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VII. Note on some Propositions in the Theory of the Tides. 

 By the Rev. T. K. Abbott, M.A., Fellow and Tutor of Trinity 

 College 3 Dublin*. 



IN the following paper it is proposed merely to give elementary 

 proofs of the following propositions : — 



I. That if there were no friction it would be low water under 

 the moon. 



II. That friction accelerates the times of high and low water. 



III. That, in addition to the oscillatory motion of the water, 

 there is a constant current produced by the moon's action. And 



IV. That the effect of friction is to increase the length of the 

 day. 



I. The theorem that if there were no friction it would be 

 low water under the moon, is common to the theories of Newton 

 and Laplace and Airy. That it is not mentioned in elementary 

 treatises is, I suppose, owing to the circumstance that the proof 

 is supposed to require a knowledge of the higher mathematics. 

 The Astronomer Royal gave a simple proof of the theorem in the 

 1 Astronomical Notices ' for 1866 ; but as it is very indirect, and 

 is open moreover to another objection presently to be mentioned, 

 it does not make another attempt at simplification superfluous. 

 The following proof is direct, and presupposes only the most 

 elementary knowledge. 



I suppose the moon to be fixed, and the earth rotating in the 



Fisr. 1. 



*l 



direction A B C D, carrying the ocean with it. It is shown in 

 elementary books (Galbraith and Haughton's l Manual of Astro- 

 nomy/ p. 66) that the tangential force alone need be considered, 

 and that this acts in each quadrant in the direction indicated by 

 the arrows, viz. always towards the line joining the centres of 

 the moon and the earth. 



Now in the course of one lunar day every particle of the ocean 

 is subjected to precisely the same forces acting in the same order 

 of succession and for the same periods, being accelerated for 

 * Communicated bv the Author. 



Phil. May. S. 4. Vol. 39. No. 258. Jan. 1870. E 



