400 Intelligence and Miscellaneous Articles. 



NOTE ON THE THEORY OF TIDAL FRICTION. 

 BY D. D. HEATH, M.A. 



To the Editors of the Philosophical Magazine and Journal. 



Kitlands, Dorking, 

 Gentlemen, April 12, 1867. 



I have only today seen a short criticism on my March paper, by 

 Mr. Stone. 



I think he will see, on reconsideration, that there is no inconsis- 

 tency in my treatment of the orders of small magnitudes. In the 

 first investigations I am dealing with linear magnitudes ; and I believe 

 I keep them in due subordination. In (11), which Mr. Stone com- 

 ments upon, I am concerned with a moment, or product of volume, 

 accelerating force, and leverage, and I compare two quantities of 

 this kind together. 



As this is the part of my paper which is most likely to be gene- 

 rally interesting, I may perhaps be allowed to restate its purport. 



On the assumptions which I have borrowed from Mr. Airy, the 

 tide-wave, with friction, will lie obliquely at a certain angle h to 

 the moon's position. If we imagine it to become momentarily rigid 

 and attached to the earth, the moon's action will tend to produce 

 rotation westward, or check the actual eastward rotation ; and the 

 moment of this action I calculate as 2gHc sin 2S(\-\-k)x. If the rigi- 

 dity continued, the obliquity, and consequently the moon's action, 

 would change. But in fact the oscillatory motion of the water is 

 such as to keep the ridge always in the same relative place. And 

 the (juestion then is, Does the reaction of friction (supposing it to 

 act as required by Mr. Airy's theory) produce the same effect on the 

 earth as would be produced at each moment by a rigid wave-shape ? 

 And my answer is that the effect of such reaction is a force whose 



moment is J- — (1+k)tt, — a quantity not only of the same order of 



magnitude, but identical with the former one, as appears by the pre- 

 vious calculation. 



In the investigation, v is essentially the oscillatory velocity which 

 produces the wave-shape. If therefore there is a permanent current 

 (which I neither affirm nor deny), the force of friction will not be 

 fv, as we have supposed, hutf(V-\-v), where V is the velocity of the 

 current ; and I fear the whole matter, when friction is taken account 

 of, may require reexamination. 



I take this opportunity of requesting the readers of my original 

 paper to strike out the latter part of the paragraph (4, a) as thought- 

 lessly written. The reason for taking the force set free as simply 

 vertical and — 2nv is, that we have just before shown that the whole 

 velocity is sensibly horizontal. 



I will also notice a misprint. In page 1 67, last line, read 



dv dv 



dt dto 



D. D. Heath. 



JT T YT7f 10/MT 



