to ascertain, if possible, the exact degree of separation of the several rays.'

With this view, the author began a series of observations, the result of which forms the subject of the present paper; and which, we think, tend to confirm his hypothesis: but he remarks, in conclusion, that it is in vain for him to pursue the subject farther, in a situation so ill adapted to astronomical observation as Somerset House; and he therefore resigns the task to those who are more favourably placed in that respect, and who possess instruments better calculated for an investigation which requires so much accuracy. We hope that this appeal will not be made in vain.

Determination of the North Polar Distances and proper Motion of thirty fixed Stars. By John Pond, Esq. Astronomer Royal, F.R.S.-This communication consists of three tables; the first bearing for title Standard Catalogue of the North Polar Distances of thirty principal fixed Stars reduced to the beginning of 1813. The second contains Observations made with the Mural Circle, compared with the Observations. of Dr. Bradley in the Year 1756,' from which the author draws the following results relative to the proper motion of

Polaris:

The N. P. D. of Polaris determined by upwards of 200 observations of Dr. Bradley, by computations made under the direc tion of Dr. Maskelyne, a short time before his death, and reduced to the beginning of the year

By my observations for

Variation in

2° 2' 17".25 I 41 21 .75

1749 1813

3.67 -0.0 57

Precession for

Difference

Annual proper

motion

i.e. The annual precession, which is itself negative, must be increased by the above quantity.'

The third and last table contains the North Polar Distances of 44 principal Stars for January 1. 1813.'

An Essay towards the Calculus of Functions. By C. Babbage, Esq.-The term function, as Mr. Babbage observes, has long been introduced into analysis, and, in its most extended signification, denotes the result of every operation that can be performed on quantity. In this sense, it has been employed to great advantage by modern analysts: but no one has yet been enterprizing enough to attempt what the present author designates by the term calculus of functions; the intention of which is clearly explained in the following paragraphs: REV. MAY, 1816.

G

• Many

Many of the calculations with which we are familiar consist of two parts, a direct, and an inverse; thus, when we consider an exponent of a quantity: to raise any number to a given power, is the direct operation: to extract a given root of any number, is the inverse method. The differential calculus, which is a direct method, naturally gave rise to the integral, which is its inverse: the same remark is applicable to finite differences. In all these cases the inverse method is by far the most difficult, and it might perhaps be added, the most useful.

It is this inverse method with respect to functions, which I at present propose to consider.

If an unknown quantity, as x, be given by means of an equation, it becomes a question how to determine its value; similarly if an unknown function, as, be given by means of any functional equation, it is required to assign its form. In the first case, it is quantity which is to be determined; in the second, it is the form assumed by quantity, that becomes the subject of investigation. In the one case, the various powers of the unknown quantity enter into the equation; in the other, the different orders of the function are concerned.'

Our readers will comprehend at once the nature of the problem which this gentleman proposes to himself; and, if they be acquainted with the higher analytical pursuits, they will also be enabled to appreciate its extent and difficulty, as well as the many useful purposes which it might be made to answer, if such a calculus could be fairly established: which, however, we should not consider it to be, if it merely depended on an artificial notation; although it might enable the operator to exhibit a solution to the eye, of which no one could form any mental conception, or submit to any known mode of computation. In making this remark, we by no means intend to insinuate that any of this false glare is observable in the present paper; it is, on the contrary, perhaps, as far as the author has carried it, in the simplest form that the subject will admit: but a regret seems to be expressed that the notation is not more general; and this, we apprehend, may be followed by come attempt calculated to produce that kind of artificial and unmeaning solution to which we have referred, and against which we should wish him to guard.

It is difficult, within the limits of a few pages, to furnish the reader with an intelligible idea of the nature of the notation and the mode of operation which Mr. Babbage has adopted in this first essay: yet we are unwilling not to avail ourselves of an opportunity of giving some publicity to the investigations; and we must therefore confine ourselves to the simplest cases of both.

We may observe that a, ß, y, &c. are known functional characteristics; V, x, &c. are unknown.

t

If in any function as 4, instead of a, the original function be substituted, it becomes ↓↓x or: this is called the second function of t. If the process be repeated, the result is or r, the third function of x; and similarly 4"x denotes the n' function of z. Suppose

↓2x = a + a + x = 2α + x

and generally ↓" x = na + x

A functional equation is said to be of the first order, when it contains only the first function of the unknown quantity; as, for instance,

If the second function enter, the equation rises to the sesond order: thus,

↓ (x+4x) + (4x—x)2 = 0,

are equations of the second order.

Having thus established his notation, the author proceeds to solve certain problems connected with his inquiry; of which we select the sixth as the easiest to comprehend without the entire essay; viz. to find or from the equations

4x=4ax=4ßx=&vx = &c.:

that is, to find a function of x, such that, if instead of r we successively substitute a x, ßx, yx, &c. the results shall be all equal to the original functions. We must dispense with the general solution: but some of the author's partial cases will be readily understood. Thus he proposes to determine the function which shall not change, when for x we substitue æ, —x, or √(x2—1); and he finds the general solution to be {

= }

In the same manner, it may be found that the function {a} will remain the same, whether the variable

1—ax2+x'

1

be x,-x, or I

Some idea, though an imperfect one, may be formed from what we have now stated, of the nature of the author's investigations; which, in all their generality, involve many important considerations, and are indicative of an enterprizing mind. Should Mr. Babbage ultimately succeed, his name will undoubtedly stand in the first rank of modern analysts, and we sincerely wish him every possible success.

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ART.

ART. VII.

(84

Clan-Albin; a National Tale. rl. 8s. Boards. Longman and Co.

4 Vols.

1815.

12mo.

THIS novel has obtained a degree of reputation that is by no means inconsiderable, and it is not unworthy of some distinction. A delineation of Scotish manners, whether Highland or Lowland, has indeed lost some of its claims to novelty: but still, as far as fictitious tale-writing is concerned, we can scarcely even now name a path of literature which is so little beaten, in which so little comparatively has been described, and so much in consequence is new to be exhibited. The present tale has the common fault of most modern novels; viz. needless expansion; by which much of the interest evaporates, and the attention to characters, in the discriminations of which this writer displays considerable acuteness, becomes distracted, and nearly lost. We are always, however, happy to meet with a well-imagined story, and a well-written tale: it forms the innocent and pleasing amusement of many fire-sides; and, if it really deserves those appellations, its moral advantages, with the young at least, are not limited to the mere innocent employment of reading, which may detain the mind from lower or worse occupation: but it may prove the foster-mother to many good and honourable feelings, in the same manner as the sickly and meretricious sentiments of much modern trash may produce results altogether the contrary. The language, in which the story before us is conveyed, is for the most part lively and unaffected; and, without laying much claim to any of the higher praises of composition, which indeed the nature of the subject renders scarcely possible, it may be said to have the requisites of natural and easy narration.

Having given this brief opinion of these volumes, we will present our readers with a short abridgement of the tale which they convey.

year

The commencement of the action is placed about the 1780; and the opening scene is the hamlet of Glen-Albin, a secluded vale in the western Highlands of Scotland. An inhabitant of this remote spot, by name Ronald, who united the profession of a blacksmith with many pastoral occupations, was returning from a neighbouring cattle-fair, on a stormy November night, when he overtook on his road a lonely young female, apparently in much distress, who inquired the distance to the nearest hamlet. Ronald would willingly have relieved her on the way with a share of his own saddle, but was repulsed with a peremptory refusal. The Highlander's softer feelings overcame his anger; and, when he had gone on to some dis

tance, he returned with a sledge to assist the stranger, who had appeared very faint and weak. He succeeded in his purpose; she was accommodated with all that his cottage could afford; and, almost immediately on her arrival, she was delivered of a male child, an event which was quickly followed by her own decease, before any account could be procured of her name or situation. This child, as may be supposed, is the hero of the tale. The highland village, in which he was brought up, formed as it were only one extensive family. Among the more prominent characters introduced is the Lady Augusta, residing in a small island called Eleenalin in the same glen, the last of a noble stock who had been the lairds of the glen, but, in combating for the house of Stuart, had lost their inheritance, and perished either in war or in foreign countries. The castle of Dunalbin and the village had passed into other hands: but, as the former was unoccupied by its proprietor, the lady, who lived in a dignified retirement, res tained every privilege over the now diminished clan, except those which are derived from wealth.. An aged matron, known by the appellation of. Old Moome, (or the Old Nurse,) rich in legendary stories, superstitious charms, interpretations of visions, and all the appendages of old-woman wisdom, was the person of the next consequence; and the next conspicuous figure was Hugh, the old family piper, a kind-hearted male gossip, active in every man's service but his own, and warmly attached to the honour and customs of his country. It some what surprized these villagers that the Lady Augusta, after having inspected a picture and a few other trifles that were found on the deceased stranger, ordered her to be buried at Eleenalin among the bones of her own ancestors, and gave the boy the name of Norman, which had been that of her own twin-brother, who had perished by a most tragical catastrophe.

The boy, however, continued to live for many years in the family of Ronald: but his education was not neglected. When the lore of Moome and of the piper was exhausted, he was placed under a neighbouring pedagogue, famed for his schoollearning; and in all the more elegant accomplishments he was instructed by Lady Augusta herself, who took uncommon in terest in his welfare, and delight in his society. He had nearly attained the age of manhood, when the proprietor of the district became desirous of raising the returns of his estates, and leased them out in large sheep-farms. The necessary consequence of this measure was the entire depopulation of the hamlet and glen, the inhabitants of which hau supported themselves by produce of small portions of land attached to their cottages. Emigration ensued; and of the former in

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habitants