## Elementary Theory of the Tides: The Fundamental Theorems Demonstrated Without Mathematics, and the Influence on the Length of the Day Discussed |

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accelerating force accelerating quadrants amount angular acceleration angular velocity centre circle circle of latitude covered with water depth of sea difference displacement diurnal tide earth earth's rotation earth's velocity east effect of friction elevating force ellipse equal equator force acting force is greatest greater greatest eastward greatest eastward velocity greatest elevation Hence horizontal force hour angle increase latitude low water lunar day maximum mean abscissa mean place meridional component middle point miles per hour moon's action moon's force motion moving faster moving slower Multiplying obstacle ocean parallel place of high point whose mean produced proportional quadrant CD radius represented Retard the Rotation retarding quadrants rising fastest SIR GEORGE AIRY solid globe tangential component tangential force theorems tidal elevation tidal prominences tide is rising tide-producing force tion velocity is diminished water is flowing water is moving westward velocity

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Page 20 - N its antipodes. rotation uorthward, and daring the other half southward, and in each case is an elevating force, which, as before, has its greatest effect 90° from the moon. At all places whose latitude is less than the moon's declination there is a permanent accumulation. In the circle abed this component is directed towards the north at a and towards the south at c, the points of change being where the great circles from M touch abed. This gives rise to a north and south oscillation. The southerly...

Page 13 - ... taken so great that any sensible disagreement with experiment would completely vanish. [To be continued.] II. Further Notes on the Theory of the Tides. By the Rev. TK ABBOTT, Trinity College, Dublin*. IN the demonstrations given in two previous papers in this Magazine (January 1870 and February 1871), we have supposed the water to be limited to an equatorial canal, the moon also being in the equator. It is desirable to consider what modifications will be introduced, first, by supposing the earth...

Page 17 - Hence, by 5 and 4, the elevation at the equator (and up to lat. 45°) will be greatest (te it will be high water) 90° from the moon. Beyond lat. 45° the depression will be greatest under the same circumstances. In these latitudes, therefore, the effect of the former component would be partially counteracted. It is easy, however, to see that the variation in the meridional force (which...

Page 14 - ... us, the water rises when the particles behind are moving faster than those before. The rate of rise is greatest when this difference is greatest; but as the effect is cumulative, the whole amount of the rise is greatest at the moment when the difference = 0, and is about to change to the opposite. Fourthly, as in 2, this difference ceases to increase (ie is greatest) when the force (or difference of forces) producing it ceases to act; but it is not reduced to 0 until the opposite force has done...

Page 15 - Making these substitutions in the equation of the projection of the path on a horizontal plane yields du _ _ g sin2 a cos a " Hence, if the cone be developed into a plane, it is seen that the orbit of the path on the surface will be the same as would be produced by a particle moving in a plane under the action of a constant central force. 3.9 REFERENCES 1. Sokolnikoff, Ivan S.: Tensor Analysis; Theory and Applications. John Wiley & Sons, Inc., 1960. 2. Bond, E.; Auslander, M.; et al.: PL/I FORMAC...

Page 19 - This limitation will not affect our results. We shall, as before, take the two components separately. With respect, then, to the component which acts parallel to the equator. Near the equator itself the considerations applied in the former paper still hold good. Next consider a place, a., whose polar distance is less than the moon's declination, to which therefore the moon is circumpolar, and (with the assumed declination) alternately north and south of the zenith. If abcd be the circle of rotation...

Page 14 - ... is desirable to consider what modifications will be introduced, first, by supposing the earth to be uniformly covered with water, and, secondly, by taking into account the moon's declination. It will save repetition if we state once for all certain general principles which we shall have to employ. First, suppose an accelerating force acts alternately in opposite directions, the effect (measured by velocity) increases as long as the force acts in either direction, and therefore the velocity in...

Page 21 - ... from M meet the circle. Hence, by 4 and 5, the tide will be lowest at the latter points and high at the former. Combining this with the former result, the effect of both components together will be to give high water at a. it is not necessary to enter into a detailed examination of the state of things at intermediate places. It is not difficult to see that, as long as the moon's declination is small, there will be an accumulation effected by the meridional component extending from the equator...

Page 15 - If the reader wishes to apply these considerations to the case of aż equatorial canal treated in the first paper, it must be observed that there the elevating force is the excess of easterly force acting on any particles of water above that which affects those in advance, ie to the east of them. This excess is positive from 45° west of the moon to 45° east (ie while the moon passes from 450 east zenith distance to 45° west), then negative for 90°, and so on.

Page 14 - ... new force) continues to be in the same direction until this counter force has undone all the work done in that direction by the previous force. When the circumstances are alike in both directions, this will be when the force has done half its work. This is precisely the case of the common pendulum. Thirdly, in the case before us, the water rises when the particles behind are moving faster than those before. The rate of rise is greatest when this difference is greatest; but as the effect is cumulative,...