Pelonax bathrodon (Marsh) Peterson.15 Unnamed form b. It will be noticed at once that Archæotherium is not listed from the Protoceras beds. Whether it is really absent or merely lacking from the collections so far examined is uncertain, nor is it yet possible to say to what extent the species listed from the Titanotherium and Oreodon beds respectively are confined to these levels. A. wanlessi, A. mortoni and A. crassum? are found in the zone of rusty nodules in the upper part of the "turtle-oreodon layer" of the lower Oreodon beds, and were certainly contemporary. The origin of the group as a whole is uncertain. The Eocene achænodonts, as Professor Osborn points out,16 are too specialized in the teeth to be regarded as directly ancestral. The European genus Entelodon and the American Archæotherium both appear in the lower Oligocene, and, as Professor Osborn suggests, may have sprung alike from an unknown northern or Holarctic form. A significant fact bearing on this general subject is the sudden appearance in the Protoceras beds of several types of entelodonts, both large and small, in which the shape of the fourth upper premolar agrees more closely with the character of that tooth in the European genus than in Archæotherium. Perhaps this is to be explained as a new faunal invasion. Megacharus zygomaticus, the small form which I have not named, and a still larger individual represented in the Princeton collection by some teeth and other fragments, all show this character. On the other hand, Megacharus latidens Troxell and Scaptohyus altidens gen. et sp. nov. have the anterior border of the tooth in question indented, as in Archæotherium. Further discussion of the affinities of entelodonts in general and the forms from the Protoceras beds in particular may be postponed until the collection of the American Museum of Natural History has been studied. This contains excellent material of one or more undescribed large 15 Protoceras sandstones? Big Badlands of South Dakota, Peterson, loc. cit., pp. 57, 58. 16" The Age of Mammals," pp. 217, 218. forms from the Protoceras beds, perhaps the same as some of the fragmentary specimens in the Princeton collection already referred to. With the kind permission of Professor Osborn and Dr. Matthew, I hope to pursue these studies farther, on the collections in their charge. The facts presented regarding habits amplify Professor Scott's published observations and have an important bearing on the supposed function of the dependent process of the jugal which projected far enough beyond the cheek to be grasped and badly injured by the teeth of an adversary. ON MEAN RELATIVE AND ABSOLUTE PARALLAXES. BY KEIVIN BURNS. (Read April 22, 1921.) In computing the mean parallax of a group of stars by comparing the radial velocities with the proper motions, it has been the custom to proceed in one of two ways. Knowing the apices of the sun's way, a great circle is passed through these points and the star. The total proper motion is then divided into two parts, one at right angles to the plane of this great circle, and the other in the direction of the circle. The former is called the tau component and the latter the upsilon component. The tau component is evidently free from any motion due to the motion of the sun, while the upsilon component contains all of the effect of the solar motion. Knowing the sun's velocity, the mean parallax of a group of stars distributed at random over the whole sky can be derived from a study of the mean algebraic upsilon component taken for each part of the sky. The formulæ used in this and the following method are found in "Stellar Motions," by W. W. Campbell, page 214 and following. It is seen that for the average of a group of stars V4.74(7/π), where V, denotes the radial velocity freed from the motion of the sun. For each star Vm=4.74(μ/ñ), Vm being the total velocity across the line of sight. Let Vr be the total radial velocity, then for the mean of a group Vm=1.57Vr. For, denoting the cross velocity freed from the motion of the sun by Vm, Campbell shows that in the mean, Vm=1.57Vr, and Sm=1.57Sr, S being the velocity of the sun with respect to any star, the subscripts denoting cross and radial motion as above. The individual values of S, and V, unite by addition and subtraction to form the values of Vr, and the quantities Sm and Vm unite in the same manner to form Vm. Hence we have I'm F(Sm, Vm) = 1.57F × (Sr, Vr)=1.57Vr. The relationship deduced by Campbell, Vm 1.57V, holds equally well if we choose a coördinate system fixed = with respect to the sun. In that case the two last-mentioned equations are identical; that is, the motion across the line of sight with respect to the sun is equal to the motion in the line of sight affected by the factor 1.57, for the mean of a large number of objects distributed and moving at random. The unit of is kiloV meters per second, and of μ is seconds of arc per year. Substituting for Vm we find In case the data can be represented by curves somewhat similar to the probability curve, the average parallax is given by the formulæ of the last paragraph by comparing average radial velocities with average proper motions. If the data fit the probability curve well, we may use mean values of μ and V, as the ratio of the mean is the same as that of the average. We may still use mean values if, as is the case with the data under discussion, the curves for μ and are of the same form and depart but little from the error curve. By "average" is meant such a value that there are as many data larger as smaller. The mean is 18 per cent. larger than the average, so we have for the mean parallax Formulæ similar to the foregoing are only strictly valid in case the objects involved are distributed at random and moving at random. Any tendency toward systematic motion places certain restrictions on the use of these formulæ. For instance, if we had accurate proper motions of all the planetary nebulæ, the parallax as derived from the tau components would be too small, for the radial velocities of these objects show that they are moving sensibly parallel to the galactic plane. The apices of the sun's way lie near to this plane, therefore the tau components are very small in comparison with the radial velocities and total proper motions. The use of the upsilon components would be free from this latter difficulty. The use of total motions also avoids this difficulty, and has the further advantage of saving a great deal of labor. By this last-named method, the average parallax was computed for each spectral class, using the data for all stars brighter than mag. 5.6. The "First Catalogue of Radial Velocities" by Voute was of great assistance in this work. It at once becomes evident from an examination of the data that the selection of the fainter stars for observation of the radial velocity has been largely influenced by the consideration of large proper motion. In the case of stars fainter than the sixth magnitude this effect is quite noticeable, and it begins to be seen in Classes F and G, even among stars of magnitude 5.2. The motions of the stars of large proper motion. are directed more nearly across the line of sight than would result from random distribution, and so these data are not suited to studies like the present investigation. The effects of this type of selection are not of importance in the case of Classes other than F and G when stars brighter than 6.0 are considered; and the results for stars brighter than 5.6 of Classes F and G are not greatly influenced by selection. To obtain the average velocity and proper motion, these data were plotted by spectral classes. The probability curve which best represented each group was found, and the average value was taken from the curve. The data in most cases fit the curve fairly well, the departure being in the sense of too many large velocities for the number of small ones. This excess may be due entirely to the element of selection among the stars fainter than 5.0. In the case of Class G the proper motions would suggest that we are dealing with two distinct groups of stars, the larger group being at about the distance of stars of Classes K and M of the same magnitude. The smaller group, about ten per cent. of the whole, appear to be much closer. Among stars of Class F there are just as many large proper motions as in Class G but there is no break in the curve. There are but few large velocities or motions among stars of Classes K and Ma-c. The data for the latter class fit the curves particularly well. The data for Class Md do not fit any smooth curve and they were not included in the discussion. Taking the average values from the curves, the average parallax was computed as outlined in a previous paragraph. The results are given in column 2 of Table 1, and may be compared with the |