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In the part just issued of the American Journal of Mathematics (vii, pp. 3, 4) Proffessor Cayley has occasion to deal with the functionwhere stands for i (i – 1)…(i – j + 1). He Points out thatexpresses {i, 1}, {i, 2},…{i, 5} each in descending powers of i, and tabulates the function from {1, 1} to {5, 5}.

In my paper on Continuants, recently communicated to the Royal Society, it was shown that the order of a continuant may be depressed if the first element of the main diagonal be unity, viz., thus:—and from the definition it is evident thatHence we have

The assumption that there is a limit in Professor Tait's problem regarding the interpretation of Limn=∞(cosnx) means that if we start with an angle x, and find the number which is its cosine, then the number which is the cosine of this number, and so on, we shall at last come to a limiting result ω, such that cos ω = ω.

Major cation and silicon budgets from a 2.3 hectare, watershed are interpreted to determine environmental factors controlling chemical denudation rates on siliceous lithologies in the alpine-subalpine ecotone, Coast Mountains, British Columbia. Ionic source areas were sought by spatial and temporal solute sampling. The data were stratified by soil-vegetation complexes and components of the hydrologic cascade. The geochemical system has two major ionic sources. Soil solutions are sodium facies and are stoichiometrically balanced with volcanic glass probably because the glass fraction of volcanic ash is the primary weathering source in soils and is dissolving congruently. Streamflow is predominantly calcium facies because bedrock is composed of calcium-magnesium silicates and groundwater flow through bedrock discharges to phreatic soil water zones. The hydrologic system has two major components. Direct stormflow is generated by saturation overland flow from large contributing areas due to extensive soil water-repellency. Indirect stormflow is generated by saturated subsurface throughflow due, in part, to strong textural contrasts among soil parent materials. The actual basin drainage density is greater than is apparent superficially because preferred subsurface pathways are common. Annual runoff was very similar in the study years 1979 and 1980 (904 mm and 1027 mm respectively) but net major cation and silicon yields varied from 1.6 metric tons km⁻⁴ to 3.4 metric tons km⁻². The low 1979 yield was caused by segregated ground ice which limited soil-water contact, soil solution shunting, and groundwater flow. Differences in denudation rates between subcatchments were equally large because of topographically induced groundwater contributions to streamflow. A similar conclusion is reached for larger adjacent watersheds. Most mass is denuded along groundwater flowpaths in bedrock rather than from soils. Denudation from alpine surficial materials during 1980 was approximately 2 tons km⁻² yr⁻¹ which is an order of magnitude less than expected. Results imply that where groundwater contributions are not defined, watershed experiments can produce highly biased estimates of surficial denudation which confound process interpretations. Hydrologic pathways are more important than the inherent chemical reactivity of soils and bedrock in regulating chemical denudation rates in this landscape. Runoff mechanisms partition water between the surface, soil and groundwater reservoirs and control flowpaths, contact-times and shunting within each reservoir. Biopedochemical controls on denudation appear to be relatively less important than hydrologic controls because soil solutions are similar among alpine soil-vegetation associations. Soil leaching losses are small because soil water-repellency, preferred subsurface pathways and the discharge regime limit soil solution shunting.

1. The existence of relations between the coaxial minors of a determinant was first discovered by MacMahon in 1893. The whole literature of the subject is comprised in three papers, viz.:—MacMahon, Phil. Trans., clxxxv. pp. 111–160.Muir, Phil. Mag., 5th series, xli. pp. 537–541.Nanson, Phil. Mag., 5th series, xliv. pp. 362–367.My present object is to continue the investigation of the relations in question, and more particularly to draw attention to an explicit expression for a determinant of the 4th order in terms of its own coaxial minors. At the outset some fresh considerations regarding determinants in general will be found useful.

So far as mathematical writers have as yet noted, a set of equations of the typehad not made its appearance in mathematical work prior to the year 1846: and it is almost absolutely certain that before that year the determinant of such a set had never been considered. It is not at all unlikely, however, that the expressionwhich is the case of the determinant for n equal to 3 had more than once turned up in other connections, and that its divisibility by a + b + c had been noted: but of this, too, there is no record.

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The conception of conjugate permutations appears to be due to Rothe. The definition may be expressed thus:—Two permutations of the numbers 1, 2, 3,…, n are called conjugate when each number and the number of the place which it occupies in the one permutation are interchanged in the case of the other.

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If a row of n elements be taken, and closely following this array, but separated by a bar from it, we write n rows of n elements each; and closely following either outside column of this square array, but separated by a bar from it, we write n columns of n elements each; and closely following an outside row of this second square array, but separated by a bar from it, we write n rows of n elements each; and so on, passing from the rows or columns of one array to the columns or rows of the next, and ending not with a square array, but, as we began, with a single line of elements, we have the matrix representation of a bipartite function.

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In Crelle's Journal, vol. x., Stern devotes fifteen pages (pp. 245-259) of his Theorie der Kettenbrüche to the examination and elucidation of the following problems:—as continued fractions of the formwhere a1, a2, a3, … are independent of x.

The present note is a continuation of a short paper which appeared in the Proceedings of the Royal Society of Edinburgh for session 1876–77. The problem in question is that referred to in Professor Tait's “Memoir on Knots,” viz.:—To find the number of possible arrangements of a set of n things, subject to the conditions that the first be not in the last or first place, the second not in the first or second place, the third not in the second or third place, and so on.

(26) Of the various determinant forms thus far obtained the most promising is that of §8 or that of §14; and to these it is desirable now to return in order to obtain an expression for the eliminant in the ordinary non-determinant notation. In doing so it will also be well to make a slight change in the coefficients of the three quadrics, viz., to write f, g, h for 2f, 2g, 2h, as in this way the diversity in the cofactors of the determinants occurring in the last three rows of either form of the eliminant disappears.

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There is a short paper in a recent volume of the Comptes Rendus of the French Academy of Sciences which deserves notice if only in order that the attention of the author and others may be drawn to previous work on the same subject and to more effective methods of treatment.

It is well known that the square root of any commensurable number N is expressible in the formand that the (z + 1)th convergent to this is,where the continuant notation K( ) is explained by the example

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(1) The process of dialytic elimination was first applied to a set of general ternary quadrics by Sylvester in 1841, his method being to form from the three quadrics u1u2u3 ten cubics, viz.:—and then eliminate dialytically the ten quantitiesThe actual work of obtaining the eliminant as a determinant of the 10th order he did not perform; and, indeed, with the notation which he used the labour would have been very irksome.

(1902). XLV. Aggregates of minors of an axisymmetric determinant. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 3, No. 16, pp. 410-416.

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(1) It is well known that if equations of the type referred to in the title be dealt with like ordinary quadrics, the eliminant obtained is marred by association with an irrelevant factor. Thus, to take the simplest case, viz.the left-hand member of which contains the irrelevant factor |α2β2|, being readily shown to be equal toThe object of the present short paper is to draw attention to other modes of procedure, and to formulate the results for n variables.

An essential part of Pfaff's method of sloving an ordinary differential equation in 3m variables consists in obtaining what he calls his auxiliary equations.

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My last paper on the “Theory of Circulants” appeared four years ago (Proc. Roy. Soc. Edin., xxxii, pp. 136–149), and brought up the history of the subject to 1880. I have now completed the research up to the close of the nineteenth century—the goal originally aimed at. It is considered that from the beginning of the present century the indexing of mathematical writings is sufficiently full and detailed to make it comparatively easy for a worker to follow up for himself the recent lines of development.

NATURE of October 4 opens with a review of a book on “Determinants” by two pupils of Prof. Valentin Balbin, whose energy and enthusiasm have done so much for mathematics in the University of Buenos Ayres. In regard to the naming of the various special forms of determinants, your reviewer says:—“The nomenclature adopted in the second book differs in some particulars from that employed by Muir. Thus, our authors do not follow him in substituting ‘adjugate’ for the more euphonious and more familiar adjective ‘reciprocal,’ and they agree with Scott and others in calling those determinants ‘orthosymmetrical’ which Muir names ‘persymmetric.’ We think that their name determinate hemisimetrica is a distinct improvement on the old ‘zero-axial skew determinant,’ but …”

PROFESSOR EVERETT, towards the close of his much-needed and exhaustive letter in NATURE (March 2) on the Measurement of Mass and Force, proposes to supply an undeniable want in Dynamics by coining the word kinit to denote that force which, acting on an avoirdupois pound of matter for a second, generates a velocity of a foot per second. He then adds, “If we substitute gramme for pound and metre for foot, we obtain a different unit which must be called by a different name.”

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The problem in question is—To find the number of possible arrangements of a set of n things, subject to the conditions that the first is not to be in the last or first place, the second not in the first or second place, the third not in the second or third place, and so on.

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IN all the five editions of Baltzer's “Theorie und Anwendung der Determinanten” there stands at the foot of the first page an historical note, in which reference is made to a work entitled, “Demonstratio eliminationis Cramerianae,” by Mollweide (Leipzig, 1811). About a year ago it became necessary to examine this demonstration for the purpose of having it reported upon in an historical work. The University Libraries in Scotland were applied to in succession, but no copy could be heard of. Inquiries made at the more important libraries in Cambridge by friends resident there, or by letter, ended in the same unsatisfactory way. Letters, followed by an actual visit, to several libraries in London, brought no better result; and after every possible biographical scrap about Mollweide had been ferreted out in the British Museum, the suspicion began to form itself that some curious error had crept into Baltzer's footnote. In order to get to the bottom of the matter, the excellent mathematical library of Gottingen University was next applied to, and the library of Giessen University, where Baltzer was Professor; but in both cases in vain. A last effort was then made about a month ago in a letter to the University Library of Leipzig, where the reputed author Mollweide had taught, and where the “Demonstratio” (or Demon, as it had for more than one reason come to be called) had been published. Even here, at first, there was failure. But Prof. Virchl, who most kindly interested himself in the matter, was soon successful in his quest. What he found, however, was not a “Demonstratio” by Mollweide; the title was simply as follows: “Ad memoriam Kregelio-Sternbachianam in auditorio philosophorum die xviii. Julii, MDCCCXI. h. ix. celebrandam invitant ordinum Academiae Lips. Decani seniores caeterique adsessores—‘Demonstratio eliminationis Cramerianae.’ ” Either, therefore, no author should have been mentioned by Baltzer, or an indication should have been given that Mollweide's name was an interpolation in the title. One or other of these courses would likewise have been less hurtful to Baltzer's reputation for accuracy; for, after all, Mollweide was not the author. In the Leipzig Library Catalogue the work is entered under the name of De Prasse, and Prof. Virchl had no doubt whatever, for perfectly conclusive reasons which he gave, that De Prasse was the author. The work extends to only 15 pages quarto, and is considered by the same authority to be very rare.

I WISH to direct the attention of the reviewer of the “New Works on Mechanics,” in No. 107 of NATURE, to the following statements which he makes while speaking of Wernicke's book:— “The symbol j is here and throughout the work used to denote an acceleration; for example j x (sic) is the acceleration parallel to the axis of x. This notation (unfamiliar to English readers) has obvious advantages when the more appropriate language of the differential calculus cannot be employed.”

When the elements of the first row of a determinant are all positive integral powers of one quantity, the elements of the second the like positive integral powers of another quantity, and so on, the determinant is called an ALTERNANT; for example, .Every alternant of the nth degree is evidently a function of n variables, viz., the n quantities whose powers are the elements. To interchange two of these variables would be the same as to interchange two of the rows of the determinant, and therefore would have the effect of merely changing the sign of the function. A function having this property, and therefore closely resembling a symmetric function, Cauchy called a symmetric function also distinguishing the two kinds as alternating and permanent.

The vertices of the original triangle beingthe conic beingand A1, B1, C1,. … being defined by the nine equations