34 REPORT — 1841. 



of the Association for the sum of 50/. to enable Mr. Ross to complete the 

 discussion of a series of tide observations made at Leith, extending from 1827 

 to 1839 inclusive, and now including 184'0. This long series of years is ad- 

 vantageous for the purpose of obtaining the declination correction ; since, in 

 consequence of the motion of the moon's nodes, the range of lunar declina- 

 tion and the mean declination is verj- different in different years ; as was 

 stated in my Report on the subject, presented to the Association last year. 



The present Report will i-efer to a new mode of presenting the corrections 

 of the height of high water for lunar parallax and declination. It has been 

 shown by me in various memoirs that the correction of the height both for 

 lunar parallax and declination is nearly the same for all the hours of moon's 

 transit. This being the case, the greater part of this correction may be ex- 

 pressed by means of a table of double entry ; the two arguments being the 

 moon's parallax and declination. Mr. Ross suggested to me the advantage 

 of such a table, and has constructed it from the Leith observations, and it is 

 laid before the Association along with the present Report. 



It appears by this table, when separated into the two parts dependent upon 

 parallax and declination, that the parallax correction varies very exactly as the 

 parallax; and that the declination correction applicable to declination , varies 

 very nearly as the square of the declination ; results agreeing both with those 

 obtained from the tide observations made at other places, and with the conse- 

 quences of the equilibrium theory modified, as I have previously shown that it 

 must be, in order to express the results of observation. As I have stated, the 

 principal part of the correction of the height of high water for lunar parallax 

 is constant for all hours of moon's transit. But there is a further term of this 

 correction, though a small one, which goes through a cycle of positive and 

 negative values in the course of a semilunation. This has already appeared 

 in the results of the London, Liverpool, Plymouth, and Bristol observations, 

 and also agrees with the theory above referred to. A like result appears in 

 the results of Leith tides by the discussions now reported, but at first sight 

 with a remarkable difference. At Plymouth it appeared (Ninth Series of Tide 

 Researches, Phil. Trans. 1838), that the correction for parallax is least when 

 the hour of moon's transit is 10*^, and greatest when the hour of moon's 

 transit is 4^ or 5*^ ; the mean parallax correction when the part depending on 

 the hour of transit disappears, occurs at transit l"* and 1'^. At Leith, on the 

 contrary, the effect of the parallax is greatest when the transit is about 6"^, 

 least when the transit is 0^ and the mean value obtains when the transit is 

 about 3"^ and 9**. But this great difference in the results, which at first ap- 

 pears to make the course of this correction nearly opposite at the different 

 places, is, in fact, the result of the difference of the time which the original 

 tide-wave employs in reaching Plymouth and Leith. This correction varies 

 nearly as the sine of the double angle of the moon from the sun, minus a 

 certain epoch. Or to be more exact, instead of the sine we may substitute a 

 circular function, which vanishes, and is positive and negative when the sine 

 is, but which does not exactly follow the law of the sine. If this function be 

 called s, the term of which we are now speaking is, in the Plymouth tables, 

 as 5, 2 — 14" ; in the Leith tables it is as s, 2 — IS**. The difference of 

 the epochs, \\^ and 18'', depends on the time of transmission of the tide 

 from Plymouth to Leith. This is further illustrated by remarking that in 

 the results of London observations this term is also represented by *, 2 — IS*" ; 

 while the Bristol observations give the term s, 2</) — 15*^. 



The agreement of these results cannot but be considered as decisive evi- 

 dence of the correctness of the tables which M'e have obtained, as to their 

 form and general law. And this is the more remarkable when we consider 



