the time there was not wind enough to do more than to swing the spider to the same angle from the vertical that he was then making above the horizon. It seemed the more surprising, as the spiders were large, and ought, by all the laws of gravity, to have fallen to the earth at once. And what was their objective point, aiming, as they did, for the clouds and stars? But I content myself with the statement of the facts, leaving to others the how, why, and whither. Hanover, N.H.

E. T. QUIMBY.

Improvement of western pasture-land.

In his article in SCIENCE, p. 186, Professor Shaler's opening sentence, "that the greater part of the United States west of the meridian of Omaha is unfit for tillage," leaves a somewhat wrong impression. The greater part of Nebraska is west of that meridian; but nearly the whole state, as far as longitude 99°, produces crops of the cereal grains, grasses, corn, fruit, and roots, more surely, even, than the middle states. This area embraces 30,000 square miles. Large sections west of the 99th meridian produce almost equally well, as our statistics show. His suggestions, however, apply to the proper management of the grasses outside of this area, and are of very great importance.

A remarkable peculiarity of our Nebraska flora is its changing character. While not confined to the grasses, it is especially conspicuous among them. When I first crossed this county (Lancaster) in 1865, buffalo-grass (Buchloe dactyloides) covered much of the uplands. By 1871 nearly all of this species had disappeared; and its place was taken by blue-joints (Andropogon furcatus, etc.), interspersed with Boutelouas, Sorghum nutans, Sporobolus, etc. Again, in 1878, the blue-joints disappeared from entire townships, and the Boutelouas usurped their place. Similar phenomena were observed in almost every county in the state, and even in sections where settlements had not penetrated. During the last two years Sorghum nutans has been gaining in eastern Nebraska over all other species. On the whole, the species indigenous to moist regions have been gaining on the buffalo-grasses to such an extent that the latter have almost entirely disappeared east of the 100th meridian, and from large areas farther west. In extreme north-western Nebraska, on tributaries of the Niobrara, I have observed, since 1865, a remarkable exchange of buffalo-grass for Boutelouas and other grasses in different years. This tendency, therefore, is common, though not to the same extent, in the drier as well as the moister portions of the state. When old Fort Calhoun, above Omalia, was occupied by the military, twenty-five years ago, Kentucky blue-grass was brought in baled hay to that post from the south. It spontaneously took root, and spread in every direction, and now it can be found on prairies thirty miles away. Many of our farmers in eastern Nebraska are looking to that species now for a grass to give late fall and early spring pasturage.

Under favorable conditions, the wild native grasses produce a remarkable amount of hay. The bluejoints range in productiveness from one to three tons and more per acre. The latter large yield has been realized even at the 99th meridian on the wide Elkhorn-river bottoms. All the facts noted in the moist as well as dry sections of the state confirm Professor Shaler's theory; namely, that the natural conditions on the plains are most favorable to a changing grass vegetation, and that it is possible, through the agency of man, greatly to improve on the native species. SAMUEL AUGHEY.

Apparent attractions and repulsions of small floating bodies.

As I thought it worth while, in the interests of clear teaching, to object (SCIENCE, i. p. 43) to certain things in Professor John Leconte's explanation of the Apparent attractions and repulsions of small floating bodies,' it seems my duty, now that Professor Leconte has replied (SCIENCE, i. p. 249) to my criticism, to justify that criticism, or, failing in that, to acknowledge my error.

A statement in his explanation of the behavior of two moistened floating bodies, to which I particularly objected, was the following: "But when brought so near that their meniscuses join each other, the radius of curvature of the united, intervening, concave ineniscus . . . is less than that of the exterior concave meniscuses, . . . and its superior tension acts upon both bodies toward a common centre of concavity."

The parts omitted from this sentence are merely references to a diagram. Professor Leconte now states that he should have said superior force instead of superior tension. I, however, objected to the statement on quite other grounds. After quoting it, I said, "We do not think physicists generally will admit that a liquid film tends to draw a solid, to which it is attached, toward the centre of concavity of the film. Indeed, if this were so, the tendency of a column of water raised between two floating bodies by surface-tension would be to lift those bodies. Similarly, a column of liquid sustained in a fine tube would tend to lift the tube."

I have quoted myself thus at length, using italics, which I did not use before, because Professor Leconte appears to understand me as denying that what he calls the 'capillary forces' such, for instance, as the force exerted upon the enclosed air by the film of a soap-bubble-are directed toward the centre of concavity of the film. I spoke merely of the force exerted upon the body to which the edge of the film is attached; and the force exerted by the film upon such a body is certainly not directed toward the centre of concavity of the film. If we coil a rope round a cask, and set a man to pull at each end of the rope, the pressure on the cask will be everywhere directed toward the centre of curvature of the coil: but the pull on the men will not be toward the centre of curvature of the coil; it will be tangential to the coil. In the same way, the action of a meniscus upon the water beneath it, or the air above it, is directed toward the centre of concavity of the meniscus; but the action of the meniscus upon the body to which it is attached is tangential to the liquid surface, and perpendicular to the bounding edge of the meniscus.

Professor Leconte, however, has chosen to make the statement I have quoted above; and to my criticism thereon he replies, "Indeed, it is obvious that the elastic reaction of the common meniscus, formed when two such floating bodies are brought near to one another, does not tend to lift them; for the vertical component of the capillary forces, directed toward the centre of concavity, is exactly counterbalanced by the weight of the adhering liquid elevated between them, while the horizontal component is free to draw them together." He makes a similar statement concerning the action in a capillary tube.

It is, indeed, obvious, that the weight of the water must be sustained; but how and where is this weight applied to the floating bodies or to the tube? If it is applied by means of the surface-film, and at the line where the bounding edge of that film meets the floating bodies, or the wall of the tube, Professor Leconte's 1 Amer. journ. sc., December, 1882.

final statement of the case of two floating bodies apparently comes to this: that the concave meniscus

acts upon both bodies toward a common centre of concavity," and also exerts upon these bodies a vertical downward force equal to the weight of the water sustained. If this is Professor Leconte's conception of the case, I do not feel to blame for not understanding him at first.

If, on the other hand, he supposes the weight of the water to be applied to the floating bodies, not by means of the surface-film, but in some other manner, it was, I submit, incumbent upon him to explain how and where he supposed it applied.

So much in explanation and support of my criticism of Professor Leconte's original statement. It is now, perhaps, worth while to examine a little further his final statement, as quoted above, beginning, "Indeed, it is obvious." Does not this statement, taken in connection with his first statement, also quoted above, lead directly to the conclusion that he supposes a column of water may be sustained between two bodies by capillary action without exerting any resultant downward force upon these bodies? that, in short, the water is pulled up without any resultant tendency to pull the bodies down?

I have written thus at great length, and with perhaps unnecessary statement of elementary principles, because I intend this letter to be final upon my part. EDWIN H. HALL.

Harvard college, Cambridge, Mass.

THE INDIANA GEOLOGICAL REPORT. Indiana: department of geology and natural history. Eleventh annual report (1881). John Collett, state geologist. Indianapolis, State, 1882. 401 p., 55 pl. 8°.

THIS volume contains some interesting scientific and economic matter, partly original, but largely in the form of useful reprints of things not accessible to the people whose needs it is meant to serve.

There is, in the first place, the report of a well-made inquiry into the transverse strength and elasticity of building-stones, principally of the excellent oolite of the St. Louis division of the sub-carboniferous limestones. The point is well made, that the resistance of hammered blocks of stone to compressive strains is very much less than that of sawed masses, owing to the unseen disintegration of the mass produced by the blows of the hammer. There is also the noteworthy suggestion, that the modulus of resistance to compression may be approximately estimated by the ring' of the mass when struck.

There are several county reports which have no general value. They contain some venturous discussions of the extremely difficult problems connected with the work of the last glacial period in the Ohio valley. Glacial rivers, glacial lakes, ice-fronts, and all the other machinery of that time, are handled with charming ease and dexterity. We only hope the observers will work past this first transpar

ent stage of the inquiry, and find how beyond imagination hard is this task of explaining the work of the ice-time, and how useless are such vague conjectures unfortified by the amplest delineation of facts.

In the report of Mr. Collett on Shelby county, we find the very interesting statement, that, in several wells sunk in one part of this county, heated waters have been struck within fifty feet of the surface. Nothing is given concerning the amount of flow of these waters or their chemical composition, nor are we told any thing concerning the goodness of the thermometers with which the observations were made, all very important points. We only have the statement that the water was not potable, and that its temperature was as high as 86° F. As this district is below the level of the carboniferous series, it may not be reasonable to suppose that the temperature is due to the decomposition of iron pyrite, the only considerable known sources of that mineral available in this district being in the coal-measures. It is perhaps more probable that the temperature is due to downward penetration and return of water in a system of faults, which we must suppose to extend to a great depth, though they do not manifest themselves on the surface. If the waters are highly sulphurous, the origin of the heat in the decomposition of pyrite is the most probable; if they are not sulphurous, their source must be sought in faults. The question merits a careful study.

Two hundred pages of the text, and thirtytwo of the plates, are reprints of James Hall's Waldron fossils, with some emendations, including four new plates.

Dr. Charles A. White gives a series of plates and descriptions of fossils from the collection of Mr. J. W. Van Cleve. Hall's monograph is well known to but few. It was originally published in the twenty-eighth report of that mysterious body corporate, the regents of the university of New York. This is the first publication of it that could have been of any use to Indianian students.

The species described by Dr. White are chiefly corals, and are not regarded by the author as new species. This part of the work is essentially of local interest. All the species have been better set forth before, but never in a form so accessible for the dweller in the rural parts of Indiana.

Although there is not much that is original in this book, it very likely has a higher measure of utility for the people who pay for it than many a survey report that has better served the purposes of pure science. The old

day when the advance of American geology seemed to depend on state surveys is passing, and will soon pass away. They did good skirmish-work, and deserve to be remembered for many gifts to science; but the problems in scientific geology are now too large to be solved within the limits of a state. Scarce a state in this country has a question that can be properly considered from work done within its limits alone. In the future the state surveys can find their best place, not in efforts to develop general scientific problems, but rather in economic questions, which can always be localized, and in the work of bringing to the notice of the people whom they serve such matters of pure science as may naturally concern them. Other forms of research would better be left to the general government surveys, or to the studies of independent geologists.

It is now pretty well ascertained that our states are unwilling to support permanent scientific establishments on such a scale as will enable them to do good scientific work, but they will pay some one or two men to keep a sharp lookout for any utilities that may be discovered. Fortunately nature so mingles the utile and the dulce,' that some good to science will come out of this care for profit, which is to be in the future the task of the state surveyor.

M. HERMITE'S LECTURES. Cours de M. Hermite, professé pendant le 2° semestre 1881-82. Redigé par M. ANOYER, élève de l'Ecole normale supérieure. Second tirage revu par M. HERMITE (Librarie scientifique). Paris, A. Hermann, 1883.

THIS work of M. Hermite fills, in great part, a decided gap in mathematical literature, and affords a means to American mathematical students, at least, of overcoming a difficulty that of late has become rather serious. With the exception of those who have had the opportunity of listening to the lectures of Hermite or Weierstrass on the theory of functions of a complex variable, all students interested in that subject must have experienced a great deal of difficulty in reading the more modern memoirs which deal with it. Some such book as Durége's, or Neumann's, on Riemann's theory, is very much wanted on what may, with propriety, be called the Weierstrass-Hermite theory of functions. The necessity for such a treatise is steadily increasing, as any one will readily see by looking over the last few volumes of Crelle-Borchardt, the Mathematische annalen, the Annali di matematica, or the

two numbers which have already appeared of Mittag-Leffler's acta mathematica. The pres

ent work by M. Hermite does not profess to be such a treatise. In fact, it is not a treatise at all, but, as its title implies, simply the course of lectures given at the Sorbonne by M. Hermite, and treating of quite an extended list of subjects. The principal topics discussed are the quadrature and rectification of curves, the determination of the areas and volumes of curved surface, the general theory of functions of a complex variable, and the application of this theory to the study of the Eulerian integrals and the elliptic functions.

The first five chapters are devoted to geometry, and contain applications which are chosen with a view to what is contained in the succeeding chapters. Since, for the rectification of conics and the quadrature of plane cubics, it is necessary to consider integrals of the form ff (xy) dx, where f (xy) is a rational function of x and y, and y is the square root of a quartic function of x, the author takes up this general integral, and gives Legendre's reduction to the normal forms of the elliptic integrals, and also some of Tcheby chef's results concerning the cases where the elliptic integrals are reducible to algebraico-logarithmic functions.

The next three chapters are taken up with an exposition of the more elementary properties of functions of complex variable, the author giving an account of Darboux's investigations relatively to the integral F(x) f (x) dx, where F(x) is, between the limits, always positive, f(x) is a continuous function of the form

(x) + i 4 (x), and where a and b are real. Another method, due to Weierstrass, for integrals of this nature, is also indicated.

In the next four chapters the immediate consequences of Cauchy's theorem are developed, and an account given of Weierstrass's and Mittag-Leffler's investigations in the theory of uniform functions, including their decomposition of a holomorphic function into prime factors, and their general expression for a uniform function with an infinite number of poles, or of essential singular points, the last being due almost solely to Mittag-Leffler.

The next three chapters deal with the Eulerian integrals, and include Prym's expression for T (x), and Weierstrass's expression for

The next two chapters refer to functions which are discontinuous along a line, - Appell's and Tannery's series, and Poincarré's example of a function having an espace lacunaire. As preliminary to Cauchy's theorem concerning the number of roots of a polynomial contained in the interior of a contour, the expression is given by a line-integral of roots of an equation contained within a given contour. Then follows Cauchy's theorem, the establishment of Lagrange's series, Eisenstein's theorem upon series whose co-efficients are commensurable, and which satisfy an algebraical equation, and the enunciation of Tchebychef's theorem upon series with rational co-efficients, which may represent functions composed of algebraic, logarithmic, and exponential functions.

The next chapter treats of multiform functions arising from the integration of uniform and of multiform functions, and of the means of reducing them to uniform functions by systems of cuts (conpures).

The remaining five chapters treat entirely of the doubly-periodic functions. After first showing the multiple values of the elliptic integrals of the first kind which correspond to the different paths traced out by the variable, and establishing the double periodicity of the inverse functions to this integral, he defines a function, (x), which conducts to the analytical expressions for the doubly-periodic functions. The function (x) is defined by the equations,

kinb

+(x+b) = 4(x) exp. [-(2.x + b)],

where k is an integer. Then follows the investigation of the elliptic functions, including, of course, Jacobi's O, H, and Z functions, the definition of Weierstrass's functions, Appell's expression for doubly-periodic uniform functions in the case where they possess essential singular points, and, finally, a demonstration by M. Goursat of Fuch's theorem concerning the definite integrals K and K', considered as functions of the modulus.

It is perhaps to be somewhat regretted that the book is lithographed instead of printed in the usual manner; but this is of comparatively little consequence, as the writing is very clear and legible. Thanks are certainly due to M. Andoyer, the editor, for the trouble which he must have taken in elaborating what would seem to have been merely a set of notes on M. Hermite's lectures. The whole matter has been revised by M. Hermite, and the aggregate result of his and M. Andoyer's labors is a book which is a decided acquisition to mathematical literature. It is to be hoped that M. Hermite will see fit to go more fully into the subject of the functions of a complex variable, and that of elliptic functions, at a future time, and give to the world a treatise which will be more satisfactory than even the present very valuable work. T. CRAIG.

ASTRONOMY.

New measures of Saturn's rings.-O. Struve gives the results of a series of measurements of the rings of Saturn at Pulkowa during August and September, 1882, compared with a similar series, also taken by himself, with the same instrument, and at the same time of the year in 1851. In a memoir on the subject in 1851, he seeks to prove, that, while the outer diameter of the rings remains constant, the inner is continually shortening, basing his conclusions on the observations and drawings from Huygens's time. If the conclusion were correct, and the contraction constant, the measures of 1882 should have given a perceptibly shorter inner diameter than those of 1851. The inner diameter of the dark ring seems to be slightly shorter than in 1851, but the difference is not nearly so large as the theory calls for. The dark ring seems, however, to have changed since 1851. Then it seemed divided by a dark streak, the inner part being entirely separate from the bright ring. In 1882, all trace of this division had disappeared, and the dark ring seemed to be merely a faint continuation of the bright ring. (Astr. nachr., No. 2498.)

M. MON.

[688

Formation of the tails of comets. - Mr. Rumford suggests that the repulsive force which is unmistakably manifested in the formation of comets' tails may be due, not to any electric action, or any imagined impulse of solar radiations, but merely to evaporation. A small particle from which evaporation is taking place on the side next the sun will be driven backward with a velocity continually accelerated; and, when more than half of the mass of the particle has been evaporated, the velocity of the residue may be much greater than the average velocity with which the gaseous molecules are driven off from the heated body. In the case of hydrogen at a temperature of 70° or 86° F., the velocity thus acquired might be greater than a hundred thousand miles a day. If we suppose the evaporating material to be gases which have been liquefied by the cold of space (carbon dioxide and volatile hydrocarbons), it becomes easy to account for a powerful repulsive action at distances from the sun even much greater than that of the earth. The writer suggests that the comet's light may be in part due to the bombardment' of precipitated particles by the evaporated molecules in the condition called by Crookes the fourth state of matter'; so that, "without electrical discharges,

the whole phenomena of the continuous and bright line spectrum in the neighborhood of the nucleus may be accounted for." He also discusses briefly some of the polarization phenomena of comets, and the envelopes which appear near the nucleus. The article is a very interesting and suggestive one; but in view of the fact that comets' tails sometimes grow, not a hundred thousand, but more than a million miles a day, it is doubtful whether the proposed hypothesis can be regarded as sufficient. (Astr. reg., March.) c. A. Y. [689

GEODESY.

Altitude of Lake Constance. - Part of the work laid out by the European geodetic commission consists in carrying an accurate series of levels across the country, and a share of this has recently been completed by the royal Prussian geodetic institute. It is published as the Gradmessungs-nivellement zwischen Swinemünde und Konstanz, by W. Seibt (Berlin, 1882), and records the altitudes of a large number of points from the Baltic, where the datum plane is the mean water-level from fifty-four years' observations, to Lake Constance, where connection is made with the Swiss triangulation. The railway station in Constance is 399.990 met. above the Baltic. - (Verh. ges. f. erdk., Berlin, 1882, 514, 538.) W. M. D.

MATHEMATICS.

1690

Symmetric functions.- Previous mention has been made of Mr. Durfee's tables for the twelfthic. By a curious coincidence, M. Rehorovsky of Prague has, almost simultaneously with Mr. Durfee, computed the same tables. M. Rchorovsky's tables differ from those of Mr. Durfee only in arrangement. The tables as arranged by the former are identical in form with those given by Prof. Cayley for the first ten orders in the Phil. trans., vol. 147; while those of Mr. Durfee are arranged symmetrically, and cannot be included in a half-square, as M. Rehorovsky's are. -(Sitzungsb. akad. wissensch. Wien, 1882.) T. c.

n

n

[691 Maximum value of a determinant. The elements of a determinant being restricted to lie between (a) and (a), Mr. Davis finds, that, for all determinants whose order is greater than 2, a numerical maximum is found by making all the elements of the principal diagonal a, and all the remaining elements of the determinant = +a. In the maximum cubic determinant Da", all of the strata are made identical, and equal to D. The value of this determinant is ±n! D2) a". Formulae are also (2) given for hyperspace determinants. — (Johns Hopk. univ. circ., No. 20.) T. C. [692 Functions of several variables.-M. Combescure seeks to develop completely the immediate conditions to be satisfied by an analytic function of several imaginary variables. Assuming Z1, Z2... Zn as the variables, these are defined by the equations Zj = x; +i yj, where j 1, 2... n. Then the function to be considered is F(Z1, Z2. In) φ + ίψ. The differential co-efficients of and of the first order are connected by relations precisely similar to those connecting these quantities when there is only one variable, z: so, when one of the functions or is given, the other may be found by simple quadratures. It is shown that the group of conditions for the determination of reduces itself to n (n + 1) the partial differential equations of the 2 second order, Ah,k = 0, where

Ah, k

for h, k = 1, 2

n, and, of course, including the cases where h k. These are the necessary and sufficient conditions to be satisfied by . A means is given of representing analytically by an exponential series, the co-efficients of which depend upon the sines and cosines of (a, x1 + . . . + ann) and (B1y + ... + ẞn?n); a ẞ, as well as the constant co-efficients of these sines and cosines, being indeterminate real quantities, to which we can give any values we please. — (Comptes rendus, Jan. 22.) T. C.

1

[693

Homologies and conics. If L and M are two fixed points on a conic, K, and P a variable point, then PH, perpendicular to L M, cuts again the circle L M P in a point, H, which describes a conic, K'. If the circle on L M as diameter cuts K again in E F, then L M and E F are the axes, and the point at infinity in the direction PH is the common centre of two of the twelve homologies which two conics in general determine. The ratio of corresponding areas of K and K' is constant, a function of the eccentricity of K and of the inclination of L M to the focal axis of K. Given, on the other hand, the centre and axes of the homology, two triply infinite systems of conics, K and K', can be determined; the conics of each system being similar and similarly placed, and the common points at infinity of one system being orthogonal to those of the other. All the conics of the plane are thus distributed into a doubly infinite number of triply infinite systems. The net of conics determined by three arbitrary points in a plane will give a doubly infinite number of conics, one out of each system, and hence will produce all the homologies of the plane, and each once only. There is therefore a (2,1) correspondence between the doubly pointed plane and the plane of the homologies. The discussion of these points by Luigi Certo is followed by an investigation of the variation of the ratio of corresponding areas, first, with the variation of the eccentricity, and, second, with the variation of the direction of the line LM. He also considers the distribution in the plane of the pairs of similar conics of which the system of conics through four points on a circle is composed. (Giorn. mat., xx.)

C. L. F.

PHYSICS. Optics.

[694

Color of water. W. Spring reviews the several explanations suggested to account for blue and greenish colors of water in lakes and seas, Bunsen's idea of inherent color, Tyndall's theory of reflection, and others, and concludes that some further study of the question is needed. Blue from reflection would imply red by transmission, but this is not observed from diving-bells. The author concludes provisionally that the color depends on the presence of certain salts, especially calcic carbonate in solution. The more complete the solution, the bluer the water. (Rev. scient., 1883, 161.) w. M. D.

(Photometry.)

[695