publication of a work on all the mosses, for we may well expect something excellent from him. The next author of importance is C. Müller, who published a synopsis of all known mosses, in two volumes. He deserves our thorough appreciation for his diligence in collecting the existing material. His views on system, however, are less happy. Led by the consideration of certain characteristics, he often classifies very different species together, and separates those closely related. Among other writings on exotic mosses, we must mentioned Dozy and Molkenboer's "Musci inediti Archipelagi Indici," and their " Bryologia Javanica," which was continued after their death by Van der Bosch, and Van der Sande Lacosta. They follow the same plan as the "Bryologia Europea," and are, therefore, of great value. The works of Sullivant, on the moss flora of North America, and those of Wilson, Mitten, and Hampe, are also of considerable importance.

In the last class, that of the ferns, a series of the most important discoveries was inaugurated by Nägeli. He observed that the antheridia, or male organs of fructification, were developed upon the prothallium, which originates directly from the germinating spore. Count Lesczyc Suminski followed up his discovery by proving that the prothallium contained also the archegonia or female organs. Through these two brilliant discoveries new prospects were opened for the morphology of ferns. We recognized that in this whole class of plans fructification was effected on the small prothallium, and that the foliage, which we had been accustomed to take for the whole plant, was developed only when fructification had taken place. Schacht, Mettenius, and especially Hofmeister, deserve great credit for following up these discoveries. The brilliant researches of the latter author in particular, have made known to us the exact process of fecundation, and we now understand that the so-called large and small spores of the selaginella and water-fern are nothing more than the female and male organs of these plants. Hofmeister has furthermore ascertained with unexampled acuteness the laws according to which the leafy plant is developed from the impregnated germ-vesicle of the archegonium, and also how the stem grows, and how the fans are formed. Although Hofmeister came to the erroneous conclusion that the latter were not true leaves, but peculiarly transformed branches, the value of the grand discoveries of this most original and thorough of all organographists of the acrogens remains unimpaired. Hugo von Mohl has drawn a masterly picture of the structure of the stem of tree-ferns, in his classical desertation, which has since been developed more in detail, partly by himself and partly by other authors. The most thorough investigation of the development of the indusium and sporangium are due to Schacht.

Besides the older works of Kaulfuss and Kunze on the classification of ferns, we must mention especially the numerous pteridographic works of Hooker, which have considerably advanced our knowledge of the subject by their excellent illustrations. The works of K. B. Presl

are of great importance, and of especial interest to us Austrians. In his "Tentamen Petridographia," this thorough scholar has studied the reticulation of ferns more accurately than any of his predecessors, introduced new names, and endeavored to divide the class into more natural genera. Although he sometimes goes too far in this direction, we cannot but appreciate his earnestness, consistency, and extensive information. Fée attempted to follow in Presl's footsteps, but he was less successful, and his works must be used with caution. Our most distinguished pteridographist, Mettenius, successfully opposed the tendency to split up the existing material into too many untenable genera and species, in his excellent work on the ferns of the Leipsic botanic garden, and in a series of critical essays, which mostly appeared in the Senkeberg Museum. May this distinguished scholar indefatigably pursue and ultimately attain his object! Moore deserves great credit for his very critcal index of all ferns, for the introduction of many tropical specimens, and for publishing (together with Newman) the first work in which nature was successfully employed to print herself. Lowe's "British and Exotic Ferns" is also a valuable illustrated work. Besides all these there are many special publications on single species. The following are among the most important: Milde's Essays on the Equisetacea and Domestic Ferns; Presl Van der Bosch and Mettenius on Hymenophylleæ; Spring's Monograph of the Lycopodiacea; and A. Braun on Isoëteæ, and Water-Ferns in General.

This then is a condensed review of the most important achievements in cryptogamy within the last few decades. Taking them altogether, we may say that this branch of botany has made more progress in this period than in all preceding times, and that it has now indeed become a science. The study of the cryptograms is no longer confined to a few isolated scholars as formerly, but it is exciting general interest, and many excellent investigators are making it their fa vorite subject. Morphology was not only founded, but even completed and established for certain classes. Numerous and highly important anatomical and physiological data have been furnished; the classification has in the last period been reformed in accordance with the latest views, and various authors have endeavored to obtain a natural arrangement of species, and have succeeded in many cases.

Although much has been accomplished, much still remains to be done, and we need the combined efforts of many. May, therefore, the interest in cryptogamous plants ever become more general and lively, and may, especially in Austria, many scholars and amateurs turn their attention to this branch of botany! The most grateful results will surely reward their exertions.

RECENT RESEARCHES ON THE SECULAR VARIATIONS OF THE PLANETARY

ORBITS.*

BY JOHN N. STOCKWELL.

The reciprocal gravitation of matter produces disturbances in the motions of the heavenly bodies, causing them to deviate from the elliptic paths which they would follow, if they were attracted only by the sun. The determination of the amount by which the actual place of a planet deviates from its true elliptic place at any time is called the problem of planetary perturbation. The analytical solution of this problem has disclosed to mathematicians the fact that the inequalities in the motions. of the heavenly bodies are produced in two distinct ways. The first is a direct disturbance in the elliptic motion of the body; and the second is produced by reason of a variation of the elements of its elliptic motion. The elements of the elliptic motion of a planet are six in number, viz: the mean motion of the planet and its mean distance from the sun, the eccentricity and inclination of its orbit, and the longitude of the node and perihelion. The first two are invariable; the other four are subject to both periodic and secular variations.

The inequalities in the planetary motions which are produced by the direct action of the planets on each other, and depend for their amount only on their distances and mutual configurations, are called periodic inequalities, because they pass through a complete cycle of values in a comparatively short period of time; while those depending on the varia tion of the elements of the elliptic motion are produced with extreme slowness, and require an immense number of ages for their full development, are called secular inequalities. The general theory of all the planetary inequalities was completely developed by La Grange and La Place, nearly a century ago; and the particular theory of each planet for the periodic inequalities was given by La Place in the Mécanique Céleste.

The determination of the periodic inequalities of the planets has hitherto received more attention from astronomers than has been bestowed upon the secular inequalities. This is owing in part to the immediate requirements of astronomy, and also in part to the less intricate nature of the problem. It is true that an approximate knowledge of the secu lar inequalities is necessary in the treatment of the periodic inequalities; but since the secular inequalities are produced with such extreme slowness, most astronomers have been content with the supposition that they are developed uniformly with the time. This supposition is suffiIntroduction to a memoir to be published in the "Smithsonian Contributions to Knowledge."

ciently near the truth to be admissible in most astronomical investigations during the comparatively short period of time over which astronomical observations or human history extends; but since the values of these variations are derived from the equations of the differential variations of the elements at a particular epoch, it follows that they afford us no knowledge respecting the ultimate condition of the planetary system, or even a near approximation to its actual condition at a time only comparatively remote from the epoch of the elements on which they are founded. But aside from any considerations connected with the immediate needs of practical astronomy, the study of the secular inequalities is one of the most interesting and important departments of physical science, because their indefinite continuance in the same direction would ultimately seriously affect the stability of the planetary system. The demonstration that the secular inequalities of the planets are not indefinitely progressive, but may be expressed analytically by a series of terms depending on the sines and cosines of angles which increase uniformly with the time, is due to La Grange and La Place. It therefore follows that the secular inequalities are periodic, and differ from the ordinary periodic inequalities only in the length of time required to complete the cycle of their values. The amount by which the elements of any planet may ultimately deviate from their mean values can only be determined by the simultaneous integration of the differential equations of these elements, which is equivalent to the summation of all the infinitesimal variations arising from the disturbing forces of all the planets of the system during the lapse of an infinite period of time.

The simultaneous integration of the equations which determine the instantaneous variations of the elements of the orbits gives rise to a complete equation in which the unknown quantity is raised to a power denoted by the number of planets, whose mutual action is considered. La Grange first showed that if any of the roots of this equation were equal or imaginary, the finite expressions for the values of the elements would contain terms involving arcs of circles or exponential quantities, without the functions of sine and cosine, and as these terms would increase indefinitely with the time, they would finally render the orbits so very eccentrical that the stability of the planetary system would be destroyed. In order to determine whether the roots of the equation were all real and unequal, he substituted the approximate values of the elements and masses which were employed by astronomers at that time in the algebraic equations, and then by determining the roots he found them to be all real and unequal. It, therefore, followed, that for the particular values of the masses employed by La Grange, the equations which determine the secular variations contain neither arcs of a circle nor exponential quantities, without the signs of sine and cosine; whence it follows that the elements of the orbits will perpetually oscillate about their mean values. This investigation was valuable as a first attempt to fix the limits of the variations of the planetary elements;

but, being based upon values of the masses which were, to a certain extent, gratuitously assumed, it was desirable that the important truths which it indicated should be established independently of any considerations of a hypothetic character. This magnificent generalization was effected by La Place. He proved that, whatever be the relative masses of the planets, the roots of the equations which determine the periods of the secular inequalities will all be real and unequal, provided the bodies of the system are subjected to this one condition, that they all revolve round the sun in the same direction. This condition being satisfied by all the members of the solar system, it follows that the orbits of the planets will never be very eccentrical or much inclined to each other by reason of their mutual attraction. The important truths in relation to the forms and positions of the planetary orbits are embodied in the two following theorems by the author of the Mécanique Céleste: I. If the mass of each planet be multiplied by the product of the square of the eccentricity and square root of the mean distance, the sum of all these products will always retain the same magnitude. II. If the mass of each planet be multiplied by the product of the square of the inclination of the orbit and the square root of the mean distance, the sum of these products will always remain invariable. Now, these quantities being computed for a given epoch, if their sum is found to be small, it follows from the preceding theorems that they will always remain so; consequently the eccentricities and inclinations cannot increase indefinitely, but will always be confined within narrow limits.

In order to calculate the limits of the variations of the elements with precision, it is necessary to know the correct values of the masses of all the planets. Unfortunately, this knowledge has not yet been attained. The masses of several of the planets are found to be considerably different from the values employed by La Grange in his investigations. Besides, he only took into account the action of the six principal planets which are within the orbit of Uranus. Consequently his solution afforded only a first approximation to the limits of the secular variations of the elements.

The person who next undertook the computation of the secular inequalities was Pontécoulant, who, about the year 1834, published the third volume of his Theorie Analytique du Systéme du Monde. In this work he has given the results of his solution of this intricate problem. But the numerical values of the constants which he obtained are totally erroneous on account of his failure to employ a sufficient number of decimals in his computation. Our knowledge of the secular variations of the planetary orbits was, therefore, not increased by his researches. In 1839 Le Verrier had completed his computation of the secular inequalities of the seven principal planets. This mathematician has given a new and accurate determination of the constants on which the amount of the secular inequalities depend; and has also given the coefficients for correcting the values of the constants for differential variations of the