78 Elementary Investigations relating to Forced Vibrations. 



In the column at right angles to this, which goes down from 

 the second point, the disturbing force is vertically downward, 

 and has the average value ^M. 



As each column has the length r, the pressure to be coun- 

 terbalanced by difference of levels at the tops of the two 

 columns is (M + ^M)r; and this must be equal to 2hg, where 

 2A denotes the height of the equilibrium tide. We have 

 therefore 



2k = ^, 



as previously found. 



12. The vertical component of the moon's disturbing force 

 at a point of the surface at angular distance from the point 

 under the moon, is found by compounding the upward com- 

 ponent of a force 2M cos 6 parallel to the line of centres with 

 the downward component of a force M sin perpendicular to 

 the line of centres, and is therefore an upward force, 



M(2 cos 2 0-sin 2 0) = M(3 cos 2 0-l) = M(i + f cos 20). 



The average force through the column which goes down from 

 the point in question to the earth's centre will be half of this, 

 or will be 



iM(l + 3cos26>); 



and to prevent the column from moving upward we must have 

 an elevation at its summit equal to 



rM 



i_ (1 + 3 cos 20). 



The elevation measured from the mean level of a great circle 

 passing through the point under the moon is 



f — cos 20, 



9 



which follows the simple-harmonic law. 



13. We have taken no account of the attraction of the ele- 

 vated water upon the general mass of water. When this effect 

 is included, it makes the height of the equilibrium tide greater 

 by about one eighth part. 



14. The principles of sections 1 to 6 have an important 

 bearing on the control of pendulums. 



Suppose that the controlling clock gives the controlled 

 pendulum an impulse in one direction at the beginning of 

 every even second of Greenwich time, and an impulse in the 

 opposite direction at the beginning of every odd second. Then, 

 if the controlled pendulum is naturally too slow in its move- 

 ments, it will adjust itself so as to receive an inward impulse 



