Prof. Challis on a Mathematical Theory of Tides. 31 



shores, or against obstacles in ocean-beds to the free propagation 

 of the tidal wave westward, may not be exactly compensated for 

 by the reverse action at low water, and that there is consequently 

 a residual action westward tending to diminish the rate of the 

 earth's rotation. Such an effect would depend on terms of the 

 second order. But whether this cause actually operates, and 

 whether currents produced by the moon's attraction have any 

 effect, we should, I think, have some means of deciding if the 

 problem of tides, as here proposed, were strictly solved to the 

 second approximation. 



I beg to conclude this communication by adverting to a ques- 

 tion which bears in an important manner on the truth of the 

 foregoing mathematical reasoning. It will be seen that I have 

 inferred from the expression for u' that it is " high water along 

 the meridian under the moon." In an article contained in the 

 ' Monthly Notices of the Royal Astronomical Society ' (vol. xxvi. 

 No. 6), the Astronomer Royal has maintained that, when fric- 

 tion is left out of account, " it is low water under the moon." 

 It is absolutely necessary, for the credit of mathematics, to ac- 

 count for such contradictory results being obtained from the 

 same premises. 



In support of his view Mr. Airy cites Newton's Principia, 

 Lib. I. Prop. 66, Cor. 19. But Laplace, in his account of New- 

 ton's theory of the tides, interprets that passage in a different 

 sense. In the Mecanique Celeste (vol. v. p. 146, edit, of 1825) 

 he states that Newton in that corollary only shows that the water 

 in an equatorial canal surrounding the earth ought, when at- 

 tracted by a luminary, to have a flux and reflux movement like 

 that of the sea, and that " he gives neither the law nor the mea- 

 sure of the movement." Laplace adds that " the true explana- 

 tion of the phenomenon is contained in Propositions 36 and 37 

 of the third book," which Mr. Airy does not refer to. There, in 

 fact, Newton says expressly that " the sea is depressed at places 

 90° distant from the sun, and elevated both under the sun and 

 in the opposite region," the same law applying, of course, to the 

 moon. 



Again, Mr. Airy asserts that, according to Laplace's theory, 

 it will be low water under the moon. But in the Mecanique 

 Celeste (vol. ii. p. 222) I meet with this sentence : — " If the ocean 

 covered a spheroid of revolution, and if it experienced no resist- 

 ance to its movements, the instant of high water would be that 

 of the passage of the sun across the superior or inferior meridian." 

 On what ground, therefore, is the opposite view attributed to 

 Laplace ? In the Essay on Tides and Waves in the Encyclopcedia 

 Metropolitana (arts. 109, &c), Mr. Airy finds that it is low water 

 under the moon by a method which he considers to be " equiva- 



