50 The Rev. T. K. Abbott on some Propositions 



about one quarter of a day, viz. while passing from B to C, then 

 retarded for a quarter, from C to D, and so on. The variation 

 in the amount of the force does not concern us, being the same 

 for every particle. 



This being so, it is obvious that those particles will be moving 

 faster which have been for a longer time acted on by an accele- 

 rating force, and the velocity will be a maximum when the acce- 

 lerating force has acted during its full period, viz. through one 

 quadrant. On the other hand, those particles will be moving 

 slower which have been longer acted on by a retarding force, 

 and the absolute velocity will be a minimum when the retarding 

 force has acted during its full period, or through one quadrant. 

 The maximum velocity is therefore at A and C, the minimum at 

 B and D. 



Secondly, it is clear that the tide will be rising where each 

 portion of water is moving faster than that just in advance of it, 

 or, in other words, where water is flowing in faster than it flows 

 out. Where this process has gone on for the maximum time, 

 the tide will be highest. On the other hand, the tide will be 

 falling where the water is moving slower than that in advance 

 of it, or, in other words, is flowing out faster than it flows in. 

 Where this has continued for the maximum time the tide is 

 lowest. 



Now consider any point s in the quadrant B C. The water 

 now passing s has been subject to an accelerating force during 

 the whole time since it passed B, longer therefore than any par- 

 ticks behind it, as at r. It is therefore moving faster ; and as 

 the water in the space rs is thus flowing out faster than it flows 

 in, the tide is falling. This is the case through the whole 

 quadrant B C. 



At C the force changes and becomes a retarding force. A 

 particle at y has been subject to this retarding force longer than 

 one behind it, as at a, and is therefore moving slower. Here, 

 therefore, water is flowing in faster than it flows out, and the 

 tide is rising; and this holds through the quadrant C D. What 

 is said of these quadrants holds also of those opposite to them ; the 

 tide is falling all through D A and rising through A B. Hence 

 it is highest at B and D, lowest at A and C. Where will the 

 tide be falling fastest ? Clearly where the difference of velocity 

 between r and s is greatest, i. e. where the amount of force to 

 which the water at s has been subject since it passed r is greatest- 

 in other words, where the force is at its maximum, viz. at/ (fig. 2), 

 45° from C* Similarly it will be rising fastest at that point in 

 the quadrant C D where the force is greatest, viz. at#, 45°from C. 

 On the whole, then, the water assumes the form of an ellipse, 

 and as it is the earth that is rotating, this ellipse does not change 



