ness of channels along which the wave has to travel before reaching the port,—their length, &c. &c., the times above distinguished would be identical. But all these causes tend to create a difference, and to make that dif ference not alike at all ports. The observation of the establishments of harbours is a point of great maritime importance; nor is it of less consequence, theoretically speaking, to a knowledge of the true distribution of the tide waves over the globe.* In making such observations, care must be taken not to confound the time of "slack water," when the current caused by the tide ceases to flow visibly one way or the other, and that of high or low water, when the level of the surface ceases to rise or fall. These are totally distinct phænomena, and depend on entirely different causes, though it is true they may sometimes coincide in point of time. They are, it is feared, too often mistaken one for the other by practical men; a circumstance which, whenever it occurs, must produce the greatest confusion in any attempt to reduce the system of the tides to distinct and intelligible laws.

(533.) The declination of the sun and moon materially affects the tides at any particular spot. As the vertex of the tide-wave tends to place itself vertically under the luminary which produces it, when this vertical changes its point of incidence on the surface, the tide-wave must tend to shift accordingly, and thus, by monthly and annual periods, must tend to increase and diminish alternately the principal tides. The period of the moon's nodes is thus introduced into this subject; her excursions in declination in one part of that period being 29°, and in another only 17°, on either side the equator.

(534.) Geometry demonstrates that the efficacy of a

*The recent investigations of Mr. Lubbock, and those highly interesting ones in which Mr. Whewell is understood to be engaged, will, it is to be hoped, not only throw theoretical light on the very obscure subject of the tides, but (what is at present quite as much wanted) arouse the attention of observers, and at the same time give it that right direction, by pointing out what ought to be observed, without which all observation is lost labour.

CHAP. XI.

THE TIDES.

339

luminary in raising tides is inversely proportional to the cube of its distance. The sun and moon, however, by reason of the ellipticity of their orbits, are alternately nearer to and farther from the earth than their mean

distances. In consequence of this, the efficacy of the sun will fluctuate between the extremes 19 and 21, taking 20 for its mean value, and that of the moon between 43 and 59. Taking into account this cause of difference, the highest spring tide will be to the lowest neap as 59+ 21 to 4319, or as 80 to 24, or 10 to 3. Of all the causes of differences in the height of tides, however, local situation is the most influential. In some places, the tide-wave rushing up a narrow channel, is suddenly raised to an extraordinary height. At Annapolis, for instance, in the Bay of Fundy, it is said to rise 120 feet.* Even at Bristol, the difference of high and low water occasionally amounts to 50 feet.

(535.) The action of the sun and moon, in like manner, produces tides in the atmosphere, which delicate observations have been able to render sensible and measurable. This effect, however, is extremely minute.

(536.) To return, now, to the plantary perturbations. Let us next consider the changes induced by their mutual action on the magnitudes and forms of their orbits, and in their positions therein in different situations with respect to each other. In the first place, however, it will be proper to explain the conventions under which geometers and astronomers have alike agreed to use the language and laws of the elliptic system, and to continue to apply them to disturbed orbits, although those orbits so disturbed are no longer, in mathematical strictness, ellipses, or any known curves. This they do, partly on account of the convenience of conception and calculation which attaches to this system, but much more for this reason,—that it is found, and may be demonstrated from the dynamical relations of the case, that the departure of each planet from its ellipse, as deRobison's Lectures on Mechanical Philosophy.

termined at any epoch, is capable of being truly represented, by supposing the ellipse itself to be slowly variable, to change its magnitude and excentricity, and to shift its position and the plane in which it lies according to certain laws, while the planet all the time continues to move in this ellipse, just as it would do if the ellipse remained invariable and the disturbing forces had no existence. By this way of considering the subject, the whole permanent effect of the disturbing forces is regarded as thrown upon the orbit, while the relations of the planet to that orbit remain unchanged, or only liable to brief and comparatively momentary fluctuation. This course of procedure, indeed, is the most natural, and is in some sort forced upon us by the extreme slowness with which the variations of the elements develope themselves. For instance, the fraction expressing the excentricity of the earth's orbit changes no more than 0.00004 in its amount in a century; and the place of its perihelion, as referred to the sphere of the heavens, by only 19′ 39′′ in the same time. For several years, therefore, it would be next to impossible to distinguish between an ellipse so varied and one that had not varied at all; and in a single revolution, the difference between the original ellipse and the curve really represented by the varying one, is so excessively minute, that, if accurately drawn on a table, six feet in diameter, the nicest examination with microscopes, continued along the whole outlines of the two curves, would hardly detect any perceptible interval between them. Not to call a motion so minutely conforming itself to an elliptic curve, elliptic, would be affectation, even granting the existence of trivial departures alternately on one side or on the other; though, on the other hand, to neglect a variation, which continues to accumulate from age to age, till it forces itself on our notice, would be wilful blindness.

(537.) Geometers, then, have agreed in each single revolution, or for any moderate interval of time, to regard the motion of each planet as elliptic, and performed

CHAP. XI. VARIATIONS, PERIODIC AND SECULAR.

341

according to Kepler's laws, with a reserve in favour of certain very small and transient fluctuations, but at the same time to regard all the elements of each ellipse as in a continual, though extremely slow, state of change; and, in tracing the effects of perturbation on the system, they take account principally, or entirely, of this change of the elements, as that upon which, after all, any material change in the great features of the system will ultimately depend.

(538.) And here we encounter the distinction between what are termed secular variations, and such as are rapidly periodic, and are compensated in short intervals. In our exposition of the variation of the inclination of a disturbed orbit (art. 514.), for instance, we showed that, in each single revolution of the disturbed body, the plane of its motion underwent fluctuations to and fro in its inclination to that of the disturbing body, which nearly compensated each other; leaving, however, a portion outstanding, which again is nearly compensated by the revolution of the disturbing body, yet still leaving outstanding and uncompensated a minute portion of the change, which requires a whole revolution of the node to compensate and bring it back to an average or mean value. Now, the two first compensations which are operated by the planets going through the succession of configurations with each other, and therefore in comparatively short periods, are called periodic variations; and the deviations thus compensated are called inequalities depending on configurations; while the last, which is operated by a period of the node (one of the elements), has nothing to do with the configurations of the individual planets, requires an immense period of time for its consummation, and is, therefore, distinguished from the former by the term secular variation.

(539.) It is true, that, to afford an exact representation of the motions of a disturbed body, whether planet or satellite, both periodical and secular variations, with their corresponding inequalities, require to be expressed; and, indeed, the former even more than the latter; seeing that

the secular inequalities are, in fact, nothing but what remains after the mutual destruction of a much larger amount (as it very often is) of periodical. But these are in their nature transient and temporary: they disappear, and leave no trace. The planet is temporarily drawn from its orbit (its slowly varying orbit), but forthwith returns to it, to deviate presently as much the other way, while the varied orbit accommodates and adjusts itself to the average of these excursions on either side of it; and thus continues to present, for a succession of indefinite ages, a kind of medium picture of all that the planet has been doing in their lapse, in which the expression and character is preserved; but the individual features are merged and lost. These periodic inequalities, however, are, as we have observed, by no means to be neglected, but they are taken account of by a separate process, independent of the secular vari

ations of the elements.

(540.) In order to avoid complication, while endeavouring to give the reader an insight into both kinds of variations, we shall henceforward conceive all the orbits to lie in one plane, and confine our attention to the case of two only, that of the disturbed and disturbing body, a view of the subject which (as we have seen) comprehends the case of the moon disturbed by the sun, since any one of the bodies may be regarded as fixed at pleasure, provided we conceive all its motions transferred

B

K

Q

M

in a contrary direction to each of the others. Suppose, therefore, S to be the central, M the disturbing, and P the disturbed body. Then the attraction of M acts on