It is a well-known problem to find for n letters the number of the arrangements in which no letter occupies its original place; and the solution of it is given by the following general theorem :— viz., the number of the arrangements which satisfy any r conditions is

where 1 denotes the whole number of arrangements; (1) the number of them which fail in regard to the first condition; (2) the number which fail in regard to the second condition; (12) the number which fail in regard to the first condition, and also in regard to the second condition; and so on: Σ(1) means (1) + (2) … + (r) : Σ(12) means (12) + (13) + (2r) … + (r − 1, r); and so on, up to (12…r), which denotes the number failing in regard to each of the r conditions.

The convenience of having only one system of numeration is so well recognised, that there is no need for any discussion. Already the numerical nomenclatures of all nations having any culture have been converted to one, namely, the decimal system, and traces of the ancient use of dozens, scores, fifteens, or sixties, can be found in only a very few of them. Although we count our hours in sixty minutes, we do not date the present as the year thirty-one sixties and seventeen. Yet, in the matters of measure,weight, and value, the old and irksome divisions continue to be used; nay, even in those departments of science which most need laborious calculation, we continue to employ the ancient scale of division.

The real utility of tables of numerical results is only secured by making them accessible to those computers who may require them; and the essence of their utility lies in this, that the labour of a single computer saves that of many others.

It is indispensable that those who use the tables be able to rely implicitly on the accuracy of the tabulated numbers, and that they have a ready means of detecting any error should the existence of one be suspected.

Cymaderma (nov. gen,). —Generis Characteres:—Corpus cylindricum, elongatum, nudum, striatum; striæ tenues, in ordinem undulatæ, et ubique corpus cingentes, ita ut cutis subrugosa videatur; linea dorsualis continua, alternasque cavaturas ovatus et vertices ostendens; nulla verorum articulamentorum indicia; vestigium tortuosum, bisulcum.

Cymaderma (new genus).—Generic Characters:—Body cylindrical, elongated, naked, striated; striæ minute, waved symmetrically, encompassing the form in all parts, and giving to the skin a subrugose appearance; a continuous dorsal line, showing alternate oval depressions and slight ridges; no traces of true articulations; track, tortuous bisulcate.

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.

Taking the harmonical progression

and denoting the sum of n terms of it by Sn we have by means of Euler's transformation,

From the point of view of the Hypothesis of Vortex Atoms, it becomes a question of great importance to find how many distinct forms there are of knots with a given amount of knottiness. The enormous numbers of lines in the spectra of certain elementary substances show that the form of the corresponding Vortex Atoms cannot be regarded as very simple.

From a boring which has recently been made about a mile west from the town of Linlithgow, water has been obtained differing so much in character from that of the usual wells and springs of the district that some notice of it deserves to be placed on record. When the fact was communicated to me I was asked to explain by what means sea-water could obtain access to underground rocks in an inland district. On visiting the ground I found the site of the bore to be among some hollows of the gravel and sand which cover the country between Falkirk and Linlithgow, its height being about 165 feet above the sea, from which it was distant about three miles. The ordinary wells of the district are situated in the superficial deposits, and supply good potable water, though the supply is necessarily limited.

FAM. EUPHROSYNIDÆ.—In Euphrosyne foliosa, Aud. and Ed., the separate nerve-cords are comparatively large, and lie quite within the body-wall, the oblique muscles, which generally bound the longitudinal ventral muscles, decussating beneath them.

AMPHINOMIDÆ.—The cords are somewhat small and flattened in Chloeia, and occupy an area bounded internally by a transverse band of fibres, and externally by the circular muscular layer and the hypodermic basement-tissue. The oblique muscles are attached at the outer border of each trunk.

The general scope of the paper was to show how by graphic methods we might find the relation between effort exerted at one part of a machine and the resistance overcome at another part. It was shown that for a given machine at any instant a linear frame might be substituted, such that the stresses in the links corresponded to the pressures at the joints between the elements of the machine, including the pressures due to driving effort and resistance. Numerous examples were given of the application of this frame, called the dynamic frame, to the solution of the above problem, the friction, inertia, and weight of all the parts being taken rigidly into account.

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.

To the foregoing Professor Cayley has kindly made the following additions:—

The investigation may be carried further: writing for shortness u3, u4 &c., in place of ψ(3), ψ(4), &c., the equations are

and hence assuming

we have

If a cord be knotted, any number of times, according to the pattern below

it is obviously perverted by simple inversion. Hence, when the free ends are joined it is an amphicheiral knot. Its simplest form is that of 4-fold knottiness. All its forms have knottiness expressible as 4n.

In this chapter Professor Heddle submitted the results of the analyses of Orthoclase from fifteen localities; of Albite from four; of Oligoclase from eight; of Labradorite from eleven; of Andesiel from five; of Anorthite from three; and of Latrobite from two.

The three last minerals being now, for the first time, recognised as British species.

This paper contained an extension to discs of any shape whatever of the principles explained in the ‘New General Theory of the Teeth of Wheels,” as applicable to circular discs.

It is with extreme reluctance that I venture to disturb the present repose of the much-contested field of ocean circulation. My object is not, however, to provoke discussion on the general theory of ocean circulation, as I am sure all will agree in thinking that the subject has already been discussed far beyond the point at which it is likely to be benefited by discussion. My object is simply to call attention to certain influences at work in the ocean, the effects of which seem to have been totally overlooked.