96 REPORT—1883. 
In the form actually prepared for the computers, the horizontal lines 
between the successive days are absent, and the place for each single 
entry is indicated by a single dot. 
The incidence of tke hours in the computation forms for the several 
series was determined by Mr. Roberts. ; 
Since the first day is numbered 1, and the first hour 04, it follows that 
the hourly observation numbered 74" 11" is the observation which com- 
pletes a period of 737 12" of mean solar time since the beginning ; in fact, 
to find the period elapsed since 0" of the first day we must subtract 1 from 
the number of the day and add one to the number of the hour. The 
73° 12" of m. s. time, inserted at the foot of the form, is very nearly equal 
to 71 days of mean lunar or M-time. For each class of tide there are five 
pages, giving in all about 370 values for the height of the water at each 
of the 24 special hours ; the number of values for each hour varies slightly 
according as more or less ‘changes’ fall into each column. 
The numbers entered in each column are summed on each of the five 
pages; the five sets of results being summed, the results are then divided 
each by the proper divisor for its column, and thus is obtained the mean 
value for that column. In this way 24 numbers are found which give 
the mean height of water at each of the 24 special hours. 
It is obvious that if this process were continued over a very long time 
we should in the end extract the tide under analysis from amongst all the 
others, but as the process only extends over about a year, the elimination 
of the others is not quite complete. 
The elimination of the effects of the other tides may be improved by 
choosing the period for analysis not exactly equal to one year. For sup- 
pose that the expression for the height of water is 
A, cos t+ B, sin n,f+A, cos not+B, sin nt. . . (61) 
where n, is nearly equal to n,, and that we wish to eliminate the 1,-tide, 
so as to be left only with the m,-tide. 
Now, this expression is equal to 
{A, FA, cos (n;—n2)t—By sin (7, —14)t} cos mt) (62) 
+ {B,+Ay, sin (m,—12)f+ By cos (nm, —n2)t} sin nyt} 
That is to say, we may regard the tide as oscillating with a speed m,, but 
with slowly-varying range. Now we want to find the mean semi-ranges 
A,, B, of such an oscillation, and these will be found if we take the 
average semi-ranges estimated over a good many periods 27/(n,—mo). 
It will be best to stop exactly at the termination of such a period, so that 
the number of positive errors may be as nearly as possible equal to the 
number of negative ones. 
It is of course impossible to choose for each tide , a period which 
shall minimise the effects of more than one of the tides of short period 
Ny in vitiating the values of mean semi-ranges of the tide n,, and accord- 
ingly the periods have been chosen so as to minimise the effect of the 
principal solar semi-diurnal tide 8, upon the principal lunar semi-diurnal 
tide M,, and of the M,-tide upon the others. 
If n, be a diurnal tide and m, a semi-diurnal one, it does not seem 
worth while to choose any particular period for the averaging process, 
because the coefficients will go through so large a number of oscillations 
(about 350) in the course of the year. Nevertheless, special periods for 
