THE ASTRONOMICAL DEDUCTIONS. 



13 







Now, this simple case illustrates a general dynamical principle, namely, that if a 

 system capable of oscillating with a certain period is acted on by a periodic force, when 

 the period of the force is greater than the natural free period of the system, the oscil- 

 lations of the system agree with the oscillations of the force; but if the period of the 

 force is less than the natural free period of the system the oscillations are inverted 

 with reference to the force. 



This principle may be ap- 

 plied to the case of the tides in 

 the canal. When the canal is 

 more than 13| miles deep, the 

 period of the sun's disturbing 

 force is 12 hours and is greater 

 than the natural free period of 

 the oscillation, becavise a free 

 wave would go more than half 

 roimd the earth in 12 hours. 

 We conclude, then, that when 

 the tide-generating forces are 

 trying to make it high water, it 

 will be high water. It has been 

 shown that these forces are tend- 

 ing to make high water immedi- 

 ately under the sun and at its 

 antipodes, and there accordingly 

 will the high water be. In this 

 case the tide is said to be direct. 



But when the canal is less 

 than 13| miles deep, the sim's 

 disturbing force has, as before, 

 a period of 12 hours, but the 

 period of the free wave is more 

 than 12 hours, because a free 

 wave would take more than 12 

 hours to get half round the 

 earth. Thus the general prin- 

 ciple shows that where the forces 

 are trying to make high water, 

 there will be low water, and vice 

 versa. Here, then, there will be 

 low water under the sun and at 

 its antipodes, and such a tide 

 is said to be inverted, because 

 the oscillation is the exact in- 

 version of what would be natu- 

 rally expected. 



All the oceans on the earth 

 are very much shallower than 

 fourteen miles, and so, at least 

 near the equator, the tides ought 

 to be inverted. The conclusion of 

 the equilibrium theory will therefore be the exact opposite of the truth, near the equator. 



This argimient as to the solar tide requires but little alteration to make it applicable 

 to the lunar tide. 



The positions of a set of "inverted" tides corresponding to the fore- 

 going set of "direct" tides are shown in fig. 2. 



Now since the rotation of the earth gives its surface an angular motion 

 greater than that of the tide-producing body, its effect must be to carry 



Fia. 2. 



