl. II, Fig. 16), we find this lateral arrangement very strongly marked, and it is easily understood that each of these parallel sets may have their own peculiar limitations, and that each may be separately and independently modified. But lest this seem, after all, a far-fetched conclusion, let us examine the maxillx of Bittacus strigosus (Pl. III, Fig. 4”), and we find almost exactly the hypothetical state of affairs actually existing! Lacinia, galea and palpifer all separated, of nearly equal length, but of quite different appearance. ‘The appearance of a transverse sec- tion made at about the middle is shown as Fig. 4". Fora generalized type this form is especially valuable, and we may fairly use it as a guide in our discussion of maxillary possibilities. There is no absolute rule in the matter, but usually the galea tends to become the dominant maxillary organ. In many Neuroptera, and especially in their larval stages, the laciniate structure is best marked, as illustrated in Pl. III, Fig. 9, representing the maxilla of a Perlid larva Here the galea is reduced to a subordinate rank, and in many predaceous Coleoptera it is truly palpiform. In many Orthoptera the development of the galea justifies the name by forming 188 AN ESSAY ON THE DEVELOPMENT an almost complete hood over the lacinia. ‘This is well illustrated in the maxilla of the oriental cockroach, Periplaneta orientalis, shown at Pl. ILI, Fig. 8. At this point a comparison of the figure just cited with the galea of Simulium (PI. I, Fig. 1*) will prove interesting and instructive. In the Hymenoptera the galea dominate throughout ; no elongated palpifer is ever developed, and indeed the maxillary palpi are sometimes almost rudimentary in the Apide, as shown at PI. III, Fig. 15. : In Polistes, illustrated at Pl. II, Fig. 18°, we find a common type of the Vespide, where the lacinia forms a small, blade-like structure, free for almost its entire length, and the maxillze as a whole shelter a large part of the labium. In those cases in which the “ maxille ” are elongated, the galea is usually the organ affected. Thus in many Meloids among the Coleoptera we have the mouth parts elongated, and a study of the maxilla of Wemognatha (Pl. III, Fig. 20) shows at once the scler- ites concerned. Here the lacinia is much reduced, and if we remove it altogether we have the normal Lepidopterous maxilla, which tends to a locking together to form a complete tube. Recently it has been found that in certain Lepidoptera the lacinia are actually present, and the figures which I have seen indicate a structure in all essentials like that of Nemognatha. While speaking of the Lepidoptera it may be well to cite Pronuba (PI. III, Fig. 21), in which the palpifer is elongated in the female and highly specialized into a sen- sory and tactile structure, though unjointed. In a well-prepared specimen the point of origin is perfectly clear, and it is entirely homologous with the structure seen in Bétta- cus. In the male (PI. III, Fig. 19) the “tentacle” is not developed, though the palpifer is enlarged to some extent. In the Apide, among the Hymenoptera, the lacinia disappear entirely in extreme cases, or are at least greatly reduced, while as already stated the palpi are sometimes scarcely visible. The galea, on the other hand, is very prominently developed, and when at rest envelopes the ligula and paraglosse almost completely. In PI. III, Fig. 15, is represented the usual appearance of all the parts separated, while at Pl. II, Fig. 15", the transverse section of the mouth structures of Xenoglossa pruinosa shows their normal relation when at rest. It is seen that the galea actually overlap somewhat at one margin, and a union along this line would be scarcely considered a violent stretch of the range of variation. Assume such a union, eliminate the paraglosse which are organs tending to obsolescence, and then compare with the transection of Hristalis fenax (Pl. I, Fig. 3"). If the palpifer be eliminated from this latter figure the cuts are practically identical. Returning to our figure of Bombus (Pl. II, Fig. 15), we note at the outer edges OF THE MOUTH PARTS OF CERTAIN INSECTS. 189 of the galea a series of ridges which, under a high power, look extremely suggestive of the structures found in the labellx of Diptera, especially where, as for instance in Bombylius, the pseudotrachea are imperfectly developed. These ridges vary much in the species; but are particularly marked in a little Andrena near vicina, if not that species itself. Here we see (PI. III, Fig. 3) the entire inner face clothed with a thin membrane which is crossed by numerous closely set fine chitinous lines! I claim that this structure is the homologue of the pseudotracheal structure in the Diptera, and that in the latter order it is in the galea that the development occurs, as it does here in the Hymenoptera. The relative differences in size are not of importance. As to the particular use of this structure in Andrena I have no suggestion to make. In the Proceedings Ent. Soc. Washington, Vol. III, Mr. Ashmead figures on Pl. III, some very suggestive mouth structures of parasitic Hymenoptera, of which that of a Pteromalid is reproduced on Pl. III, Fig. 18. The central labium with its attached structures is much reduced in size, and the maxille, bearing the well-devel- oped palpi, are reduced to a single structure, the galea, resting upon what may be con- sidered the stipes. Now if we bring these two parts of the maxille a little more closely together, we have almost the exact structure seen in Libio (Pl. II, Fig. 11’). The basal ring, bearing the palpi, corresponds almost exactly to the basal ring of Pteromalus except for size, while except that the surmounting galea are two-jointed, the correspondence with the upper portion of the structure is equally marked. ‘The labium in Bibio is much like that figured in Pl. III, Fig. 14, for Hermetia, and in PI. I, Fig. 12, for Huparyphus. I am making no very risky statement when I assert that the sclerite to which the maxillary palpi are attached must of necessity be maxillary; and further, it is equally safe to say that no maxillary sclerite can bear a labial appendage: and certainly not a labial palpus. It would be an absurdity, contrary to all the laws of a natural deyelop- ment, for a modified labial palpus to become attached to the sclerite bearing also the maxillary palpus; while if we consider it the two-jointed galea, its position is normal, requires no assumption of change or character, and does not differ in any essential points from the gale of the roach (PI. III, Fig. 8). Yet these two joints in Bzbio will, with a ridged membrane thrown over them, represent the labellate tip of the Muscid proboscis. That such a ridged membrane is well within the range of galear variability we found in the Andrena near vicina (Pl. II, Fig. 3). The structure in Huparyphus -bellus (Pl. I, Fig. 12) resembles Pteromalus yet more closely, in that a single ring only surmounts the segment bearing the palpus. In this instance the maxilla is reduced to exactly the same segments seen in the Hymen- opteron, and logic demands that we recognize them as the same. In this case, how- 190 AN ESSAY ON THE DEVELOPMENT ever, the lower ring is complete—7. ¢., the two halves of the stipes have become united. That it must be stipes is shown by the fact that it bears the palpus, and again the surmounting sclerite must be maxillary also. There are other species allied to those already cited in which similar structures occur; but I need for the present call attention to only one more; a species of Olfersia (Pl. I, Fig. 19). Here the ring is complete in front, but broadly open behind, and bears the chunky, single-jointed palpus. Surmounting is a single sclerite, very much resembling in appearance that of Pteromalus, and undoubtedly homologous with it. Of course Olfersia is parasitic in habit, and the mouth parts are specialized for blood- sucking ; but the sclerites composing them are nevertheless derived from the same source as in the “higher” types. I have several times referred incidentally to Simulium, and of this the galear structures are figured (Pl. I, Fig. 1"). Dissecting the parts out carefully we find an almost complete ring at the base, the stipes, to which the palpus and palpifer are attached. Surmounting this is a pair of sclerites, each almost a half cylinder, repre- senting the subgalea, and bearing the two-jointed galea. Here again I claim that the three joints just referred to must be maxillary because they are directly articulated to the sclerite bearing the maxillary palpi, and the labial structures are all shown at Rigel” A step in the direction of union we find in the Anglesea gnat or midge—also a Simuliid, to which reference has been already made. Here we see (Pl. I, Fig. 2") the subgalea united most of their length at one side, while the galear joints are yet free. The basal stipes is not figured because none of my specimens showed it clearly ; but the palpifer, palpus and lacinia, as they are connected with it, are shown in the specimen. In the Aszlidw we find another suggestive structure, studied in the light of the facts already set out. Here we see, as illustrated Pl. II, Figs. 1* and 1’, the basal stipes well developed, united posteriorly, but separated in front. The palpifer and its attached palpus are situated at the sides, clearly articulated to the stipes, whose char- acter is thus fixed. Attached to this stipes is a broad, infolded structure, united be- hind but open in front; maxillary because of its attachment to the stipes, and sub- galea from its location. It bears in orderly sequence the two-jointed galea of which the terminal joints are free. The species of the Asiléde are large and easily dissected, and the figures were drawn from a species of Laphria. The attachments are but little different in the species, and as the figures illustrate the structure from both front and rear, the position of the joints should be clear. These figures will be again referred to in another connection. OF THE MOUTH PARTS OF CERTAIN INSECTS. 191 Jn all the species heretofore cited the galear joints were more or less distinct and the pseudotracheal system was little or not at all developed. As the face of the joints becomes covered by a ridged membrane the texture of the entire structure changes. It becomes less chitinized, and the chitine is not evenly distributed, causing sutures to become indistinct and poorly marked. Yet, keeping in mind the general line of varia- tion, we can usually reach a correct conclusion. In a Leptid, species unknown, we find the appearance shown in PI. II, Fig. 1. Here there is a united basal plate, covered on one surface with a membrane, and from the chitinous portion arises the palpifer with its attached palpus. Surmounting the chitinous base are two joints, the galea, the chitinous parts of which only are skown in outline, the balance of the space being covered by membrane. Here again the attachment of the maxillary palpus to the basal sclerite determines the maxillary char- acter of all the sclerites directly articulated to it. In Hermetia mucens (P|. II, Fig. 17) the entire structure is much more membran- ous, yet the basal chitinous plate is paired, and while the parts are shown in a dis- torted position, the two galear joints and their relation to the basal, palpus-bearing structure is yet perfectly obvious. The other maxillary structures have completely disappeared, while what is left of the labium is seen at PI. III, Fig. 14. The mouth parts in some species of 7vpula are interesting, and a fair illustration of one of the “snub-nosed” species is seen at Pl. I, Fig. 5. Dlere the origin of the palpus at the immediate base of the chitinized part of the labella indicates its character, and if we divest the chitine of the surrounding membrane we get the appearance shown at Fig. 5%. Practically we have a completely paired organ, the relations of which are perfectly simple when the confusing and unimportant membrane is removed. The peculiar relation of labrum and labium in the Hmpide has been already noted, and this makes it easy to separate off all the other parts adhering to the margin of the head, but not in any way connected with the labium. The relation of the parts to each other in Empis spectabilis is shown on Pl. II, Fig. 13, while on PI. IT, Fig. 2’, are shown the maxillary structures of Rhamphomyia longicauda. In this latter figure we note that the parts, except palpifer, are entirely membranous. From the basal sclerite the palpi arise so as to form only a continuation of the membrane itself with an extremely slight attachment to the chitinous palpifer ; and to this very same mem- brane there is articulated by a slightly thickened suture the subgalea, united poste- riorly, but separated in front; and this bears in turn the indistinctly segmented galea. This entire structure obviously belongs together and is one organ—necessarily the maxilla. A very similar structure is found in Chrysops (Pl. II, Fig. 14) and in other species 192 AN ESSAY ON THE DEVELOPMENT of the Tabanide. Now it will be remembered that in this genus I showed the con- nection of all the labial parts with the mentum, where they normally belong; hence all the other parts must be, of necessity, maxillary. So we find also in Pl. II, Fig. 14, that the central labellate structure, two of the piercing structures and the maxillary palpi all arise from a single united basal sclerite, the stipes. In Fristalis tenax (Pl. I, Fig. 3) these labellate structures are shown, turned aside to expose the labial structures. Here also I showed the presence of labial palpi in close connection with the ligula and hypopharynx, normally attached to the men- tum, and again it follows that the other structures must be maxillary. Again also I must call attention to the fact that the palpi are mere continuations of the enveloping membrane, and that this membrane continues without break to the tip of the labella. Unless we are to believe that a continuous membrane may give rise to both the maxil- lary and labial palpi, we cannot possibly consider the labella as labial structures. I have now traced out what seems to me a continuous development of the modifi- cations of the subgalea and galea, and have shown, I think, that from Pteromalus in the Hymenoptera to /ristalis in the Diptera, a continuous chain may be constructed, requiring nowhere any change of character, function or location. No disassociation from other maxillary structures and no connection with labial structures. ' In taking up the modifications of the palpifer I am confined almost entirely to the Diptera, in which this sclerite is best developed. In Bittacus I showed its devel- opment to an elongated structure of no particular type or function and of about the same texture as the galea. In Pronuba I showed its development into a highly spe- cialized “ tentacle,” tactile and sensory as well as mechanical in character. In the Diptera it is quite usually present as an elongated, rigid, chitinous organ adapted for piercing. It occurs in all the piercing types and is present as a rudiment in many others. It undergoes a curious and interesting change in function as the Dipterous mouth changes from the piercing to the scraping or lapping type, and as it becomes flexed. The simplest form occurs in those piercing Diptera in which the proboscis is not flexed. Thus in the Buffalo gnat (PI. II, Fig. 9) it is a stout, semicylindrical piercing organ, enlarged both at base and at tip, at which latter point it is also toothed. The connection of the palpus with the subgalea was already shown on PI. I, Fig. 1%, and this shows how the chitinous palpifer forms part of the combination. The palpifer arises, normally, outside of the galea; yet at the tip it is found in connection with all the other piercing structures inside of that organ. How it gets there is illustrated in the Anglesea Simuliid (Pl. I, Fig. 2"), where all the maxillary parts are shown in proper connection, and it is seen that the palpifer enters the galear envelope in the OF THE MOUTH PARTS OF CERTAIN INSECTS. 195 incomplete articulation between galea and subgalea. By separating off the galear structures, the relation of palpifer and lacinia in Simulium is illustrated (on PI. I, Fig. 1°), and the convergence of the two at tip is not distortion, though perhaps a little exaggerated by pressure. The result of this change of position is that a section made near the base of the proboscis would show as illustrated on Pl. I, Fig. 2’, while one made nearer the tip would show as in Fig. 1”. Incidentally it will prove interest- ing to compare these sections with that of Bittacus strigosus (Pl. III, Fig. 4"), leaving out of consideration the abnormal labium of the latter. The resemblance is perfect, and the resemblance expresses fully the actual condition of the matter. A very simi- lar state of affairs exists in the Asilide (PI. III, Fig. 1°). Here the palpifer is the only maxillary piercing organ, and the figure itself shows clearly how easily it would swing inside the ample space left in the subgalea for its entrance. The curvature of the organ is such, also, that when in place it meets the central ligula so as to form a solid puncturing organ. So in Chrysops (PI. II, Fig. 14) the structure is seen to be similar to that ip Simulium ; but here, as almost everywhere else in the order, it is cylindrical or nearly so, in marked contrast with the lacinia, which is always flattened. As we get into types that have lost the piercing habit, the function of the palpifer fails or changes. If the species have a short, nonflexed proboscis, it simply dwindles from disuse. So in Stratiomyia and in Leptis (Pl. I, Figs. 1 and 2) it simply forms a little chitinous appendage to the palpus—a mere remnant without function. If, on the other hand, the species are able to flex the proboscis, another change takes place. There is needed then some lever to which muscles for flexing can be attached, and no structure seems to have been so easily adaptable as the palpifer. So we find in the Empide, where only slight flexion is required, only a small basal extension, shown at Pl. U, Figs. 4 and 3, for Empis spectabilis and Hulonchus tristis, and at P|. IL, Fig. 2’, for Rhamphomyia longicauda. In the Bombyliide is a step forward. The insects are not predaceous, have the habit of hovering over flowers and using the proboscis in feeding in that position. This requires a much better control, and as a result the basal extension is much better developed, as shown in PI. II, Figs. 6 and 7, illustrating Bombylius and Anthrax. As we get into types like Hristalis and other Syrphide, the basal extension be- comes the most prominent and the piercing portion diminishes in size (PI. IH, Fig. 5), and keeping step with this modification is a gradual separation of the palpus itself from the palpifer. This is well illustrated both in Hristalis and Spherophoria, and this tendency continues until in Lucilla (Pl. I, Fig. 10) the separation is complete, though the piercing portion of the palpifer is yet distinguishable. In ¢ ‘alliphora even A. P. S—VOL. XIX. Y. 194. AN ESSAY ON THE DEVELOPMENT this disappears and the chitinous rod is entirely disassociated from the palpus. Finally in Stomoxys calcitrans (P1. II, Fig. 12) there remains nothing to indicate the existence of any relation between the slender chitinous rod and the distant maxillary palpus. It is not in the least strange that guesses as to the character of this structure in Musca domestica should have been so often wide of the mark; though with a proper series as now shown, its origin is clear. There remains to be accounted for the lacinia, and this in the Diptera is the flat, blade-like structure generally identified as the mandible. It has been shown that while the lacinia is often the dominant organ in many mandibulate insects, the tendency is, on the whole, to a decrease in size, ending in the Hymenoptera in its entire elimina- tion. In the Diptera it is present in the blood-sucking species only, and it may be identified by its position and its relation to the other maxillary structures. It has been several times referred to incidentally, and in the Anglesea Simuliid (Pl. I, Fig. 2") its relation to the other maxillary parts is shown. In PI. I, Fig. 1°, is illustrated the connection between the palpifer and lacinia in the Simuliwn sent me by Mr. Aldrich. This connection is not fanciful but actual, and no sclerite so intimately con- nected with an admitted maxillate structure can be anything but maxillary. Again in Chrysops (Pl. I, Fig. 14) I have illustrated the fact that all the struc- tures which I consider maxillary have a common origin. At Fig. 14° I show the lacinia alone, and it is to be noted that at the base it is modified for attachment with reference to the palpus. Now unless this is a maxillary sclerite, why should it be modified to accommodate the maxillary palpus? Does it not seem rather absurd to believe that this can be a mandible brought to originate from one point with the palpifer and modi- fied to allow it to envelope at base the maxillary palpus ? One of the most serious difficulties in the way of the proper understanding of the mouth parts of haustellate insects has been the desire to provide for the mandibles on the theory that they are among the permanent structures. Yet I cannot understand why this should necessarily be the case. When functional, mandibles are essentially chewing or biting organs, and when the insects do not require such structures, it seems to me most natural that they should become obsolete: and that is exactly what has occurred according to my reading of the facts. Their functional character never changes; they simply dwindle from disuse and gradually disappear. So we find them in the Lepidoptera as mere rudiments, connected with a highly specialized maxilla ; and in the Rhynchophora they are sometimes mere remnants, occasionally reversed in position—exactly as I pointed them out in Simulium. I think that in view of all the evidence presented by me, none of the piercing organs of the Diptera can be consid- ered mandibles, and I cannot even yet, after carefully weighing all that Dr. Packard OF THE MOUTH PARTS OF CERTAIN INSECTS. 195 has written, see any reason why the rudimentary structures at the tip of the labral extension in Stmulium are not mandibles. If we refer back again for an instant to the Panorpids we note (Pl. ITI, Fig. 4") that in Bittacus strigosus the origin of the mandibles form an extension of a lateral head sclerite, with the labrum-epipharynx between them. In /anorpa the mouth structures are much shorter, set on an immensely elongated stipes, and at the tip of the frontal extension of the head we again have the mandibles, much reduced, with a small, lappet-like labrum-epipharynx between them. Now the situation of the rudi- ments in Simulium corresponds almost exactly with that of the undoubted mandibles in Panorpa rufescens (Pl. III, Fig. 4°); but in the Hmpide we find a yet more closely allied structure. I have already called attention to the peculiar elongation of the front of the head in this family, and now if we examine this at tip, in Mmpis spectabilis (Pl. I, Fig. 13") its very close resemblance to Panorpa is at once evident. We find a central lappet-like structure with a sensitive surface, which looks like and logically should be the epipharynx, and moying below it is a pair of appendages which, in my opinion, represent mandibles. They are membranous and probably not functional; but this is no argument against their character. I believe that the similarity in the appear- ance between Pl. III, Fig. 4°, and Pl. I, Fig. 13%, is the expression of a true homol- ogy, and that mandibles in the Diptera exist in no other form or situation. It is likely that other species, showing them much more perfectly, will yet be discovered ; but so indeed do I believe that labial palpi, properly connected with the mentum, will yet be found, so distinct in character that, even if not functional, their homology cannot be mistaken. Labrum and epipharynx have been frequently referred to in the course of this paper, and in the introduction the general relation of these two parts has been ex- plained. Both structures occur in many families of the Diptera. As in the case of the hypopharynx, the epipharynx has always connected with it a salivary duct. In its intimate connection with the labrum it, is shown on PI. I, Fig. 10%, illustrating the epipharynx of Libellula. Here the chitinous tube giving passage to the duct is fully shown. As an example of a highly developed structure, the epipharynx of Copris carolina is shown (PI. I, Fig. 4), and here the salivary duct opens among the dense central mass of spinous processes. The epipharynx of /’olistes was referred to in the description of the labium, as was that of Andrena in the connection. In the Hemip- tera the labrum and epipharynx are usually well developed and the salivary duct is in many cases very well marked. Among the Diptera some of the larger Syrphide have the labrum quite distinct, and on the under surface is a sensitive surface into which an obvious duct, with chit- 196 AN ESSAY ON THE DEVELOPMENT inous protecting margins, is led, as shown on PI. III, Fig. 10. A much better devel- oped organ, strongly resembling that in some of the Hemiptera, we find in the Aside (Pl. LI, Fig. 1”), and here also the salivary duct is obvious. The structure in S7mu- lium has been already referred to, as has that in the Hmpide. To recapitulate concerning the maxille: The sclerites form three series, each of which has its own possibilities of development. The lacinia never develope into any- thing other than a chewing or piercing organ and always arises inside of the galea. The galea varies in the direction of forming an enveloping organ for all the other mouth parts, and the subgalea eventually unites along one margin for that purpose. There is a tendency to develop a ridged membrane on the inner surface of the galear joints which culminates in the pseudotrachea of the muscid labella. The palpifer has a small range of development, from an unjointed, flexible, tactile organ, to a rigid, piercing structure; and as this becomes useless, to a process for the attachment of muscles used to flex the proboscis. It remains only to acknowledge the assistance received from my entomological friends. Dr. S. W. Williston has from time to time sent me such specimens as I thought might help me; Mr. C. W. Johnson has given me numerous species of fami- lies selected because of apparent differences in the mouth structure; and to Mr. J. M. Aldrich I owe many other species in some numbers, among them the Simuliid already referred to. Mr. H. P. Fell kindly sent me specimens of Panorpa and Bittacus, which enabled me to make a much more complete study of these insects than would have been otherwise possible. T’o all these gentlemen, as well as to the others who have in any wise aided me, I desire to express my thanks. Concerning the figures—most of them are camera lucida drawings. A few are drawn from micro-photographs, assisted by the specimens themselves. The figures of transections are largely made from actual preparations; some are redrawn from other sources, while a few are ideal. OF THE MOUTH PARTS OF CERTAIN INSECTS. 197 EXPLANATION OF THE PLATES. The lettering of the parts, the same throughout, and the abbreviations, are as follows: Lor, labrum; epi, epipharynx (the two sometimes combined as lbr-epi) ; md, mandible; car, cardo; st, stipes; pfr, palpifer; mp, maxillary palpus; gal, galea; sg, subgalea; lac, lacinia; dig, digitus; sm, submentum; m, mentum; gl, ligula or glossa; par, paraglossa ; Jp, labial palpi; Ayp, hypopharynx. Plate J. Fig. 1. Buffalo gnat. 14, galear structures with palpi attached ; 15, labial structures ; 1¢, lacinia and palpifer of Simulium from Aldrich ; 14, labrum and Jabium of Simulium from Aldrich; 1¢, transverse section through middle of mouth of Buffalo gnat. 3 & wo Simulium from Anglesea, N. J. 2¢ the maxillary structures in their actual relation to each other ; 2b, transverse section of mouth parts toward the base of subgalea. Fig. 3. Mouth parts of Hristalis tenax. 34, transverse section of same at the middle of subgalea. Fig. 4. Copris carolina, epipharynx. Fig. 5. Mouth structures of Tipula sp.; 54, the chitinous parts of the same. Fig. 6. Copris carolina ; labial structures dissected out and seen from side. Fig. 7. Copris carolina ; chitinous part of under side of head. Fig. 8. Copris carolina; mandible with the sclerites named and homologized. Fig. 9. Andrena vicina ; labial structures, with part of epipharynx attached. Fig. 10. Libellula sp. a, the epipharynx ; }, the hypopharynx. Fig. 11. Stomoxys calcitrans ; transverse section through the middle of the ligula. Fig. 12. Mouth parts of Zuparyphus bellus. Plate I. Palpifer of Chrysops vittatus. Fig. 1. Maxillary structure of Leptis, sp. Fig. 2. Palpifer of Stratiomyia. Fig. 3. Palpifer of Hulonchus tristis. Fig. 4. Palpifer of Empis spectadilis. Fig. 5. Palpifer of Spharophoria cylindrica. Fig. 6. Palpifer of Bombylius. Fig. 7. Palpifer of Anthraz. 8. 9. Palpifer of Stmulium. Fig. 10. Palpifer of Lucillia. Fig. 11. Palpifer of Calliphora. Fig. 12. Palpifer of Stomozys. Figs. 10 to 12 inclusive were accidentally reversed in making up the plate. Fig. 13. Mouth parts of Zmpis spectadilis. 134, elongated head structure at tip, showing mandibles and epipharynx ; 13>, transverse section at middle of subgalea. Fig. 14. Mouth parts of Chrysops vittatus showing maxillary structures attached together. 144, the lacinia ; 14”, pal- pifer and palpus ; 14¢, transverse section at middle of galea. Fig. 15. Labial structures of Xenoglossa pruinosa. a, transverse section at about middle. Fig. 16. Labial structures of Periplaneta orientalis. Fig. 17. Maxillary structures of Hermetia mucens. Fig. 18. Mouth structures of Polistes metricus. 184, ligula, paraglossa and mouth opening ; 182, labium as a whole, with epipharynx attached ; 18¢, maxilla. Fig. 19. Maxilla of Olfersia. 192, seen from front; 19, seen from behind or below. 198 Fig. Fig. AN ESSAY ON THE DEVELOPMENT OF THE MOUTH PARTS OF CERTAIN INSECTS. Plate ITI. Mouth structures of Asilide—Laphria sp. a, maxilla from front ; }, same from behind ; ¢, labium ; d, lab- rum ; é, transverse section of mouth at junction of galea and subgalea. Mouth structures of Ramphomyia longicauda. a, the labium ; }, maxilla; c, extension of front of head ; d, relation of this extension to the labium. Galea of an Andrena allied to vicina. Mouth parts of Bittacus strigosus. a, mandibles and labrum; 4, maxilla and labium; ¢, mandibles and labrum—epipharynx of Panorpa rufescens. Labial structures of Hristalis tenaxz. 54, transverse section at about middle ; 5%, same at about tip. Labial structure of Bombus sp. 64, transection at about middle ; 6%, same made near tip. Labium of Harpalus calignosus. Maxilla of Periplaneta orientalis. Maxilla of Perlid larva. Epipharynx of Eristalis tenax. Mouth parts of Bibiosp. a, maxilla from behind ; 6, same in front ; c, transection made near the base. Labium of Bombus fervidus ; the transections are lined to the portions referred to. Labium of Chrysops vittatus ; the transections are lined to the parts referred to. — Labium of Hermetia mucens. Maxille and labium of Bombus, showing the relation of the parts to each other. Maxilla of Sialis. Maxilla of Hydrophilus from upper and lower surface, redrawn from Comstock. Maxilla and labium of Pteromalus, redrawn from Ashmead. Maxilla of Pronuba, male. Maxilla of Nemognatha. Maxilla of Pronuba, female. Mouth parts of Locusta from Kolbe. 7, labrum ; 7, mandibles ; @7, maxillz ; 7, labium. Trans. Am. Phil. Soc., N.S xIxX Trans. Am. Phil. Soe. a ~ ue ipee ee ee en en ee se : VEAL LUE N.S. XIX Trans. Am. Phil. Soec., 3 ’ , } aD ira ” \ j :/ iy oe ' 1 Ph ‘ ‘ : Pian 4 i 5 . } A - a. ARTICLE III. SOME EXPERIMENTS WITH THE SALIVA OF THE GILA MONSTER (HELODERMA SUSPECTUM). BY JOHN VAN DENBURGH, Pu.D., CURATOR DEPARTMENT OF HERPETOLOGY, CALIFORNIA ACADEMY OF SCIENCES. Read before the American Philosophical Society, September 3, 1897. I. INTRODUCTION. When, in 1651, Franciscus Hernandez published his Historia animalium et minera- lium Nove Hispanie he gave to Europe the first account of a curious reptile native to those far-western lands which the Spaniards had won beyond the sea. This was a large lizard, said to grow three feet long, thick-set, heayvy-jawed, protected by an armor of wart-like bony plates, gaudily colored in orange and black—withal so repulsive that Wiegmann, nearly two hundred years later, christened it /e/oderma horridum. For many years, this name was applied to these lizards wherever found, but in 1869 Prof. Cope discoyered that those which had been caught within the borders of the United States and Sonora differ in many details from their more southern relatives. He named the smaller, northern species Heloderma suspectum. It is this species which, because of its former abundance near the Gila river, in Arizona, has become popularly known under the name Gila Monster. The Indians and Mexicans claimed for these lizards power to inflict a bite even more deadly than that of the rattlesnake, but, since they claimed like powers for other reptiles known to be quite innocent of yenom, their evidence was of little value. It received some confirmation, however, when the herpetologists of Europe found that the teeth of the Heloderma bear grooves similar to those which in some poisonous snakes serve to introduce venom into the wound. Since this was discovered the question of the poisonous nature of the bite of the Gila Monster has attracted considerable attention and many opinions have been published. LAE UES ASN ONE PCD. 200 SOME EXPERIMENTS WITH THE In 1857, Dr. J. E. Gray, of the British Museum, wrote : “« This lizard is said to be noxious, but the fact has not been distinctly proved.”’ Seven years after this there appeared a popular account of the habits of the Mexican species (H. horridum), in which M. Sumichrast, after dwelling at some length upon the general habits of the animal, wrote : ‘« In support of this pretended malignity, I have been told of a great number of cases in which ill effects were produced by the bite of the animal, or by eating its flesh in mistake for that of the Iguana. I wished to make some conclusive experiments on this point; but, unfortunately, all the specimens which I could procure during my stay in the countries inhabited by it were so much injured that it was impossible to do so. Without giving the least credit to the statements of the natives, I am not absolutely disinclined to believe that the viscous saliva which flows from the mouth of the animal in moments of excitement may be endowed with such acridity that, when introduced into the system, it might occasion inconveniences, the gravity of which, no doubt, has been exaggerated.”’ Prof. Cope, in 1869, stated : ‘« That though the lizards of this genus could not be proven to inflict a poisonous bite, yet that the sali- vary glands of the lower jaw were emptied by an efferent duct which issued at the basis of each tooth, and in such a way that the saliva would be conveyed into the wound by the deep groove of the crown.”’ Six years later Dr. Yarrow said : “Tt is believed to be very poisonous, but such is not the case; for, although it will bite fiercely when irritated, the wound is neither painful nor dangerous. . . . . The Pueblo Indians of this place said they were quite common, and were regarded by the Mexicans as poisonous; the poison being communicated by the breath as well as by the teeth. This has no foundation in fact.’’ The same year, M. Bocourt published some notes which he had received from M. Sumichrast, who, having finally been able to make a few experiments, concludes : “ Quoique ces expériences soient insuffisantes pour prouver que la morsure de I’ Héloderme est véritable- ment venimeuse, elles me paraissent assez concluantes pour faire admettre qu’elle ne laisse pas de causer de trés-rapides et profonds désordres dans |’ économie des animaux qui en sont l’objet. . . . . “* Je ne doute pas que des expériences, faites avec des individus adultes et nouvellement pris, ne pro- duisent des effets beaucoup plus terribles que ceux qu’ont pu occasionner la morsure d’un individu jeune et affaibli par une captivité de prés de trois semaines.’’ In 1882, several opinions were published on each side of the question. A Helo- derma, which had been received at the Zodlogical Gardens in London, bit some small animals, and because these died several English writers—as Giinther, Boulenger, and Fayrer—concluded that the Monster was poisonous, while some American authors have thought that death in these cases might have resulted from the mechanical injuries received. The American Naturalist noted that “ Dr. Irwin, U. 8. A., experimented with the H. suspectum in Arizona, fifteen years ago, and concluded that it was harmless.” OR te ni li ee i a in ee ee SALIVA OF THE GILA MONSTER. 201 Dr. R. W. Shufeldt had a personal encounter with an active Gila Monster, of which he wrote : “* On the 18th inst., in the company of Prof. Gill of the [Smithsonian] Institution, I examined for the first time Dr. Burr’s specimen, then in a cage in the herpetological room. It was in capital health, and at fst I handled it with great care, holding it in my left hand examining special parts with my right. At the close of this examination I was about to return the fellow to his temporary quarters, when my left hand slipped slightly, and the now highly indignant and irritated Heloderma made a dart forward and seized my right thumb in his mouth, inflicting a severe lacerated wound, sinking the teeth in his upper maxilla to the very bone. He loosed his hold immediately and I replaced him in his cage, with far greater haste, perhaps, than I removed him from it. “* By suction with my mouth, I drew not a little blood from the wound, but the bleeding soon ceased entirely, to be followed in a few moments by very severe shooting pains up my arm and down the corre- sponding side. The severity of these pains was so unexpected that, added to the nervous shock already experienced, no doubt, and a rapid swelling of the parts that now set in, caused me to become so faint as to fall, and Dr. Gill’s study was reached with no little difficulty. The action of the skin was greatly increased and the perspiration flowed profusely. A small quantity of whiskey was administered. This is about a fair statement of the immediate symptoms; the same night the pain allowed of no rest, although the hand was kept in ice and laudanum, but the swelling was confined to this member alone, not passing beyond the wrist. Next morning this was considerably reduced, and further reduction was assisted by the use of a lead-water wash. “ Tn a few days the wound healed kindly, and in all probability will leave no scar; all other symp- toms subsided without treatment, beyond the wearing for about forty-eight hours so much of a kid glove as eovered the parts involved. .... “ Taking everything into consideration, we must believe the bite of Heloderma suspectum to be a harmless one beyond the ordinary symptoms that usually follow the bite of any irritated animal. 1 have seen, as perhaps all surgeons have, the most serious consequences follow the bite inflicted by an angry man, and several years ago the writer had his hand confined in a sling for many weeks from such a wound administered by the teeth of a common cat, the even tenor of whose life had been suddenly interrupted.”’ Only a few months had passed after the publication of Dr. Shufeldt’s article when there appeared an account of the first carefully conducted series of experiments with the saliva of the Heloderma. This was by Drs. 8S. Weir Mitchell and Edward T. Reichert, who conclude that : ‘© The poison of Heloderma causes no local injury. “‘ That it arrests the heart in diastole, and that the organ afterwards contracts slowly possibly in rapid rigor mortis. ‘* That the cardiac muscle loses its irritability to stimuli at the time it ceases to beat. ‘That the other muscles and the nerves respond readily to irritants. “« That the spinal cord has its power annihilated abruptly, and refuses to respond to the most powerful electrical currents. 202 SOME EXPERIMENTS WITH THE ‘« This interesting and virulent heart poison contrasts strongly with the venoms of serpents, since they give rise to local hemorrhages, and cause death chiefly through failure of the respiration, and not by the heart, unless given in overwhelming doses.’’ For a time, it seemed that the experiments of Mitchell and Reichert had answered the question of the poisonous power of the Heloderma once and for all. But five years later, Dr. Yarrow, then Honorary Curator of the Department of Reptiles in the United States National Museum, performed some equally careful experiments upon rabbits and chickens. These, he says, ** Would seem to show that a large amount of the Heloderma saliva can be inserted into the tissues without producing any harm, and it is still a mystery to the writer how Drs. Mitchell and Reichert and himself obtained entirely different results. Were it not for the well-known accuracy and carefulness of Dr. Mitchell, it might be supposed possibly that the hypodermic syringe used in his experiments contained a certain amount of Crotalus, or cobra venom, but under the circumstances such a hypothesis is entirely untenable.’ Notwithstanding Yarrow’s results, Dr. Mitchell still held his original opinion in 1889. The following year, Prof. Samuel Garman, of the Museum of Comparative Zodlogy of Harvard University, published an account of experiments in which he caused an active Gila Monster to bite the shaved legs of kittens without serious effect. He con- cludes that “* The results of the experiments suggest danger for smal] animals, but little or none for larger ones. Large angle worms and insects seemed to die much more quickly when bitten than when cut to pieces with the scissors.”’ Thus while in England the Heloderma was unanimously held to be venomous, Dr. Shuteldt, in 1891, summarized American opinion as follows: “* Here in America the evidence would seem to be rapidly leading to the demonstration of the now entertained theory that the saliva of this heretofore much-dreaded reptile is possibly entirely innocuous.”’ “* Thus the matter seems to stand at the present time—perhaps the vast majority of physicians who _ followed Drs. Mitchell and Reichert in their experiments fully believe to-day that the bite of a ‘ Gila Monster’ will very often prove fatal even in the case of man; while, on the other hand, naturalists almost universally believe that the saliva of this saurian is hardly at all venomous, and then only under certain conditions. ”’ W. THE MOUTH FLUIDS. In the winter of 1896-97 I began a series of experiments with the saliva of the Gila Monster, the results of which are given in the subsequent pages. My object was to answer the following questions : (a) Is the bite of the Gila Monster poisonous ? SALIVA OF THE GILA MONSTER. 203 (b) If poison is present what are its physiological effects ? (c) What are the causes of such diversity of opimion ? My Heloderma was the sole survivor of eight or ten brought from Arizona in 1892 and, although seemingly fat and healthy, was not very active. It was of moderate size, being about eighteen inches long. The amount of saliva obtainable from it was so small that it could be gathered satisfactorily only by causing the reptile to bite absorbent paper wrapped around a piece of soft rubber and afterwards dissolving out the saliva in water. For this purpose filter paper was used. It would not do to let the Monster bite the pigeons, because if this were done and the pigeons died the skeptics might justly claim that death was due to the mechanical injury inflicted by the powerful jaws, with their long, curved fangs, rather than to any poison haying been inserted. Even when the Heloderma’s saliva solution was injected hypodermically and death could not have been occasioned by the severity of a wound there might be some doubt as to the effect of a quantity of water suddenly placed under the skin, or it might be claimed that some substance was present in the water or the paper used quite poisonous enough to cause a pigeon’s death irrespective of any venom from the Monster. So samples of all the materials used had to be subjected to careful tests to show that they were harmless.* i Mvcvs. A greater or less quantity of thick mucus is present in the back part of the mouth of the Gila Monster. Some of this often adheres to the filter paper in stringy masses. It is entirely without poisonous properties and need not be mentioned again. THe Portsonous Satya. The water solution of saliva when extracted from the paper is a slightly yellow- ish or opalescent liquid, often more or less stained with blood owing to injury to the gums. It is faintly alkaline, and ordinarily possesses a pungent and highly characteristic though not unpleasant odor. This odor becomes less and less noticeable when the Monster is caused to bite every day, but its strength seems to be no indication of the lethal power of the saliva. That the solution of saliva thus obtained contains a yery powerful poison is shown in the following experiments : ExpPERIMENT I.—Noy. 11, 1896. The Heloderma was caused to bite on paper three times. The *In order to test my materials, and some other things as well, the following preliminary experiments were performed, the first repeatedly : EXPERIMENT.—A sample of filter paper was soaked in water, which was then injected subcutaneously in front of the wing ofa pigeon. During two hours there was no effect, and the next day the bird was still well. EXPERIMENT.—Mixed human saliva with an equal quantity of water and injected about twenty minims in Wing of pigeon at 12.01 P.M. No effect. Next day well. EXPERIMENT.—Mixed blood of horned toad (Phrynosoma frontale Van D.) with water and injected wing of pigeon. No effect. 204 SOME EXPERIMENTS WITH THE water solution—about twelve minims—was then injected subcutaneously in front of the shoulder of a pigeon at 3.18 P.M. In three minutes the pigeon was no longer able to stand, and fell over on its side with eyes closed. At the end of the tenth minute the bird was unable to hold up its head when raised by its wings. During the eleventh minute respiration was in gasps, and at the end of the eleventh minute the pigeon was dead. [No local effects; heart beating regularly. ] ExPERIMENT IIJ.—Novy. 12, 1896. Monster was caused to bite seven times during about as many minutes. Saliva then dissolved in about seventy minims of water, of which ten minims were injected under the skin in front of right shoulder of pigeon, at 11.24 A.M. 11.28. Pigeon barely able to walk. 11.29. Not able to walk. 11.3¢ 11.31. Head nods; respiration is forced. 11.32. Muscular straining; head drawn back between shoulders. 11.33-38. Respiration greatly forced; bill opens and shuts with each breath. bo ob Cannot stand; lies on side; eyes closed. Co w — 2 oD 11.59. Violent contractions of caudal muscles. 11.40. Violent contractions of head and wings. 11.40}. Head falls forward onto table. 11.404. Death. No local effects; ventricles empty, auricles full of clots; blood almost black. Tf these experiments leave any room to doubt that the bite of the Gila Monster is poisonous it is entirely removed by the results of a large number of experiments which I afterwards performed and in which death followed the injection of Heloderma saliya quite as certainly and almost as quickly as when rattlesnake venom is used. It now became of interest to learn whether this powerful poison is affected by boiling or decay, or the presence of alcohol, ete. The Effect of Boiling—TYwo experiments were performed which show that the poisonous properties of the saliva are not injured by boiling. The solution becomes opalescent and, if boiling be prolonged, loses its odor or gives off one similar to that of boiled barley. EXPERIMENT IIT.—Noy. 12, 1896. The Heloderma was caused to bite seven times during about as many minutes. Saliva then dissolved in about seventy minims of water. Ten minims of this solution, having been boiled a few seconds, were injected under the skin of the right shoulder of a pigeon, at 2.21 P.M. The temperature of the pigeon before injection was 104° F. 2.22. Sits down, but is able to stand when frightened. 2.26. Sits down. 2.27. Sits down immediately after being caused to stand, seems dizzy. 2.29. Lies on side; temperature 100°. 2.34, Cannot stand; temperature 98°. 2.36. Violent respiration; temperature 96°. SALIVA OF THE GILA MONSTER. 205 2.38. Violent respiration; temperature 98°. 2.39. Violent respiration; temperature 100°. 2.424. Violent respiration; temperature 1014°. 2.45. Violent respiration; temperature 100°. 2.48. Respirations about 108 per minute; temperature 99°. 2.50. Temperature 100°. 2.53. Respiration more labored; temperature 99°. 2.54. Temperature 98°. 2.55. ‘VYemperature 97°. 2.56-57. Temperature 95°; respiration short and forced, 39 per minute. 2.58. Wheezing; vomits. 2.584. No motion except quivering of wings; temperature 90°. 2.59. Wings and tail flapped twice. 3.00. Dead. No local effect; small clot of blood in base of right Jung; ventricles full of black clots; auricles beating; arteries empty; veins dilated with blood. This experiment would seem to show that the action of the poison is slightly delayed by boiling. Experiment IV shows that such is not the case. Exeerimenr [V.—Nov. 14, 1896. Ten minims of the solution used in experiments II and ITI were boiled about five minutes on Nov. 12, and again Noy. 13 and 14, and then were injected under the skin of a pigeon’s wing at 3.30 P.M. 3.34. Respirations 32 per minute. 5.37. Staggers about with peculiar circular motion. 3.39-40. Respirations 48, becoming constantly more forced, so that at end of minute tail moves up and down. 3.42. Cannot stand. 3.44-45. Respirations 49. 3.46. Falls on side. 3.47. Head nods; pupil seems slightly dilated. 3.52. Respirations 47, irregular. 3.53. Bill begins to open and shut. 3.54, Convulsive action of wings and head, head drawn under to breast. 3.55. Death. The Effect of Decay.—When a solution of saliva is allowed to stand for a few days it soon begins to decay, and this process continues until a strong odor of putrescence is given off and a muddy sediment appears at the bottom of the liquid. After this had occurred, very large doses of the solution were injected into pigeons without producing the slightest ill-effect. Decay, then, appears to destroy the lethal power of the saliva, but my experiments are not absolutely conclusive because the solution was not tested while fresh. 206 SOME EXPERIMENTS WITH THE EXPERIMENT V.—Saliva of several bites was collected, November 14, and dissolved in about ten minims of water per bite. November 16 there was a marked odor of decay. November 23 the odor of putrescence was yery strong and the liquid appeared muddy with a slight sediment. At 2.31 P.M., ten minims were injected under the skin in axilla of pigeon whose temperature at 2.29 (when frightened ) was 106°. 2.35-36. Respirations 35. 2.40. Temperature 105°. 2.44-45. Respirations 32. 3.09. Temperature 104°. 9 .10-11. Respirations 32. 32 oo © 9 ¢ . 28-29. Respirations 32. 31. Temperature 104°. Repeated injection. 33-34, Respirations 34. 56. Respirations 32. 4.21-22. Respirations 33. November 24, ete. Still perfectly well. 0. oO. oo oO Ww or oO | o SD ExpEeRIMENT VI.—December 1, 1896. Injected forty minims of solution used in experiment V under skin of legs and wing of pigeon at 12.45 P.M. 4.30. Still no effect. December 2. Well. The Effect of Drying —That drying does not affect the power of the venom was shown by the following experiment, although the dose was too small to cause death. ExperRIMENT VII.—December 1, 1896. A small quantity of the solution used in experiments II, III and IV, having been dried, was redissolved in water and injected subcutaneously in a pigeon at 3.40 P.M. 4.10. Respiration slightly forced. 4.30. Cannot walk well. ’ 4.45. Very ‘‘ tame;’’ respiration forced. December 2. Pigeon recovered. The Effect of Alcohol—When alcohol is added toa water solution of saliva, the solu- tion becomes opalescent, as when boiled. This change in color is probably due to the formation of a finely divided albuminous coagulate. It is not removed by filtration through paper. Alcohol does not influence the action of the venom. ExpertmMent VIII.—About twenty minims of the solution used in experiments IJ, III, IV and VII was mixed with an equal quantity of ninety-five per cent. aleohol, November 14. About half of this had evaporated when ten minims of the remainder were mixed with ten of water and thrown down the throat of a pigeon at 11.25 A.M., November 18, 11.46. Seems well. 2.15 P.M. No effect. SALIVA OF THE GILA MONSTER. 207 2.26. Injected the other ten minims in left axilla. 2.29. Shows uneasiness of left wing and cannot always control it. 2.291. Sits; cannot walk. 2.30. Pupils contracted; cannot stand. 2.31. Lies on side; respiration convulsive. 2.32. Respiration still more labored. 2.33. Seems unable to feel pinching of legs. 2.37. Rate of breathing very greatly increased. 2.38-39. Respirations 62. 2.40-41. Respirations 84. 2.4344. Respirations 64. 2.45-46. Respirations 55. 2.46-47. No respiration; convulsions. 2.48. Death. Auricles beating; ventricles still; blood black, clotted; auricles and veins full; ventricles and arteries empty; slight extravasation in coat of smal! intestine near head of pancreas; no local effect. Ninety-five per cent. aleohol when added to undiluted saliva does not injure its poisonous properties, nor does the alcohol act as a solvent of the venom, although its solubility in water is unaffected. ExprerIMenr IX.—November 23, 1896. a. Filter paper containing saliva was washed in about one ounce of alcohol for about twenty hours. The alcohol was then poured into an open dish. As soon as evaporation began a thin white scum appeared on the surface of the alcohol, but did not increase much as evaporation proceeded to dryness. This scum was not soluble in water, even after the addition of salt (NaCl). Placed under the skin of a pigeon, it produced no effect. b. ‘The alcohol-washed paper was soaked during a few minutes in sixty minims of water. Twenty minims of this water were injected under the skin of each wing of a pigeon at 3.25 P.M., November 24. Half an hour later twenty minims were injected into the left leg. 4.07. Pigeon sits down. 4,12-13. Respirations 45. 4.15-21. Stands on right leg only. 4,.22-23. Respirations 54. 4.2324. Respirations 49. 4,25. Temperature stil] normal, 102° 4.35. Temperature 99°. 4.39—40. Respirations 48. 4.42. Temperature 98°. 4.4446. Respirations 35 per minute. 4.47. Temperature 96°. Slides along on breast when trying to walk. 4.47-48. Respirations 44, very weak. A, P. 8.— VOL. XIX. 2 A. 208 SOME EXPERIMENTS WITH THE 4,52, Temperature 96°. 4.53-54. Respirations 44. 4.56-57. Respirations 31, 4.58. Temperature 96°. 5.00-OL. Respiration, wheezing pants. 5.0102. Respirations, wheezing pants, 21. 5.02. Temperature 96°. Death without struggles. The Effect of Glycerine.—Glycerine seems to dissolve the poison and to partly destroy its effectiveness, though this seeming injury may be due to the slowness with which the glycerine is absorbed, preventing the poison from reaching the circulation rapidly enough to result fatally. EXPrRIMENT X,—Paper containing saliva of four bites was placed in about forty minims of glycerine and Jeft for some hours. The glycerine, having been extracted, was injected in the breast muscles of a pigeon at 12.10 P.M., December 4, 1896. 1.00. Still no effect. 5.15. Still no effect. December 5. Well, but with yellowish-white swelling on breast. Decemher 17. Well, but breast muscles sloughing. Used in experiment XII. EXPERIMENT XI.—December 4, 1896. Since it was quite possible that the poison had not been dissolyed by the glycerine, the paper used in the last experiment was well washed in alcohol to remove glycerine, and then, after the aleohol had been removed by pressure and evaporation, was placed in water (thirty minims). This water was injected into a pigeon at 3.15 P.M. 3.30. No signs of poison. 5.15. No effect yet. December 5. Well. December 8. Well. Expertment XII.—December 17. Saliva of the lower jaw from about three bites was collected and divided into two parts, one slightly larger than the other. The larger part was then soaked in glycerine, a little more than one-half of which was afterward injected in leg of pigeon used in experiment X. 4.35 P.M. Injected subcutaneously. 5.30, Seems slightly drowsy ; ‘otherwise well. December 18. Found dead.* EXPERIMENT XIII.—December 17, 1896. To test the power of the saliva used in experiment XITJ the smaller portion of the saliva-soaked paper was placed in a small quantity of water, and one-half of the resulting solution injected in the breast muscles of a pigeon, December 18. 4.07. Injected. 4.50. Bird sitting; staggers when raised. * Death may have been due to the rather extensive sloughing of the pectoral muscles, but that this was the case does not seem probable. SALIVA OF THE GILA MONSTER. 2O9 4.31-32. Respiration still normal, 7. ¢., 35. 4.35. Can still stand. 4.56-37. Respirations 30. 439-40. Respirations 31. 4.46-47. Respirations 29; sits with eyes closed. 4.53. Does not notice loud noises, as stamping on floor; cannot stand. 4.55-56. Respirations 31. 4.58. Head moves from side to side, slightly. 4.59-5.00. Respirations 30. 5.03-04. Respirations 34, slightly forced. 5.09-10. Respirations 34, slightly forced. 5.13-14. Respirations 43, a little more forced; head nodding. 5.15-16. Respirations 36, nearly normal. 5.18-19. Respirations 32, slightly forced. 5,21-22, Respirations 50, much forced. 5,23-24. Respirations 32, convulsive. 5,24-25. Respirations 23, convulsive. 5,254. Raises tail and flaps wings. 5.26-27. Respirations 13, weak. 5.28. Heart still beating strongly and regularly. 5.30. Death. Heart irritable and nerves of pectoral muscles, etc., likewise; blood very dark, semi-liquid, coagu- lating quickly; no local effects. Tor Harmurss SAniva. There is, then, in the saliva of the Gila Monster a very powerful poison which may be subjected to very rough treatment without impairing its lethal vigor. This poison is present in the saliva of one jaw only. Tf, when collecting the mouth fluids, the rubber be properly placed between two layers of paper, the saliva from each jaw may be readily obtained unmixed with that of the other. When thus obtained and dissolved in water, the saliva of the upper jaw is a yellowish liquid, usually more or less tinted with blood, slightly alkaline, without any odor, and absolutely harmless at the very time when the lower jaw is flooded with deadly venom. The quantity of saliva which may be collected from the upper jaw at any one time is only a little less than is obtainable from the lower ; but in one ease all of the saliva from the upper jaw was injected into a pigeon without causing the slightest ill effect, while one-fifth of that obtained at the same time from the lower jaw caused death in fifty-two minutes. The following experiments are quite numerous enough to show beyond doubt the difference in effect between the two kinds of saliva. 210 SOME EXPERIMENTS WITH THE EXPERIMENT XIV.—November 24, 1896. Saliva of upper jaw from four bites was dissolved in water one-half of which (ten minims) was injected into a pigeon at 11.40 A.M. 3.08. Still no effect; repeated injection. 5.40. Still no effect. November 25. Well. ExperRmMeNntT X V.—November 24, 1896. Same as last experiment, but with saliva of lower jaw in another pigeon. 12.15 P.M. Temperature 104°. 12.17. Injected. 12.20-21. Respirations 31. 12.27—28. Respirations 31. 12.35. Temperature 100°. 12.36-37. Very “*‘ tame.’’ Respirations 38. 12.38. Sways backward and forward. 12.39-40. Respirations 52. 12.42. Temperature 98°. 12.47-48. Respirations 30. 12.50. Very drowsy. Temperature 97°. 12.54-55. Respirations 34, irregular. 1.05-04. Respirations 28, labored. 1.06. Temperature 95°. Can still stagger when placed on feet. 1.09-10. Respirations 38, very irregular. 1.11. Temperature 96°. 1.16. Temperature 95°. 1.17-18. Respirations 42, greatly labored. 1.28. Temperature 95°. 1.24—25. Respirations 46, bill opening and shutting. Can stil] walk slowly. 1.28-29. Respirations 55. 1.30. Temperature 96°. 1.33-34. Respirations 52. Can barely walk. 1.36. Temperature 96°. 1.37. Cannot walk. 1.37-38. Respirations 54. 1.46. Temperature 94°. 1.47-48. Respirations +9, head nods. 1.53. Temperature 94°. 1.54. No respiration. 1.55. Temperature 93°. 1.56. Death with convulsions. Exprerment XVI.—November 25, 1896. At 2.15 P.M., injected a pigeon with all of solution of saliva of upper jaw from four bites. 2.30. Still no effect. SALIVA OF THE GILA MONSTER. ZAI 2.40. Still no effect. 3.07. Still no effect. 5.05. Still no effect. November 26. Well. Experiment XVII.—November 25, 1896. Injected one-half of the solution of lower-jaw saliva from same bites as last experiment. 3.02-03. Respirations 37; temperature 104°. 3.06. Injected as above stated. 5.14. Temperature 102°. 25-24. Respirations 38. ge 27. Very ‘‘ tame,’’ temperature 98°. 28. Cannot stand. 5.285-294. Respirations 53. 3.30. Temperature 98°. 5.52-35, Respirations 45. 2 33. Temperature 98°, 3.37. Temperature 98°. 3.38-39. Respirations 45. 3.40. Temperature 96°. 3.40-41. Respirations 45. 3.51. Temperature 94°. 3.93-54, Respirations 45. 3.56. Temperature 94°. 5.58-59. Respirations 45. 4.07. Temperature 93°. 4.15-16. Respirations 51. 4.21. Temperature 93°. +.27-28. Respirations 26. 4.29. No respiration. 4.30. Death. Heart (auricles and ventricles) beating strongly when exposed at 4.31 and until 4.36; blood in veins; arteries and ventricles empty; no local effect. Exprerment XVIII.—Noyember 28, 1896. Injected all of solution of saliva from upper jaw, in pigeon, at 11.55. No effect. EXPERIMENT XIX.—November 28, 1896. Injected all of solution of saliva from lower jaw (same bites as last experiment) in pigeon at 12.15 P.M. 12.19. Tips forward on legs, therefore cannot stand still. 12.20. Seems dizzy. 12.20}. Sits. 12.22. Can walk well. lo ft bo 12.24. 12.27. 12.34. 12.39. 12.40. 12.41. 12.42. SOME EXPERIMENTS WITH THE Very ‘‘tame;’’ hardly able to walk. Can stagger with help of wings. Respiration terribly labored, loud, wheezing pants, about 28 per minute. Head drawn far back; still panting. Still panting, but more slowly and weakly, 24 per minute. Struggles, lies on side with head on floor. Respiration practically stops. 12.421. Dead. ExpERIMENT XX.—December 1, 1896. Injected solution of saliva of upper jaw from two bites, at 12.30 P.M. 1.30. 3.30. 4,30. 5.00. Pigeon has shown no signs of poisoning. Still no effect. Still no effect. Still no effect. December 2. Well. Experiment XXI._ Injected solution of saliva of lower jaw from same two bites (experiment XX) at 2.25 P.M., December 1, 1896. 3.25. 4.00. 4.10. 4,20. 4.50. Totters; lies down when set on feet. Totters, leaning forward. Can still totter. Cannot rise or stagger. Muscles all tense; bill opens and shuts. 4.504. Respiration ceases. 4.31. Death. EXPERIMENT X XII.—December 2, 1896. All of the solution of saliva of the upper jaw from three bites was injected under the skin of the wing of a brown pigeon at 3.05 P.M. without any effect. ExeprrimMent XXIII.—December 2, 1896. Two-fifths of the solution of lower-jaw saliva from the same three bites as last experiment were injected under the skin of wing of a pigeon at 3.15 P.M. 3.25. 3.28. 3.32. 3.36, 5.40. 3.45. No effect yet. Staggers slightly; sits immediately; respiration slightly forced. Respiration very rapid—forced. Respiration very slow but labored. ‘« Skates ** on breast when trying to walk. Convulsive auivering of wings. 3.44-45. Convulsive quivering of wings. 3.40. 3.48. Lies stretched out on floor; convulsive respiration; wheezing with each breath. No respiration. 3.483. Death. SALIVA OF THE GILA MONSTER. 213 ExPeRIMENT XXIV.—December 2, 1896. Two-fifths of the solution used in the last experiment (XXIII) were injected in the breast muscles of a slate-colored pigeon at 3.16 P.M. 3.25. -Barely able to walk. 3.26. Not able to stand; respiration forced. 3.28. Lies on side with head drawn back. 3.34. Respiration very rapid and convulsive, bill opening and shutting; head twisted on side. 3.39. Respiration ceases. 3.394. Apparently dead. 3.40. Heart still beating. Experment XX V.—December 2, 1896. One-fifth of solution used in experiments XXIII and XXIV was injected in a gray pigeon at 3.20 P.M. 3.25. Respiration deeper. 3.42-45. Respiration very rapid and shallow, 148 per minute. 3.51-52. Respirations 167; can still walk, but sits immediately. 3.58-59. Respirations 168, 4.02. Cannot stand. 4.04. Slight trembling. 4.05-06. Respirations 149. 4.08. Head drawn back; bill opens and shuts. +.09-10. Respirations 62. 4.10. Slight general contractions of muscles. 4+.114-114. Respirations 4. 4+.114-12. No respiration. 4.12. Death. ExprermmMent XX VI.—December 8, 1896. Solution of upper-jaw saliva from one bite injected in breast of a gray pigeon at 3.08 P.M without effect. Experiment XX VII.—December 8, 1896. One-half of solution of lower-jaw saliva, same bite as experiment XX VI, was injected in breast muscles of a gray pigeon at 3.16 P.M. 3.26. Pigeon very quiet. 4.00 Drowsy. December 9. Well. December 18. Well. THE SouRCES OF SALIVA. We have seen that two very different fluids are present in the mouth of the Helo- derma; the one—from the lower jaw—capable of causing profound disorder when intro- duced into the circulation of pigeons, the other—from the upper jaw—producing no more effect than so much water. What are the sources of these fluids ? 214 SOME EXPERIMENTS WITH THE In Heloderma suspectum, there are two large glands, one on each side of the anterior part of the lower jaw between the skin and the bone. When one of these glands has been freed from its outer sheath it is found to be not a single gland but a series of three or four glands, each perfectly distinct from the others and emptied by a separate duct. These glands increase in size posteriorly, so that the last is very much larger than the first. They vary in number because of the occasional union of the first and second elands, or the presence, posteriorly, of a small, isolated, ductless portion. Their duets open between the lower lip and gum, as described by Stewart. It is shown later on that these are the yenom-producing glands. No glands have yet been described as existing in the upper jaw ; indeed there seems to be no room there for a well-developed gland. Nevertheless, paper which comes in contact with the upper jaw during the bite collects almost as much fluid as is obtained from the lower jaw. This, however, is true only when the paper is bitten a very few times. The saliva of the upper jaw is exhausted much more quickly than that of the lower. This fact, taken in connection with the absence of known glands, might lead one to suspect that the upper jaw receives its saliva from the lower and holds it in the compli- cated folds of its gums. This might perhaps be true if one or more segments of the sub- labial glands secreted a harmless fluid, but the following experiments show that all are specialized for the production of venom. I believe that the harmless saliva is secreted by minute glands which lack of material has preyented me from finding—that it is in fact the ordinary buccal liquid of lizards. That it is present in the lower jaw as well as in the upper would seem to be shown by the fact that the fluids of both jaws are decidedly alkaline, while a solution of the poison gland itself is quite neutral. The following experiments were performed to show that each part of the sublabial glands is deyoted to the production of yenom : Experimenr XXVIII.—January 5, 1897. Soaked the first portion of the right sublabial gland in water and injected the resulting solution (three minims) into the breast muscles of a small finch, at 12.26 P.M. 12.28. Respiration forced; eyes closed. 12.29. Respiration greatly forced. 12.31. Flutters. 12.314. Convulsions and death. 12.33. Heart beating weakly; blood dark but lightens quickly. Experiment XXTX.—January 5, 1897. Soaked the second portion of the right. sublabial gland in water and injected solution (four minims) into breast muscles of a small finch, at 12.00 M, 12.04. Eye nearly closed; respiration normal, 12.05. Respiration slightly forced, bo pear 5) | SALIVA OF THE GILA MONSTER. 12.054. Bill begins to open and shut. 12.07. Respiration greatly labored. 12.08. Convulsions followed by death. 12.10. Heart still beating; blood dark, lightens slowly. Experiment XXX. Treated the third portion of right sublabial gland as the first and second were treated in experiments XX VIII and X XIX, and injected four minims into a small finch at 11.34 A.M. 11.35. Wheezes; sitting down; eyes closed; tail moving up and down with each breath. 11.36. Same, but bill opening and shutting. 11.37. Does not open eyes when handled. 11.374. Respiration very short and jerky. 11.38. Respiration ceases, followed by convulsions and death. 11.41. Heart still beating, empty; blood dark brown, reddening very slowly. Expermment XXXI.—January 5, 1897. Injected four minims of solution of fourth portion of right gland into a small finch, at 11.07} A.M. 11.084. Unable to stand erect; head drooping. 11.09. Respiration labored. 11.094. Respiration greatly labored. 11.10. Bill opens and shuts. 11.11. Bird falls on side. 11.124. Respiration in gasps. 11.13. Convulsions and death. Heart responds to mechanical stimuli; blood black but becoming red on exposure. Experiment XXXII. _ Injected five minims solution of first portion of left sublabial gland into a small finch, at 2.41 P.M. 2.42. Hyes closed. 2.45. Respiration labored; bird leaning on side. 2.46. Almost unconscious; bill opening and shutting. 2.47. Convulsions. 2.471. Death. Exprerment XXXIII. Injected six minims of water into the breast muscles of a small finch without effect. Ill. THE PHYSIOLOGICAL ACTION OF HELODERMA POISON. When a pigeon has received an injection of Gila Monster saliva it at first shows no ill effects, and feeds or fights with its fellows as before. Soon, however, it begins to wink very frequently, and ceases to show interest in anything about it. It stands thus for a A. P. S.— VOL. XIx. 2B. 216 SOME EXPERIMENTS WITH THE longer or shorter time and then sits down. If now it be frightened into attempting to walk, it appears dizzy and staggers about, or, if unable to stand, slides along on its breast. If not caused to arise, it never does so of its own accord, but becomes more and more drowsy and sits with eyes closed. The rate of respiration now becomes very rapid for a time, but soon the breaths are shallower and then gradually fewer and fewer.* The legs become more or less paralyzed, but the wings retain their power, although the codrdina- tion of their motions sometimes is destroyed. The temperature falls as the respiration becomes slower. The bird rolls over on its side. The head is drawn down oyer the back. Respiration becomes nothing more than a series of wheezing gasps, with each of which the bill opens and shuts. The head falls forward to the floor. The pigeon is unconscious. Breathing ceases. There may be slight convulsions followed by death, or death may come quietly. If the pigeon now be opened, it is found that the blood is very. dark—often almost black instead of red or blue. The heart either is beating or responds readily to mechan- ical stimuli. The arteries and usually the ventricles of the heart are empty, while the veins and auricles are full of blood which usually is more or less clotted. There is no trace of discoloration about the point of injection, nor is the slightest extrayasation of blood to be found in any of the organs. With all these facts in view, it is very evident that death is due to asphyxiation ; to the failure of the blood to proyide the various tissues of the body with the oxygen neces- sary for their welfare. But, although we may say that death is due to asphyxiation, we have not really answered our question, for there are several ways in which this failure on the part of the blood might be brought about : 1. If the poison acted upon the nerve centres which control the movements of respi- ration in such a way as to interfere with the action of the lungs, the blood would be unable to procure its usual supply of air. We have seen that there is a yery decided dis- turbance of the respiratory function.y It may, perhaps, be due to direct nerve-poisoning ; but I am inclined to believe that it is entirely a secondary phenomenon. 2. If the poison caused a breaking down of the capillaries of the lungs—such as Martin{ claims to have found in certain cases of death from the venom of the Australian black snake—the same effect would be produced, but there appears to be no such change. as Mitchell and Reichert have stated of their experiments—the flow of blood would be diminished and the tissues » 3. If the action of the heart became gradually weaker *This is normally true, but respiration sometimes stops suddenly, even nearly at the time when it is most rapid. + The table upon the opposite page shows the effect upon the number of respirations and the temperature. t Martin, Jour, and Proce, Royal Soc. N. 8. Wale 3, XXDX, 1895, 146-276. MINUTE.|} RESP. TEMP. 104 100 Bee epee eH Shy Len FSS We WD ODED EWM DMNWW WWW Pee eS SSS eS SESS SIS SS ESRSSSSGESSESSNARES ES SmSsaRaeSsnr]|anoe e e = ad sak dene SIR SE4RESBRSRSSESSS Pipi htt rts he es RESP. | TEMP. 31 BL Bl 30 28 42 46 55 52 54 49 SALIVA OF 104 100 98 97 98 95 96 95 95 96 94 | THE GILA RESP. TEMP. 45 54 49) 99 98 RESP. 37 38 45 45 43 51 MONSTER. TEMP. 102 98 98 98 96 94 94 93 93 D. RESP. | 48 4s) 47 RESP. RESP. 35 35 30 31 29 31 380 34 RESP. 148 168 218 SOME EXPERIMENTS WITH THE would not receive their normal amount of oxygen. In all my experiments the heart con- tinued to beat regularly long after respiration had ceased, so that this cannot have been the cause of death. 4. If the poison acted upon the blood in such a way as to destroy its power to carry oxygen—as Cunningham * says is true of cobra yvenom—or, j 5, if the poison caused the formation of clots in the veins, thus stopping the flow in either of blood—as Martin tells us the venom of the Australian black snake does case the effect would be the same as if the action of the lungs were to cease. The sudden death of my Gila Monster prevented me from testing these possible causes of asphyxiation from its poison, but I shall not be surprised if it be found that in one or both of them exists the explanation of the phenomena exhibited. But perhaps I should limit this statement somewhat, for Mitchell and Reichert state very positively of their experiments that death was occasioned by the action of the poison upon the heart. Here is an apparent contradiction of my results, and by the highest American authority upon reptile poisons; but the seeming contradiction disap- pears, perhaps, when we recall that Dr. Mitchell’s Gila Monster saliva was less dilute than mine, and that it is known of some serpent poisons that “ with higher concentration of venom the heart is the more rapidly affected, but the continuous operation of the poison in small concentration more quickly affects the respiratory ” system. IV. SOME CAUSES OF DIVERSITY OF OPINION. We have now reached our last question: Why has the bite of the Gila Monster so often been considered harmless ? Several reasons must, I think, already have suggested themselves. Dr. Shufeldt, it will be remembered, was severely bitten on the thumb, and concluded that the bite of the Gila Monster is no more poisonous than that of other angry animals; for example, a cat. 3ut Dr. Shufeldt expressly states that the wound was made by the upper teeth pene- trating to the bone, and we have already seen that the saliva of the upper jaw is harmless at all times, the venom being confined to the lower jaw.+ So it well may be that Dr. Shufeldt owes his life to the circumstance that the injury to his thumb was inflicted by the upper instead of the lower teeth of the Monster. This same fact will account for the experiences of other authors who have thought the bite of this reptile harmless, but there are other reasons for the occasional failure of the Heloderma to inflict a deadly wound. The teeth, although sharp and long, are very weakly fastened to the jaws, and often so many of them have been broken out that the *Cunningham, Set. Mem. Med. Officers Army India, IX, 1895, pp. 1-54. | It would be interesting to know why the teeth of the upper jaw are grooved. SALIVA OF THE GILA MONSTER. 219 Monster is unable to inflict a wound at all. Even if the teeth are in working order the chances of the poison finding its way into the wound are yery few, for the teeth are not directly connected with the poison glands, and the latter are below the fangs instead of above as in poisonous snakes. The poison simply flows out onto the gums below the teeth, and, to be effective, has to be forced wp into the wound. Unless the flow of saliva be abundant and the teeth all present and forced into the bitten flesh so deeply as to press it down upon the poison ducts where they open between the lip and the gum, it is difficult to see how even the smallest quantity of poison could enter the wound, eyen though the teeth are grooved to afford it a passage. The strange thing, then, is not that bitten animals should sometimes survive, but that they should sometimes die. Neyertheless, small animals often do die from the bite of this, the only poisonous lizard, and we must believe that a venom which can kill a pigeon in seven minutes and a ‘abbit in less than two might easily under favorable circumstances cause a wound to prove fatal even to man—a belief which is rendered far from improbable by the extra- ordinary virulence of the poison and the lizard’s habit of holding like a bulldog to what- ever it bites. VW. BIBDIOGRAPREY: 1651. Hrernanpez, F.—Historixe animalium et mineralium Nove Hispaniz, p. 315. 1829. Wrromann, A. F.—Ueber das Acaltetepon oder Temacuilcahuya des Hernandez, eine neue Gattung der Saurer, Heloderma. Jsis, pp. 627-629. 1857. Gray, J. E.—On the Genus Necturus or Menobranchus, with an Account of ItsSkulland Teeth. Proc. Zool. Soc. Lond., p. 62. 1864. Sumicurastr, F.—Note sur les Mceurs de quelques Reptiles du Mexique. Bibl Univers. et Reowe Suisse (Ar- chives des scien. phys. et nat.), XIX, pp. 45-61. [Reprint pp. 1-5.] 1864. Sumicurast, F.—-Notes on the Habits of Some Mexican Reptiles. Annals and Mag. Nat. Hist. (3), XIII, pp. 497-500. 1869. Corr, E. D.—[Remarks.] Proc. Ac. Nat. Sei. Phila., p. 5. 1873. Grrvars, P.—Structure des dents de l’Héloderme et des Ophidiens. Comptes Rendus Acad. des Sciences, LXXVII, pp. 1069-1071. 1875. Bocourt, F.—Observations sur les murs de l’Heloderma horridum, Wiegmann, par M. F. Sumichrast. Comptes Rendus Acad. des Sciences, LXXX, pp. 676-679. 1875. Yarrow, H. C.—Report upon the Collections of Batrachians and Reptiles made in Portions of Nevada, Utah, California, Colorado, New Mexico and Arizona during the years 1871, 1872, 1873, and 1874. U. 8. Surv. W. 100th Merid., V, pp. 562, 563. 1878. Bocourt, F.—Mission Scientifique au Mexique et dans l’ Amerique Centrale, III, Reptiles, 5e livr., pp. 296- 806, Pls. XX E, Figs. 1-12, XX G, Figs. 1, 3,6, 7, 8, 9, 10, 11. 1880. Sumrenrast, F.—Bulletin de la Société Zoologique de France, p. 178. 1882. AmeRrIcAN NaruRAList.—[Note.] Am. Nat., XVI, p. 842. 1882. Bounencer, G. A.—[Remarks.] Proc. Zodl. Soc. Lond., p. 631. 1882. Fayrer, J.—[{Remarks.] Proc. Zodl. Soc. Lond., p. 682. 1882. Fiscner, J. G.—Anatomische Notizen tiber Heloderma horridum Wiegm. Verhandl. des Vereins Hamburg, Bd. V, pp. 2-16, Pl. III. 220 1882. 1883. 1883. 1883. 1883. 1885. 1885. 1887. 1887. 1888. 1888. 1889. 1890. 1890. 1891. 1891. 1891. 1891. 1891. 1891. 1991. SOME EXPERIMENTS WITH THE SALIVA OF THE GILA MONSTER. Suuretpt, R. W.—The Bite of the Gila Monster (Jeloderma suspectum). American Naturalist, XVI, pp. 907, 908. American Narurauist.—[Review of Mitchell’s and Reichert’s article.] XVII, 7, July, p. 800. Garman, S.—Reptiles and Batrachians of North America. Memoirs Mus. Comp. Zool. Cambr., VIII, 3, Ds eXd- M , H. N.—The Physiological Action of Heloderma Poison. Science, I, 18, May 4, p. 372. Mircnety, S. W., and Rercuert, E. T.—A Partial Study of the Poison of Heloderma suspectum (Cope)— the Gila Monster. Medical News, Phila., XLII, No. 8, Feb. 24, pp. 209-212. Bov.encer, G. A.—Catalogue of the Lizards in the British Museum, Vol. IJ, pp. 800-302. Guntuer, A. C.—Biologia Centrali-Americana, Reptiles, pp. 48, 44, Pl. XXVI. Benpire, C. E.-Whip Scorpion and the Gila Monster. Forest and Stream, XXIX, 4, Aug. 18, pp. 64, 65. SuHuFrELpDT, R. W.—The Gila Monster. Forest and Stream, XXIX, 2, Aug. 4, p. 24. Yarrow, H. C.—Bite of the Gila Monster. Forest and Stream, XXX, 21, June 14, pp. 212, 213. Lussock, J.—[Letter from 8. A. Treadwell.] Proc. Zool. Soc. Lond., p. 266. MircHet, S. W.—The Poison of Serpents. Century Magazine, XXXVIII, 4, p. 505. GarMANn, S.—On the ‘‘Gila Monster”’ (Heloderma suspectum). Bull. Essex Inst., XXII, pp. 60-69. Suuretpt, R. W.—Contributions to the Study of Heloderma suspectum. Proc. Zool. Soc. Lond., pp. 148-244, Pls. XVI-XVIII. BouLENGcER, G. A.—Notes on the Osteology of Heloderma horridum and H. suspectum, with Remarks on the Systematic Position of the Helodermatide and on the Vertebrie of the Lacertilia. Proc. Zool. Soc. Lond., pp. 109-118. BouLENnGmER, G. A.—The Anatomy of Heloderma. Nature, XLIV, p. 444. SHuUFELDT, R. W.—The Poison Apparatus of the Heloderma. Nature, XLIII, p. 514. SHuFELD?, R. W.—Further Notes on the Anatomy of the Heloderma. Nature, XLIV, p. 294. SHuFELDT, R. W.—Some Opinions on the Bite of the ‘Gila Monster’’ (Heloderma suspectum). Nature's Realm, I, 4, April, pp. 125-129. Suuretpt, R. W.—Medical and Other Opinions Upon the Poisonous Nature of the Bite of the Heloderma. New York Medical Journal, LIM, 21 (651), May 28, pp. 581-584. Srewart, C.—On Some Points in the Anatomy of Heloderma. Proc. Zool. Soc. Lond., pp.119-121, Pl. XI. . ARTICLE IV. RESULTS OF RECENT RESEARCHES ON THE EVOLUTION OF THE STELLAR SYSTEMS. (Plates LV and V.) BY T. J. J. SEE, A.M., Pa.D. (BERLIN), ASTRONOMER AT THE LOWELL OBSERVATORY, Read before the American Philosophical Society, January 7, 1898. It is now two hundred and eleven years since Newton published the Principia, embodying his grand generalization of the law of gravitation, and the proof of this law for the most obyious and fundamental phenomena of the solar system. Geometers have since been occupied with the development and extension of the principle discovered by the illustrious Newton, and have finally explained with almost entire satisfaction the motions and attractions of the planets, satellites, comets, and other bodies which revolve about the sun. This great development can hardly fail to excite the admiration of those who contemplate the history of scientific progress, and must be accounted one of the most noble and enduring monuments of the human mind. So sublime an achievement has required the combined labors of a long series of men of transcendent mathematical and mechanical genius, each building upon the foundation laid by his predecessors. Though many distinguished geometers haye borne an honorable part in this remarkable develop- ment of Physical Astronomy, it will not be inappropriate to point out the great credit for the perfection of the Newtonian theory due to Clairaut and Euler, Lagrange and La- place, Gauss and Hansen, Adams and Leverrier. Among living investigators in mathe- matical astronomy the names of Hill and Neweomh, Darwin and Poincaré occupy the foremost place. These great men have brought the mechanics of the heavens to so high a state of perfection that in almost every case we may now predict the heavenly motions as accurately as we can observe them. In view of the rapid perfection of telescopes and other instruments of precision, this achievement, from the intricacy of the analysis required in the problem, and the abstruseness of the methods used in the reduction of DL RESULTS OF RECENT RESEARCHES ON THE observations, must be ranked as incomparably the most profound yet attained in any branch of Physical Science. Notwithstanding these splendid triumphs of the science of Celestial Mechanics, an eyen greater and more recondite work remains to be done in a closely related field. This is the investigation of the origin and cosmical history of the planetary and other systems observed in the immensity of space. Even if some credit for pioneer work on this problem be assigned to Kant, or, more remote still, to the Greeks of the pre-Socratie age, it yet remains true that Laplace is the real discoverer to whom we are indebted for the first ideas which proved fruitful for the advancement of science. About a century ago this great geometer outlined for the solar system the celebrated Nebular Hypothesis, upon which nearly all subsequent investigation has been based, and which has since been sub- stantially confirmed, though but very little modified until within the last twenty-five years. Passing over as irrelative in the present discussion the early work of Herschel and Rosse, Helmholtz and Kelvin, Newcomb and Lane, we come down to the modifica- tions introduced by Darwin about 1880. In establishing the theory of gravitation, Newton assigned also the true cause of the tides of the seas, though his explanation carried with it all the defects of the equilibrium theory. More than a century passed before the dynamical character of the problem of the oceanic tidal oscillations was clearly perceived, when Laplace developed and applied the true theory with all the penetration characteristic of that great mathematician. Yet in spite of the profundity which marks his treatment of the tides of the oceans, it seems neyer to haye occurred to him, or at least he made no record of the fact, that the attraction of the moon necessarily produces tides in the body of, as well as in the aqueous layers covering, the earth. We need not be surprised at this omission on the part of Laplace and those who followed him, if we recall that for many years after the perfection of Analytical Mechanics by D’Alembert and Lagrange, the subject was treated wholly from the point of view of material particles, and the resulting system was what is now called Rigid Dynamics. Little attention was bestowed upon the theory of fluid motion, partly because of its intricacy, and partly because there were no obvious applications of the results except in the case of the tides, already treated by Laplace with great penetra- tration and extreme generality. As mathematicians since the time of Newton had been occupied chiefly with the development of the theory of planetary perturbations along the line of rigid dynamics, it did not occur to them that they were building on a false premise, that in reality the heavenly bodies so far as known are not solid, but fluid, though Laplace with his usual sagacity had long foreseen that in the case of our planets the nuclei are covered with fluid layers held in equilibrium by the pressure and attraction of their parts. His grand treatment in the Mécanique Céleste recognizes the fluidity of — EVOLUTION OF THE STELLAR SYSTEMS. Ds} the envelopes of the planets, and exhaustively examines the oscillations that will arise therein. Nor did he fail to consider fully the deviations from spherical form and the probable laws of density for the layers which compose the bodies of the planets. The effect of so monumental a work as the JMécanique Céleste was twofold: on the one hand it brought Physical Astronomy to’an unexpected state of perfection, while on the other it produced the impression on the less creative minds that there were no great problems untouched by the master-mind of Laplace. His work had indeed well-nigh exhausted the theory of Celestial Mechanics, so far as it could be built upon the assumptions of rigid dynamics ; at least subsequent work has been for the most part little more than refinement or perfection of the methods and processes given in the Mécanique Céleste. The work of Laplace was designed for the solar system, and the idea that the universe is really composed of fluid bodies, self-luminous stars and nebulee in space, seems neyer to haye occurred to him, or he would have foreseen that however adequate Rigid Dynamics may be for effecting a first approximation, the true theories of ultimate Celestial Mechanics must be founded upon the laws of viscous fluids in motion. So great is the influence of tradition that it is difficult for us to realize fully that the stars and nebule are viscous fluids, self-luminous liquid or gaseous masses, and that even in the solar system the bodies are all fluids of various viscosities. This new point of view respecting the actual facts of the universe has brought about an important modification in the nebular hypothesis and in the ultimate theories of Celestial Mechanics, of which we shall now give some account. ; About 1875, G. H. Darwin, who had qualified himself for the Law and been called to the Bar, on account of ill-health, abandoned his profession to undertake for Lord Kelvin some scientifie work, which among other things included the reduction of a great mass of Indian tide observations with a view of throwing light upon the problem of the rigidity of the earth. This work, besides leading Lord Kelvin to the celebrated conclusion, that the earth as a whole is “‘ probably more rigid than steel, but not quite so rigid as glass,” was the oceasion* of the younger Darwin developing the theory of bodily. tides, or the theory of the tides which would arise in the earth on supposition that it is not rigid as at present, but a viscous fluid, as it must have been, according to Laplace, at some past age. While some allusions to bodily tides can, be found in scientific literature as far back) as Kant, and especially in the papers of Delaunay on the secular acceleration of the moon’s *In the Atlantic Monthly. tor April, 1898, ProfDarwin remarks: “It was very natural that Mr: See should find in certain tidal investigations which I undertook for Lord Kelyin the souree of my papers, but as 4 fact the subject was brought before me in a somewhat different manner, Some unpublished experiments on the viscosity of pitch induced me to exténd Lord Kelvin’s beautiful investigation of the strain of an elustic sphere to the tidal dis tortion of a viscous, planet. This naturally led to the consideration of the tides of an ocean lying on Such a planet, which forms the subject of certain paragraphs now incorporated:in Thomson, and Tait’s Natural Plilosaphy. A. PB. S.— VOL. XIX. 2c. 294 RESULTS OF RECENT RESEARCHES ON THE mean motion, it is yet indisputable that Darwin was the first writer to treat the problem in a systematic, thorough-going and original way. Recognizing that at some epoch in the past, the earth was probably a mass of viscous fluid, he set for himself this problem : To determine the bodily tidal distortion of the earth, and the effects of this alteration of figure upon the orbital motion of the moon, and upon the earth’s rotation. His papers were communicated to the Royal Society between 1878 and 1882, and are celebrated con- tributions to the general theory of tides. In these papers he has traced the moon back to close proximity to the earth, when the two, at the breaking off of the moon, were most probably revolving in about 2h. 41m. The moon has since receded from the earth under the action of tidal friction, while the rotation of the earth has been slowed up in correspond- ing degree. It was rendered certain that in the origin of the Lunar-Terrestrial System, the action of tidal friction had played a prominent, if not a paramount part, and the question naturally arose whether it had not been equally potent in the development of other parts of the solar system. When, however, Prof. Darwin came to apply the results to other satellite systems and to the solar system as a whole, it was found that here the effects had been much less considerable than in the case of the earth and moon, owing chiefly to the small masses of the attendant bodies. Thus the major axes of the orbits had perhaps been very slightly increased, and the rotations correspondingly exhausted, but no radical change had taken place. Under these circumstances it was natural that Darwin should drop the subject without further search for extension of the principle he had developed. About November 1, 1888, while I was still an undergraduate at the Missouri State University | became much interested in the origin of the double stars. The immediate cause of my taking up the subject was the Missouri Astronomical Medal, occasionally awarded by the University to a graduate of highest standing in the Mathematical and Physical Sciences. Having been informed by Prof. W. B. Smith that I was eligible to write for the medal, by virtue of my standing in the Physical Sciences, our conyersa- tion drifted on to the probable subject of the Thesis, and in this way he was led to suggest a criticism of Darwin’s work on the origin of the moon. He remarked: “You may find this only a pocket, already worked out, and not a continuous vein of rich ore, but it seems to me worth thinking of. At any rate I would not advise you to write on the orthodox Laplacean Nebular Hypothesis, for that subject is worn threadbare.” The suggestion of a critique of Darwin’s work did not quite meet my approval, for I feared the subject was already exhausted and would leave no field for future progress. As I had been observing various double stars for the past two years, and had seen no suggestion regarding their mode of development, it occurred to me that perhaps the tidal theory might find application among the stars. When I had collected such orbits as were EVOLUTION OF THE STELLAR SYSTEMS. 225 available in the books at my disposal (Humboldt’s Cosmos, Herschel’s Outlines, ete.), I discovered to my surprise that unlike the orbits of the planets and _ satellites, they are very eccentric, though not so eccentric as those of the periodic comets. It was at once evident that it would be hopeless to attempt to explain the origin of the stellar systems, if we could not explain the cause of the high eccentricities of the orbits. The next day I called on Prof. Smith and told him of the discovery that the orbits are very eccentric, and asked whether he thought [ might explain this peculiarity on the tidal theory ; rub- bing his head for a moment in quiet reflection, he replied: “Oh! I see what you mean ; you think the dragging of the tides in the bodies of the stars has produced the elongation you find in the orbits. Such an idea can hardly be discussed off-hand, but it is at least worth examining; it may prove fruitful.” “That is exactly what I mean,” said I, “and you have correctly interpreted my line of thought.” After this conversation, which is here reported exactly as it occurred,* there was nothing else before my mind for several days, as I was wholly occupied with finding out whether the problem undertaken was soluble, and, if so, whether it would result in any important Physical Truth. Having established the fact of high eccentricity as thoroughly as the published orbiis at my dis- posal would admit, I set about that same day the problem of explaining the cause of the eccentricities ; and as I worked the impression continued to grow on the mind that since the stars are not solid, but self-luminous fluid bodies like our sun, and the two members of a system comparable in mass, the action of each body would produce tides in the other, and the lagging of the tides in the two stars would gradually expand and elongate the orbits as now observed in space. And before I had obtained access to the learned papers which Darwin had communicated to the Royal Society, or eyen to his article “Tides” in the Encyclopedia Britannica, I proved by an elementary process that when the bodies rotate more rapidly than they revolve, the eccentricity of the orbit would gradually increase. Here then was a result confirmatory of the happy intuition, and for the past nine years my energies haye been largely devoted to the extension and generalization of the theory of bodily tides in relation to cosmical evolution. After concluding my undergraduate studies at the University of Missouri, I con- tinued the work at the University of Berlin. It is particularly of that work and the extension which I have since made of it that I shall speak to-night. The theory of tidal friction developed in the Jnaugural Dissertation presented to the Faculty of the Uni- versity of Berlin is essentially a special treatment of the general theory as it occurs in nature, while that previously developed by Darwin in connection with the moon and planets is restricted by the condition that the perturbing body is very small. I shall therefore discuss the general case as presented in my own researches. *As the occasion of my beginning this work has never been published, I trust it will not be thought inappropriate for me to recall it in this paper to the American Philosophical Society. 226 RESULTS OF RECENT RESEARCHES ON THE Suppose we denote an element of the mass of a spheroid by m, and its distance from the axis of rotation by ¢; then the moment of inertia is L = pm? If the spheroid be rotating with an angular velocity y, then Jy will be the moment of momentum of the body about its axis. For a second body whose moment of inertia is /’, and angular yelocity z, the moment of momentum is /’z. Foilowing the analogy of Darwin’s procedure, we choose a system of units designed to simplify the resulting equations. Let us take as the unit of mass M M’ M+ mM and as the unit of length a space P such that the moment of inertia of the spheroid about its axis of rotation shall be equal to the moment of inertia of the two spheroids treated as material points, about their common centre of inertia when distant apart [. Then we have . sat 12 (oe ae iM | ; + M ; \ M+ we} — VE ox pi of LM we) a i) MM’ j Let the unit of time be the interval in which one spheroid describes 57°.3 in its orbital motion about the other when distant [. In this case, 1 is the orbital angular velocity of the body. The generalization of Kepler’s law gives 6-1? =u (+ IW), and ; “(12 (M+ MM) \3 eS <3 pw (MM f Now suppose the two stars to revolve about their common centre of inertia in a cir-' cular orbit, with an angular velocity ©, when the radius yector is p. Then the orbital moment of momentum is M (ar) 2 Me (aor) Ow Cea 20, In a circular orbit the law of Bene gives (?o% = ua (MM + FP); “and Op" = « (M+ IM) 9’; and on inserting for Op” its value, we have 2 WW (M+ WM’) —? 6, EO it Sete Oe. oe EVOLUTION OF THE STELLAR SYSTEMS. 227 which in special units is p3. Now the total moment of momentum of the system is con- stant, and is given by The kinetie energy of orbital motion is : MW Gar) oO +5 WG i soap). On ( MM’ ) eer z u MM’ The kinetic energy of rotation is to ergy a sel Wes ld Ce il » -The potential energy of the system is uM a By adding all these energies together we get the total energy of the system : E 9 Lk 35 » MM’ pel ae : 2p ? where / is twice the whole energy. In the system of special units, J, wIZI/, are equal to unity. If we put 4 = =, we shall get E=y+ke—! p Let « = OF, and then O* = 6}, « = 9}, and we have finally If we suppose the two stars to turn on their axes in the same time in which they revolve in their orbits, so that they show always one face to each other, the motion of the system will be as if the masses were rigidly connected. This condition is given by 228 RESULTS OF RECENT RESEARCHES ON THE Accordingly we haye the system of fundamental equations : H=y-+ kz + x, plane of momentum, | 9 9 ] : 4 1 TO I ep a a surface of energy, r Re eae ae oe (4). 9 2 > oe 0 70 1 ey = 1,2 =A curve of meiaihye 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 ‘ase 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 plane of momentum to a particular line of that plane : a! — He + (1+ &) = o* — Ae + 2 =0, since & = 1. The equation of the energy surface passes into the form foe Se ee 9 L te The curve of rigidity becomes H—«x : DS where 7 = / es Every point in the plane of momentum represents one configuration of the system, 7. e., 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 ease 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 Entwickelung des Doppelsternsysteme, Berlin, 1893, R. Friedliinder & Sohn. ] As the tides raised in the stars are subjected to frictional resistance, energy 1s TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE IV. Pea ey V, — =) = Le om I ae 4, Le a9 | NJ | } Su | a | } a= =| —_—_—_—_— t \! / Axis of orfital moment) of momentum | mm | | ; . | | 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. Ty 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 /7 = 4. If the guiding point is set at @ it may move either of two ways: it may slide down the slope ac, in 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 @ is 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 6, 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 1 de I fp alates Cel Se) 8 edz —%% | a(H—a)—2 S$’ which, on integration, is put into the form H == IT8 y—a (@ Mm a)? exp. [ ” | TGR STE) Ce ee os ale (5) a Ha, | i} ii 1 — 9 a) | @o dj [@ —a +P lary where B is an arbitrary constant; a, 6, a + 31, 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 curyes 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 0.3 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 event 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 preyent 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. Pe, S.— VOL. XIX. 2D: Zar RESULTS OF RECENT RESEARCHES ON THE unseen tides in every part of the heavens. In a fluid universe tides necessarily result from grayitation, and are as universal as this great law of nature. In my later researches [ 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 contrae- tion homogeneous incompressible fluid masses may divide into equal or comparable parts. The next question was: Are there nebule 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 nebulee 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 aun unsymmetrical pear-shaped figure which I have called the Apioid. 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 Systems, Vol. I: On the Universality of the Law of Grae- itution and on the Orbits and General Characteristics of Binary Stars (Tne Nichols Press, Lynn, Mass., 1896). EVOLUTION OF THE STELLAR SYSTEMS. 2509 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 Fig. 1 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 nebulse 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. 3urnham and Barnard had previously assured me that the interpretation of the figures of double nebulze 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 nebulze 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 nebulee by any means incompressible ; yet many considerations lead us to believe that in most cases the density of 254 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 is 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 TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE Vv. Drawings of double nebule according to Sir John Herschel iy 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. ; ae Ber St fe on ies = = ap a PM iN Sere : arse fy a aa . wes = ts ms, 4 yl Sqoe 7 thei ae 10 iss : gia: tone 5 ' " “ent © if as eet, aft ep tone / i > < ~~ A 7 ARTICLE V. ON THE GLOSSOPH AGIN At (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 Phyllostomididie, 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 chaernycterine and the phyllonycterine. 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, * Zoodlogists are indebted to Prof. W. Peters (M. 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 ; asa 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. 938 ON THE GLOSSOPHAGIN &. is probably also of Vampyrine origin. The third division contains but a single genus, viz., Phyllonycteris. Tt is so near Brachyphylla that it would be easy to effect the transition and remoye 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, | have not seen. I have coneluded from the published descriptions of Glossonycteris 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 Anwra. 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 G'lossophaga 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 cheernyeterine. 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. Interspace. Interspace. Interspace. Interspace. Interspace. Glossophaga soricina ...... 2 12 17 LONChOG 1088. ..- 1s veeeneene 2 16 23 Glossophaga truei..........- 2 11 15 LEO R@scosencronsnscneces2s39¢ 3 15 30 Leptony cteris ....-cecceeeeee 3 15 25 Phyllonycteris .. 3 13 25 Cheronycteris........-..++- 2) 11 20 Tnough 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 distinet, while in Lyroderma and Luvia they ave supplanted by a skin-fold which becomes an integral part of the nose leaf, ON THE GLOSSOPHAGIN ®. 239 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 moyements 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 deviated from the axial positions and thus appear to correlate with increase of wing strain. Lonchoglossa is intermediate between Anura 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 las the weakest development in Chaernycteris, in which form it supplies the first and fourth digits only. In Phy//onycteris it omits only the second, while in Lonchoglossa and Glossophaga it supplies all the digits. * The stomach in the Glossophaga villosa Rengyer ( Naturgesch. der Saugcthiere von Paraguay, Basel, 1530, 50) was found to contain blood with remains of insects. It is not known what forms would now be included under this title. See remarks on Anvvic. A. P. S—VOL. XIX. 2 E. 240 ON THE GLOSSOPHAGIN®. 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 Phy/lostoma is the type, I have imper- fect knowledge—having 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 Gervais (Eep. du Sud.) as characteristic of Tylostoma and Monophyllum (Dolichophyllum). V inter that Trachyops, Phylloderma and Mimon are members of this group from Dobson’s statement (Br. Cat. Chir.) that they resemble Phy/lostoma. 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 Glossophaginw 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 aboye 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 concave periotic region, but its other characters, taken as a whole, connect the form intimately with the Glossophagi. The taxonomic yalue 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 ime 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 gr ups of species than is now the case. (GLOSSOPHAGA. Upper incisors in a continuous row. Length of forearm not exceeding 36 min.; thumb, Simon; 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. +— ¢. 4 — prm. }—m. 3 = 21. 35 ON THE GLOSSOPHAGIN®. QAI The Flexor 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 G'lossophaga soricina as haying uo tail (Mise. Zoblog., 1766, 48), the type being a female. He subsequently deseribed and measured a second speci- men (Spicil. Zoil, 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) tor the assumed new species possessing a tail. Gray (Ann. and Mag., N.S., 1838, IT, 490) acting on these erroneous premises proposed the name Phy/lophora for Glossophaga amplexicaudata. Gervais (Kepn. 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 conforms 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’ obseryations. 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 bad : : peek one cannot gainsay the yalue of the figure of the fimbriated and elongated tongue. 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 Glossophagina vera. along the tongue. * Gervais (/. ¢.) believes the form is not G/ossophaga at all, but Hemiderimu. 249 ON THE GLOSSOPHAGINA. a. Median upper incisors larger than lateral; premolars 4; 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 }; thumb’ one-fourth the lenethof forearm) (SI=S4 mM) even sa-nerlnrgceee eer ators seneeeeanasewceeesestans Glossophaga. b. Upper incisors with wide interval between centrals; molars 3; thumb one-sixth the length of forearm (45 mm. ).....+.+eeeeeeeseeeeeeeeee eee scbecoi2a0 Leptonycteris. a’. Median upper incisors smaller than lateral; premolars +; crown of lower canine with base not lying inside position of lateral incisor; median incisor fora- men well in adyance of paired foramina ; upper incisors vertical. ce. Lower canine compressed, with cingulum; metacarpal bone of thumb exceeds length of phalanges. d. No phalanx to second digit of manus; premolars 4; tail present; thumb one-seventh the length of forearm (Gris 119) prornaceopecnencacs Cosdoosoos do booscd nossBabtes sceodhemaes once Chernycteris. ce’, 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.)..-...-.-...:...+. Lonchoglossa. d'’, No phalanx to second digit of manus; no tail; thumb one-sixth the length of forearm...............cseseeseeeeeeeeeees Anura. tympanic bulla almost touches postglenoid process; occipito-squamosal suture with large foramen; ethmoid hone not convex in brain case; an ectopterygoid lamina. In ( | Palatal portion of premaxilla not rostrum-like; gland mass crosses muzzle back of nose leaf ; ite , 2 third to fifth manal digits first and second phalanges equal; premolars 37; molars $; fim- briw of tongue at tip only. 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, retroyerted, internal lappet a mere projecting nodule. Tragus straight on inner, conyex 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 &. 245 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 papille. The first phalarix 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—heing 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 conyex 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, yomerine ridge. The lateral depres- sions on the basioccipital are conspicuous, the mastoid process is obtuse. 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—The 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, 944 ON THE GLOSSOPHAGIN#. for grinding. In the upper jaw, with the exception of an interval on either side of the eanine, 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 concave 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 fluting on the para- conid, rudimental ; the metacone is wnited 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 distinct 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 alveolar processes. The third molar slightly overlaps the second at the buccal border. The lower incisors are provided 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 less 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. In 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 advance of the lateral teeth. I am inclined to believe these are variations due to advanced age. “The upper incisors as represented by Leche (Studier dfver Mijolkdentionen och Tiindernas Homologier hos Chiroptera, 1876, Tab. II, VII) do not touch. ON THE GLOSSOPHAGIN &. DAD Glossophaga truet, W. s. In the Proc. U.S. Nat. Mus., XVIII, No. 1100, 1896, 779, IT described a new species of Glossophaga under the name G. villosa. Since Rengger (/. ¢., p. 80) described in 1850 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 G'lossophaga, 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 have presented degrees of variations so low as never to have permitted the recognition of more than a single species. The complicated synonymy successtully 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 careful study of two specimens (Nos. 9522 and 9523) belonging to the United States National Museum has convinced me of the necessity of recognizing two species of Glossophaga—namely, Glossophaga soricina and the one which I here name Glossophaga true. 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 every- 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 removed in preparing the skin, 246 ON THE GLOSSOPHAGIN ®. Based on skins of two adults: No. 9523, U. 8. N. M., La Guayra, Venezuela ;* and No. 9522, U.S. N. M., co-types. No. 9523, U.S. N. M., fur soft, shrew-like; 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 shghtly 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 nerye 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. truer. In Geoffroy’s figure { the measurements of the nose leaf agree with those of G. sorteina, 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. frwer 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 * It 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 byzthe 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 calear 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 Glossophaga truer. Millimeters. Head and body (from crown of head to base Of tail) .......-.csseeeeensccseceeeteeneeseeee ceeeeeweeseeeee 45 ELGAGE ANG! TOLCAT Mees ence chas ccs acces eetase docs sore sestsencessdessetuctslatsasceseveshcvedvessvscacssectsacdssecseses 32 First digit : Menpthrotfirsbimetacanpall DOUCsecressesencscresacscns-acannacsnensceacucseessacrsesncceeusrerseussceaee 4 Length of first phalanx 4 Second digit : Length of second metacarpal bone.......:c+-ceccceeeeee ececeeenessseeetseseeseetcceceseerseneneserees 25 Menpthvon urstiphalanxtect-caseccoc-cescoonsavassntnnsecmetanntassccsws=ahassucasasecacssssesssacaassesene 2 Third digit : Length of third metacarpal bone... 30 eneth’ of first phalanx.-...c...ccsccssesssconscnnssenesecnuctssacascastsanceanscasssavsraneenaescnncnas ell : Tenethronsecond phialanxrecsnrccsenacsncscecesseneseesnseerevecconeecas+nsscreasrnasvesenssankenrsnicsaes 14 Men pros thinda pislankieseesewcneessececsncseecdres +: oneceysceqescenrseceseencacndado=mnsseAcchan'enntien 6 Fourth digit : Length of fourth metacarpal bone 27 ene Hito mtirstap ia lanxe eee sasa-sacksescecncanecucarsanrascanearastnsrsteccnsnsntscecesvsseresctorrs sss 9 Length of second phalanx .......-..... SUAS ES ADAGER OULCCOCE COOL EE Sq SPO NC ADACIO REO EHOCIDOCRIE 9 Fifth digit : Length of fifth metacarpal bOne.....0.....-e.cseccscesesescneecssccccnscesrescnsseretecescrsscseccessecs 27 Length of first phalanx ......... 8 en ot hvomsecond ip Nala kesenscespiesnceseueriracccsandanseacnsusnrecasnecesserinesnusccvsssonendcaresess 8 WSN POM NCA scan es exnpellcceacticsdGapnensoscnssansauecssedusavecstsdadcccessassscescratsccccsrecnccnversscn-nsbeesses> 21 ENGI MG OL CAlnana cuter rec enacncstsdeccvanvecatenpscaractandeanecessconacsecashseucesecaerencasseecnceesscncccnse--=9<0 11 Height of tragus 3 Length of tibia.......... 11 Went OF L006 <2 2 8 Length of interfemoral MeMbrane....-....-.c2ececeeseesereceececcecececeersecscecaucncescnaesensaessecscueseese 9 MonopHyLuus. Upper incisors not in a continuous row. The first and second upper molars with hypo- cone. Length of forearm, 37 mm.; length of thumb, 10 mm. The tail projects from the margin of the short interfemoral membrane. The proencephalon does not create an eminence on the brain case. No vertical line is found on any of the interdigital spaces. Dental formula: i. #— ec. +— prm. ? —m. 3 = 21. * The measurements of No. 9523, U. S. N. M., are the same asin No, 9522, U. S. N. M., excepting in the second phalanx of the third manal digit, which is but 12mm. long. A. P. S.—VOL: XIx. 2 F, 248 ON THE GLOSSOPHAGIN®. The single specimen of Monophyllus which was available was that of a skin of an adult (No. 83347, °, U.S. N. M.) obtained by exchange from the Berlin Museum. The genus is in close alliance with Glossophaga—closer, indeed, than any two genera of the group. The retention of the hypocone in the first and second upper molars, the presence of a keel on the symphysis of the lower jaw and absence of the vertical line in the inter- digital spaces, separate the two forms. Other characters if they existed unassisted by those just named would be those of relation and proportion. The presence or absence of the calear could not be determined. Monophyllus redmani Leach. Auricle with blunt tip, scarcely emarginate on outer border. Wing membrane from basal third of the tibia: terminal cartilage of the fourth digit, spatulate. Marked rudi- ment of ascending process from the zygoma, Nose leaf, upper lip and membrane much as in. Glossophaga truer. The auricle resembles G. ¢ruei nearer than G. soricina. It is blunt at tip, scarcely at all concave on the outer margin. A faint emargination is noted on the inner margin which may be exaggerated in the dried skin. The external basal lobe was everted by the method used in preparing the specimen. The parts do not differ from those studied in Glossophaga. The tragus is blunt, presenting two coarse sinuations at the outer side and two denticulations at the base. The nose leaf, upper lip and mentum almost precisely the same as in G. true? No warts are anywhere present. Fur above is dark brown; the head, neck and shoulders a lighter shade than the back of thorax and loin. Examined with a lens, the fur has an admixture of fine gray hairs, which are more numerous on head, neck and shoulders than elsewhere. The fur’ beneath is gray and brown, about equally admixed. Both above and below the hair is unicolored. Sparse gray hairs extend below on arm to elbow and slightly over the endo- patagium. The legs are naked. There is no vertical line on the membrane of any of the interdigital spaces. The endopatagium exhibits a few coarse vertical lines. The fourth interdigital space is obscurely areolate. The skull was mutilated at occiput and posterior third of the base. It closely resem- bles G'lossophaga. The fronto-temporal crest is more defined, while the fronto-maxillary inflation is less defined than in that genus. The posterior palatine notch, narrow. Seen from above, the posterior border of the infraorbital foramen appears as a blunt spine. A narrow but well-defined groove extends the entire length of the face, beginning at a foramen near the pretemporal ridge. The ascending process from the zygoma is greatly in excess of the same character in Glossophaga. The external auditory opening ON THE GLOSSOPHAGIN ©. YAY is smaller than in the genus just named. The thick skull does not admit of the divisions of the brain being discerned. The lower jaw is more robust—the depression in advance of the angle most marked of any genus in the group; the angle is raised high above the level of the lower border of the high ramus as in the Lobostomina; the symphysis is pro- vided with a large keel. On the whole the skull is more robust in texture and is of a larger animal than Glossophaga, but the face structures more extended, and presumably from the symphysal modifications, a longer and more prehensile tongue. The Upper Teeth—TVhe incisors are not arranged in a continuous row or in pairs, but intervals* are found between the teeth. The space between the central incisors is wider than that between these teeth and the laterals. The central incisors are obscurely hatchet-shaped, while the laterals are conical. Wide intervals also exist between the canine and the first premolar and between the first and second premolars. The other upper teeth are contiguous. The premolars are aciculate, compressed, with prominent base conules. The first and second molars are quadrate with conspicuous hypocone. The third molar is more triangular and resembles the first and second molars of Glossophaga. The Lower Teeth —The incisors are reduced to tubercles, arranged in pairs, which are widely separated both from the symphysis and the canine tooth, though nearer the latter than the former. The central incisor is larger than the lateral. All the other teeth are contiguous, except the second and third premolars, which are separated by an interval equaling that in the upper series. The first premolar is distinctive. It closely resembles the homologous tooth in Glossophaga and anteriorly overlies the base of the canine. The second and third premolars are similar to those in the upper jaw. The molars are of the same type as in G'lossophaga, but elongated and compressed in advance of the protocone and paracone as in Leptonycteris. The comparison of the skull and lower jaw seen from in front with G'lossophaga is instructive in the differences in the shapes and relations of the shapes of the teeth already noted. The upper canines are observed to be longer-and more trenchant in J/onophy/lus than in G'lossophaga. Rug ten in number, the anterior five undivided and the posterior five divided. Measurements of Monophyllus redmani. Millimeters. Head and body (from crown of head to base of tail).........c.csccceesisceeseccececesaseeescensenensaneveens 24 RSP LIN Ob: AU or ceeicey ieee eey ete rione a tives 65 First digit: Length of first metacarpal bone..........-:.scscceseeceeeeeeeeeenescneeeneerscuseseseceeecceesens penoccnaaoe! WEN GH OH PHALAN PCS -s..c-s0c0-ne enone seca sce lsesenn detect msiencassemeencsaaelecd=6een=nyaddexenehvsess==naien 12 Second digit: Length of second metacarpal bone Length of first phalanx.......-..---.csssccccrsscecsseeseesccenscserccccnsssecceesscssccetacccconsscrsevecrens Third digit: Length of third mefacarpal bone..........00.-.ccescscseereesnveceensecanssaccneeccnnsesecnnesscnssesansees 55 AGSTI ES TON OUELES tap Heb UX caneeeenesiecs oosunsnncavesiaceserarss=remseeea=needescienereecdec uch senderes)saeneemss . Length of second phalanx... Benes Length of third phalanx...........2....-2..ccsccecercaseccnsercneenscoeeseanes Sagndoansone shotuaeseoqcosnW8e 11 Fourth digit: Length of fourth metacarpal bone...........s.scsseesseceesencceeeccesecnscceesernseaseessenessneeenseas 51 Benet hyo titurstyp lr leat Xcescncscs seccasacecosapaccenss=-encseedersnaneanccseyenscestedaccuss=ssss agrecenasose 15 Length of second phalanx Fifth digit: Length of fifth metacarpal bone......-....sscceseccsecssceseeeceecesceeeccueencessccssessessessaasseveees 55 WSN SEM OR MISH PNALANK cca cccenecaescescasscesaccettuercusestdseuckapecuanvasuns=acasabacsmnsiserssduvccesass 15 Length of Second phalavx.....:......:-.sccoeccecnseeceesentecnnsenscnecrevsrscsesseccsstencsesascbeusesansas 14 Length of head... meer)! THOIGHGOM CAT asce-esencsantxsan 0 csccauannss24ss1ca(aenecsiessesncccusnavecacearosesus cates ea-eero=aeasersa-Fasnsaassenn 12 Height of tragus............ ou RbeEw sceasaesess0s «nwa an capeasiacbeataancaseasnacescaslfecutsy tere = cing Cenc sonatas 9 TBST AN 8 8 Fa ea eece = oondc un Te NonSnCCLOOe BRERRERCOS USEC OCD CHES nOSCOSIEBSSS. Saonandccosdssoococeacussssose esas 23 Bengt, Of GU laens: ecasccve tect <¢.-.csvsdaces «~~ n+a-Abuavneusd savasasenrdce ac duscceecs san sesaneenes=ssansenenensn== 27 Length of foot.... Length of interfemoral membrane Length of tail 7 ON THE GLOSSOPHAGIN®. 261 PHY LLONYCTERIS. Upper incisors separated from the laterals by wide intervals; naked skin-fold defining nostrils laterally ; nose leaf not reaching above 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. Fimbrive not arranged in rows, but form a uniform covering 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: 1. + — c. + — 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 (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 Phy/lonycteris at first hand. Phyllonycteris sezecorni Gund. 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 leaf 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 leaf, 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 GLOSSOPHAGIN®. 2Ve2 nerve markings in the fourth interspace as in G'lossophaga except that from the fourth digit there are three instead of one nerve. 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. ¢., 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 groove 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 vamerine 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—TVhe 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 caleaneum. 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 rugze 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. 2H, 264 ON THE GLOSSOPHAGIN®. Table of Measurements (in millimeters). | | x : x] Z ¥ =| = & Bal | = 2a|82)]88) 8 |S Be| Be] PS] eS] Sa) = |es Sie Shel cata sein arses Be ees oa) 87 | a7 88 lee Sl on: eee o G 4 |i) die ioe Head and body (from crown of head to base of tail)..--...-....-..ssseeeeceeeeee 45 15 57 55 40 42 32 Length of arm 19 2 20 20 20 25 Tiength Of £0rearmd...-.-....-.cesccceovenrsereeesnscessecannnnerernnscenstecstessunnseess ins 36 39 50 42 35 38 45 First digit : Length of first metacarpal bone «...--...-.---++---sesesenerecnnseneecnen seers 4 4 ay a. 3 3 5 PET GHY Of ATs typ Lela Kg nee en ss siee ee eene ee eaten tee enriesee ena eceeeteeliar 4 4 4 3 3 3 7 Second digit : | Length of second metacarpal bone......----.+.s+e:-sseeseeeeeseeeeeeneeeens 30 25 40 40 29+! 33 33 Length of first phalanx...............-.s0se Sache else s|ate cee a os eee eee 1 2 3 0 2 Or te eS Third digit : Length of third metacarpal bone 3: 30 47 45 37 38 38 Length of first phalamx.................+- Length of second phalanx Teng th) ofsblundsphiallanixsssce. cone=- ae on¥ sae een ce eteemsesanes=spienesatteaneceatene "7 6 | g 9 9 | 11 | 8 Fourth digit : Tength of fourth metacarpal bOne......-..--....0ce--e-.-cereeesseeteesenaee 33 27 | 49 40 34 | 37 | 35 Length of first phalanx ... oe 10 Bye) sill 12 9 10 13 Wength of second! phalanx...:--.-20-.cn--scs-+-locenn-prrmansese=neeanaqinnsacnren } 10 9 16 15 12 13 11 Fifth digit : Length of fifth metacarpal bone.....-.....-----.sesssseeeeseeesesweeeeeeeeees 30 D7 40) Bal} .ekh) 30 35 TREN EHVOE TSH phalarixe-weseeesesteeeetege ees snsase eames anscee=seeeeeneeeeeanra 9 8 10 | 10 i 8 | 11 Length of second phalanx. 9 8 10 | 13 | ae alee Length of head 23 21 a7 | 32 25 29 25 18 (erred N88 heb oaacocoboasbonacaccsaeceodicomccoccuseccs cagnsogabeseccnoonecareeeeesroJbotscon 14 ret! 12 13 13 14 | 11 Perot 108 Gta Cas ees ener sneauerseaen mene ates =nect omnes =se seers cent cecesnsces sae eeeeeeees 4 3 4 ue |New | 4h | 5 Length of thigh............-2s.c.csscs0se Sceetiessfas denesssavewsseve sansseaedCabaenesttep as 10 2 15} 95) 13° «14 19 Length of tibia -.. 14 11 20 17 13 13 | 20 Tier PHN OL OO beeen ce cceee emacs nee tetates mere aeeeese mb aes ne mena senna Canes eee eet 8 Sealine 10 7 he) a8} Length of interfemoral membrane in median line............-..eeeeseese eee eeee 10 9 20 | 4\> | S36 Aa, Lengthy of tail f:c..5.10 4: ee ee 5 2 8 4 | 0 10 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. 7 Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15. right. X 8. to 39. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. Glossophaga soricina. ON THE GLOSSOPHAGIN-E. EXPLANATION OF THE PLATES. PLATE VI. Head seen from in front. X 2. Skull vertex. x 3. Skull profile. x 3. Skull base. X 3. Jaws with incisors and canines seen from in front. MSs Upper teeth. x 10. Lower teeth seen from above. X 10. 265 Left lower molars seen in profile from lingual aspect. The first molar is to the Prate VII. Glossophaga truet. Head seen from in front. X 2. Glossophaga truet. Skull vertex. X 3. Glossophaga truet. Skull profile. X 3. Glossophaga truet. Skull base. X 3. Glossophaga truet. Upper teeth. X 8. Glossophaga truet. Lower teeth seen from above. X 8. Glossophaga truei. Lett lower molars seen in profile from lingual aspect. Monophyllus redmani. Monophyllus redmani. Monophyllus redmani. Monophyllus redmani. Monophyllus redmani. Monophyllus redmani. PLATE VIII. View of head from in front, showing ear and nose leaf. X 2. 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. X &. Teeth of the same as seen from the surfaces of crowns. X &. PLATE IX. Brachyphylla cavernarum. View of head showing ears and nose leaf. Brachyphylla cavernarum. Skull of same. Norma verticalis. X 3. Brachyphylla cavernarum. Skull of same. Norma lateralis. X 3. Brachyphylla cavernavum, Skull of same. Norma basilaris. X 3. Brachyphylla cavernarum, Upper and lower jaws seen from in front. X 8. PLATE X. Brachyphylla cavernarum. Teeth of same seen from the surfaces of crowns. X 2. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Leptonycteris nivalis. Brachyphylla cavernarum. Terminal cartilages of the fourth and fifth digits. PLATE XI. Head seen from in front. xX 2. Skull vertex. X 3. Skull profile. x 3. Skull base. xX 3. Jaws with incisors and canines seen from in front. X &. Upper teeth. x 8. The first molar is to the Fig. Fig. right. right. 16. 47. <0; 0! . 62. ig. 63. right. Leptonycteris nivalis. Leptonycteris nivalis. Chernycteris mexicana. Charnycteris mexicana. Charnycteris mexicand. Charnycteris mexicana. Charnycteris mexicana. Charnycteris mexicana. Charnycteris mexicand. Charnycteris mexicana. Lonchoglossa caudifera. Lonchoglossa caudifera. Lonchoglossa caudifera. Lonehoglossa caudifera. Lonchoglossa caudifera. Lonchoglossa caudifera. Lonchoglossa caudifera. Lonchoglossa caudifera. x 10. Anura wiedii. Anura wiedit. Anura wiedii. Anura wiedii. Anura wiedit. Anura wiedit. Anura wiedii. Anura wiedii. Phyllonycteris sezecorni. Phyllonycteris sezecornt. Phyllonycteris sezecorni. Phyllonyecteris sezecorni. Phyllonycteris sezecorni. Phyllonycteris sezecorni. Phyllonycteris sezecorni. sezecorni. Phyllonycteris Lower teeth, Left lower molars seen in profile from lingual aspect. Head seen from in front. Skull vertex. Skull profile. Skull base. Upper teeth. Lower teeth. Left lower molars seen from lingual aspect. ON THE GLOSSOPHAGIN &. 4 tek The first molar is to the PLATE XII. Head seen from in front. x 2. Skull vertex. xX 3. Skull profile. XX 3. Skull base. X 3. Jaws with incisors and canines seen from in front. X bd. Upper teeth. X 10. Lower teeth. x 10. Left lower molars seen in profile from lingual aspect. The first molar is to the PLATE XIII. Head seen from in front. X 2. Skull vertex. X 3. Skull profile. x 3. Skull base. X 3. Jaws with incisors and canines seen from in front. xX 8. >< teh x 8. First and second right lower molars seen from lingual aspect. Upper teeth. Lower teeth. The first tooth PLATE XIV. xX 2. Gk x 3. X 3. Jaws seen from in front showing incisors and canines. X 8. x 8. x 8. The first tooth is to the right. x 10. PLATE XV. Head from in front. xX 2. Skull vertex. x 3. Skull profile. x 3. Skull base. X 3. Upper teeth. x 10. Lower teeth. X 10. x 8. Left lower molars seen from lingual aspect. The first tooth is to the right. x 10. Jaws seen from in front showing incisors and canines. TRANS. AM. PHILOS. SOC., N. S. XIX. GLOSSOPHAGA SORICINA. PLATE VI. PLATE VII. TRANS. AM. PHILOS. SOC., N. S. XIX. 14 13 GLOSSOPHAGA TRUEI. TRANS. AM. PHILOS. SOC., N.S. XIX. PLATE VIII. MONOPHYLLUS REDMANT. TRANS. AM. PHILOS. SOC., N.S. XIX. PLATE IX. HW aga en E a BRACHYPHYLLA CAVERNARUM. TRANS. AM. PHILOS. SOC., N. S. XIX, PLATE x. BR ACHYPHYLLA CA VERNARUM. TRANS. AM. PHILOS, SOC., N.S. XIX. PLATE XI. LEPTONYCTERIS NIVALIS. PLATE XIil. TRANS. AM. PHILOS. SOC., N.S. XIX. CHGERNYCTERIS MEXICANA. ca pas mt \ ASL as ts ; 7 - _ gui - > 19 - ad ¥ i - Os | A eet ead, i Ae a ee a ae rire e TRANS. AM. PHILOS. SOC., N. S. XIX. PLATE XIil. LONCHOGLOSSA CAUDIFERA. PE aN ahah . PLATE XIV. TRANS. AM. PHILOS. SOC., N. S. XIX. zh ay i ‘ | » IN i \ (i PLT Vs deer eat ANURA WIEDII. TRANS. AM. PHILOS. SOC., N.S. XIX. PLATE XV. PHYLLONYCTERIS SEZEGORNI. ARTICLE VI. THERE 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 Eetophylla 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 verticalis 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 basilaris 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 flat. The tympanic bone is small, leaving 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 vyentre 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: i. 2 — ce. 1 — prm. 3 — m. $ K 2 = 28. roto 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 concave. 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 interyal 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. Ectophylla 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 Ectophylla 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 and the lower jaw. In Chiroderma, Vampyrops and Eetophylla 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 “ye 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 Kefophylla 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 Stenoderminz constitute, with the exception of the Heamatophillia, the most aberrant group of the Phyllostomidide. I recognize, therefore, the following natural arrangement of the gener: Subfamily SreNoDERMATIN ®. | STO 117 0 a a ee Brachyphylla. Artibeus. ang aie Uroderma. JIB OPA TU tease Re cabe alte Rete aoe eee ee 4 | Dermanura. | Sturnira. | Chiroderma. Oho cermin 4 ere eee sees ee oes aaais. ooo Vampyrops. | Ectophylla. Stenoderma. | Pygoderma. NOLCIO COMMU Neeser. Ss eISE IAS be Jee a , ee "3 Trichocorys. | Ametrida. | 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 aboye, not deflected, ascending ramus defined. Hard palate oblong, palatal bones produced. Upper incisors coni- Group Brachyphyllini.... ; : | cal, molars $ ; crowns coarsely ridged ; all cusps of the first lower molar subequal... { Brachyphylla. * Chiroderma is not as near Vampyrops and Hetophylla as the members of other groups are to each other, 270 Group Artibeini. ........ 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- H mental. J i CG; Molar 2icsecsecccctivocsncccecewseccaces seccucecucewscncnenscvacreusccedesevercevcccdeseas Artibeus. CPS MOLATS raven cte se cscutennacaiscte erp tele casteisecersiustucsie-capsesncneavactsciusnassenss 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 Cuspidate ; MOlATS F....-....ceeeeeesevcresecevevencnscscecescssescavcesserssaceseesaes Sturnira. II. First lower molar subquadrate without paraconid. Group Vampyropini....-- Group Stenodermini....-. { 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 lower molar with protoconid and mesaconid subequal. Molars 3... Chiroderma. e/, Angle of lower jaw acuminate, not deflected. Protoconid of first | lower molar aciculate, enormous. Jf. Hypoconid first lower molar rudimental ; molars 3... Vampyrops. jf’. Hypoconid first lower molar none ; molars §..........-. Ectophylla. { d’. 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 touching the panic bones; upper incisors conical; protoconid of first lower molar scarcely larger than other cusps ; hypo- | conid of the same tooth marginal, rudimental molars 3... | Pygoderma. } 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. q''. Frontal bone in orbit scarcely inflated ; hard palate with posterior margin excised; pterygoids not produced. enormous. h. Palate excised to first molar ; hypoconid of first lower molar inside contour. Molars $......-....Stenoderma. i | | Upper incisors flat; protoconid of first lower molar | h'. Palate excised to middle of first molar ; hypoconid of | first lower molar marginal. Molars 3 ... Trichocorys. *Mr. O. Thomas (Ann. and Mag, Nat. Hist., 1889, p. 70) first employed this character to separate this group from the foregoing. ee THE SKULL AND TEETH OF ECTOPHYLLA ALBA. 271 Measurements of Ectophylla alba (in millimeters). SA le hae 5 r=) Head and body (from crown of head to base Of tail).......-ccceccceseceesecesccrececsccceestencarecseteeaccsererees 36 36 TABLAS AP) TP PAT ba soem aeans « cect eng gg OOE CES CONNEC TES SALAS HINO ISOS SEI EEED TSA SAAD HORDAESO nS inbs EE sOSE edn Toads 17 ILSIN GT OORT ITT eoomscateicaaccds oggoccoeachecondcosccaDcene eae ebasoaon¢et acdc enonde odin sos nc SoccuaSsheS coos EcboeeCE AD OaC One 25 26 First digit : een peters tne kACAL pal DOUG. se. phien ane scenes na ecknatnns che cens cals neacbeetateest anes scansicespiesaersaneea =a ; 3 3 TLTa pa TG} TAR POLAT DTS. chosenacachonbeoseestoct oe | osboe Conca oso oncconneaeriotibecd WoncRncse once onurcag ee casecraa scree 3 3 Second digit : henpiblon second me tacni pall DONG: ccrneseecass.ccnsses-aceeuseancnteseeetawsntanrdees decesteru=veesrercessneeen. 21 20 Deen) gra NRTA EAI AE reer acnrniaetatinet oe nis trata teanns “ona naan once eenemncer seas Sncverecsecnatecrienssecan> 3 Third digit : Length of third metacarpal bone... ence ae 25 25 Length of first phalamx.....-....0--.+ 9 8 RNP HO OM SECOUGN DUA LATIN decane: atactsecpanesasnassonsarsconeons ences cn varsinesccussensnust asaveacacurserisccuens cs 12 13 Tene HOO ULLCOS p Ue lke teinnnae: seuss eveaensnccncavenknccsies seins sreabinne ners rlcnronacesastianccusecpecsonnr ss 6 6 Fourth digit : Benpth of fourth metacarpal DONE... .--.---.c0<.-senca-nsasencocsrcensesecnsasanuansssee vecccnecausenseyse ress 25 25 Noese De CRO LPUS LUE erates eee sere atennans «peers tee clewseicelce eter ssaales acleiets tv evetoencicsaraveinssen sreent 7 Rg eT P i OMBECOUG ND Hel li Kavecsnenaccaestsdusceus ccnanse-nedeavereesclesuen-sederscsnnnsme=-rattuvearccearassiececsest 8 7 Fifth digit : en eth One CAEN G LA GAN pal OU Gres cena iworss scesacnssmseniccnse(scussecatebesuesurmaceachssteretsismcecessvsepe-p ete 25 z Gn Ray Om rsp Nall Ker anenaessnees tar seneteasanedcerstessanslanasescun esceeuchstenmanannenesranccnanencressccssse te 6 6 Length of second phalanx. ........0..00:csssccsssscnsccesccsscecsstecnsceacscavconcrssnncesecsercnssrentees a eae 7 7 TSS iy Gh LET ond cesceoacondpsancegnonsancogHitsds Joa ocenconiuo soy ¢cancosAecnOaEsogo INTO soos aac HOnNOanESOSe cod Ac ssnedecooTBSASREE 14 14 Let helye (2 (Ge Ts ncescaacc cb coe CONE REDD ACOEEDOCCORE NS O- QI TOORO AS EOC OC RS CBO C AoE bpIOg nt CUO CHOC E SSO. nC SSBC CUS OnSnT SO SUOOne ees 10 10 TCO GO Pe ECAC TIS Cnenenannanuvnsaassaeucescerncndnsliesssonnsatasssioccaan is separated by a short space from the canine, while pz 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 yariation 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 mz or my. It is less commonly found on m x and only in the very large 2. 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, however, the tip of p + 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 ® 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 eas 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 eyen simpler than the upper; dp 5 and » 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 4. mortoni; LT have not seen these teeth in the larger species. Measurements. No. 11156 No. 10885 No. 11009 No, 11440 upper dentition, Length l dito Ms. <5 oc «oceans mreeisie 70.270 fs molar series, IGHIGIN SES. Sanoobods pA RGOOUe Aa pee xatate 118 .104 064 .065 Sem LE HLOIATUSELICS yl OD ODM or storie ye sie) e)elal alsivele]syetele ta) « ofatetesele oles -238* 175 124 113 PE COMULG AM t eH POSus OLIN LOL mers sie feievaie +e siete eles) eichsucie) ole .048* 046 032 fs Ht: HANS VOLS GIMM CLO L eye ran w «aie elas wieva'eias slopes otalet .0885* 042 .022 “JP UL USS cothado sccdh< dpodade ca enndnosusaaoeoUsne .080* 024 O19 Mee CO eo 8a NGGn CoRR GOOD DUE Rens CUE taty se .038* 038 025 023 Vo MR} y Nese 6 Se on be ntolte 5 UE SOD GOOLE See reo emer. .041* .028 028 7) 1 Re cet eee "kor haste ae ee 035% 031 | 0195 018 Seem NA Ghee] OYTO emneraets ecc clave cfs ayche cele tier 0,0. idee os was. < aoe 085 033 020 a CO” CGI hess oe Sg .ct SUS OCHCE CR TGS CORT TO EE Se eaenae 036 O19 MS ame re fl namin tes cielos ai neltetaaeis aie oeinis chic ciecce + «x .042 035 025 023 e SCNT ehae es lets Cero Resid Share! Stars sree te felss cele te Coste yas 039 | 0235 024 O0- ILROUETINE 2 gation aocuesibodsscedodcodreges So obo eamore 087 084 021 .021 CUS EArt BS ee pele 5 co a Set ney acy A Se ede ee oe ee .083 .022 0215 Lower dentition, length [1 to M3.......... OC rItS DEA ceo meee .4382* 261 es siiYol ie tees, WOMVa INS oo5.caade SOUnDOutS ObdHn Sots GAA SOer -121* 108 070 ENTE MO SI SCLICS s LEME Ula. ki-reterete ctereisferst si-reisratsioe = ects este se .211* 192 126 Me de aL enya oS os oon. obackos SC UEOOUCU Como tOnena Heme aOoD .028* 026 O17 Lo GEL) OREN Guana Solas Ole Cian Aon Sd boheee sec .026* O19 Be TEP TGCS FNS SE aie aie pao bicels Geta CRIB COTE MEA cris .081* 033 .020 is OG UT GITOMEgirighs 5 pone, See AOE RECO da AITO RODE ECORI .038* 023 CG IPSN Gagan ocedac oneenee penenor ods DOduaEan wae .046* 048 .027 ue 3 TIGW SHIT en cic tieesOpp Gob icon Sano IU TUUO eT ODDO OCRb nee .061* 0381 Exe ea OMY OUI ei Meraie ot face ONT WIN AN DLEAAGD eeescs cascecccucccdocessceecGceesscvbacsslcssicessecscrescevesccsviesceadavecessecedcevcsssccecelcecsclld 036 TLTRTER, ANARTEEES, coneesce conpoa na sosbO ceo SGunO OOS NCoO TONDOSEDOAESESSESSSoIS524 EREDDO SECONDS MSDS IBSATOIGT 050 Re VLAN AUS OID teceaslnceanttatenes seananetecneascuscelssmcsarasaniasenseolnsssievis-rsvacarancienastsessecsnsunssnoohis 033 PRAT PALL, RETIN oot ancbeccodercoocasto deed soo radas cae ccDon Aas SOLOS HoCCO OCDE SCaC OSE OS SaSSRCG CosecE SESSION) 027 Reseres IRLC ADIT CRStrou nt estes samen atic =p estes sac natewamaencee wees ecestaeneeanthiecessbesenascesinsesseses=sas 039 Metacarpal iv, length...............0-seseseeee np ssa LOL MetACAC PAL TVs p WIC til) POO XI ONG ease sas asnanscevacerss nests decout spsceenmseassannqve= 060 Proximal phalanx, width proximal end...............s:ssccsesceseessenerecneeccensenssssnnsceseovsscvenssens 032 Proximal phalanx, width distal end...s......2..2.cssscscscsoesensscccnscescsece seseescscceecosecenstecscone 027 Second phalanx, length - .042 SecOneMNAlAn .005 006 | ob “1 aie 008 0085 0095 DNR i Maas ocecastdstons ath co Wes roaversucs-sescaceoecs 009 2.010 009* | 1 De GSO EEE Ve ae oe ee 2014. | .0145 | .015 ‘oien | | SMMIRIAY epi l Eline tatu ss hoo tap beeen evi edeNer covdpodecceses | .0085 | .008 | .0105 £0105 | SME Tots. feet 2S 18 evoke hoc ees 012 Oi || Sem or | “OWT al gy 7G a dt aoceencecceecenoosasacsace cee COLO en ire OLO: O15 016 | “M2, levgth.. LOO |= 2007 118 *:007 007 SE Mp De wadltiiee sas das-n> ace [O10 sec OLd SO1de eh OLL Lower dental series, length C to M 3 ...:-e1..sceeeeeeeeee .078 | | -090* | .090 “(premolar series, length ....0.........sceeeeeeeeeeeesens 036 | .041* -040 | .0315 © molar series, length .....sseeesesceee eee .026 0245 | .031* .030 emne CATIITIE CLIP CH seven es sce swccess woe cn deaces codnrotweswenss O11 | .O11* .012 .010 vi # ROLCLGHoNs course ercskaedensvanss Corps tet socmonwses .0085 |. 009% .008 007 MES UIGNIDE Neesavtnsassiecasacebrsenenaercnasasesencerenese -0045* | .005* 004 003 rs PQ: = 0085 008 .009* 010 006 Sebo? 0095 2010 ©] ee OLOR -O11 -008 Cs Oy a .012 012 | .012* 012 O11 Sse ME St O14 013 .014* 017 O14 “M1, width 007 007 009 008 em VS em LONP UN ieckaredenca..cvoretaKercdvnsceveccucsbusvetenese 0085 008 | .0095* .0095* MTN fii to coiscts vice han ste h cv sepdond on access 006 0055 Pe MG SALON GUD, oases ccvevccencnevasbocvoevesctsavasscnseees .003* -004* | 006% 004% AVP GREW Gilowesauccee sscunsevwisersacuansstavibneaccacnqes> .002* 003% *Alveolus. 350 NOTES ON THE CANID# OF THE WHITE RIVER OLIGOCENE. IJ. Tue Sxuru (Pl. XIX, Figs. 1-7). The skull of Daphawnus is exceedingly primitive in character and plainly shows many traces of the creodont ancestry of the genus. Unfortunately, well-preserved skulls are exceedingly rare and none of the species is represented by an altogether complete specimen. However, several more or less imperfect specimens haye been recovered, which together give us information concerning nearly all parts of the skull. As in the creodonts generally, the cranial region, reckoning from the anterior edge of the orbits backward, is exceedingly elongate, while the face in front of the orbits is very short, slender and tapering. The elongation of the cranium is not due to an enlarge- ment of the cerebral fossa, which on the contrary is short, narrow and of relatively small capacity. The postorbital constriction, which marks the anterior boundary of the cerebral fossa, is notably deep and is remoyed much farther behind the orbits than in Canis. On the other hand, the cerebellar fossa is long, and the postglenoid processes occupy a more anterior position than in the existing species. In consequence of the elongate cranial region, the zygomatic arches are yery long, as in the more primitive types of creodonts. The upper contour of the skull is nearly straight, the descent at the forehead being very slight and gradual, which gives to the skull an alopecoid rather than a thooid aspect. This resemblance is, however, entirely superficial, for the frontal sinuses are large and well developed, as in the thooid series of the modern Canidae. The sagittal crest is low, but varies in the different species, being decidedly thicker and more prominent in the larger and heavier D. vetus than in the smaller and lighter D. hartshornianus. Turning now to the more detailed study of the elements which make up the skull, we shall find a number of striking and significant differences from the existing repre- sentatives of the family, though the general aspect of the whole is distinctively canine. The basioccipital is broad and quite elongate and has a much more decided median keel than Canis. All the occipital bones are firmly ankylosed in the specimens at my disposal ; hence, in the absence of sutures, it will be necessary to describe the compound bone as a whole, without much reference to the elements of which it is made up. the occiput is of quite a different shape from that found in the existing members of the family, being broader, lower, and with a wide, gently arched dorsal border or crest (see Pl. XIX, Fig. 3); in Canis this crest is pointed and somewhat like a Gothic arch in shape. The occipital crest is thin, but much more prominent than in Canis, which is due to the larger and deeper depressions of the cranial walls behind the occipital lobes of the cerebral hemispheres, the shape of which is plainly visible externally. The foramen magnum has much the same low and broad outline as in Canis. The condyles are low, but well extended transversely, and on the ventral side they are sepa- NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. 351 rated by a wider notch than in Canis. The depression, or fossa, external to the condyle is very much deeper and more conspicuous than in the modern genus, in consequence of which the condyles project more prominently backward from the occiput than in the modern dogs. The paroccipital processes’are short, but quite stout and bluntly pointed ; they project much more strongly backward and less downward than in the living forms, and are less compressed laterally. Another difference from the modern genus consists in the fact that, while in the latter the paroccipital process has quite an extensive sutural contact with the tympanic bulla, in Daphenus there is no such contact, the minute bulla being widely separated from the process. The direction taken by the paroccipital process in its course is thus evidently not determined by the size of the bulla, for in the John Day genera, Temnocyon, Hypotemnodon and Cynodesmus, in which the tympanic is greatly inflated, the shape and direction of the paroccipital are the same as in Daphenus, with its insignificant bulla. A considerable portion of the mastoid is exposed on the surface of the skull, but it is rather lateral than posterior in position, a difference from Canis, in which the mastoid is hardly yisible when the skull is viewed from the side. The mastoid process is slightly larger than in the existing genus and is channeled on the inner side by a grooye leading to the stylo-mastoid foramen. The limits of the dasisphenoid are not clearly shown in any of the specimens, but this element appears to have much the same broad and flattened form as in the recent dogs. The presphenoid is long and narrow and, as in the existing species, is almost concealed from view by the close approximation of the palatines and pterygoids along the median line. The ali- and orbito-sphenoids are not well displayed in any of the speci- mens, but so far as they are preserved, they differ little from those seen in the more modern members of the family. The auditory bulla of Daphenus is very remarkable and differs from that of any other known carnivore. Its principal peculiarities were observed and noted by Leidy, but the material at his command was insufficient to enable him to describe these peculiarities with confidence. The tympanic is exceedingly small, and is but slightly inflated into an inconspicuous bulla, the anterior third of which is quite flat and narrows forward to a point. There is no tubular auditory meatus, the external opening into the bulla being a mere hole, but the anterior lip of this opening is drawn out into a short process, some- what as in existing dogs. Behind the bulla is a large reniform yacuity or fossa, of which Leidy remarks: “ At first, it appeared to me as if this fossa had been enclosed with an auditory bulla and what I have described as the latter was a peculiarly modified auditory process” (769, p. 35). Several specimens representing both the White Riyer and John Day species of Daphenus show that the fossa is normal and was either not enclosed in bone, or, what seems less probable, that the bony capsule was so loosely attached that it 302 NOTES ON THE CANIDE OF THE WHITE RIVER OLIGOCENE. invariably became separated from the skull on fossilization. At the bottom of the fossa (7. e., When the skull is turned with its ventral surface upward) is seen the exposed periotic, or petrosal, which is only partially overlapped and concealed by the tympanic. Such an arrangement is far more primitive than that which is found in any other known member of the canine series, and is not easy to interpret. A clue to its meaning may, however, be found in the mode of development of the bulla in the recent Canide. Here, as is well-known, the structure consists of an anterior membranous and_ posterior carti- laginous portion, which eventually ossify and coalesce into a single bulla. Reasoning from this analogy, we may infer that in Daphenus the bulla was also composed of two portions, but that only the anterior chamber was ossified, the posterior one remaining cartilaginous. Communication between the two chambers was provided for by the space which separates the hinder edge of the anterior chamber from the petrosal. If this interpretation be correct, it supplies an interesting confirmation of the results derived from the ontogenetic study of the recent genera. At all events, it seems much more probable that we have to do here with a primitive rather than a degenerate structure. The parietals are large and roof in most of the cerebral fossa; they are much less convex and strongly arched than in Canis, in correspondence with the smaller size of the cerebral hemispheres, and posteriorly the depressions behind the hemispheres are much larger and deeper. As already remarked, the sagittal crest varies in the different species, and is much thicker and more prominent in D. vetus than in D. hartshornianus. The frontals are more or less damaged in all the specimens and in none of those at my disposal is it possible to determine the posterior limits of these bones, though from the position of the postorbital constriction we may confidently infer that they formed a smaller proportion of the cranial roof than in the modern members of the family. The supraciliary ridges are feebly developed, especially in D. hartshornianus, and the postorbital processes are like- wise much less prominent than in most of the recent dogs; from this process a ridge de- scends downward and backward to the optic foramen, which, though not prominent, is yet more so than in Canis. The frontal sinuses are large and yet in spite of them the forehead is nearly flat, both longitudinally and transversely, with a very shallow depression along the median line. The nasal processes of the frontals are long, narrow and pointed, and are separated by only a short interyal from the ascending rami of the premaxillaries. The squamosal is of moderate size and differs only in subordinate details from that of Canis. One such difference is the presence of a broad shelf-like projection, the pos- terior extension of the root of the zygomatie process, which overhangs the auditory meatus and is doubtless to be correlated with the lesser breadth and conyexity of the brain. The glenoid cayity is like that of the recent species, but has a much more distinct internal boundary, due to an elevation of the squamosal at that point. The NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. 339 zygomatic process is stout and well-developed, especially in D. vetus, which has heayier arches than a large wolf, while in D. hartshornianus the zygoma is lighter and more slender, much as in the coyote. The juga/ is strongly curved upward, as well as out- ward, and is shaped quite as in Canis, forming nearly the whole anterior and inferior boundary of the orbit ; the postorbital process is very feebly indicated, being even less prominent than in the modern genus, so that the orbit is more widely open behind. ‘The lachrymal is rather larger than in Canis, forming more of the anterior orbital border, and has a quite well-developed spine. The nasal/s haye a general resemblance to those of Canis, but, in correspondence with the shortness of the whole facial region, they are considerably shorter, and somewhat broader and more convex transversely ; their posterior ends are more simply rounded and have a less irregular suture with the frontals, while the anterior, free ends are much less deeply notched. The maxillary is somewhat peculiar in shape, corresponding to the remarkably constricted, narrow muzzle. The facial portion of the bone is relatively higher than in existing representatives of the family, especially in front, its anterior border rising in a steeper and bolder curve. Just in advance of the orbits the maxillaries expand quite suddenly in the transverse direction, much more abruptly than in Canis. The infra- orbital foramen occupies nearly the same position, with reference to the teeth, as in the latter genus, being above the front edge of the sectorial, but it is very much nearer to the orbit, which occupies a more anterior position. The palatine processes of the maxil- laries follow the shape of the muzzle, and are long, narrow for most of their length, but broadening much behind; anteriorly they are emarginated in an unusual degree to receive the long premaxillary spines. The premacxillaries, especially their alveolar portion, are somewhat narrower than in Canis, and behind the external incisor the alveolar border is constricted on each side, forming well-marked grooves for the reception of the lower canines. The exposed part of the ascending ramus is much varrower than in the modern genus, forming a mere strip on the side of the narial opening. At the same time, this ascending ramus is relatively longer than in existing dogs and extends almost to the nasal process of the frontal. The anterior narial opening is somewhat larger proportionately than in the recent members of the family, especially in the vertical direction, and its borders are less inclined ; the floor, formed by the dorsal surface of the horizontal rami of the premaxille, is more simply and deeply concaye, and the horizontal rami themselves are less massive. The palatine processes of the premaxillaries are distinctly smaller than in Canis, while the spines are relatively longer and more slender. The incisive foramina are large and from them quite deep grooves are continued forward to the alveolar border, while in the modern genus these grooves are very shallow and feebly marked. 5334 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. The palatines are shaped yery much as in Canis. As a whole, the bony palate differs from that of the latter genus in the greater and more abrupt expansion of its posterior half, beginning at p ®; it is also somewhat more concave transversely and has a more prominent ridge along the median line, The palatine foramina are likewise somewhat different from those of recent dogs; one conspicuous opening on each side occupies the same position as in the latter, opposite the middle of the sectorial, but instead of a single opening opposite m ?, is a group of two or three minute foramina. The Cranial Foramina. Unfortunately, none of the specimens are sufficiently well preserved to permit a complete account of the cranial foramina, though the more important facts concerning these structures may be determined. Leidy states that in D. vetus “the anterior condyloid, Eustachian and oval foramina present very nearly the same condition as in the Wolf” (’69, p. 33). The specimen upon which Leidy’s descrip- tion was founded, belonging to the Academy of Natural Sciences of Philadelphia, has been mislaid and is not at present available for comparison, but the description cited above does not altogether apply to the cranium of D. hartshornianus, of which an account has been given in the foregoing pages. In this specimen the condylar foramen is widely remoyed from the condyle, much more so than in Canis, and is placed near the edge of the reniform fossa which lies behind the tympanic bulla. The existence of this fosse remoyes the necessity for a distinct foramen lacerum posterius, which is indicated only by a notch in the hinder margin of the fossa ; similarly, the stylomastoid foramen is an open groove, only partially enclosed by bone. The postglenoid foramen is large and conspicuous and is not concealed by the anterior lip of the auditory meatus as is the case in the John Day Cynodesmus. The foramen lacerum medium appears to occupy a somewhat more internal position than in Canis, though this is not altogether certain, because of the unfavorable condition of the fossil just at this point. The Eustachian canal is more concealed under the long anterior process given off from the tympanic bulla than in the existing genus, and the foramen oyale is separated from the entrance to the canal by a much more prominent bony ridge, so that the foramen presents forward instead of downward. By a curious coincidence all the crania of Daphenus in the Princeton museum are damaged in such a way that none of them displays the alisphenoid canal, the foramen rotundum or the foramen lacerum anterius, though there is no reason to doubt that all of these foramina were present and corresponded in position to those of Canis. The optic foramen is overhung by a ridge, already deseribed, which is much more prominent than in the latter, and the lachrymal foramen is decidedly larger and more conspicuous. The parietal is perforated by a venous foramen which opens in the depression behind the cerebral hemispheres ; this foramen, the postparietal, is not found in the modern genus. a NOTES ON THE CANIDE OF THE WHITE RIVER OLIGOCENE. ae The mandible differs considerably in the yarious species, though the comparison between them can as yet be but partially made, for the only specimen known to me in which the angle and coronoid process are preserved, is that figured by Leidy (/. ¢., Peale Fig. 2), which belongs to D. vetus. In ? D. dodgei (Pl. XIX, Figs. 6, 7.) the horizontal portion of the mandible is thick, heavy and relatively short; the inferior border is very far from straight, rising beneath the masseteric fossa almost to the level of the molars and descending forward from this point in a bold, sweeping curye, quite as in the modern Canis aureus ; the masseteric fossa is very deep and its ventral border forms a prominent ridge, distinct from the lower border of the jaw; the symphysis is short and the chin abruptly rounded and steeply inclined. In D. vetus the horizontal ramus is of an entirely different shape (see Pl. XIX, Fig. 5) being longer, more compressed and slender and with a decidedly straighter ventral border; the symphysis is longer and the chin more gently rounded, rising more gradually from the inferior margin of the ramus. The masseteric fossa is quite deeply impressed, though less so than in ? D. dodgei, and is very large, extending far up upon the ascending ramus. The angle is a stout hook, which is less elevated aboye the general level of the horizontal ramus than in modern wolves or foxes. The condyle also has a low position, below the level of the molars, while in recent species the condyle is raised above the molars, and in some species very much so. The ascending ramus has great antero-posterior extent, by which the condyle is removed far back of the last molar. This is a primitive feature which recurs in most creodonts and is evidently correlated with the characteristic elongation of the cranium and zygomatic arches. The coronoid process is high and wide, and has a bluntly rounded end; it inclines much more strongly backward than in Canis and has a much more concaye posterior border. The condyle resembles that of the recent dogs, but is set upon a more distinct neck, is more extended transyersely, and is less cylindrical in shape, tapering more toward the outer end. In D. hartshornianus the mandible, so far as it is preserved in the various speci- mens, resembles that of D. vefus, save that the horizontal ramus is somewhat shallower and more slender. The Brain. Very little can be said concerning the brain, since no complete cast of the cranial cavity is available for study. The general shape and development of the brain are, however, indicated in the specimen of D. hartshornianus already described (Pl. XIX, Fig. 1). Its proportions are very different from those found in existing members of the family, a difference which may be briefly stated as largely consisting in the much greater relative size of the cerebral hemispheres and smaller size of the olfac- tory lobes in the modern species. In Daphenus the brain is narrow and tapers rapidly toward the anterior end; the cerebellum and medulla oblongata are long, the A. P. S—VOL. XIX. 2Q. 336 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. hemispheres narrow and short, and the olfactory lobes very large. The partially exposed east of the cerebral fossa shows that the cerebral conyolutions are fewer, simpler and straighter than in any known species of Canis, and are even more primitive than those of Cynodesmus (see Scott, 94, Pl. I, Fig. 2). The only suleus visible in the specimen is apparently the suprasylyian, which is short and pursues a nearly straight course, but curving downward slightly at both ends. From the external character of the skull it is clear that the hemispheres overlap the cerebellum but little. Measurements. No, 11421. | No. 11424. No. 10538. No, 11423, | No. 11425. | No, 11422, | | Siegidlll, Tyo abe soos soouasoeinnoccoaon ndconoopecadanodsanasecasioco | 20.151 Cranium, length fr. oce. condyles to preorbital border | -108 | | Face, length in front Of Orbits .-----...0sses-seseesesssoees 065 | ?.050 073 | | | Zygomatie arch, length -O80 | RAL Ate wen pt itescasavassacwenceenssase cur seniecaecusmenesesm aster 076 | 3 092 | | SSopemwaid that pl rccsentececceanoccenmetacpeteceses sh eaeaane | .044* | .047* .052 | Mandible, length from chin to masseteric fossa...--.--- .084 | | 093 | .096 | 2.079 CALSy oH PE Fee =e coscce soncaoenssaspecssnsrsnsesnoganssS 020 018 023 | .025 | .025 « Doh Ons Pet oR ei Gah, 10175) ON Oro. sony’ 117)" 020) Si saaatg cs Hhickness Abin aedesssiesaeseasetaetesseesaaraes 010 | .009 | 010 | .012 | .012 | | * A pproximate. Ill. Tue Verresrat Conumn. The vertebral column is remarkable in many ways. All the regions of the column are well represented by seyeral specimens of D. vetus and D. hartshornianus, but no com- plete backbone belonging to a single individual has as yet been recovered. Cervical Vertebre. The collection contains only a single imperfect specimen of the atlas and this belongs to D. vetus. Imperfect as it is, this atlas displays some important differences from that of Canis and most of these differences are approximations to the feline and yiverrine types of structure. In Daphenus the atlas is elongate in the antero-posterior direction, the anterior cotyles are small and only moderately concaye, and are somewhat more widely separated on the ventral side than in Canis. When viewed from above, the cotyles are seen not to project so far in front of the neural arch as in the cats, but farther than in the dogs. The posterior cotyles for the axis are small, nearly plane, and but slightly oblique in position, with reference to the fore-and-aft median line of the yertebra. These cotyles are more distinctly separated from the articular surface for the odontoid process of the axis than in the modern dogs, in which NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. 357 all three facets are confluent. The neural arch is low and broad, considerably elongated from before backward, and without ridges of any kind, save an inconspicuous tubercle, which represents the neural spine. Near its anterior border the arch is perforated by the usual foramina for the first pair of spinal nerves. The inferior arch is very slender, forming a more curved bar and has a much less antero-posterior extension than in Canis. Wortman (’94, p. 137) has pointed out that the foramina of the atlas display certain characteristic features in the various carnivorous families. “In all of the Fe/ide which I have had the opportunity of studying, the [vertebrarterial] canal pierces the transverse process at its extreme posterior edge, where it is thickened and joins the body of the bone. The superior edge of this posterior border slightly overhangs the inferior edge. .... This character appears to be very constant in the /e/ide@ and so far as we know the structure of the atlas in the more generalized Nimravide [Machairodonts], it is true of them also. In the Canida, upon the other hand, the foramen for the vertebral artery is situated well in adyance of the posterior border of the process, and instead of having a fore-and-aft direction, as in the cats, pierces the process almost vertically from above. In the Viverride and Hyenide the position of the foramen is very much as in the cats. There is, however, an important difference between these two families and the felines where the artery enters the suboceipital foramen in the anterior part of the atlas. The difference consists in the formation of a bony bridge in this situation, which gives to the suboccipital foramen a double opening in the hyzenas and civets, whereas it is single in the cats.” In Daphenus, it is interesting to observe, the foramina of the atlas are in all respects like those characteristic of the cats and thus depart in a very marked way from the arrangement found in the recent.Canide. The transverse processes are broken away, so that their shape is not determinable, but enough remains to show that the atlanteo-diapo- physial notch is not conyerted into a foramen, thus agreeing with the canines and felines and differing from most of the hyzenas and ciyets. The azis is likewise feline rather than canine in its general character and appear- ance. The centrum is elongate, narrow and depressed, with a thin and inconspicuous hypapophysial keel, running along the ventral surface, and has a slightly concave posterior face. The articular facets for the atlas are convex and rise higher upon the sides of the neural canal than in Canis, and on the ventral side they project below the level of the centrum, so that they are separated by a broad notch, which is not present in the modern dogs, and is not well marked in the eats. The odontoid process is a long, slender, bluntly pointed peg, with a heayy, rounded ridge upon its dorsal surface, which is continued back along the floor of the neural canal. The transverse processes are quite long and relatively very stout; they are shorter and heavier than in Canis, and keep more nearly 3388 NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. parallel with the centrum, not diverging so much posteriorly. As in the felines, the ver- tebrarterial canal is longer than in the modern dogs, and its posterior opening is not vis- ible when the vertebra is seen from the side; the anterior opening is larger and is placed farther forward than in the recent Canide. The neural canal is proportionately larger than in the latter, both vertically and transversely, nor does it contract so much toward the hinder end. The neural spine forms the great, hatchet-shaped plate usual among the Carnivora, and in its details of structure it is feline rather than canine. In the latter group, the spine is not continued back of the postzygapophyses into a distinct process, but its hinder borders curve gently into them. In Daphenus, as in nearly all the cats and viverrines, the spine is drawn out into a blunt and thickened process behind the zyga- pophyses, from which it is separated by a deep notch. The zygapophyses are rather small and do not project so prominently from the sides of the neural arch as they do in Canis. The other ceryical vertebree are more slender and lightly constructed than in the existing Canidae of corresponding stature. The centra are long, narrow, depressed and very feebly keeled in the ventral median line; in most of the species this keel does not terminate in a posterior hypapophysial tubercle, such as is found in the existing dogs. In the largest species, however, D. felinus, the keels are more prominent, especially on the third and fourth vertebrae, and there is some indication of the tubercle. The centra are slightly opisthoccelous and the faces are somewhat oblique in position. In very few of the specimens are the transverse processes sufficiently well preserved to require description, and in such cases as they are present (as, for example, on the fifth and seventh cervicals of one individual of D. hartshornianus) they display no noteworthy differences from the corresponding processes of Canis. The vertebrarterial canal is, however, somewhat longer than in the latter. The neural arches are very different from those seen in the modern representatives of the family. In them the dorsal surface of the neural arch is very broad and on each side projects outward as an oyerhanging ledge, which connects the prezygapophysis with the postzygapophysis of the same side ; ridges and rugosities for muscular attachment are well marked and in the large species often very prominent; the zygapophyses, and especially the posterior pair, project but little in front of and behind the arches, and those of each pair are separated by notches of only moderate depth. _ In consequence of this arrangement, there are but small interspaces visible between the successive arches, when the vertebrze are in position. In Daphenus, on the other hand, the dorsal surface of the neural arch is relatively narrow, somewhat convex transyersely and usually smooth, with- out ridges or tubercles ; the overhanging ledge which gives such an appearance of breadth to the arch in Canis is little developed ; the zygapophyses project far in adyance of and ee ao q) NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. oo behind the arch, and between each transverse pair is a deep notch which greatly reduces the antero-posterior length of the bony arch in the median line. When the vertebrae are placed in position, the openings between the successive arches, on the dorsal side, are very large and are longer antero-posteriorly than broad transversely. In these peculiari- ties of the cervical yvertebre of Daphenus we find no approximation to the structure of the cats or the viverrines. The neural spines are also quite differently developed from those of the recent dogs. The third cervical has no spine, merely a very faintly marked keel, the overhanging spine of the axis leaying no room for the deyelopment of one on the third vertebra. The fourth cervical has a very low spine, and on each successive vertebra the spine becomes higher and more pointed; that of the seventh is very high and slender, very much more prominent than in Canis, being almost as high, though not nearly so stout, as the spine of the first thoracic vertebra in the modern genus. The length of the spines in the neck constitutes another similarity to the structure of the felines. Thoracic Vertebre—The number of trunk vertebrae characteristic of Daphanus ‘annot as yet be definitely determined for any of the species, for no specimen has been found with complete backbone. In one specimen of D. vetus are preserved twelve thoracic and five lumbar yertebrxe and the type of D. fedinus contains six lumbars. — It is altogether probable that the extinct genus agreed with the existing dogs in having thirteen thoracics and seyen lumbars. The first thoracic has a broad, very much depressed centrum, with anterior face convex and posterior face deeply concaye. The prezygapophyses project forward yery strongly and, as in the cervicals, the notch between them is very deeply incised, invading the base of the spine, a very different arrangement from that seen in Canis; these processes are relatively larger and more concave in D. vetus than in D. hartshornianus. The postzygapophyses are much smaller, but project prominently from the hinder end of the neural arch, extending both laterally and posteriorly ; the articular faces are somewhat convex transversely and have an oblique position, presenting outward rather more than downward. The neural spine is high and compressed, shaped very much as in Canis, but somewhat more slender. The transverse processes are very long, prominent and heavy, especially in the large species, D. felinus ; at the distal end of the process is a large and deeply concave facet for the tubercle of the first rib. The second thoracic very much resembles the first, but has a smaller, narrower, lighter, and much less depressed centrum ; the prezygapophyses are smaller, less concave and less widely separated, while the postzygapophyses are larger and present downward, instead of obliquely outward, as they do on the first. The transyerse processes are much smaller in eyery dimension than those of the first thoracic, and spring from the neural 540 NOTES ON THE CANID.E OF THE WHITE RIVER OLIGOCENE, arch at a higher leyel, though they are still very prominent and carry large, concave facets for the second pair of ribs. The neural spime is somewhat heavier than on the preceding vertebra, and was probably higher, as well, but in none of the specimens is the spine preserved for its entire length. The other vertebrze in the anterior part of the thoracic region haye rather small centra, and in general character are very much like those of Canis. The (?) sixth vertebra has a curiously shaped spine, which exaggerates the condition seen in the modern genus; its proximal portion is inclined very strongly backward, while the distal portion is curyed so as to project upward ; the other thoracics, as far back as the (?) tenth, have similar spines. One very marked difference from the recent Canide consists in the deep noteh which, in Daphwnus, separates the two prezygapophyses. The anticlinal vertebre is probably, as in the existing dogs,.the tenth, and at this point the thoracic vertebrae undergo an abrupt change of character, assuming more the appearance of lumbars. In Canis the spine of the tenth thoracic is exceedingly small and much lower than those of the ninth and eleventh, but in Daphenus, on the other hand, the spine is much better developed, both in length and thickness; the postzygapophyses are small, somewhat conyex and placed high up upon the neural arch, presenting outward. The (?) eleyenth thoracic is not preseryed in any of the specimens. The (?) twelfth and thirteenth are much like lumbars, except for the smaller and lower spines, thickened at the distal end, and for the entire absence of transverse processes, which in Canis are present, though very short, even on the thirteenth; the anapophyses are remarkably long and stout, being much heayier and more prominent than in the recent dogs, and high, massive metapophyses rise above the prezygapophyses. The lumbar vertebrae (Pl. XTX, Fig. 8) were probably seven in number, though not more than six have been found in connection with any one specimen. These vertebree are remarkable for their relatively great size and massiveness, and for the length of all their processes, being in these respects feline, rather than canine in character and appear- ance. Assuming that seven is the full number, the missing one will then be the third, and the following description is made upon that assumption. ‘The centra increase in length posteriorly, reaching a maximum in the fifth and sixth, but the seventh is no longer than the first, though much broader and heavier. Compared with those of Canis, these centra are longer, stouter, less depressed and more rounded. The transverse pro- cesses are longer and heavier than in Canis and less so than in the large species of Fe/is. The neural spines are likewise intermediate in character between those of the recent dogs and of the larger felines; they are much higher, more extended antero-posteriorly, more thickened at the distal end and more steeply inclined forward, than in the former. In D. felinus especially, the great height of these spines is very striking and the resemblance NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. 341 of the lumbar yertebrie to those of the contemporary Machairodont Dinictis is very great. Another similarity in the structure of the lumbar vertebrie between Daphanus and the felines consists in the great height and heaviness of the metapophyses, which are much better developed than in the recent Canide ; on the last lumbar these processes become yery much reduced and are, in fact, almost rudimentary. The anapophyses are smaller than on the thoracic vertebree and diminish in size on each successive vertebra posteriorly ; only on the first and second are they very large and prominent. In the existing representatives of the Cunidw these processes are rudimentary, except on the first lumbar, where they are small. This constitutes another point of resemblance between Daphenus and the cats, and emphasizes the statement already made, that the posterior thoracic and lumbar vertebree of this Oligocene dog, for as such it must be regarded, are decidedly more feline than canine in appearance, using those terms only with reference to their modern application. The sacrum (Pl. XX, Fig. 14) consists of three vertebrae, and, in correspondence with the great development of the tail, it resembles that of the larger cats in many respects. Only the first sacral vertebra has any contact with the ilium and bears massive pleurapophyses. -The centra are much larger and heayier than in the modern dogs and the postzygapophyses much more prominent. The resemblance between the sacrum of Daphenus and that of the large cats is not very close, and the following differences may be noted: (1) the neural spines are much lower and weaker; (2) the neural canal is smaller ; (5) the transverse processes of the second, and especially of the third vertebra, are decidedly shorter, so that the posterior portion of the sacrum appears much narrower. From the sacrum of the recent dogs that of Daphenus differs particularly in its greater proportionate length and massiveness. Caudal Vertebre (Pl. XTX, Figs. 9, 10).—In none of the specimens of the collection is the tail completely preserved, the largest number of yertebree found being thirteen of one individual and eleven of another, but enough remains to satisfactorily demonstrate its character. The tail is remarkably long and stout and is, in fact, almost as well deyeloped as in the leopard or tiger, and, consequently, is much longer and thicker than in any of the existing Canidae. The first caudal vertebra is quite like that of the lion, but is relatively lighter and more slender in all its parts, and has a short but distinct neural spine; the zyga- pophyses are yery prominent, and even the metapophyses are distinctly shown ; the transverse processes are very long, but are not so broad proportionately as in the lion, and are quite strongly recurved. Posteriorly the caudal vertebrae become successively more and more slender and elongate, while all of the processes are gradually reduced in size. The middle region of the tail is made up of extraordinarily elongate yvertebrie, 342 NOTES ON THE CANID® OF THE WHITE RIVER OLIGOCENE. which are very much like the corresponding caudals of the long-tailed cats, but are decidedly longer and more slender proportionately. Near the tip of the tail the vertebree become Very small. The ribs are represented only by fragments, which, so far as they are preseryed, do not differ materially from those of the modern Canidew. From the character of the pos- terior thoracic vertebrie, it may be inferred that the eleventh, twelfth and thirteenth pairs of ribs did not possess tubercles. Of the sternum very little is preserved. One segment of the mesosternum is asso- ciated with the type specimen of D. felinus ; it has much the same shape as in modern dogs, but is somewhat thicker transversely and shallower vertically, in proportion to its length. Another segment accompanies a specimen of D. vetus (No. 11424) and is much wider and more depressed than in any of the existing fissipedes, except certain hyzenas. As the association of this weathered fragment with the skeleton of Daphenus may be accidental, no great stress can be laid upon it. Measurements. 4 No, 11421, No, 11423, | No. 11425. St ee eS = So = |- | = Atlas, length 52066 Sneseccosaecs ie | 0.031 | AGIs dengthu(excl’ of Od OnGOId))\acccsees coceccoce cose cece dosrsscesoseaweseseeaestnee steer eoneteer eae | | covet ee Sem OL/ OC ONCOL DLOCESS wecitasece scenes seater mle tert ences ]ctnreciesterieltatceeteee mais setaskieneet te | heer Os | -014 WACGM aNteriOLtACce\accccceseccnsseavoetsdeccccsesecascadees sea snasseactieetaceneaeenentenerectseteeees | | .028 | 031 Mbindicenvicalsvertebra, Wen oth wcccsancseccatsesssecenccsecewose tweens c= 50% NOTES ON THE CANID& 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 Cunis, 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 Canide 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 haying a lateral curya- ture which becomes very 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. NOTES ON THE CANID OF THE WHITE RIVER OLIGOCENE. 359 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 Daphanus. That an animal with the skull and dentition of a primitive dog should prove to pos- sess eyen 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 phalane is hardly less peculiar than the second, being short, very much compressed laterally, and bluntly pointed ; it is very little decuryved and has a plainly marked groove on the plantar face near the distal end. The narrowness, compression and straightness of this claw are in yery decided contrast to the heavy and strongly decuryed ungual phalanges of the modern Canidae, though among the latter there is con- siderable yariation in these respects. The articular surface for the second phalanx is much more strongly concaye than in Canis, permitting a greater freedom of motion in this joint, as was necessary in order to proyide 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 Daphanus, and has a decidedly larger subungual process, in correlation with the more complete retractility of the claws. The A. PB) S:—VOL. XIX. 27. 360 NOTES ON THE CANID® 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, haying much the appearance of the unguals in the viyerrine 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 yery likely. Measurements. No, 10546. No. 11421. No. 11424. No, 11423. No. 11425. Caleaneum, length 0.045 0.044 | 0.051 | 0.055 4 dorso-plantar Giameter....-.-cecseesecereseesseneeeeee essere O16 O15 -020 -020 “ Jength Of tuber <-..........22.0esceoscaveesecnsccreeeceveceses 03 029° | 036 040 s¢ Extreme Cistal bread th.........2+.0s.cccesecseseeceereuseene 017 017 .022 | .022 Astragalus, length <....5:...sccccerevsrccccescascccessecne see PCPEEEOCCobc pO 027 Aipstiy len ces eS PFOXIMA] Hreadth.........-.sscececeeeceeceescreeeeerseereeeees .O18 O21 | 022 Gi width of head. 014 O16 | -019 GRID DIO MMBGIO Mineetecetsencneciesscurenonsecestasna sera esccnastnncackerscen ssc O15 016 ULC UN concn tansteeaanetteteresscs dace rec ctsessoeeedcoccendes da ccnesees COLT TROLS Navicular, WiGth...--.2-.0...1..sccssscnseeceseecsracen-enescascouercessaescay O17 Ie e019 HIGLOCUNENTO“M pny Ulicsssrevetacsstcan sex cessecesncreerccsscdesntascsrenasers 010 | 010 | Meta teatad Miesleniriincccccaressattcsanccosccranscseradsusun testers sesacnenc= O31 | 4s breadth prox. end 009 010} = SS Gist. 007 | Metatarsal ai! Wengthtc-:..<.cs | O10 010 O11 -010 -010 O10 Wis TEL bd Peirrg eRe ee Se | 0085 003 003 003 US! ya NC Lae | Bare ee ceed €or ee Se 005 | | 0045 2.004 0045 | .003 CS ICE aac cer Cree CEPR EEE eee ee 0055 005 .0055 RSE A i ea NSO RN ge cnec csavave co eseees | .010 009 009 0095 .0085 | .009 SMB P YA Mibread tlie. seat secreccessesesdctsszseeessesacecs aces: | .006 2.004 0055 005 ‘“ M1, length...... | 0065 007 006 006 .006 | .006 Same NGl Tea Ui sceseedscatactccersenes te nccatccacsecccsoscbe- .009 008s) .|Pe- 008 Se VI MLE CE itoaincencnchasscedes Bre stssiacaqseuesevaue-cnds -0035 003 004 004 003 063 SM MERIOSE Dread tates teststevor-e Seeceeencaacaes ae eselet aes .006 006 .0055 | .004 .005 Lower premolar series, length.....-+....cce00.ceeseeeeeeeeees areO2r .019 Sheeler: Series) len oth, ¥2-. 25. seeyetsoseace essnadsseeee O17 O16 O17 O15 | “P11, length.... -003 003 SOE GIRO Ga sd er Ree doe eo ee 005 005 004 oe ER ee Eu webteicatiwusceateckadctectcrs cocsctverbecectaer -0055 005 006 -005 Semele LN SCBRL. « watinbads sereeciy. car as. scebnasacres sscvs~<> 057 O61 “* ant. post. diam. prox. end.....-.........seeese-seeeeercese renee 005 006 007 005 005 7 transv. 007 007 .009 OOF 007 “breadth of distal end -012 O13 -013 009 ~ SCAU PERU SACOL a eegaacenghesensicnessneserccassabsaaaxcene .0055 06 007 -0055 Was WLC PEN: oe cnccoxscvcncewsscccvcestveuresuvacrercrosuasdcensvanassvjeshacses 072 se OP OL GCP Oieecte man eeaeet ernectnect sae iaeec ose reste sceese ens 007 010 0095 009 SES ENIGRICHS Ole DICCTAL Gite oc bass seasanss | O13 | Ischium, length fr. acetabulum........ 027 026 | Acetabulum, fore-and-aft diameter 008 O11 PENT MleEN Sb ic pessceshrecesecchtencucsest-ncesebes-cscitccrcccsssacevocerseeess 095 -085 OR6 See HLEACUDION DIOX CN seas -n-h nesasseoneeesnasennpoan-aveeesincasnn 0s. O17 020 -O15 016 + Ag SUSE RUE GNC mason worsen saeunen waste kn eccncelscsn--.¢¢s= un sa O16 017 O14 O14 | Tre JIGH oi Mercere paoongce sto: ce ahosscnpacneeebcsioase Heosesecieee roeaeaee -OR9 -099 Smee DFEAUTHIOL PLOX ENC eve seaateaes) scuatendccacccidesusersscservecss++ O15 O18 O14 -O14 ** thickness of prox. end .... O13 O16 O12 O12 Bread bi OU IN tA WOU Ge rneececarcacesedsetesstcuncwscccarseccsgaerec= 009 OL -012 009 | .009 Erbola, stoi CKNESS Of LOX. CNG a+. 2ssascaccaverscsseencscarccnecssessnssne-- 007 | s Re Mdigtal! onder seats ee anes 0065 009 | Willies ae wpm (PT. XOX, Bie. 24). The general appearance of the hind foot recalls that of the viverrines. The astra- galus is quite like that of Daphenus, ut with some differences which tend in the direc- tion of the modern Canida, this bone in Cynodictis standing intermediate in structure between the two extremes, though somewhat nearer to Daphanus. The proximal or tibial trochlea is but little more deeply grooved than in the latter genus, and is therefore much shallower than in Canis, but its borders have the same clean-cut angularity as in the modern forms, instead of curving gradually into the facets for the tibial and fibular malleoli. In Canis the tibial trochlea is extended oyer upon the dorsal side of the neck, but this is not the case in either of the White River canines. The neck of the astraga- S94 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. lus is relatively longer than in Canis or eyen than in Daphenus, resembling that of such viverrine genera as Paradoxurus, but is not directed so strongly toward the tibial side of the foot as in Daphenus. The head with its convex nayicular facet is shaped much as in Canis, except that it is more depressed in the dorso-plantar dimension. In Daphe- nus there is a distinct facet for the cuboid, which meets the nayicular facet nearly at right angles; in Cynodictis this cuboidal facet is very much smaller and sometimes it is altogether wanting, while in Canis the astragalus and cuboid are not in contact. As in Daphenus, the external caleaneal facet is more oblique in position and more simply con- cave than in Canis, but the sustentacular facet is different from that of both the genera mentioned ; it agrees with that of Daphenus in being shorter and wider than in the modern forms, but while in the former this facet is separate from that for the navieu- lar, in Cynodictis, as in Canis, it is confluent with it, but at a different point; 7. e., more toward the tibial side. The interarticular sulcus is somewhat deeper than in Daphenus, but shallower than in Canis. In the latter we find a third calcaneal facet which forms a narrow band upon the fibulo-plantar side of the head and is connected at one end with the sustentacular facet. This accessory caleaneal facet does not occur in either of the White River genera. ; The caleaneum, like the astragalus, is more yiverrine than canine in general appear- ance and quite closely resembles that of Paradoxurus, but the resemblance to Daphenus is eyen more marked. The tuber is slender, compressed and proportionately much shorter than in Canis; in the latter the tuber makes up more than two-thirds of the total length of the caleaneum, while in Cynodictis it is about two-fifths of this length. The free end of the tuber is moderately thickened and club-shaped and is deeply grooved by the sulcus for the plantaris tendon. As in Daphenus, the dorsal and plantar borders of the tuber are nearly parallel and its dorso-plantar diameter is thus almost uniform throughout, not increasing toward the distal end as it does in Canis. Near the distal end of the caleaneum and on the fibular side is a very prominent process for the attachment of the lateral ligaments. This process is not present in the recent Canidae, but is very conspicuous in the primitive carnivores, such as Dinictis and Daphenus, and it recurs among modern plantigrade and semiplantigrade forms, such as Procyon, Gulo, Para- doxurus, etc. Usually, however, it is smaller and less prominent in the fossil than in the recent genera. The facets for the astragalus are somewhat different from those of both Daphenus and: Canis.- Tn the latter the external astragalar facet is in two parts, one of which presents distally and the other dorsally, the two meeting at an angle which does not much exceed 90° ; in the former the whole facet forms one continuously curved con- vexity, not divided by an angulation. In Cynodictis the two parts are distinguishable as in Canis, but they meet at a much more open angle. The sustentaculum is of moderate NOTES ON THE CANID#® OF THE WHITE RIVER OLIGOCENE. 39D prominence and, as in Daphenus, it carries a subcircular facet for the astragalus ; in the modern genus this surface is narrower and more elongate. The sustentaculum also agrees with that of Daphenus in not being so obliquely placed, with reference to the long axis of the ecaleaneum, as in the existing members of the family. On the plantar side, between the sustentaculum and the body of the bone, is a groove, the sulcus flexoris hal- lucis, which is better marked in Canis than in either of the White River genera. ‘This is curious, in view of the fact that the latter possess a well-developed and functional hallux, while in the former this digit is reduced to the merest rudiment. In Canis we find a third facet for the astragalus, a small plane surface distal to the sustentaculum, from which it is separated by a narrow suleus; continuous with this accessory facet, but at right angles to it, is a small facet for the navicular. Neither of these articular surfaces is to be found in Cynodictis. The facet for the cuboid, which in the recent dogs is almost plane and semicircular in shape, is quite deeply concave and of nearly circular outline. The cuboid is relatively high and narrow, differing from that of Canis principally in the smallness of its transverse and dorso-plantar diameters. The proximal surface is oceupied by a large facet for the caleaneum, which, as in Daphenus, is much more con- vex than in the existing dogs. The hook-like projection from the plantar side, which in Daphenus is yery large and prominent and in Canis is even more massive, in the present genus is quite inconspicuous and is continuous with the projection from the fibular side which overhangs the deep tendinal suleus. The astragalar facet is small and is confined to the dorsal side of the cuboid, being much less extensive than in Daphenus. The facet for the nayicular is not so prominent as in Canis or eyen as in Daphenus, and is con- tinuous with that for the ectocuneiform. The distal end of the cuboid resembles that of Daphenus iv haying quite a concave facet for the head of the fourth metatarsal, while that for the fifth is lateral in position. In Canis, on the other hand, the surface for mt. iv is almost plane and that for mt. vy occupies an entirely distal position ; the plantar portion of the facet for mt. iv is much narrower than in the two White River genera, and has thus quite a different shape and appearance. The navicular is almost a miniature copy of that of Daphanus and presents the same differences from that of Canis. Seen from the proximal end, it is of more regularly oval shape and is less contracted on the plantar side than in the modern genus. The position of the navicular in the tarsus is likewise different. In Canis this bone has been somewhat rotated, so that its principal diameter is the dorso-plantar one, and on the plantar border it has been brought into contact with the caleaneum, for which it has acquired a special facet. It is of interest to observe that a similar but more extensive rotation of the tarsal elements has been carried out in the horses, as Riitimeyer has shown. In the White River genera, on the other hand, the principal diameter of the 396 NOTES ON THE CANID&® OF THE WHITE RIVER OLIGOCENE. navicular is transverse, and owing to the elongation of the neck of the astragalus, it is carried so far distally that it can have no contact with the caleaneum, the astragalus articulating with the cuboid. The astragalar surface is concave, but somewhat less so than in Canis, and the facet for the cuboid is small and confined to the dorsal moiety of the fibular side. The distal end displays the usual facets for the three cuneiforms, which do not require any particular description. The entocuneiform has much the same shape as in Canis, elongate in the proximo- distal diameter, but very narrow and much compressed. The navicular facet is rela- tively smaller than in the modern genus and there is no such distinct facet for the meso- cuneiform. The distal surface, for the head of the first metatarsal, is no wider but much more deeply concave than in Canis. The mesocuneiform is a minute bone and, as in the fissipede Carnivora generally, its vertical or proximo-distal diameter is much less than that of the adjoining ento- and ectocuneiforms, forming a depression or recess in the distal row of the tarsus, into which the head of the second metatarsal is tightly wedged. The only articular surfaces visible on the mesocuneiform are the proximal and distal, for the navicular and the second meta- tarsal respectively. The ectocuneiform is much the largest of the three. Compared with that of Canis, it is narrower in proportion to its height and is also less extended in the dorso-plantar dimension, but the projecting process from the plantar surface is eyen more prominent, and is more thickened and club-shaped at the free end. On the tibial side is a minute facet (not double as in Canis) for the side of mt. ii. The facet for the cuboid is much smaller than in the modern dogs and is confined to the dorsal border, while at the infero- external angle of the bone is a minute facet for the head of mt. ivy, which is not repre- sented in Canis. The distal end of the ectocuneiform is taken up by a facet for mt. 111, which is less concave and has a shorter plantar prolongation than in the modern genus. The metatarsus consists of five well-developed members. Unfortunately, there is not a single complete metatarsal preserved in connection with any of the specimens, but enough remains to show that these bones were much longer and stouter than the meta- carpals, and that the disproportion in size and length between the fore and hind feet was much greater than in the recent dogs and quite as great as in many viverrines, such as Herpestes and Paradoxurus or as in Daphenus. The first metatarsal is sufficiently well preserved to indicate that the hallux was well developed and functional, though somewhat more reduced than in Daphanus, or in such recent yiverrines as Cynogale or Paradoxurus. The head bears a narrow, convex facet for the entocuneiform and upon its tibial side is a large, rugose prominence for the attachment of the lateral ligament. The shaft is very slender and is arched slightly NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. 397 toward the fibular side of the foot, making the tibial border somewhat concave. The length of the bone, as already intimated, is not determinable, but the portion preserved in one specimen is nearly as long as the entire fifth metacarpal of the same individual. The second metatarsal is much stouter than the first and more slender than the third. The head is yery narrow, being slightly excavated on the tibial side. Owing to the shortness of the mesocuneiform, the head of mt. ii rises above the leyel of mt. i and iii and is firmly held between the ento- and ectocuneiforms, though there are no such distinct lateral facets for these tarsals as we find in Canis ; a stout prominence occupies the plantar side of the head. The shaft is slender and of oval section, not haying acquired the trihedral shape characteristic of the recent dogs. The third metatarsal is the stoutest of the series; the head is broad dorsally but very narrow on the plantar side, where there is a large, projecting process, more promi- nent than in Canis. The facet for the ectocuneiform is convex (in the recent dogs it is slightly concave) and oblique in position, inclining downward toward the tibial side. Deep sulci invade the head on both sides; on the tibial side the sulcus is narrow, but that on the fibular side is broad. A deep pit on the fibular side of the head receives a corresponding prominence from mt. iv, and an additional facet for the same metatarsal is found on the plantar projection, so that the two median metatarsals are very firmly inter- locked. The shaft, for most of its length, is of transversely oval section, very different from the squared, prismatic shape seen in Canis, though an approximation to this shape occurs in the proximal portion of the shaft, where mt. iii and iy are closely appressed. The distal end is broadened and antero-posteriorly compressed ; the trochlea resembles that of the corresponding metacarpal, save that it is larger and relatively somewhat lower. The fourth metatarsal is of nearly the same thickness as mt. ili, though a trifle more slender. The head is narrow and the facet for the cuboid is slightly convex in both directions ; the plantar extension is neither so broad nor so prominent as in) Canis. On the tibial side is a rounded protuberance, which is received into the depression already mentioned, in the head of mt. iii, while on the fibular side is an excayation for a prominence on mt. vy, and proximal to this excavation is a narrow but well-defined facet for the same metatarsal. Very little of the shaft is preserved, and this proximal por- tion has much the same tetrahedral shape as in the recent dogs. Doubtless, however, the distal part of the shaft assumes a transversely oval section, as does that of mt. iii, though the digits of the pes evidently diverge less distally than do those of the manus. The fifth metatarsal is entirely missing from all of the specimens, so that the inter- esting question regarding the reduction of the external ascending process cannot be answered, 398 NOTES ON THE CANIDZ OF THE WHITE RIVER OLIGOCENE. The phalanges of the pes do not differ from those of the fore foot, except in their considerably greater size. Measurements. No. 10493. | No. 11012. No. 11381. Tarsus, Mephitis exCl sical CANO UIM) lene soe ctee eeem esis ei nevloste ates Bacall les lolaciseis eae iioetsieleniss learn O21 G@alcaneum, Lenp thy .c...2c-.ccceccessccssecarecsonensscauscsavecscesacsscoseseisnnns=suesesanceaseesscessseeess 0195 .020 ss Ven Pt ONOPMMDEM ssaecesnere cee occasectecae. suaicsetan cesta sss@n cenmeeaeantenben aes =maese seh veceseaind 012 .012 dorso-plant. diam... -007 -008 Astragalus, Len gth)...e---crarseersescnnsevseccuecreemratcncoesanaesssnarncmaessiansdnassursuiesruseeneeneeens 013 .013 “014 we aE RA SIRCET ah ho’) 91 (<*: eRe PTE SR eae OC rere ACd a cee ain 46 Soon etibacn 005 | 0055 006 length) Ob Meck? .:-.cncr--ree=s-cctasaresversecvnss+orsecceosseucsnasesesevesesssrscancesrmstcs 006 .006 .006 WL tH Of MIGAG pcccs.cte woah uwartan secede casjnmevancuie decades cape venttede nate Cee eagias caee mean catia 007 007 .008 INGMIGH ariel eh tiecnecessteadeaeterace teeaee antes acs caste cise tals ees onal alae aitetacs sets Manet aren toanetoistaeetae .003 as WLC Lis swestis ta cieve ese ceules clencee vaclinowe ce vickisiesie rctismndac celiac » : Ate seg ee te, ? —— +7@a) Gua Sel igitib ach sabe S «tq oe del ‘gun =4 tik. le Haile “ig, s ti set Z eae jigs. ule a Sak pe ‘ ‘ Wa ae) aie at : ~ sc ngouh icine iegel Meal aye alee San yes) ee Te Fal : ys. ERE Ie VF yd, WAG i eid. Gnr.Dedl. Tote / ei Wot HE Ai y, AX. 7 Yo i eu . ' = 7 i i i fe t F ‘ . > i ‘ 2 rat te ae / } _ sy | ve uf ° oe : 4 » ; 7 iA ie e ARTICLE IX. CONTRIBUTIONS TO A REVISION OF THE NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. (Plates XXI-XXYV.) BY SAMUEL N. RHOADS. Read before the American Philosophical Society, May 6, 1898. An unusually fine series of the skins and skulls, with reliable data and measure- ments, of the beavers, otters and fishers of the United States and Canada having lately come into the custody of the writer, it is thought advisable to publish the results of a study of the various nominal forms of these mammals and briefly discuss the nomencla- ture involved. Owing to a lack of specimens from some regions whose faunal condi- tions are known to produce in many other mammals well-recognized geographic varia- tions, this paper must be considered rather as a contribution to the subject, and in no sense a complete synopsis. The area covered by this study comprises solely that part of North America north of Mexico, no attempt being made to discuss the relationships of the tropical species. To Mr. Outram Bangs the author acknowledges his gratitude for a most valuable loan of skins and skulls of nearly every species and race recorded in these pages. To the kindness of Mr. F. W. True, of the National Museum, is due the loan of a series of skulls of the Alaskan otter. The North Carolina Department of Agriculture has courteously loaned two skins and four skulls of beavers recently killed in Stokes county of that State through the kind offices of Mr. H. H. Brimley, the Curator of the State Museum. Aid has likewise been generously given by Dr. J. A. Allen, Dr. C. Hart Merriam, Dr. T. 8. Palmer, Mr. Gerrit 8. Miller, Jr., Dr. M. W. Raub and Mr. C. 8. Brimley. THE BEAVERS OF NORTH AMERICA. Contrary to evidence which must eventually be accepted by all zodlogists, the Ameri- can beaver, Castor canadensis Kuhl, is still considered by many eminent authorities as 418 CONTRIBUTIONS TO A REVISION OF THE specifically the same as the Castor fiber Linnzeus of Europe. In 1897, Dr. E. A. Mearns deseribed* a subspecies of the typical Canadian animal, naming it Castor canadensis frondator and assigning its habitat to the “southern interior area of North America, ranging north from Mexico to Wyoming and Montana.” This appears to be the first attempt in literature to formally subdivide the American beaver, a species whose con- stancy of characters over the yast and varied habitat which it frequents had hitherto been unquestioned. There can be no doubt as to the tenability of Dr. Mearns’ “ Broad-tailed Beaver ” as distinguished from the Hudson bay animal, whose habitat Kuhl designated as “ad fretum Hudsoni” in his original description of canadensis. It is probable that the beavers inhabiting the Carolinas, Georgia, Alabama, Missis- sippi and Tennessee are equally entitled to subspecific rank. So rare has the beaver become in these States, however, it would probably be impossible to verify such a predic- tion with specimens now in our museums. From what we know of the relationships of the representatives of our eastern species inhabiting the Pacific slope, we are led to expect that the beaver of that region would also prove separable from canadensis. A very complete series of skulls, with three adult and three young skins from the Cascades of Washington and Oregon, shows this to be the case. Fortunately the synonymy of the American beayer is not involved and requires no elucidation in this connection, as is shown by reference to Dr. J. A. Allen’s Monograph of the North American Rodentia. A synopsis of the American forms is herewith pre- sented. CanapiIAN Beaver. Castor canadensis Kuhl. Plate XX1; Fig. 3. Plate SOX Bic. 3: Castor canadensis Kuhl, Beitr. Zool., 1820, p. 64. ?“ Castor americanus F. Cuvier, Hist. des Mam. du Mus., 1825 ” ( fide Brandt in Kennet. Sdugt. Russl., 1855, p. 64). Castor fiber americanus Richardson, Faun. Bor. Amer., 1, 1829, p. 105. Castor fiber var. canadensis J. A. Allen, Monog. N. Amer. Rod., 1877, p. 444. Type Locality.—Hudson bay (“ad fretum Hudsoni” Kuhl). Geographic Distribution —Northeastern North America, from the northern limit of trees south to the United States and west to the Cascade mountains ; intergrading east of the Mississippi river into subspecies carolinensis, south-centrally into subspecies fron- dator and westwardly into subspecies pacificus. * Proc, Nat. Mus., Vol. XX (adv. sheet, March 5, 1897). } As will be seen later, such specimens have since come to hand and are described as Castor canadensis carolinensis. NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 419 Color.*—Winter pelage, above, including sides, dark bay or blackish brown, tip- ped with chestnut or russet, becoming pure chestnut on top and sides of head and on chin, jaws and sides of neck. Rump and thighs purer chestnut. Ears black. Hair of feet, legs and under parts seal brown. Anatomical Characters.—Size, smallest of the American forms. Scaly portion of tail more than twice as long as wide; hind foot with claw about 175 mm. Skull wide for its length ; maximum size of skull 136 by 99 mm. ina New Brunswick example, No. 31, collection of E. A. and O. Bangs. Rostrum and nasals relatively short and wide, the nasal bones averaging more than half as wide as long and extending but little behind the premaxillaries. Upper molar dentition wide and heayy, the crowns oblique, triangular and very wide anteriorly. Measurements—Of a large, typical, adult male specimen from Quebec, No. 3825, collection of E. A. and O. Bangs (measurements made by collector from newly killed specimen). Total length, 1130 mm.; tail vertebrae, 410 mm.; scaly portion of tail (dry meas. from skin), 263 by 122 mm.; hind foot, 176 mm.; length of skull, 152 mm.; breadth of skull, 93 mm.; length of nasal bones, 46 mm.; breadth of nasals, 21.4 mm.+ Remarks. of the authentic measurements and superior condition of the skin and pelt. The aver- The above diagnosis is taken mainly from the Quebec specimen, because age beaver from the Hudson bay regions, however, is somewhat lighter colored than this specimen, which, in its darkness and richness of shade, rivals the best examples of paci- ficus. In size, and ratio of length to width, the skull of the Quebec specimen is typical, but the nasals are too narrow to serve as a standard for canadensis, whose nasals average wider than pacificus and narrower than frondator. In general terms, canadensis differs from frondator in smaller size, narrower tail, much darker coloration and narrower nasals. It differs from carolinensis in smaller size, narrower, longer nasals and somewhat darker coloration. From pacificus it differs in smaller size, lighter coloration, wider nasals and broader skull. Subspecies pacificus differs from frondator in larger size, greatly nar- rowed and lengthened tail-paddle, rostrum and nasals, and in its dark coloration. In color frondator is decisively and uniformly lighter than eastern canadensis and carolinen- sis and western pacificus, but darkened canadensis (not melanistic) are nearly as dark as pacificus. In size, pacificus is much the longest of the three, with very long hind foot and tail. Its skeleton is slenderer and weaker in every part as compared with the massive frame of canadensis and frondator of same age. Cuarolinensis is nearly of the color of * Ridgway’s Nomenclature of Colors is the standard used throughout this paper. + The narrow nasals of this specimen are an exception, the average of several east Canadian specimens showing the ratio of leagth to breadth as less than two to one. {20 CONTRIBUTIONS TO A REVISION OF THE lighter hued canadensis, but agrees with all the other characters of frondator, to which it seems most nearly allied in cranial and caudal characters. Specimens Examined.—New Brunswick, 1 skull; Quebec, 1 skin with skull ; Canada (?), 3 skulls, 1 skeleton, 2 mounted skins; Ft. Simpson, N. W. T., 1 mounted skin; Idaho, 1 skin with skull. CAROLINIAN BEAVER. Castor canadensis carolinensis subsp. noy. ’ Plate XXIII; Figs. 1 and 2. Type Locality.—Dan river, near Danbury, Stokes county, North Carolina. Type No. z.607, old ad. %, in the collection of the North Carolina State Museum, Raleigh, N.C. Collected by a trapper in flesh for the Museum, April, 1897. Geographic Distribution —Carolinian fauna, south into the Austroriparian. Color—Of type and topotype: Oyerhair of upper head, neck, back and sides, bright hazel. Underfur of same parts, seal brown. Hinder back and rump lightening from hazel to cinnamon rufous and then to tawny olive near base of tail. Vent and under base of tail, dark, rich burnt umber. Ears pale blackish. Sides of head below eyes light hair brown, shaded with pale cinnamon rufous. Feet bistre. Below, from throat to vent, dark broccoli brown with wood-brown tips to oyerhair. Anatomical Characters.—Size large, larger than canadensis, with relatively much broader tail, as in frondator. Skull large and broad, with very short, broad nasals. In the type the base of nasals does not reach back to the line connecting the anterior walls of the orbits. Ros- trum very short and broad. Audital bulle remarkably contracted laterally, with a strongly developed osseous column on the outer wall and the transverse diameter less than the longitudinal. Incisors weak, narrowed; molars large, with triangular crowns. Pelage short and harsh as compared with canadensis. Measurements. —Of the type, from carcass: Total length, 1130 mm.; scaly portion of tail, 279 by 158 mm.; hind foot, 184 mm.; ear, from crown, 21 mm.; length of skull, 148 mm.; breadth of skull, 107 mm.; length of nasals, 43.5 mm.; breadth of nasals, 29 mm. Of the topotype (ad. ¢): Total length, 1080 mm. ; scaly portion of tail, 260 by 146 mm.; hind foot, 174 mm.; ear from crown, 23 mm. Remarks.—The two skins and four skulls upon which the above diagnosis of caroli- nensis is based were secured, just before the completion of this paper, from the authorities of the State Museum of North Carolina. They are intended to form a group exhibit in the State Museum, and have been carefully measured by the curator, Mr. H. H. Brimley, while yet in the flesh. The old male which forms the type had lost one of its fore feet, NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 421 apparently in a trap, some years previous to its final capture, but its evident health and great size show that it had suffered little inconvenience from the loss of the member. The strong cranial and caudal affinities which this beaver shows to frondator as dis- tinguished from canadensis indicate that it is more closely related to the western form. In color, however, it shows a nearer approach to canadensis, as, in fact, do many other animals of similar distribution and racial differences. The Mississippi and Louisiana beavers are undoubtedly, from what I can hear from the furriers, the darkest and thin- nest pelted of our American beavers, but their separability from what I have named carolinensis is not probable. They may be considered as belonging to carolinensis rather than to frondator. Specimens Examined.—Stokes county, North Carolina, 4. Sonoran Beaver. Castor canadensis frondator Mearns. Plate X XI; Fig. 2. Plate XXIT; Fig. 2. Castor canadensis frondator Mearns, Proc. U. S. Nat. Mus., XX, ady. sheet, Mar. 5, 1897. Type Locality—San Pedro river, Sonora, Mexico, near monument No. 98, of the Mexican boundary line. Geographic Distribution Southern interior of North America from Mexico to Wyoming and Montana, intergrading northwardly into canadensis, southeastwardly into the trans-Mississippian carolinensis and westwardly into pacificus. Color.—Much paler than canadensis or carolinensis. “ Above russet, changing to chocolate on the caudal peduncle aboye and to burnt sienna on the feet ; toes reddish chocolate. Below grayish cinnamon, brightening to ferruginous on the under side of caudal peduncle. Sides wood brown enlivened by the tawny-olive color of the over- hair.”* A specimen from Red Lodge, Montana (No. 32, collection of E. A. and O. Bangs), taken in November, is wood brown aboye and below, the longer oyerhair of upper pelage washed with pale rusty. Anatomical Characters.—Size large, exceeding average of Hudson bay beaver, with a longer foot and broad tail. Scaly portion of tail less than twice as long as wide, hind foot with claw about 185 mm. Skull massive, large, with short rostrum and very wide, short, tumid nasal bones, the average skull probably exceeding canadensis in size, cer- tainly exceeding it in relative width to length and in the relative breadth of the nasals. Upper molar dentition as in canadensis. Measurements.—Of the type: Total length, 1070 mm.; tail vertebrae from anus, 360 oo» mm.; scaly portion of tail, 290 by 125 mm.; hind foot, 185 mm.; length of skull, 155 * Quoted from Dr. Mearns’ original description (/. c.) of type. 4922 CONTRIBUTIONS TO A REVISION OF THE mm.; breadth of skull, 99 mm. Maximum length of old males, measured by Dr. Mearns, 1150 mm.; of the tail paddle, 285 by 155 mm. Remarks.—Dr. Mearns’ comparisons of frondator with canadensis were evidently not made with the largest specimens of the latter, as I haye examined some whose cra- nial and body measurements are about equal to the maximum recorded by him for frondator. Nevertheless, there is little doubt that the larger size of average frondator is well established. Its long hind foot, broad tail and light coloration distinguish it immediately from canadensis. Its approach to pacificus is solely along the line of great size as indicated by the length of body and hind foot, but in cranial characters, as also in color, it is farthest removed from that race. The close anatomical relation of frondator to carolinensis has been mentioned. Specimens Examined.—Montana, 1 skin with skull; Wyoming, 1 skull. Pactric Beaver. Castor canadensis pacificus, subsp. nov. Plate XXI; Fig. 1. Plate XXII; Fig. 1. Type Locality.—Lake Kichelos, Kittitass county, Washington ; altitude about 8000 feet. Type, No. 1077, ad. 2, in the collection of 8. N. Rhoads; collected in April, 1895, by Allan Rupert. Geographic Distribution —Pacific slope, of America, from Alaska to California. Colg.—Ahbove with very uniform, dark and glossy reddish chestnut overhair, almost concealing along dorsum the seal-brown underfur. Top of head like back ; sides of head, throat, rump, thighs and yent not decidedly lighter than back and belly as in the other forms, these parts paling to walnut brown. Ovyerhair of sides and under parts, between seal brown and broccoli brown ; under fur of belly drab gray at the roots ; hind feet dark seal brown ; fore feet and limbs, dark wood brown. Ears black. Anatomical Characters.—Size, largest of the canadensis group, but of more slender build, the skeleton throughout being of much greater longitudinal and lesser lateral dimensions than in the other forms. Tail and hind foot relatively long. Skull large, relatively narrow, with long, narrow rostrum and nasals, the latter with outer margins nearly parallel and reaching basally decidedly beyond the premaxillaries. Upper molar dentition weak, the crowns of molar teeth rectangular. Measurements.—Of the type from carcass: Total length, 1143 mm.; tail vertebrae, 330 mm.; (from relaxed skin) scaly portion of tail, 295 mm. by 122 mm.,; hind foot, 185 mm.; length of skull, 142 mm.; breadth of skull, 101 mm.; length of nasals, 53.6 mm.; breadth of nasals, 24 mm.; average length and breadth of five skulls from Tacoma and Lake Kichelos, Washington, 144 mm. by 99 mm.; average nasal length and breadth of same, 54 mm. by 23 mm. NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 423 Remarks.—Reliable measurements of only one adult skin specimen (the type) of pacificus were accessible. An adult mounted specimen from Josephine county, Oregon, in the Wagner Institute, Philadelphia, confirms the color and measurements of the type so far as the latter can be ascertained from the stuffed animal. Pacificus, like its associates, Mustela americana caurina and M. canadensis pacifica of the Pacific slope regions, is distinguishable by its rich and deep coloration from its darkest trans-Cascadian representatives. No specimens have come to hand from Alaska, but undoubtedly, from what we know of other species found there as well as from the accounts of trappers and furriers, the Alaskan coast beaver represents the maximum of size* and the greatest richness and depth of fur coloration seen in American beayers. Specimens Examined.—Washington, Tacoma, 1 skeleton, 1 skull; Lake Kichelos, 1 adult skin with skull, 3 young skins with skulls, 1 skeleton, 12 separate skulls ; Ore- gon, Josephine county, 2 mounted specimens; British Columbia, (?) Sumas, 1 skull; + Victoria, 1 skull. THE OTTERS OF NORTH AMERICA. As Mr. Oldfield Thomas has shown in his “ Preliminary Notes on the Species of Otter,” published in 1889 in the Proceedings of the London Zoblogical Society, the charac- ters and nomenclature of the North American species are in great need of study. Dr. Elliot Coues has elucidated with sufficient clearness, in his Monograph of the Mustelide, the habits and characters, and, to some extent, the synonymy of the typical Canadian otter, Lutra hudsonica Lacéptde. Its relations, however, to other nominal species, especially to the otters of the Pacific slope of America from California northward, demand investigation. As in the case of the American beaver, just treated, this paper has to do solely with one central Canadian type and its subspecies found in America north of Mexican terri- tory. Avoiding a general preliminary discussion of the rather perplexing questions of nomenclature and geographic variations and distribution, I will present these in order in the more formal and detailed synopses which follow. * Dr. Allen’s measurements of Alaskan skulls, page 447 of the Monograph of N. A. Rodentia, do not indicate unusual size, but as we have no precise locality given they may not have come from the coast region, and, therefore, do not represent pacificus. 108mm. The next in size is No. 2146, U. S. Nat. Mus., from Nebraska, recorded by Baird. Its size was 147 by 105.5 mm. Unlike all my pacificus specimens, No. 5545 has very wide convex nasals. A. P. S.—VOL. XIX. 3 B. 124 CONTRIBUTIONS TO A REVISION OF THE Hupsontan Orrer. Lutra hudsonica (“ Lacépede,” Desmarest). Plate XXIV; Figs. 1 and 2. Mustela lutra Linn., canadensis Schreber, Stugt., II, Pl. CX XVI, B. (dated 1778 on title-page, but, according to Sherborn, the text of Vol. III was published in 1777 and this plate in 1776). Mustela (lutra) canadensis Kerr, Linn. An. Kingd., I, 1792, p. 173 (see Thomas, Proc. Zoil. Soc. Lond., 1889, p. 197, and Allen, Bull. Amer. Mus. N. Hist., VII, 1895, p: 188). “ Mustela hudsonica Lacép.[ede],” Desmarest, Nouv. Dict. d’ Hist. Nat., XIII, 1803, p. 384; (Nouv. Hd.) 1817, p. 219. Lutra canadensis J. Sabine, App. Frankl. Jour., 1823, p. 653, and of nearly all subse- quent authors (not LZ. canadensis F. Cuvier, Dict. Sci. Nat., 1823, p. 242; see O. Thomas, /. c., p. 197). Lutra hudsonica F. Cuvier, Suppl. Buff., 1, 1831, p. 194; Merriam, N. Amer. Fauna, No. 5, 1891, p. 82. Lataxina mollis Gray, List Mamm. Brit. Mus., 1843, p. 70. Lutra destructor Barnston, Canad. Nat. and Geolog., VIII, 1863, p. 147, Figs. 1 to 6. Type Locality.—* Ou la trouve au Canada sur les bords de la mer.” Geographic Distribution —Northern North America from the Arctic ocean south- ward into the United States and from the Atlantic ocean to the Cascade mountains ; intergrading southeastwardly into subspecies Jataxina F. Cuvier and vaga Bangs, south- centrally into subspecies soronw Rhoads, and westwardly into subspecies pacifica Rhoads.* Color (taken from two specimens in the Bangs collection, No. 5638, yg. ad. 3, Annapolis, Nova Scotia, November 23, 1896, and No. 4190, ad. 2, Upton, Me., Octo- ber 25, 1895).—Above, dark seal brown from nose to tip of tail, darkest posteriorly, below from breast to tail between broccoli ‘and yandyke brown in the Nova Scotia speci- men and between seal and yandyke brown in the Maine specimen. Head and neck below a line running from nose to lower base of ear and base of foreleg light Isabella color anteriorly darkening on lower neck to wood brown in the Nova Scotia animal. In the Maine specimen the neck is Prout’s brown. Feet, legs and tail corresponding to darker shades of upper and lower body. A summer specimen from New Brunswick is dark, vandyke brown, but little paler below than on back, and darker than winter specimens of dataxina from Maryland. * The otters of Louisiana and Mississippi are stated by furriers to be very dark and light-pelted, resembling South Florida and Gulf-coast skins. No specimens having been examined, they are referred to vaga. . NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 425 Anatomical Characters.*—Size, medium (exceeded by vaga, sonora and pacifica). Tail relatively short. Inferior webs of feet and interspace between posterior and ante- rior callosities of manus, densely haired. Hind foot with claw about 125 mm. in old adults ; but so variable as to have little diagnostic value. Total length rarely exceeding 1100 mm. Skull—size, medium (greatly exceeded by vaga and pacifica). Teeth large, crowded longitudinally upon each other and obliquely overlapping. Postorbital neck of frontals relatively short and wide, its superior ridge on a plane with nasals and occi- pital crest. Mastoid width much less than zygomatic width. Postorbital processes short and stout. Audital bull large, tumid, rising abruptly from the sides of basioccipital. Measurements.—See tables. Remarks.—Variations in the size of adult otters from apparently the same region seem remarkable at first sight, but I find that these are not always to be attributed to sex (for the female otter sometimes reaches near to the average size of the males), but to environment. The otters of the Alleghany mountain streams are uniformly smaller than those of the tide-water creeks and rivers of the Atlantic seaboard. This rule applies from Labrador to Florida and is undoubtedly the result of the relative difficulty of obtaining food and securing shelter from enemies in the two kinds of habitat. On the other hand, this difference lies wholly within the limitations of individual variation and in no sense affects the well-defined cranial and other characters which distinguish the races and species hereafter defined. It has to do solely with size, not with propor- tions. In a letter from Mr. C. 8. Brimley, of Raleigh, North Carolina, the same feature is alluded to where he states: “ A trapper of our acquaintance says that otters from the saltmarshes of eastern North Carolina average considerably larger than the otters of the small streams of the central part of the State.” There is rarely to be found a case in mammalian nomenclature more puzzling than that of the first tenable name of the Hudsonian otter. Its synonymy involves that of the mink and the fisher as well as the questions of priority of publication of Erxleben’s and Schreber’s great works on the Mammalia, and the tenability of plate names. TI have consulted Drs. C. H. Merriam and T. 8. Palmer at length on these questions and have accepted their ruling as to the first tenable name of the Hudsonian otter being Lutra hudsonica Lacépéde and that of the northeastern mink to be Putorius vison Schreber. In regard to the name of the fisher, however, I prefer to abide by Canon XLII of the Code of the American Ornithologists’ Union, which accepts, under certain conditions, the names of species originally published on plates, which Drs. Merriam and Palmer and Mr. Sherborn do not accept. Returning now to the abstract of synonymy as given above for the Hudsonian otter, the case may be concisely stated thus: Mustela lutra * The diagnostic value of the nose pad has no significance in this study of the relationships of a monotypic group. {26 CONTRIBUTIONS TO A REVISION OF THE canadensis Schreber is a plate name published (fide Sherborn) in 1776, and is the ear- liest applied to this otter. It would stand (A. O. U., Canon XLITI) were it not unques- tionably applied and intended by Schreber merely as a geographic name without refer- ence to its specific relations to ‘‘ Mustela lutra Linn.” For this reason alone it should be discarded. Furthermore, the name J/ustela canadensis was used by Schreber on a pre- vious plate in the same volume (Pl. No. 126) in the specific sense for the fisher. This plate was also (fide Sherborn) published in 1776, one year before the text, which was published in 1777, and the bound volume of text and plates were dated 1778. In 1777, Erxleben published a description of the fisher and named it Mustela pennantii, by which name it has been since designated by authors generally. As this name is antedated by the tenable plate-name Justela canadensis of Schreber by one year, I adopt it as the name of the fisher of Pennant from the northeastern United States. Erxleben pub- lished in the same work a description of an animal which he named Mustela canadensis, and which Baird and Coues have considered applicable to the mink, and the accept- ance of the dates on the title-pages of Schreber’s (1778) and Erxleben’s (1777) works would give priority to Erxleben’s name and displace Mustela vison of Schreber. But Sherborn’s emendation of these dates makes J/. canadensis of Erxleben for the mink untenable, it being preoccupied by Schreber’s plate-name J/. canadensis for the fisher, as stated above. Besides this fact, Dr. Merriam considers that Erxleben’s description of J. canadensis also applies to the fisher and the marten in such a way as to make it untenable for any species. Returning to the search for a first name for the otter, we find Kerr’s name, J. cana- densis of 1792, to be unavailable because he placed it under the old genus Mustela. Next in order appears to be the name hudsonica, which is accredited to Lacépede, in an article on the Canadian otter in the first edition of the Nowvelle Dictionaire d Histoire Natur- elles, which is signed “ Desm.”” I have not examined this reference personally, but am indebted to Dr. J. A. Allen for a transcript of these facts from the only known copy of the work in America which appears to be available, belonging to the library of the American Museum of Natural History. In agreement with my previous rendering of manuscript names, and on the supposition that Desmarest was the real author and pub- lisher of this name and description of hudsonica, I cite it as Lutra hudsonica (“ Lacé- pede,” Desmarest). I agree with Dr. Merriam that this name should stand for the otter of eastern Canada. Frederick Cuvier seems to have been the first to place this animal in the genus Lutra under the Lacépéde-Desmarest name hudsonica in 1831. The Lataxina mollis of Gray and the Lutra destructor of Barnston are no doubt synonyms of hudsonica. Specimens Examined.—Labrador, Okak, 1 skull; Grand river, 1 skull ; New NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 427 Brunswick, Restigouche river, 1 skin; Nova Scotia, Annapolis, 1 skin with skull; Maine, Upton, 1 skin with skull; Bucksport, 1 skull; Massachusetts, Kingston, 1 skin with skull; Westford, 1 skull; Canton, 1 skull; Missouri, 1 skull; British Columbia, Vernon, 1 skull; Alaska, Tanana river, 1 skull. CaroumtAn Orrer. Lutra hudsonica lataxina (F. Cuvier). Plate XXIV; Fig. 4. Lutra lataxina F. Cuvier, Dict. des Sci. Nat., 1825, p. 242. Type Locality.—South Carolina. Geographic Distribution —Carolinian faunal region, intergrading through the Tran- sition region northward with hudsonica and southward through the Austrariparian into vaga of southern Florida. Color.—Much lighter than hudsonica. Above (from a specimen taken at Liberty Hill, Conn., No. 4252, ad. 3, Nov. 19, 1895, collection of E. A. and O. Bangs*), dark vandyke brown, tipped on upper head, neck and shoulders with wood brown, darkening posteriorly. Upper feet and limbs dark bistre. Below, from lower breast to end of tail, between Prout’s brown and broccoli brown. Head, neck and breast, including ears, below a line connecting nose, upper eyelid, upper ear and upper base of fore leg, grayish wood brown, lightest on head, darkening posteriorly to color (/. ¢.) of breast. The aver- age Carolinian winter specimens from Maryland southward are somewhat lighter and some are Prout’s brown above, the wood brown of lower head and neck becoming a pale grayish buff. Anatomical Characters.—Size, smallest of the hudsonica subspecies. Inferior webs of feet and interspace between callosities of manus, sparsely haired. Hind foot with claw about 120 mm. ‘Total length rarely exceeding 1100 mm. Skull relatively small, with very large teeth, and weak postorbital processes. In other respects like the /ud- sonica type. Measurements.—See tables. Remarks—The relations of this subspecies to northern hudsonica on the one hand and to the southern vaga on the other are rather peculiar. It is without question a nearer ally to hudsonica than vaga in the territory between Connecticut and South Caro- lina, but, as Mr. Bangs has implied in his remarks on vaga, there is a tendency in the Georgia (and we may infer in the South Carolina) otter to the large size and peculiar * This specimen comes from the northern edge of the Carolinian region. No equally good skins from more southern localities being available, it is used as typical of the Carolinian race. It corresponds closely to two fine 1897-8 winter pelts of Maryland otters, examined through the courtesy of Mr. S. E. Shoyer, of Philadelphia. 428 CONTRIBUTIONS TO A REVISION OF THE skull and color characters of the south Florida animal. There is so much evidence of the intergradation of dafaxina both north and south that the specific separation of vaga from it is not permissible. On the other hand it is impossible to ignore the decided racial differences of the Carolinian otter from the Hudsonian type. Cuvier’s original description of Jataxina gives “Caroline du Sud” as the locality where the type was taken ; it is, therefore, permissible to restrict this name to the Caro- linian form as typified in the otters found in the Carolinian lowlands of the eastern United States from south of the “ Transition Zone” of Dr. C. Hart Merriam, as far as middle South Carolina, Alabama and Mississippi, where it merges into vaga of the Gulf or southern “ Austroriparian Realm ” of Dr. J. A. Allen. I know of no restricted synonyms of Jataxina. Dr. Coues quotes in his Fur-bear- ing Animals a“ Latax lataxina Gray, Ann. Mag. N. H., 1, 1837, p. 119.” The work referred to contains no such name. Cuvier’s description of /ataxina gives its color as “dark blackish brown, a little paler beneath. Cheeks, temples, lips, chin and throat pale brownish gray, and under side of tail grayish brown, the hair tips reddish.” He compares the skull of dataxina with his Lutra enudris, “ Loutre de Guiane ” of the pre- ceding page and remarks on the “straight line, even concave or depressed,” joining the nasals and occiput. This is significant, as one of the peculiarities separating vaga from lataxina and hudsonica is the convexity of the frontal plane in the former. Specimens Examined.—Connecticut, Liberty Hill, 1 skin with skull; Pennsylva- nia, Clinton county, 2 mounted specimens; Monroe county, 3 skulls; New Jersey, Tuckerton, 1 skull; Mickleton, 2 disarticulated skeletons ; Maryland, 2 fresh cased winter furs; North Carolina, Raleigh, 2 skulls. Froripa Orrer. Lutra hudsonica vaga Bangs. Plate XXV ; Fig. 2. Lutra hudsonica vaga Bangs, Proc. Bos. Soc. Nat. Hist., XXVIII, 1898, p. 224. Type Locality—Micco, Brevard county, F lorida. Geographic Distribution —Florida, southeastern Georgia and the Gulf regions of Alabama, Mississippi and Louisiana, intergrading (?) northwardly into dataxina. Color —Dark ; less black than hudsonica, darker and redder than lataxina. Breast and belly nearly unicolor with back. Paler area of head and neck, scarcely reaching breast. Above and below, dark, rich chestnut, scarcely paler on belly. Lower head and anterior throat below line from nose to and behind ears, strongly tipped anteriorly with tawny Isabella color darkening to raw umber on throat, the underfur darker than over- fur, instead of lighter as in lataxina. NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 429 Anatomical Characters.—Size, large. Tail relatively long (fide Bangs). Inferior webs of feet and interspace of palms nearly naked. Hind foot with claw reaching maximum (No. 4998 Bangs Coll., yg. ad. #, Citronelle, Florida) of 130 mm. Total length (maximum of No. 4998, / ¢., 1285 mm.) exceeding 1200 mm. Skull large, teeth relatively small, not crowded longitudinally. Postorbital neck of frontals long and narrow, suddenly constricted at base. Frontal plane strongly upraised above a line con- necting occipital crest with base of nasals and above the leyel of postorbital processes. Mastoid width nearly equaling the zygomatic width in very old specimens, in young adult skulls the mastoid width is the greater. Wings of mastoid processes strongly developed and flattened laterally. Audital bullee as in hudsonica and latavina ; well developed, tumid at basioccipital margins. Postorbital processes relatively weak and_ slender. Underfur short, sparse. Measurements.—See tables. Remarks.—This subspecies just described by Mr. Bangs in his most valuable paper on Florida and Georgia mammals is, as already noticed, quite different from lataxina, its nearest geographic ally. In color it comes nearer hudsonica intermediates from New England. In size and color and lack of hair on the webs and palms it shows approach to the remote pacifica, but its peculiar long-waisted and broad-based skull dis- tinguishes it from all other American forms except, perhaps, those of the northern Cen- tral American and South American otters which I have examined. The yellowish and reddish shades of south Florida vaga suggest affinity with what we find published of the characters of the otters of the Caribbean coasts. In essential respects Mr. Bangs’ diagnosis of this animal is very good. He, however, used the skull of a young adult male for cranial comparisons, and while it is true that the ratio of the mastoid to the zygomatic width is much greater in vaga than hudsonica it is not as great as would appear by Mr. Bangs’ figure. In crania of old adult vaga in my collection the mastoid and zygomatic widths are about equal, the latter slightly wider. In hudsonica, however, the excess of zygomatic width and slight development of the mastoid wings is marked. Specimens Examined.—Florida, Tarpon Springs, 1 adult pelt, 5 young skins with skulls and 2 extra skulls; Salt Run, St. John’s river, 1 skull. Pactric Orrer. Lutra hudsonica pacifica, subsp. nov. Plate XXIV; Fig. 3. Plate XXV; Figs. 1 and 3. LIutra paranensis and aterrima Thomas, P. Z.S., 1. ¢., p.199; Trouessart, Catal. Mamm., 1897, pp. 286, 287 (not of Pallas, Zoogr. Ross. Asiat., 1811, p. 81). Lutra californica Baird, Mamm. N. Amer., 1857, p. 187 (not of Gray, Mag. Nat. Hist., I, 1835, p. 580, which is L. felina ; see Thomas, /. c., p. 198). 430 CONTRIBUTIONS TO A REVISION OF THE Type Locality—lLake Kichelos, Kittitass county, Washington ; altitude about 8000 feet. Type No. 616, yg. ad. 3, in the collection of 8. N. Rhoads ; collected in fall or winter* of 1892-93, by Allan Rupert. Geographic Distribution —Pacific slope of North America, from Alaska to Cali- fornia. Color.—Of type: Lighter than hudsonica, with a browner cast, approaching nearly to dataxina. Average of coast specimens from Puget Sound northward, ruddy seal brown, sometimes very dark in Alaskan coast specimens. Lower parts from breast to end of tail much lighter (Mars-brown) than back. Ventral region conspicuously lighter. Lower head, neck and breast very pale wood brown, almost dirty gray. Anatomical Characters.—Size, very largest Tail normal. Inferior webs of feet and palmar interspaces nearly naked. Hind foot not recorded in type, the caleaneum missing ; no measurements of other specimens available. Skull largest of the North American otters (reaching a maximum of 119 mm. in occipito-nasal length and 83 mm. in zygomatic expanse in an Alaskan coast example); teeth relatively weak, less crowded longitudinally than in hudsonica. Interorbital width relatively very great, nearly 12 times postorbital constriction ; postorbital processes long and stout. Mastoid and zygo- matic proportions as in Audsonica. Audital bulle remarkably flattened. Measurements.—See tables. Remarks.—The type specimen, though taken in the mountains and not fully mature, is large and has a skull which would have, perhaps, eventually equaled the maximum size recorded aboye for an Alaskan specimen of much greater age. A yery old female skull from the vicinity of Puget Sound confirms fully the diagnostic characters of pacifica as given. In treating of the otters of the Pacific slope of America we are confronted with two nominal species to which they have been doubtfully referred by authors. In point of time the first to be considered is the Viverra aterrima of Pallas,{ described from a hunter’s skin, lacking skull and feet, taken in northeast Siberia, “ between the Uth and Amur rivers.” Schrenck and Middendorff listed this animal in their works on Siberian Zodlogy with the remark that they were unable to verify its existence or clear up the mystery of its strange characters as given by Pallas. Mr. Thomas (P. Z. 8, /. c., p. 199) queries, on the basis of a mistaken suggestion of Dr. Coues, whether it may * The season of capture was not recorded, but the pelt indicates that it was taken in full winter fur. {Ihave no measurements of Alaskan otters, but judging by the great size of the skulls from there they must greatly exceed any known species of Lutra. On the basis of the skull they must attain a maximum length of over 1400 millimeters. t Zoog. Rosso. Asiat., l. c. NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 43 not prove to be the same as the so-called Dutra paranensis Rengg. which he assumed might occur throughout the whole Pacific coast regions of America. The close relation- ship of our Pacific coast otters to hudsonica will effectually remove them from any com- plication with paranensis, but as regards aterrima we must devote sufficient space to show the impossibility of referring the Alaskan land otter to that animal, as Trouessart has lately done.* A careful study of Pallas’ original description, together with the fact that no later author or explorer has been able to explain or rediscover the animal, conyinces me that it is either unidentifiable or will prove not to belong to the Lutrine but to the Musteline. Pallas states it to be intermediate in size between the European otter and the European mink. He states the length of the skin to be 19 inches, 3 lines, and of the tail 5 inches with a drush of 14 inches! The color of the animal is said to be very black and shin- ing, except the sides of the head between the eyes and ears, which change from black to “subrufescent.” The absurdity of applying such a description to the animal which I have named pacifica, or, indeed, to any member of the genus Lutra, is certainly eyident. So far as any animal now known to zodlogists is concerned, the Viverra aterrima of Pallas should be consigned to oblivion. Another name which has given trouble to those who had to deal with the Pacific coast otter is the Lutra californica of Gray. Fortunately, Mr. Thomas has effectually exposed the history and at the same time the inapplicability of that name to a North American animal of the hudsonica type. He has shown in his paper in the Proceedings of the Zoilogical Society (l. ¢., p. 198) that Gray’s type of californica did not come from California, but most likely from Patagonia, in which case he makes it a synonym of Lutra felina Molina. Specimens Examined—Washington, near Tacoma, 3 skulls ; Lake Kichelos, 1 skin with skull, 1 skull ; Oregon, 1 skull; British Columbia, Sumas, 1 skull ; Alaska (coast?), 3 skulls; Kodiak Island, 2 skulls; Mission, 1 skull; Queraquinay Island, 1 skull. Sonoran Ortrer. Lutra hudsonica sonora, subsp. nov. LIutra canadensis Mearns, Bull. Am. Mus. Nat. Hist., 11, 1891, pp. 253-256. Type Locality —Montezuma Well, Beayer creek, Yavapai county, Arizona. Type, ad. 9, No. 312 in the collection of the American Museum of Natural History. Col- lected December 26, 1886, by Dr. Edgar A. Mearns. * Catalogus Mammatium, |. c. } It is conjectured that this skull came from the North Pacific. It has Capt. T. J. Turner’s name on it. I cannot find an island of this name on the maps. ASP ——VOl. kx oC. 432 CONTRIBUTIONS TO A REVISION OF THE Geographic Distribution —Arid southern interior of North America, from Mexico, probably to Wyoming. Color.—Of type, fide Mearns, /. ¢.: “ Above dark brown, without reddish tinge ; this color changing gradually to a light grayish brown below, being palest (almost whitish) upon the sides of the head below the level of the eyes and upon the under side of the head and neck as far back as the fore limbs. . ... The long hairs of the lighter por- tions of the body are pointed with yellowish gray and upon the upper surface of the head and neck the tips of the hairs are yellowish brown, giving a paler cast to that part of the dorsum.” Anatomical Characters.—Size, large, with a very long hind foot, the body length measurements exceeding those of any other specimen of North American otter exam- ined or recorded.* Webs of feet not densely haired beneath. Hind foot,145 mm. Total length reaching 1300 mm. Skull—size, large, nearly as great as in largest Alaskan pacifica, but small for the great relative length of body, “less massive, broader, with more evenly rounded zygomatic arches and with the brain case more conyex or bulging in its outlines.” “‘ Arizona skulls differ from all others in the slender, attenuated postor- bital processes and in the greater height of the lower jaw from angle to condyle, or to summit of coronoid process. From its geographically near neighbor, L. felina of Cen- tral America, it presents many cranial and dental differences; in fact, skulls of the lat- ter are so very distinct [in their inferior concayity, frontal depression, short muzzle, narrow postorbital constriction and absence of the heel in front of the antero-internal cusp of the last upper molar] from any known specimens from North America, north of Mexico, as to be distinguishable from them at a glance.” Measurements.—Of type: “Total length, 1300 mm.; head and body (measured from tip of nose to anus), 815 mm.; tail measured from anus to end of vertebrae, 472 mm. ear, height above crown, 15 mm.” No skull measurements given. Remarks.—I have accepted Dr. Mearns’ yery full and satisfactory diagnosis of the Arizona otter, given in the Bulletin of the American Museum of Natural History, as conclusive evidence of the existence of a recognizable race in arid interior America, south of Montana. Its great size and light color together form a combination not found in any other known or named otter. It has been thought unnecessary to examine the type, as, owing to the author’s removal from Philadelphia during the completion of this paper, such an examination would have caused a greater risk to the type specimens than the facts warranted. * The great size of the type, as compared with an adult male also recorded by Dr. Mearns from Arizona, indicates that the sex of the type may have been wrongly determined. If correct, the size to be expected of a full-grown male sonora would be extraordinary. NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 435 NEWFOUNDLAND OTTER. Lutra degener Bangs. Plate XXIV; Fig. 5. Jutra degener Bangs, Proc. Biol. Soc. Wash., XII, 1898, p. 35. Type Locality —Bay St. George, Newfoundland. Geographic Distribution —Confined to Newfoundland (?). Color —Of type, ad. 7, taken April 22, 1897 : Above, black with seal brown reflec- tions. Ears, seal brown. Lower head and neck areas grayish wood brown, becoming seal brown on breast; the remainder of lower parts nearly as dark as back. Tail uni- color. Feet seal brown and densely haired on under side of webs and palmar interspaces. Anatomical C haracters.—Size, much smaller than any of the /udsonica group. Hind foot small, with claw averaging about 112 mm.* long in the two specimens exam- ined. Total length about 1000mm. Tail relatively short. Skull very small, narrowed, weak and fragile; the brain case wide anteriorly ; the frontal and interorbital widths narrow and the postorbital processes weak and slender, strongly grooved on their supe- rior face. Sagittal crest not developed eyen in old specimens. Interorbital constric- tion about equal to postorbital constriction. Teeth weak, with normal cuspidation. Audital bulle normal. Measurements.—See tables. Remarks.—The type specimens of degener, so generously loaned to me by Mr. Bangs, when compared with the large series used in the preparation of this paper, con- vince me that this depauperate insular form has no intercourse with the larger typical hudsonica of Labrador and New Brunswick. A skull from Grand river, Labrador, shows no approach to the degener type, and another from Okak, Labrador, agrees in the same differences. A young adult skull and skin of hudsonica from Nova Scotia, and an adult summer skin from New Brunswick, show that the maritime otter of the mainland some- times attains a size nearly one-third larger than the largest known specimens of old, adult degener. Specimens Examined.—Newfoundland, Bay St. George, 2 skins with skulls, 1 extra skull. THE FISHERS OF NORTH AMERICA. Apology must be made for the inferior series of skins and skulls which form the basis of the subjoined remarks on the Pekan. They serve, however, to elucidate some * The collector’s measurement of the hind foot of type is given on label as ‘‘ 126 mm.”’ This is certainly incorrect, - as the length determinable by feeling the caleaneum in the dry skin could not have exceeded 115 mm, This accords with the small size of the hind foot and the length of other specimens of degener, 434 CONTRIBUTIONS TO A REVISION OF THE questions sure to be soon brought up in the active advance of monographic work in American mammalogy. The synonymy of Pennant’s Fisher has already been discussed under Lutra hud- sonica, and I have there given reasons for my adoption of the plate-name canadensis of Schreber as having priority over the long-accepted name pennanti of Erxleben for this animal. Pennant’s Fisuer. Jlustela canadensis Schreber. Mustela canadensis Schreber, Saugt., U1, p. 492, Pl. CX XIV. Text published in 1777, plate in 1776 (fide Sherborn). Mustela pennantii Erxleben, Syst. An., 1777, p. 470. Mustela melanorhyncha Boddaert, Elench. An., 1784, p. 88. Viverra piscator Shaw, Gen. Zodl., 1, 1800, p. 414. Mustela nigra Turton, ed. Linn. Syst. Nat., 1, 1802, p. 60. Mustela godmani Fischer, Syn. Mamm., 1829, p. 217. Type Locality—* New York and Pennsylvania,” Pennant. Geographic Distribution —Northern North America, east of the Cascade moun- tains, from the northern limit of trees to Colorado and North Carolina in the mountains. Intergrading on the Pacific slope into subspecies pacifica, and probably in the southern Rocky mountain region into a paler race. Probably represented in the Hudsonian faunal region by a subspecies.* Color—From an adult, male, winter specimen taken near Lancaster, Pa., March 11, 1896, and in the possession of Dr. M. W. Raub, of that city, who furnished the description : “ Head and one-half of the length of body, gray and black mixed, gray predominating ; throat darkest, with snout from tip to line of eyes dark brown. The hinder half of body gradually darkens into a deep chocolate color until it reaches the tail, which is almost black with a tip entirely black. Hind legs and tail, viewed at a distance of six feet, look very dark, almost pure black. The fore legs are black but not so deep. Tips of ears, darkest.” Two specimens from the Bangs collection, one from Moosehead lake, Maine, the other from Idaho county, Idaho, seem to answer closely the above description. The light upper and forward portions of body are a grizzled grayish brown, the long hairs black tipped. The basal half of hairs of anterior back are hair brown. I can discover no color characters to separate the Idaho specimen from the one from Maine, nor do the skulls indicate any reliable differences. The Maine skin (of an animal two-thirds grown) * Typical canadensis must be restricted to the Alleghenian form. NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. 435 has white patches on lower fore leg, breast and vent, and an immature specimen of pacr- fica has white spots on throat, arm-pits and vent. The four adult specimens examined are not thus pied. Dr. Coues, in his Fur-bearing Animals, says that the fisher is an exception to the marten, mink and weasel in not having these patches. They may dis- appear with age in the fisher, but they do not in the other species. Anatomical Characters.—Size, smaller than subspecies pacifica. Skull small ; nasals relatively short, less elongate at basal apex. Posterior upper molar relatively small, its inner lobe not greatly developed longitudinally so as to only slightly exceed the breadth of outer lobe ; neck of crown of same tooth but slightly constricted. Measurements—Of Dr. Raub’s Pennsylvania specimen, old ad. 3, /. ¢.: Total length, from end of nose to end of tail hairs, 965 mm.; tail vertebrie, 318 mm.; hind foot, 115 mm.; ear from crown, 27 mm. A mounted specimen, No. 507, Academy Natural Sciences, adult ~, from “ Pennsylvania,” has a total length of 1000 mm., with tail (minus brush), 390 mm., and hind foot, 112 mm., taken from the dry mount. The Idaho specimen, No. 6964, young adult ©, coll. of E. A. and O. Bangs, is 978 mm. long, with tail, 369 mm., and hind foot, 117 mm. Skull of No. 7437, yg. ad. 3, Greenville, Me., total length, 117 mm.; zygomatic width, 63 mm.; mastoid width, 54 mm.; mesial nasal length, 22 mm. Remarks —TVhe characters of the Pennsylvania fishers above enumerated, so far as they are based on reliable measurements and color diagnoses, may be considered as repre- senting typical canadensis, based on Pennant’s original notice of the animal. Whether a series of Alleghenian fishers will show the Hudsonian animal to be separable is an interesting question probably to be decided in the affirmative. The Idaho and Maine specimens examined, though not contrasted by me with Dr. Raub’s specimen, must be very close to it. No skulls of Pennsylvania fishers haye been examined, but the close resemblance of the Idaho skull to those from Maine, as indeed to pacifica also, strongly indicates that no cranial differences exist between the east American fishers of the north and south. The “saturated” color characters of pacifica are alone sufficient to distin- guish it from all fishers found east of the Cascades. Specimens Examined.—Pennsylvania, 1 mounted specimen (fide Dr. Raub, 1 mounted specimen) ; Maine, Mooseland lake, 1 skin with skull; Greenville, 2 skulls ; Lincoln, 1 skull ; Idaho, Idaho county, 1 skin with skull. Other specimens from east- ern North America, 1 mounted, 2 old ad. skulls. Pactric Fisner. Mustela canadensis pacifica, subsp. nov. Type Locality—Lake Kichelos, Kittitass county, Washington ; altitude about 8000 {56 CONTRIBUTIONS TO A REVISION OF THE feet. Type, No. 1074, old ad. 9, in the collection of 8. N. Rhoads; collected in the fall or winter of 1892—93, by Allan Rupert.* Geographic Distribution.—Pacifie slope of America, from Alaska to California. Color.—Above, from between eyes to middle back, grizzled, grayish ochraceous heavily lined with black, becoming hazel black on hind back and dark black on rump, thighs and tail. Whole head, behind eyes clove brown basally, strongly grizzled with dirty white. Snout to eyes blackish seal brown. Chin, throat, breast and belly between dark chestnut and hazel, obscured with black. Legs and feet black, the fore legs show- ing the yandyke brown bases of hairs. Basal half of hairs of anterior back are Prout’s brown as contrasted with the hair brown of canadensis. Anatomical Characters.—Size, large, skull very large, with relatively long nasals. Posterior upper molar large, with spreading inner lobe much wider longitudinally than outer section of same tooth; the crown suddenly, constricted at the middle. Measurements—Of type from relaxed skin: Total length, 1090 mm.; tail, 350 mm. without brush ; hind foot not determinable, as the bones are missing. Measure- ments of a specimen two-thirds grown, No. 295, coll. 8S. N. Rhoads, from near Tacoma, Wash.: Total length (relaxed skin), 970 mm.; tail, 400 mm.; hind foot, 112 mm.; ear from crown, 21 mm. Skull of type: Total length from hinder end of sagittal crest to front end of premaxille, 125 mm.; zygomatic expansion, 73 mm.; mastoid expansion, 54 mm.; interorbital constriction, 28.5 mm.; postorbital constriction, 20 mm.; mesial length of nasals, 27 mm. Remarks.—The dimensions of the type skull, when we consider it was from a female, show that the fishers of the Cascade mountains attain a much greater size than those of the Appalachian chain. Young adult skulls of the same age from western Washington and Maine show the same distinctions. The younger specimen from Tacoma, while approaching nearer to Idaho and Maine specimens in grayer color, is very much darker than they, the difference in shade between the anterior and posterior dorsal areas of the former being slight, while in the latter it is striking. The tawny suffusion so deeply marked in the type of pacifica and which separates it at a glance from canadensis is also noticeable in the Tacoma specimen. Specimens Examined.—Washington, Lake Kichelos, 1 skin with skull, 2 skulls ; near Tacoma, 1 skin, 1 skull ; British Columbia, Sumas, 1 skull. * Mr. Rupert, whose business is hunting and trapping, first sent me the fresh skull of a very old Q fisher, which was entered in my catalogue as No. 621. I wrote him immediately that I would like to have the pelt belonging thereto, and in a later shipment the skin, which forms the type of pacifiea, was sent on without label. As it is also from a female and a very old animal, I consider the skin and skull as belonging to the same individual. 437 NORTH AMERICAN BEAVERS, OTTERS AND FISHERS. Skull Measurements of North American Otters (in millimeters) { | | | | laa5a/ aq | ‘ se | had | 3 B2aa E le. a 5h] | Collection. 22 | Sex. Locality. | Species, é 4" Fil a8 82/3 g #3 ce =e | Remarks. ee E a: Agee S (3 [8 |= | es] 85] E. A. and O. Bangs| 5638 | yg. ad. g'| Nova Scotia, Annapolis L. hudsonica (‘‘La- | 113.5 | 72 | 68 | 27.7) 23 35 | 15 Large, coast form. } cép.,’’ Desm.) do. 7431 old ad. | Labrador, Okak do. 74.5) 67 | 23 | 19 | 35 | 13.5| Coast form. Acad. N. Sci. Phila.| 3150 old ad. | Labrador, Grand River do. | 105 72.5 65 20.8) 20 | 29 10.5} Inland form. Smithsonian Inst. | 21483 old ad. | Alaska, Tanana River do. 102 72 | 63.5) 24 | 18 | 32 | 12.5) Inland form. E. A. and O. Bangs} 4238 | old ad. Maine, Bucksport do. | 109 73.5) 66 25.51 21.5, 37 | 14 | Coast form. do. 4188 | old ad. ¢ Massachusetts, Canton do. 112 76! 1°69) | 26) | 22. |°88) | 15: | Intermediate. Acad. N. Sci.Phila.) 3569 | old ad. | Pennsylvania, Monroe Co. | Lh. lataxina(P.Cuy.)| 100 | 69.5 65 | 22.8 20 | 31 | 13 | Inland interm., prob. Q. S. N. Rhoads | 1840 | yg. ad. | do. do. 104.5| 68 | 61 21.5) 19 | 28.6 12 | Probably <. do. | 1565 | yg. ad. | New Jersey, Tuckerton do. 104 70 | 63.5) 24.5) 23 | 33.5) 11 | do. 3896 yg. ad. | New Jersey, Mickleton do. 107 | 70 | 63 | 93 | 12 E. A. and O. Bangs) 3537 old ad.