1800.] Harmonie Analysis of Tidal Observations. 301 



These have to be divided between H. and L.W. For the sake of 

 convenience, I suppose that we always begin the series with a H.W., 

 then when the integer is odd we put in one more H.W. than L.W., 

 and thus have the following rule : 



No. of semi-lunation 1 2 3 4 5 6 



No. of last H.W. in the 



semi-lunation 29 57 86 114 143 171 



No. of last L.W. in the 



semi-lunation 28 57 85 114 142 171 



The H.W. and L.W. are here supposed to be numbered consecu- 

 tively from 1 onwards in separate tables. 



The other rules of partition giveu in Appendix E are found in the 

 same way. 



11. On the Over Tides. 



Observations of H. and L.W. are very inappropriate for the deter- 

 mination of these tides (of which the most important are M 4 , M 6 , 

 S 4 , S 6 ), because they express the departure of the wave from the simple 

 harmonic shape, and we are supposed to have no information as to 

 what occurs between two tides. These tides make the interval from 

 H. to L.W. longer than from L. to H.W., and there is no doubt that, 

 assuming the existence in the expression for h of a term of the form 

 A-2m. cos 2 V m -f- B 2M sin 2 F^, we shall get an approximation to A^ and 

 B 2 w by finding the mean of hcos2V m and /Jsin^F^. But the com- 

 putation of the F, G, f, g, coefficients for the perturbation of M 4 by M 2 

 would be essential, and thus the amount of additional computation, 

 would be very great, whereas in the analysis of continuous observa- 

 tion the overtides are found almost without any additional work. I 

 am inclined to think that it would be best to obtain hourly observa- 

 tions for several days at several parts of a lunation, and by some 

 methods of interpolation to construct a typical semi-diurnal tide- 

 wave, from which, by the ordinary methods of harmonic analysis, we 

 could find the ratio of the heights of the over-tides to the funda- 

 mental, and the relationship of their phases. 



I make no attempt at such an investigation in this place. 



12. On the Annual and Semi-annual Tides. 



These tides are frequently of much importance, so that they ought 

 not to be neglected from a navigational point of view. It is obvi- 

 ously impossible to obtain any results from a series of observations 

 of less than a year's duration. 



Bules for the partition of tides into months or 12 th parts of a year 

 are given in the Appendix E. The mean of all the H. and Lt.W. ob- 

 servations for each month may be taken as the height of mean water 



