ON THE COASTS OF IRELAND. 11 



Di, D2, D3, D4, D5, &c. Then the means of the adjacent numbers were taken, 



D1+D2 r>2+r>3 D3+D4 D4+D5 o,„ 



2 ' 2 ' 2 ' 2 ' ^^- ' 



and the means of the numbers in this series were taken, forming 



Di + 2D,+D3 D2+2D3+D, D3+2D4+D5 „ 



4 ' 4 ' 4 ' ^^' 



Then the number — ' ^ ^ was considered to be the just difference from mean 



for the second day in the series: it was applied to the mean of times for that day, 

 and gave the adopted time for high or low water for that day, at the station under 

 consideration ; and so for the succeeding days. In regard to the legitimacy of this 

 process, it is to be observed that it does not suppress the inequalities affecting, in 

 different degrees at different stations, the semidiurnal or diurnal tide, provided the 

 period of such inequalities is of several days. Nor does it suppress any accidental 

 inequality which affects the whole tide-wave coming from the Atlantic upon a large 

 extent of coast. The only failure is, that, as 



D. + 2D2+D0 -r. , D,-2D2+D3 TA . 1 ,^ J v^ 



' ^^^ ^ =1)2+ 4^^^-'=D2+4(2nd difference) ; 



when the second difference of D is large, an error is introduced. So long as the tides 

 at the different stations follow anything like similar laws, there is no fear that this 

 error will be perceptible. The only place where there is any probability that it can 

 become sensible is Ballycastle ; and here it will be very far below the irregularities 

 of observation. 



Section III. — Theory of diurnal tide as related to observations only ; and deduction of 

 the principal results for diurnal tide given immediately hy these observations. 



The remarks with which I shall immediately proceed apply equally to times and 

 to heights, and equally to high waters and to low waters ; but, to avoid unnecessary 

 repetitions, I shall speak only of heights at high water. 



Suppose then that, for any station, the heights at high water, both of the First 

 Division and of the Second Division, have been collected and intermingled in the 

 order of times. It is evident that the diurnal tide at any one of these heights will be 

 found approximately by taking half the excess of that height above the mean of the 

 two heights immediately preceding and immediately following. The number thus 

 found will, however, be in error by one-fourth of the second difference of the semi- 

 diurnal tide. This error may be eliminated, leaving only an error depending on 

 fourth differences, by taking half the algebraical excess of that apparent diurnal 

 tide above the mean of the diurnal tides next to it. 



The process may however be put in the following algebraical form :— Suppose the 

 successive high waters to be affected with inequalities represented hy a.cosn—dj, 



c2 



