1890.] Harmonic Analysis of Tidal Observations. 339 



ence in time with the diurnal tides as with the semi-diurnal. The 

 errors of P fall under the same category as those of Kg. 



Lastly it is probable that all these errors would have been 

 sensibly diminished if I had subtracted 103 inches from the heights 

 all through instead of 99, and I know that this is to some extent the 

 case. 



(x.) Verification. 



In a calculation of this kind some gross error of principle may have 

 been committed, such, for example, as imputing to some of the *'s a 

 wrong sign ; and this is the kind of mistake which is easily over- 

 looked in a mere verification of arithmetical processes. It is well, 

 therefore, to test whether the tide heights and times are actually 

 given by the computed constants. This is conveniently done by 

 selecting some three or four tides from amongst those from which the 

 reductions have been made, and it makes the calculation much shorter 

 if we pick out cases in which it is H. or L.W. within a few minutes 

 of noon. 



For example, in the present case it was L.W. on February 16 

 (day 46) at O h 7 m P.M., and the height was 4 ft. in. ; again, it was 

 H.W. on March 25 (day 83) at O h 3 m P.M., and the height was 

 13 ft. 3 in. 



Now, if U denotes the value of any equilibrium argument whose 

 value at the epoch, O h , January 1, was denoted in (q) by w, and if 

 A denotes the height of mean sea-level above datum, the expression 

 for the heiht of water is : 



h = 



+ !! cos (Cr,,-ic,,)+fHi cos (Z7i-*,)+f'H'cos (ZT-') 

 + f H cos ( U Kg) + Hp cos (UpKp). 



The time of H.W. depends on a formula involving the sines of 

 the same angles in place of cosines. 



Since we have chosen cases where it is H. or L.W. at noon, the 

 Z7's exceed the 's by an exact number of days' motion. 



The evaluation of the separate terms may be conveniently made by 

 means of an ordinary nautical traverse table, where (neglecting the 

 decimal point) fH is represented by the " Distance," and 

 fHcos(I7 K) is given by "Latitude," and fHsin(7 *) by "De- 

 parture." 



If we know the time of H. or L.W. within 20 or so, the follow- 

 ing calculation will give the true time and height. The computa- 

 tion applies to the first of the two cases where we know that there 

 should be a L.W. at about O h of day 46. The increments of 

 argument are computed from the Table G, and the /c's are subtracted 

 either by actual subtraction or by addition of 2w *. 



