78 REPORT—1883. 
one year for prediction in future years can only be made by dividing up 
the compound tide into several parts, according to its theoretical origin. 
In order to do this it is necessary that the law should be known which 
connects the heights of a summation and a difference compound tide. A 
like difficulty arises from the fact that MSf and 2SM are also variational 
tides. 
In practice, however, the compound tide will generally be so small 
thatswe may probably treat it as though it arose entirely in one way: 
and accordingly it is proposed to treat the tides 3y—2c or MK, and 
3y—4c or 2MK, as though they arose entirely from M,+K,, M,—K, 
respectively, and MSf and 25M as though they were entirely compound 
tides. ; 
§ 5. The Method of Reduction of Tidal Observations. 
THE printed tabular forms on which the numerical harmonic analysis 
of the tides is carried out are arranged so that the series of observations 
to be analysed is supposed to begin at noon, or 0", of the first day, and 
to extend for a year from that time. It has not been found practicable 
to arrange that the first day shall be the same at all the ports of 
observation. 
Supposing 2» to be the speed of any tide in degrees per mean solar 
hour, and ¢ to be mean solar time elapsing since 0" of the first day; then 
the immediate result of the harmonic analysis is to obtain A and B, two 
heights (estimated in feet and tenths) such that the height of this tide 
at the time ¢ is given by 
A cos nf +B sin ut. 
The question then arises as to what further reductions it will be con- 
venient to make, in order to present the results in the most convenient 
form. 
First, let us put R= JV (A?+ B?), and tan [=F then the tide is repre- 
sented by 
R cos (nt—Z). 
In this form Ris the semi-range of the tide in British feet, and ¢ is 
an angle such that ¢/7 is the time elapsing after 0" of the first day until 
it is high-water of this particular tide. 
It is obvious that may have any value from 0° to 360°, and that the 
results of the analysis of successive years of observation will not be com- 
parable with one another, when presented in this form. 
Secondly, let us suppose that the results of the analysis are to be pre- 
sented in a number of terms of the form 
fH cos (V+u—r). 
Here V is a linear function of the moon’s and sun’s mean longitudes, 
the mean Jongitude of the moon’s and sun’s perigees, and the local mean 
solar time at the place of observation, reduced to angle at 15° per hour. 
V increases uniformly with the time, and its rate of increase per mean 
solar hour is the v of the first method, and is called the ‘speed’ of the 
tide. 
It is supposed that w stands for a certain function of the longitude 
of the node of the lunar orbit at an epoch half a year later than 0" of the 
