504 REPORT—1868. 
24, The shallow-water tides referred to above depend on the rise and 
fall of the tide, amounting to some sensible part of the whole depth of the 
water, or, which comes to the same, the horizontal velocity of the water 
being sensible in comparison with the velocity of propagation of a long wave, 
through some considerable portion of the sea which sensibly influences the 
tides at the point of observation. Helmholtz’s explanation of compound 
sounds, according to which two sounds, each a simple harmonic, having 
mt, nt for their arguments, give rise, if loud enough, to sounds having for 
their arguments (m-+n)t, (m—n)t, suggest that the compound action of the 
solar and lunar semidiurnal tides, must give rise to shallow-water tides 
whose arguments are 2(¢—y)t and 2(2y—n—<)t. It is intended with the 
least possible delay to perform averagings with a view to determine these 
tides*, The great influence of the British Channel, and the large extent of 
it through which the shallow-water condition specified above is fulfilled, 
makes it probable that the new tidal constituents now anticipated will be 
found sensible. 
25, The step next undertaken has been to find mean solar daily averages, 
and to purify these of lunar-diurnal and semidiurnal influence; and in a 
few days more I hope to determine the lunar fortnightly declinational and 
the solar semiannual tides; also the annual variation indicated by Mr. 
Ebenezer Maclean’s calculations ($ 10), and of the 2(¢—») luni-solar fort- 
nightly shallow-water tide suggested (§ 24) by Helmholtz’s theory of com- 
pound sounds. This work is now in progress?. 
26. Observations made every quarter hour during several periods within 
four days at a station in the Fiji Islands, supplied to me through the kind- 
ness of Lieut. Hope, R.N., have been partially reduced, by a rigorous appli- 
cation of the method of least squares. This somewhat laborious process has 
been undertaken not only for the sake of the results to be obtained, which, 
considering the chaotic mass of statements constituting our present informa- 
tion regarding tides in the Pacific, we may regard as not without value in 
themselves, but also to show how much may be done by applying the 
harmonic analysis to a very short series of observations such as may be made 
in the course of a few days in any part of the world by surveying officers. 
I expect to be able in the present case to obtain somewhat accurate deter- 
minations of the mean lunar semidiurnal, the mean solar semidiurnal, the 
lunar diurnal, and the solar diurnal tides, each of which is probably sensible 
in the series of observed heights which have been supplied; but it has been 
necessary to defer this work to allow the full reduction of the Ramsgate 
1864 series to be pushed on as far as possible towards completion before the 
present Meeting of the British Association. 
27. The work requisite to obtain the results stated above has been, as 
may readily be conceived, very heavy; but a large part of it is available for 
other years and other places. It has been almost all performed by Mr. E. 
Roberts, who has devoted himself to it with most satisfactory zeal, ability, 
and perseverance, in intervals of his laborious duties for the Nautical 
* [Note added Dec. 1868.] This has now been done by Mr. Roberts for the 2(¢—m) 
(or synodie fortnightly) tide, and a very notable result has been obtained (see § 28 below); 
but as yet I cannot feel much confidence in it, because the period is that of the spring to 
neap and back to spring tides, and the Ramsgate instrument did not work well through 
the longer ranges (the buoy, for instance, sometimes rested on the bottom, and the failing 
curve-register was supplied by guess). The calculation xecessari/y includes instrumental 
errors depending on the gauge not working equally well through long and short ranges. 
+ Mr. Roberts has since completed it. For results see § 28 below. [Note added 
Dec. 1868.] 
