The American Journal of Science and Arts

2. The euphotides of Mt. Rose according to my observations are composed of smaragdite (a pyroxene containing chrome and nickel,) in a base of saussurite, which is a compact zoisite, or lime-alumina epidote, containing portions of magnesia and soda, and having a hardness of 70 and a specific gravity of 3.333-38; characters which at once distinguish it from the feldspars. These euphotides also contain as accidental minerals, talc, actinolite and occasionally a vitreous cleavable feldspar resembling labradorite.

3. While the minerals analyzed as saussurite by Stromeyer and Delesse are feldspars, that from Mt. Genèvre examined by Boulanger has the composition and specific gravity of meionite, a species which is isomeric with zoisite; the saussurite from Orezza according to the same observer has a like composition but a density intermediate between these species. The saussu rite examined by Thompson is apparently a petrosilex.

4. By its great density and its composition, the euphotide of Mt. Rose is related to certain rocks in which a white garnet, resembling saussurite, is mixed with serpentine, with hornblende, and with a feldspathic mineral. These aggregates associated with ophiolites, albitic diorites, and a rock made up of epidote and quartz, occur in the form of beds in the crystalline schists of the altered Silurian series in Canada.*

ART. XXXVIII.-The Dynamic Theory of the Tides; by Maj. J. G. BARNARD, A.M., Corps of Engineers, U. S. A.

IN his treatise on "Tides and Waves," Mr. Airy uses in reference to Laplace's investigation of the tides, the following language:

"If now, putting from our thoughts the details of the investigation, we consider its general plan and objects, we must allow it to be one of the most splendid works of the greatest mathematician of the past age. To appreciate this, the reader must consider, first, the boldness of the writer who, having a clear understanding of the gross imperfection of the methods of his predecessors, had also the courage deliberately to take up the problem on grounds fundamentally correct, (however it might be limited by suppositions afterwards introduced); secondly, the general difficulty of treating the motions of fluids; thirdly, the peculiar difficulty of treating the motions when the fluids cover an area which is not plane but convex; and, fourthly, the sagacity of perceiving that it was necessary to consider the earth as a revolving body, and the skill of correctly introducing this consideration. The last point alone, in our opinion, gives a greater claim for reputation, than the boasted explanation of the long inequality of Jupiter and Saturn."

*See my Contributions to the History of Ophiolites, this Journal, [2], vol. xxv, 217, and xxvi, 234.

The equilibrium-theory, manifestly false in treating the problem simply as one of statics, disregarding the motions of the fluid which must accompany the changes of its superficial form, is, at least, an explanation of the phenomenon, though not a true theory.

Mr. Airy remarks of it;

* "it must be allowed that it is one of the most contemptible theories that was ever applied to explain a collection of important physical facts. It is entirely false in its principles, and entirely inapplicable in its results. Yet, strange as it may appear, this theory has been of very great use. It has served to show that there are forces in nature following laws which bear a not very distant relation to some of the most conspicuous phenomena of the tides; and, what is far more important, it has given an algebraic form to its own results, divided into separate parts analagous to the parts into which the tidal phenomena may be divided, admitting easily of calculation and of alteration, and thus at once suggesting the mode of separating the tidal movements, and affording numerical results of theory with which they are to be compared. The greatest mathematicians and the most laborious observers of the present age have agreed equally in rejecting the foundation of this theory, and comparing all their observations with its results. And till theories are perfect (a thing scarcely to be hoped for in any subject, and less in the tides than any other,) this is one of the most important uses of theory."

If we could indeed grasp the conditions of the problem-bring into our analysis the expression of the actual form (or even a tolerable approximation to that form) of the solid nucleus whose depressions form the ocean beds, then indeed the solution would be that which we seek, not a mere explanation, but a true expression for the phenomena, as they actually occur.

While we are utterly incapable of doing this-when such a mind as Laplace's is found unable to grasp the conditions of a "Dynamic Theory," it seems to me that Mr. Airy wastes epithets upon the "equilibrium theory" which, after all, I presume no physicist ever regarded as a real theory of the tides, but rather a mere putting into mathematical form of their obvious immediate cause. If, to get over the difficulties of the true theory, and bring the problem within the grasp of our mathematics, we are obliged to make assumptions, entirely at variance with the facts which really govern the question-which cannot even approximate to them-we might as well, so far as the solution we seek is concerned, go one step further, and suppose there is no motion at all-or that the fluid is destitute of inertia; in other words fall back upon the equilibrium theory, for the problem is no longer that which we propose, but a mere mathematical study which may yield us some curious results.

"It was found necessary, however, (Airy Tides and Waves,") in order to make the application of mathematics practicable, to start with two suppositions, which are inapplicable to the state of the earth. These are: that the earth is covered with water; and that the depth of this water is the same through the whole extent of any parallel of latitude."

If the actual configuration of the ocean's bed is, as I have before remarked, the very basis of a dynamic theory of the tides, then a theory which is obliged to reject entirely this actual configuration, and instead of ocean beds of limited areas, isolated from each other by dry land in those parallels where the tidal effects are greatest, substitute an imaginary ocean covering the whole globe, and of the same depth following each parallel of latitude, the problem can be only a mathematical one of more or less. interest, from which nothing of any practical value, as to the actual phenomena of the tides, can be expected.

Such is, in fact, the dynamic theory of Laplace; it has furnished no result nor been of the slightest use to physicists in their investigations of the tidal phenomena. Mr. Airy remarks, "under these suppositions (the arbitrary assumptions as to the ocean's extent and depth) it is evident that the theory is far from being one of practical application;" but when we consider that, in the very effort to make the theory a dynamic one, by introducing the motions of the fluid particles, the real motions as governed by the actual configuration of the ocean beds are discarded and purely imaginary ones substituted, we may well hesitate in giving assent to the proposition which finishes the sentence; "though it clearly approaches much nearer to truth than the theory of equilibrium which we have already described."

In the eye of the mere theorist it may be so, but to one who seeks a knowledge of the tides as they actually are, the equilibrium theory is far more useful; and of two things neither of which possess any claims to be called true, one may be considered as true or the other.

The differential equations which determine the elevation and motion of the water, when the question is limited by these arbitrary assumptions already mentioned, are obtained with no great difficulty. In fact, the equilibrium theory gives the elevation of the water as it would be were the water destitute of inertia; in other words, the forces of attraction of the earth and of the disturbing bodies are alone considered, while the forces of inertia in the water itself are disregarded. We have only to introduce these forces to convert the equilibrium into a dynamic theory; and thus considering the effects of the fluid motions only in the forces of inertia developed, it follows from the general equations of equilibrium of fluids, that the total fluid pressure resulting will be the sum of the pressures due to the separate existence of each class of forces.

Calling p the total fluid pressure at any point, arising from the action of all the forces, p' the pressure due to the earth's attraction, were its surface undisturbed, p" the pressure due to the attractions of the disturbing body, p"" the pressure due to the forces of inertia in the fluid, we shall have

p=p'+p"+p"".

If we desire to have the value of p at the undisturbed surface of the earth, put p'=0 and we have p=p"+p"".

If we call w the height of the fluid column due to the pressure P, and q the height due to p", we shall have (considering the density as unity) p=gw, p"=gq, and

in which w is the total tidal elevation due to the disturbing attractions, and to the inertia of the fluid, and q is the elevation due to the disturbing attractions alone; in other words, it is the height due to the equilibrium theory.

Confining the investigation, for simplicity, to the attractions of the sun alone, we shall find from the equilibrium theory (vide Airy's "Tides and Waves" par. 44,)

(2) q=S' ()[(2 3 (cos 20-1) (1-3 sin2 )+ sin 24 sin 2σ cos (-s) +

P m

+cos 2 cos 20 cos 2 (-s).

In which Sando are the celestial right ascension and declination of the sun; and 7 the terrestrial latitude and longitude of the place, (the latter referred to a meridian fixed in space); P the actual and Pm the mean parallax of the sun and S" a coefficient which (vide par. 41 and 42)

Sb2 P. 3

2gD3 P

2D) (the density of the water being considered insignificant compared to that of the earth) in which S and D are the sun's mass and distance, b the earth's polar radius,* and g the force of terrestrial gravity.

In the equation (2) the angle (-s) is the difference in longitude of the point of observation and the sun, referrred to a meridian fixed in space. If we consider the earth a revolving body and call the longitude of the point, referred to a meridian on the earth's surface, and n the velocity of rotation, then the variable longitude of the point of observation, at the end of the time t, referred to a meridian fixed in space, will be represented by nt+, and the angle l-s, by nt+-S.

If instead of the latitude & we use the polar distance of the point of observation we shall have

cos sin, and sin 21-sin 20.

*The spheroidal form is disregarded, as the tidal displacements are very nearly the same whether the earth is regarded as a sphere or spheroid.

Making the substitutions in equation (2), and then substituting the value of gg in equation (1), we have

3Sb2

4D3

cos 2o.sin 20.cos 2 (nt+w-s) +p".

If, now, we suppose a particle of water running towards the south and call u the arc (in latitude) passed over at the end of will be the actual velocity of the particle, in this

the time t, b

direction, and b

d2 u dt2

its acceleration. If is the angular polar distance of the initial position of the particle, bo will be the actual

dp""
bd0

lineal polar distance, and will be the differential coefficient

of the pressure arising from a variation of 9, and by a slight and admissible extension of the fundamental equations of hydrody