[ 138 ] 



VII. Note on the Theory of the FortniahtJy Jide. 

 By Lord Rayleigh, O.M., F.RJS* 



AN adequate calculation a priori o£ the tide of fortnightly 

 period — that which depends upon the moon's motion 

 in declination — would be of great interest as affording the 

 means, by comparison with observation, of determining the 

 extent to which the solid earth yields to the tide-generating 

 force. On the assumption that the fortnightly tide over an 

 absolutely rigid earth would be sensibly equal to its " equi- 

 librium value/' Prof. G. Darwin | has estimated that the 

 actual rigidity must be at least as great as that of steel, in 

 accordance with the earlier surmises of Lord Kelvin. 



But is an " equilibrium theory " adequate ? The known 

 properties of a system vibrating about a configuration of 

 thoroughly stable equilibrium would certainly suggest an 

 affirmative answer, when it is considered that a fortnight is 

 a long period in comparison with those of the more obvious 

 free oscillations. It is to be remembered, however, that a 

 tidally undisturbed sea is not in equilibrium, and that in 

 virtue of the rotation of the earth the tides are really oscil- 

 lations about a condition of steady motion. In Laplace's 

 theory the rotation of the earth is taken fully into account, 

 but the sea must be supposed to cover the entire globe, or at 

 any rate to be bounded only by coasts running all round the 

 globe along parallels of latitude. The resulting differential 

 equation was not solved by Laplace, who contented himself 

 with remarking that in virtue of friction the solution for the 

 case of fortnightly and (still more) semi-annual tides could 

 not differ much from the " equilibrium values." 



The sufficiency of Laplace's argument has been questioned, 

 and apparently with reason, by Darwin J, who accordingly 

 resumed Laplace's differential equation in which frictional 

 forces are neglected. Taking the case of an ocean of uni- 

 form depth completely covering the globe and following the 

 indications of Lord Kelvin §. he arrives at a complete evalu- 

 ation of Laplace's " Oscillation of the First Species." A 

 summary of Darwin's work has been given by Lamb || from 

 which the following extracts are taken. The equilibrium 



* Communicated by the Author. 



t Thomson & Tait's ' Natural Philosophy/ 2nd ed. vol. i. pt. ii. p. 400 

 (1883). 



X Proceedings of the Royal Society, vol. xli. p. 3-37 (188G). 



§ Phil. Mag. vol. 1. p. 280 (1875). " 



|! Hydrodynamics, § 210, Cambridge, 189-5. 



