relating to Forced Vibrations. 75 . 



tions are slower than the free vibrations, />ti-f /^ must be less 

 than yLtj, and the external force must always act outward. 



6. If a denote the amplitude, or maximum value of s, and/ 

 the maximum acceleration due to the external force, which is 

 supposed to be a known quantity, we have 



/=/* 2 a, 



from which equation we can obtain a determination of a in 

 terms of the free and forced periods, as follows: — ■ 



lAi+fi* : /ii : : Ti 



rp2 



to ■ ft : = T?-T 2 



T2 







7. We shall now apply some of the above results to the 

 lunar tides in a uniform canal encircling the earth at the 

 equator, the moon being supposed to be always vertically over 

 the equator, and to revolve round the earth at a constant dis- 

 tance with constant angular velocity. 



Let denote the difference of longitude of any point of the 

 canal from the point vertically under the moon, and M the 

 moon's attraction upon a unit of mass on the earth multiplied 

 by the ratio of the earth's radius to the moon's distance. 

 Then, as is well known, the moon's disturbing force at any 

 point of the canal can be resolved into a component M sin 

 perpendicular to the line of centres of the earth and moon, 

 and a component 2M cos parallel to the line of centres — the 

 direction of the former component being towards the line of 

 centres, and the direction of the latter component being from 

 the medial plane of the earth which separates the hemisphere 

 enlightened by the moon from the other hemisphere. By 

 resolving each of these components along a tangent to the 

 equator, we get 3M sin cos 0, or | M sin 20, as the horizontal 

 force urging the water along the canal, due to the moon's 

 action, its direction being always that which would bring the 

 water nearer to the line of centres. As the value of for a 

 given point of the canal increases at the uniform rate of 2ir 

 per lunar day, the expression § M sin 20 represents a simple- 

 harmonic function of the time with a period T of half a lunar 

 day. As the amplitude of vibration of a particle of the water 

 will be insignificant in comparison with a quadrant of the 

 equator, we may identify the actual disturbing force on a par- 

 ticle with that which would be exercised if the particle were 

 in its mean position, and may therefore adopt | M sin 20 as 

 the expression for the disturbing force on a particle. 



G2 



