in the Theory of the Tides. 



51 



its absolute position except with the moon's monthly motion ; 

 only the water accompanying the rotating earth moves fastest 

 at A and C, and is there shallowest : and slowest at B and D, and 

 is there deepest. Relatively to the earth it is moving westward 

 at B and D, eastward at A and C. 



Fig. 2. 



M 



II. The theorem that the effect of friction is to accelerate the 

 time of high and low water admits of an equally simple proof. 

 As the water approaches C, the tangential force diminishes gra- 

 dually to zero at C. Therefore it must have been equal to the 

 force of friction at some point n (fig. 1), after which friction prevails 

 and the velocity diminishes. It is therefore low water at n. Ap- 

 proaching D, the ocean is moving slower than the earth ; there- 

 fore here friction tends to accelerate it, while the retarding force 

 is decreasing to zero. The two forces, then, must be equal at 

 some point o, after which the velocity again increases. It is 

 high water therefore at o. 



It is proper to observe that the preceding proof assumes (as 

 does Mr. Airy's) that the ocean is carried round by the earth in 

 its rotation. This amounts to supposing that it has not assumed 

 a position of equilibrium*. 



Mr. Airy assumes further, that when the tide is rising fastest 

 the water is flowing in from both sides. This is by no means 

 evident. The rate of rise depends on the difference in velocity 

 between two successive parts of the ocean, and this may be greater 

 when the two velocities have the same sign than when they have 

 different signs. Taking into consideration the rotation of the 

 earth, the assumption amounts to this — that the tide is rising 

 fastest where the velocity of the ocean is just equal to that of 

 the earth. This is certainly not evident : in fact it would not 

 be true if the tangential force did not decrease at the same rate 

 on both sides of each of the four maxima. It ought not, there- 

 fore, to be assumed, but deduced. 



III. We now come to the question whether, in addition to the 



* Mr. Airy indeed supposes, not* only that the ocean has some move- 

 ment of rotation, but that its mean movement is the same as the earth's. 



E2 



