939, REPORTS ON THE STATE OF SCIENCE, ETC. 
Now suppose that ¢,? contains a quarter-diurnal portion that can be repre- 
sented by 
=Q, cos (¢,t—K,). 
Then, analysing as above for A and B will give 
A= %9,Q, cos (K,+7,) 
B=%9,Q, sin (K, + 7,7) 
where g, and 7,, being dependent only on o;,, are the same as before. 
If we now suppose that 
K,—k-=x 
and Gr = 2cQ, 
and that ¢ and x are independent of 7, we have 
a cos x—b sin x= 2cA, 
asin x+b cos x=2cB, 
whence 
The value of x can alternatively be found as 
tan-1 B — tan 12 5 
A a 
The definition of ¢ and x is such that if we write R cos (ct+a) for the semi- 
diurnal tide, then the quarter-diurnal tide is given by cR* cos (206 + 2a+ x). 
In the case of Newlyn two such analyses were made on the results for 
February 15 and March 28, on which days the quarter-diurnal tide was prominent 
and free from serious perturbation by other constituents. The results were: 
2c = 0222, 0229; ~=95°, 96°, respectively. 
These values were in accordance with reductions from the first method, and the 
mean values 
2c =0225, x=96° 
were adopted. 
Further experience with observations at Liverpool has shown that the second 
method should be used with some caution; it is necessary to choose days when (¢, 
is prominent and free from disturbance by other constituents, and several such days 
should be taken. The values of c and x, however, do not vary much more than do 
the results of analyses for M, from year to year. 
§ 15. Calculation of the quarter-diurnai tide.—(1) The first method of calculation 
tried used the values of ¢? direct. An interpolation formula was evolved for use 
with the hourly values of ¢,”. It is easily shown that 
0091 (¢:7) +0164 (47). 
gives the value of (¢,),, where suffixes outside the brackets indicate the relative 
hourly values. The formula is based upon the assumption of a mean speed of 58° 
per mean solar hour, and it dces not correspond exactly to constant values of 
ce and x; the variations in ¢ and x are small, however, and of no importance. It is 
easy to calculate the precise value of any constituent. 
Unfortunately ¢,? contains constants and long-period constituents, so that the 
formula gives a part L which has to be removed. It is easy to show, however, 
